DELETED r34.1/lib/Makefile.tmp Index: r34.1/lib/Makefile.tmp ================================================================== --- r34.1/lib/Makefile.tmp +++ r34.1/lib/Makefile.tmp @@ -1,173 +0,0 @@ -# -# Makefile for REDUCE User Contributed Library (PSL Version) -# -# Author: James H. Davenport . -# -# Modified by: Anthony C. Hearn. -# -# This Makefile may be used to build fast loading versions of all the -# PSL REDUCE User Contributed Library packages, or any particular -# package. It assumes that the relevant source files are in the -# $reduce/lib directory. It is specific to PSL, and of course those -# systems (e.g. UNIX) that support a make mechanism. However, it can -# be easily used with other systems with a make facility once a -# suitable mkfasl script has been written. - -REDUCE= /tresor/dagobert/cons/reduce3.4.1/dec3100 -FASL = b -BINDIR= $(REDUCE)/fasl -SRCDIR= $(REDUCE)/lib -TSTDIR= $(REDUCE)/lib -MKFASL= $(REDUCE)/util/mkfasl - -PACKAGES= assist camal changevar cvit desir fide gnuplot laplace \ - linineq numeric physop pm reacteqn reset rlfi showrules \ - symmetry tri wu - -UNCOMPILEDPACKAGES= odeex - -TSTPACKAGES= assist camal changevar cvit desir fide laplace linineq \ - numeric physop pmrules reacteqn rlfi symmetry tri wu - -all: $(PACKAGES) - -assist: $(BINDIR)/assist.$(FASL) - -$(BINDIR)/assist.$(FASL): $(SRCDIR)/assist.red - $(MKFASL) assist lib - - -camal: $(BINDIR)/camal.$(FASL) - -$(BINDIR)/camal.$(FASL): $(SRCDIR)/camal.red - $(MKFASL) camal lib - - -changevar: $(BINDIR)/changevar.$(FASL) - -$(BINDIR)/changevar.$(FASL): $(SRCDIR)/changevar.red - $(MKFASL) changevar lib - - -cvit: $(BINDIR)/cvit.$(FASL) - -$(BINDIR)/cvit.$(FASL): $(SRCDIR)/cvit.red - $(MKFASL) cvit lib - - -desir: $(BINDIR)/desir.$(FASL) - -$(BINDIR)/desir.$(FASL): $(SRCDIR)/desir.red - $(MKFASL) desir lib - - -fide: $(BINDIR)/fide1.$(FASL) $(BINDIR)/fide.$(FASL) - -$(BINDIR)/fide1.$(FASL): $(SRCDIR)/fide1.red - $(MKFASL) fide1 lib - -$(BINDIR)/fide.$(FASL): $(SRCDIR)/fide.red - $(MKFASL) fide lib - - -gnuplot: $(BINDIR)/gnuplot.$(FASL) - -$(BINDIR)/gnuplot.$(FASL): $(SRCDIR)/gnuplot.red - $(MKFASL) gnuplot lib - - -laplace: $(BINDIR)/laplace.$(FASL) - -$(BINDIR)/laplace.$(FASL): $(SRCDIR)/laplace.red - $(MKFASL) laplace lib - - -linineq: $(BINDIR)/linineq.$(FASL) - -$(BINDIR)/linineq.$(FASL): $(SRCDIR)/linineq.red - $(MKFASL) linineq lib - - -numeric: $(BINDIR)/numeric.$(FASL) - -$(BINDIR)/numeric.$(FASL): $(SRCDIR)/numeric.red - $(MKFASL) numeric lib - - -physop: $(BINDIR)/noncom2.$(FASL) $(BINDIR)/physop.$(FASL) - -$(BINDIR)/noncom2.$(FASL): $(SRCDIR)/noncom2.red - $(MKFASL) noncom2 lib - -$(BINDIR)/physop.$(FASL): $(SRCDIR)/physop.red - $(MKFASL) physop lib - - -pm: $(BINDIR)/pm.$(FASL) $(BINDIR)/pmrules.$(FASL) -# $(BINDIR)/pmrules2.$(FASL) - -$(BINDIR)/pm.$(FASL): $(SRCDIR)/pm.red - $(MKFASL) pm lib - -$(BINDIR)/pmrules.$(FASL): $(SRCDIR)/pmrules.red - $(MKFASL) pmrules lib - -# $(BINDIR)/pmrules2.$(FASL): $(SRCDIR)/pmrules2.red -# $(MKFASL) pmrules2 lib - - -reacteqn: $(BINDIR)/reacteqn.$(FASL) - -$(BINDIR)/reacteqn.$(FASL): $(SRCDIR)/reacteqn.red - $(MKFASL) reacteqn lib - - -reset: $(BINDIR)/reset.$(FASL) - -$(BINDIR)/reset.$(FASL): $(SRCDIR)/reset.red - $(MKFASL) reset lib - - -rlfi: $(BINDIR)/rlfi.$(FASL) - -$(BINDIR)/rlfi.$(FASL): $(SRCDIR)/rlfi.red - $(MKFASL) rlfi lib - - -showrules: $(BINDIR)/showrules.$(FASL) - -$(BINDIR)/showrules.$(FASL): $(SRCDIR)/showrules.red - $(MKFASL) showrules lib - - -symmetry: $(BINDIR)/symmetry.$(FASL) - -$(BINDIR)/symmetry.$(FASL): $(SRCDIR)/symmetry.red - $(MKFASL) symmetry lib - - -tri: $(BINDIR)/tri.$(FASL) - -$(BINDIR)/tri.$(FASL): $(SRCDIR)/tri.red - $(MKFASL) tri lib - - -wu: $(BINDIR)/wu.$(FASL) - -$(BINDIR)/wu.$(FASL): $(SRCDIR)/wu.red - $(MKFASL) wu lib - -test: $(PACKAGES) - for i in $(TSTPACKAGES) ; do \ - rm -f $(REDUCE)/log/$$i.log ; \ - echo \ -'load_package '$$i';on errcont;in"'$(TSTDIR)/$$i'.tst";showtime;bye;' \ - | reduce > $(REDUCE)/log/$$i.log ; \ - done - -check: $(PACKAGES) - - for i in $(TSTPACKAGES) ; do \ - echo 'comparing '$$i'...' ; \ - diff $(REDUCE)/log/$$i.log $(TSTDIR) ; \ - done - DELETED r34.1/plot/docs/gnuplot.dvi Index: r34.1/plot/docs/gnuplot.dvi ================================================================== --- r34.1/plot/docs/gnuplot.dvi +++ r34.1/plot/docs/gnuplot.dvi cannot compute difference between binary files DELETED r35/cslsrc/helpdata Index: r35/cslsrc/helpdata ================================================================== --- r35/cslsrc/helpdata +++ r35/cslsrc/helpdata @@ -1,11555 +0,0 @@ -\item[Contents] -Help is available on the following - -Algebra Arithmetic Booleans -Commands Declarations Functions -InputOutput Library Matrix -Operators Specfns Switches -Syntax Variables - -There are help windows for each of these topics - -To select a help page double click on the word in the help window or -use the Help Selection option on the menu. A backspace/delete will -return to this Index window. - -An alphabetical list of all topics follows - -. -# -ABS ACOS ACOSH -ACOT ACOTH ACSC -ACSCH ADJPREC ALGEBRAIC -Algebraic mode ALGINT ALGINT(Package) -ALLBRANCH ALLFAC ANTISYMMETRIC -APPEND ARBCONST ARGLENGTH -ARNUM ARRAY ASEC -ASECH ASIN ASINH -ATAN ATAN2 ATANH -AVECTOR -BALANCED_MOD BEGIN...END BERNOULLI -BESSELI BESSELJ BESSELK -BESSELY BETA BFSPACE -BINOMIAL BOUNDS BYE -CARD_NO CEILING CENTERED_MOD -CHEBYSHEV_FIT CHEBYSHEVT CHEBYSHEVU -CLEAR CLEARRULES COEFF -COEFFICIENT COEFFN COFACTOR -COLLECT COMBINEEXPT COMBINELOGS -COMMENT COMP COMPACT -Compiler COMPLEX CONJ -CONT COS COSH -COT COTH CRAMER -CREF CSC CSCH -DECOMPOSE DEFINE DEFN -DEFPOLY DEG DEMO -DEN DEPEND DET -DF DFPRINT DILOG -DISPLAY DIV DOT -E ECHO ED -EDITDEF END EPS -Equation ERF ERRCONT -EULER EULERP Euler Numbers -EVAL_MODE EVALLHSEQP EVEN -EVENP EXCALC EXP -EXPAND_CASES EXPANDLOGS EXPINT -exterior calc exterior df EZGCD -FACTOR FACTORIAL Factorization -FACTORIZE FAILHARD FIRST -FIRSTROOT FIX FIXP -FLOOR FOR FORALL -FOREACH FORT FORT_WIDTH -FORTRAN FREEOF FULLPREC -FULLROOTS -G GAMMA Gamma Function -GC GCD GEGENBAUERP -GENTRAN GosperAlg -Hankel Functions HANKEL1 HANKEL2 -HERMITEP HIGH_POW HORNER -HYPOT -I Identifier IF -IFACTOR IMPART IN -Indefinite integration INDEX INFINITY -INFIX INPUT INT -INTEGER INTERPOL INTSTR -ISOLATER -JACOBIP -KERNEL KORDER Kummer Functions -KUMMERM KUMMERU -LAGUERREP LCM LCOF -LEGENDREP LENGTH LESSSPACE -LET LHS LIMIT -LIMITEDFACTORS LINEAR LINELENGTH -LISP LIST List(operation) -LISTARGP LISTARGS LN -LOAD_PACKAGE LOG LOGB -Lommel Functions LOMMEL1 LOMMEL2 -LOW_POW LTERM -MAINVAR MASS MAT -MATCH MATEIGEN MATRIX -MAX MCD MIN -MKID MODULAR MSG -MSHELL MULTIPLICITIES -NAT NERO NEXTPRIME -NOARG NODEPEND NOLNR -NONCOM NONZERO NOSPLIT -NOSPUR NOXPND NULLSPACE -NUM NUMVAL NUMBERP -NUM_INT NUM_MIN NUM_ODESOLVE -NUM_SOLVE -ODD ODESOLVE OFF -ON ONE_OF OPERATOR -ORDER ORDP ORTHOVEC -OUT OUTPUT OVERVIEW -PART PAUSE PERIOD -PF PI POCHHAMMER -POLYGAMMA PRECEDENCE PRECISE -PRECISION PRET PRI -PRIMEP PRINT_PRECISION PROCEDURE -PROD PRODUCT PSI -QUIT -RANK RAT RATARG -RATIONAL RATIONALIZE RATPRI -REAL REDUCT REMAINDER -REMFAC REMIND REPART -REPEAT REST RESULTANT -RETRY RETURN REVERSE -REVPRI RHS RLISP88 -RLROOTNO ROOT_OF ROOT_MULTIPLICITES -ROUND ROUNDALL ROUNDBF -ROUNDED Rule_lists -SAVEAS SAVESTRUCTR SCALAR -SCIENTIFIC_NOTATION SCOPE SEC -SECH SECOND SET -SETMOD SHARE SHOWRULES -SHOWTIME SHUT SIGN -SIN SINH SOLVE -SOLVESINGULAR SPDE SPLIT_FIELD -SPUR SQRT STIRLING1 -STIRLING2 String STRUCTR -STRUVEH STRUVEL SUB -SUM SYMBOLIC SYMMETRIC -T TAN TANH -TAYLOR TAYLORAUTOCOMBINE TAYLORAUTOEXPAND -TAYLORCOMBINE TAYLORKEEPORIGINAL TAYLORORIGINAL -TAYLORPRINTORDER TAYLORPRINTTERMS TAYLORREVERT -TAYLORSERIESP TAYLORTEMPLATE TAYLORTOSTANDARD -THIRD TIME TP -TPS TRACE TRALLFAC -TRFAC TRIGFORM TRINT -TRNONLNR -VARNAME VECDIM VECTOR -WEIGHT WHEN WHERE -WHILE WHITTAKERW WRITE -WS WTLEVEL -XPND -ZETA - -\endsection -\item[Algebra] -Algebra Index - -Algebraic operators about which there is help are: - -APPEND ARBINT ARBCOMPLEX -ARGLENGTH COEFF COEFFN -CONJ DECOMPOSE DEG -DEN DF EXPAND_CASES -EXPREAD FACTORIZE HYPOT -IMPART INT INTERPOL -LCOF LENGTH LHS -LTERM MAINVAR NPRIMITIVE -NUM PART PF -REDUCT REPART RESULTANT -RHS ROOT_OF SHOWRULES -SOLVE STRUCTR SUB -WS - -\endsection -\item[Arithmetic] -Arithmetic Index - -This section considers operations defined in REDUCE that concern numbers, -or operators that can operate on numbers in addition, in most cases, to -more general expressions. - -Arithmetic operations about which there is help are: - -ABS ADJPREC CEILING -DILOG FACTORIAL FIX -FIXP FLOOR GCD -LN LOG LOGB -MAX MIN NEXTPRIME -REMAINDER ROUND SIGN -SQRT - -\endsection -\item[Booleans] -Booleans Index - -Boolean operations about which there is help are: - -EVENP FREEOF NUMBERP -ORDP PRIMEP - -\endsection -\item[Commands] -Commands Index - -Commands about which there is help are: - -BYE CONT DISPLAY -LOAD_PACKAGE PAUSE QUIT -RETRY SAVEAS SHOWTIME -WRITE - -\endsection -\item[Concepts] -Concepts Index - -There is help on the following basic concepts: - -Identifier Kernel String - -Also there is a simple editor, described by - -ED EDITDEF - -\endsection -\item[Declarations] -Declarations Index - -Declarations about which there is help are: - -ALGEBRAIC ANTISYMMETRIC ARRAY -CLEAR CLEARRULES DEFINE -DEPEND EVEN FACTOR -FORALL INFIX INTEGER -KORDER LET LINEAR -LINELENGTH LISP LISTARGP -MATCH NODEPEND NONCOM -NONZERO ODD OFF -ON OPERATOR ORDER -PRECEDENCE PRECISION PRINT_PRECISION -REAL REMFAC SCALAR -SCIENTIFIC_NOTATION SHARE SYMBOLIC -SYMMETRIC VARNAME WEIGHT -WHILE WTLEVEL - -\endsection -\item[Functions] -Functions Index - -Elementary functions about which there is help are: - -ACOS ACOSH ACOT -ACOTH ACSC ACSCH -ASEC ASECH ASIN -ASINH ATAN ATANH -ATAN2 COS COSH -COT COTH CSC -CSCH ERF EXP -EXPINT SEC SECH -SIN SINH TAN -TANH - -\endsection -\item[HighEnergy] -High Energy Physics Index - -The High-energy Physics package is historic for REDUCE, since REDUCE -originated as a program to aid in computations with Dirac expressions. -The commutation algebra of the gamma matrices is independent of their -representation, and is a natural subject for symbolic mathematics. -Dirac theory is applied to beta decay and the computation of -cross-sections and scattering. The high-energy physics operators are -available in the REDUCE main program, rather than as a module which -must be loaded. - -Arithmetic operations about which there is help are: - -DOT EPS G -INDEX MASS MSHELL -NOSPUR REMIND SPUR -VECDIM VECTOR - -\endsection -\item[InputOutput] -Input and Output Index - -Input and Output actions about which there is help are: - -IN INPUT OUT -SHUT - -\endsection -\item[Library] -Library Index - -The external modules that are included in your REDUCE system are the -first members of the REDUCE User's Library. They have been -contributed by REDUCE users from various fields for the convenience -and pleasure of the REDUCE user community. Future releases of REDUCE -will include other packages as they are developed. The packages in -the User's Library are unsupported; any questions or problems should -be directed to their authors. - -Each package comes with its own documentation, which you can find, -along with the source code, in the subdirectories lib of you REDUCE -directory (with suffix .txt, .tex and .red). The LOAD_PACKAGE command -is used to load the files you wish into your system. There will be a -short delay while the module is loaded. A module cannot be unloaded. -Once it is in your system, it stays there until you end the session. -Each package also has a test file, which you will find under its name -in the lib directory with suffix .tst. - -The following paragraphs, provided by the authors of each module, -briefly introduce packages which have not yet been described in more -detail in other sections of this document. Please refer to the -documentation for each module for detailed information on its use. -Each of them have their own switches, commands, and operators, and -some redefine special characters to aid in their notation. - -Libraries about which there is help are: - -ALGINT ARNUM AVECTOR -COMPACT EXCALC GENTRAN -NUMERIC ODESOLVE ORTHOVEC -SCOPE SPDE TPS - -\endsection -\item[Matrix] -Matrix Index - -Matrix operations about which there is help are: - -COFACTOR DET MAT -MATEIGEN MATRIX NULLSPACE -RANK TP TRACE - -\endsection -\item[Operators] -Operators Index - -Operations about which there is help are: - -LIMIT SUM PROD - -\endsection -\item[Specfns] -Special Functions - -The REDUCE Special Function Package supplies extended algebraic and -numeric support for a wide class of objects. This package is released -together with REDUCE 3.5 (October 1993) for the first time, therefore -it is far from being complete. - -The functions included in this package are in most cases (unless -otherwise stated) defined and named like in the book by Abramowitz and -Stegun: Handbook of Mathematical Functions, Dover Publications. - -The aim is to collect as much information on the special functions and -simplification capabilities as possible, i.e. algebraic -simplifications and numeric (rounded mode) code, limits of the -functions together with the definitions of the functions, which are in -most cases a power series, a (definite) integral and/or a differential -equation. - -What can be found: - A variety of Bessel functions, special polynomials, the Gamma -function, the Zeta function and integral functions. - -What is missing: - Airy functions, Mathieu functions, LerchPhi, etc.. The information -about the special functions which solve certain differential equation -is very limited. In several cases numerical approximation is -restricted to real arguments or is missing completely. - -The implementation of this package uses REDUCE rule sets to a large -extent, which guarantees a high 'readability' of the functions -definitions in the source file directory. It makes extensions to the -special functions code easy in most cases too. To look at these rules -it may be convenient to use the showrules operator e.g. - - showrules Besseli; - -Note: The special function package has to be loaded explicitly by calling - load_package specfn; - -Help is available on: - -BERNOUILLI BESSELI BESSELJ -BESSELK BESSELY BETA -CHEBYSHEVT CHEBYSHEVU EULER -EULERP GAMMA GEGENBAUERP -HANKEL1 HANKEL2 HERMITEP -JACOBIP KUMMERM KUMMERU -LAGUERREP LEGENDREP LOMMEL1 -LOMMEL2 POCHHAMMER POLYGAMMA -PSI STIRLING1 STIRLING2 -STRUVEH STRUVEL WHITTAKERW -ZETA - -\endsection -\item[Switches] -Switches Index - -Switches are set on or off using the commands ON or OFF, respectively. -The default setting of the switches described in this section is -OFF unless stated otherwise. - -Switches about which there is help are: - -ALGINT ALLBRANCH ALLFAC -BALANCED_MOD BFSPACE COMBINEEXPT -COMBINELOGS COMP COMPLEX -CREF CRAMER DEFN -DEMO DFPRINT DIV -ECHO ERRCONT EVALLHSEQP -EXP EXPANDLOGS EZGCD -FACTOR FAILHARD FORT -FULLPREC FULLROOTS GC -GCD HORNER IFACTOR -INT INTSTR LCM -LESSSPACE LIMITEDFACTORS LIST -LISTARGS MCD MODULAR -MSG MULTIPLICITIES NAT -NERO NOARG NOLNR -NOSPLIT NUMVAL OUTPUT -OVERVIEW PERIOD PRECISE -PRET PRI RAT -RATARG RATIONAL RATIONALIZE -RATPRI REVPRI RLISP88 -ROUNDALL ROUNDBF ROUNDED -SAVESTRUCTR SOLVESINGULAR TIME -TRALLFAC TRFAC TRIGFORM -TRINT TRNONLNR - -\endsection -\item[Syntax] -Syntax Index - -Syntax about which there is help are: - -BEGIN...END COMMENT CONS -END EQUATION FIRST -FOR FOREACH GOTO -IF List PROCEDURE -REPEAT REST RETURN -REVERSE RuleSet SECOND -SET THIRD WHEN - -\endsection -\item[Variables] -Variables Index - -Variables about which there is help are: - -CARD_NO E EVAL_MODE -FORT_WIDTH HIGH_POW I -INFINITY LOW_POW NIL -PI ROOT_MULTIPLICITIES T - -\endsection -\xitem[><] ->< 3-D vector and diphthong (page 356) - -\endsection -\xitem[*] -* - 3-D vector, 356 - algebraic numbers, 224 - power series, 422 - vector, 232 - -\endsection -\xitem[**] -** - power series, 422 - -\endsection -\xitem[+] -+ - 3-D vector, 356 - algebraic numbers, 223 - power series, 422 - vector, 232 - -\endsection -\xitem[-] -- - 3-D vector, 356 - power series, 422 - vector, 232 - -\endsection -\item[.] -. (CONS) (page 50) - -The CONS operator adds a new element to the beginning of a LIST. Its -operation is identical to the symbol DOT (dot). It can be used -infix or prefix. - - CONS(item,list) or item CONS list - -item can be any REDUCE scalar expression, including a list; list -must be a list. - -Examples: - -liss := cons(a,{b}); {A,B} - -liss := c cons liss; {C,A,B} - -newliss := for each y in liss collect cons(y,list x); - NEWLISS := {{C,X},{A,X},{B,X}} - -for each y in newliss sum (first y)*(second y); - X*(A + B + C) - -If you want to use CONS to put together two elements into a new list, -you must make the second one into a list with curly brackets or the -LIST command. You can also start with an empty list created by {}. - -The CONS operator is right associative: A CONS B CONS C is valid if C -is a list; B need not be a list. The list produced is {A,B,C}. - -\endsection -\xitem[/] -/ - 3-D vector, 356 - algebraic numbers, 224 - power series, 422 - vector, 232 - -\endsection -\xitem[@] -@ - partial differentiation, 271 - tangent vector, 271 - -\endsection -\xitem[@ operator] -@ operator, 251 - -\endsection -\item[#] -# (pages 256, 271) - -# is the syntax for the Hodge-* operator in the EXCALC package. - -\endsection -\xitem[^] -^ - 3-D vector, 356 - exterior multiplication, 250, 271 - -\endsection -\item[ABS] -ABS (page 72) -The ABS operator returns the absolute value of its argument. - -ABS(expression) - -expression can be any REDUCE scalar expression. - -Examples: -abs(-a); ABS(A) -abs(-5); 5 -a := -10; A := -10 -abs(a); 10 -abs(-a); 10 - -If the argument has had no numeric value assigned to it, such as an -identifier or polynomial, ABS returns an expression involving -ABS of its argument, doing as much simplification of the argument -as it can, such as dropping any preceding minus sign. - -\endsection -\item[ACOS] -ACOS (pages 76, 78) - -The ACOS operator returns the arccosine of its argument. - - ACOS(expression) or ACOS simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -acos(ab); ACOS(AB) -acos 15; ACOS(15) - 2 2 - SQRT( - X *Y + 1)*Y -df(acos(x*y),x); ---------------------- - 2 2 - X *Y - 1 -on rounded; -res := acos(sqrt(2)/2); RES := 0.785398163397 -res-pi/4; 0 - -An explicit numeric value is not given unless the switch ROUNDED is -on and the argument has an absolute numeric value less than or equal to 1. - -\endsection -\item[ACOSH] -ACOSH (pages 76, 78) - -ACOSH represents the hyperbolic arccosine of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of -ACOSH is known to the system. Numerical values may also be found by -turning on the switch ROUNDED. - - ACOSH(expression) or ACOSH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix or -vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -acosh a; ACOSH(A) -acosh(0); ACOSH(0) - 4 - 2*SQRT(A - 1)*A -df(acosh(a**2),a); ------------------ - 4 - A - 1 - -int(acosh(x),x); INT(ACOSH(X),X) - -You may attach functionality by defining ACOSH to be the inverse of -COSH. This is done by the commands - put('cosh,'inverse,'acosh); - put('acosh,'inverse,'cosh); -You can write a procedure to attach integrals or other functions to -ACOSH. You may wish to add a check to see that its argument is -properly restricted. - -\endsection -\item[ACOT] -ACOT (pages 76, 78) - -ACOT represents the arccotangent of its argument. It takes an -arbitrary scalar expression as its argument. The derivative of ACOT is -known to the system. Numerical values may also be found by turning on -the switch ROUNDED. - - ACOT(expression) or ACOT simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. You can add functionality -yourself with LET and procedures. - -Examples: -acot a; ACOT(A) - PI -acot(0); ---- - 2 - - - 2*A -df(acot(a**2),a); -------- - 4 - A + 1 - - 2 - 2*ACOT(X)*X + LOG(X + 1) -int(acot(x),x); --------------------------- - 2 -on rounded; -acot(1); 0.785398163397 - -\endsection -\item[ACOTH] -ACOTH (pages 76, 78) - -ACOTH represents the inverse hyperbolic cotangent of its argument. It -takes an arbitrary scalar expression as its argument. The derivative -of ACOTH is known to the system. Numerical values may also be found -by turning on the switch ROUNDED. - - ACOTH(expression) or ACOTH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. You can add functionality yourself -with LET and procedures. - -Examples: -acoth(0); 0 - - - 2*X -df(acoth(x^2),x); -------- - 4 - X - 1 - -int(acoth(x),x); ACOTH(X)*X + ACOTH(X) + LOG(X - 1) - -\endsection -\item[ACSC] -ACSC (pages 76, 78) - -The ACSC operator returns the arccosecant of its argument. - - ACSC(expression) or ACSC simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -acsc(ab); ACSC(AB) -acsc 15; ACSC(15) - 2 2 - - SQRT(X *Y - 1) -df(acsc(x*y),x); -------------------- - 2 2 - X*(X *Y - 1) -on rounded; -res := acsc(2/sqrt(3)); RES := 1.0471975512 -res-pi/3; 0 - -An explicit numeric value is not given unless the switch ROUNDED is on -and the argument has an absolute numeric value less than or equal to -1. - -\endsection -\item[ACSCH] -ACSCH (pages 76, 78) - -The ACSCH operator returns the hyperbolic arccosecant of its argument. - - ACSCH(expression) or ACSCH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -acsch(ab); ACSCH(AB) -acsch 15; ACSCH(15) - 2 2 - - SQRT(X *Y + 1) -df(acsch(x*y),x); -------------------- - 2 2 - X*(X *Y + 1) -on rounded; -res := acsch(3); RES := 0.327450150237 - -An explicit numeric value is not given unless the switch ROUNDED is on -and the argument has an absolute numeric value less than or equal to -1. - -\endsection -\item[ADJPREC] -ADJPREC (page 133) - -When a real number is input, it is normally truncated to the PRECISION -in effect at the time the number is read. If it is desired to keep -the full precision of all numbers input, the switch ADJPREC (for -adjust precision) can be turned on. While on, ADJPREC will -automatically increase the precision, when necessary, to match that of -any integer or real input, and a message printed to inform the user of -the precision increase. - -Examples: -on rounded; -1.23456789012345; 1.23456789012 -on adjprec; -1.23456789012345; *** precision increased to 15 - 1.23456789012345 - -\endsection -\item[ALGEBRAIC] -ALGEBRAIC (page 191) - -The ALGEBRAIC command changes REDUCE's mode of operation to -algebraic. When ALGEBRAIC is used as an operator (with an argument -inside parentheses) that argument is evaluated in algebraic mode, but -REDUCE's mode is not changed. - -Examples: -algebraic; -symbolic; NIL - 2 -algebraic(x**2); X -x**2; ***** The symbol X has no value. - -REDUCE's symbolic mode does not know about most algebraic commands. -Error messages in this mode may also depend on the particular Lisp -used for the REDUCE implementation. - -\endsection -\item[Algebraic mode] -Algebraic mode (pages 191, 197, 198) - -Most REDUCE calculatuons take place in Algebraic mode. The -alternative is Symbolic mode, which is a syntactic form of LISP. See -the commands ALGEBRAIC and SYMBOLIC for mor details. - -\endsection -\item[ALGINT] -ALGINT - -When the ALGINT switch is on, the algebraic integration module (which -must be loaded from the REDUCE library) is used for integration. - -Loading ALGINT from the library automatically turns on the -ALGINT switch. An error message will be given if ALGINT is -turned on when the ALGINT has not been loaded from the library. - -\endsection -\item[ALGINT(Package)] -ALGINT(Package) (page 178) - -Author: James H. Davenport - -The ALGINT package provides indefinite integration of square roots. -This package, which is an extension of the basic integration package -distributed with REDUCE, will analytically integrate a wide range of -expressions involving square roots. The ALGINT switch provides for -the use of the facilities given by the module, and is automatically -turned on when the package is loaded. If you want to return to the -standard integration algorithms, turn ALGINT off. An error message is -given if you try to turn the ALGINT switch on when its module is not -loaded. - -\endsection -\item[ALLBRANCH] -ALLBRANCH (page 89) - -When ALLBRANCH is on, the operator SOLVE selects all branches of -solutions. When ALLBRANCH is off, it selects only the principal -branches. Default is ON. - -Examples: - -solve(log(sin(x+3)),x); {X=2*ARBINT(1)*PI - ASIN(1) - 3, - X=2*ARBINT(1)*PI + ASIN(1) + PI - 3} -off allbranch; -solve(log(sin(x+3)),x); {X=ASIN(1) - 3} - -ARBINT(1) indicates an arbitrary integer, which is given a unique -identifier by REDUCE, showing that there are infinitely many solutions -of this type. When ALLBRANCH is off, the single canonical solution is -given. - -\endsection -\item[ALLFAC] -ALLFAC (pages 102, 104) - -The ALLFAC switch, when on, causes REDUCE to factor out automatically -common products in the output of expressions. Default is ON. - -Examples: 3 -x + x*y**3 + x**2*cos(z); X*(COS(Z)*X + Y + 1) -off allfac; - 2 3 -x + x*y**3 + x**2*cos(z); COS(Z)*X + X*Y + X - -The ALLFAC switch has no effect when PRI is off. Although the switch -setting stays as it was, printing behaviour is as if it were off. - -\endsection -\xitem[ansatz of symmetry generator] -ansatz of symmetry generator, 386 - -\endsection -\item[ANTISYMMETRIC] -ANTISYMMETRIC (page 93) - -When an operator is declared ANTISYMMETRIC, its arguments are -reordered to conform to the internal ordering of the system. If an -odd number of argument interchanges are required to do this ordering, -the sign of the expression is changed. - - ANTISYMMETRIC identifier {,identifier} - -identifier is an identifier that has been declared as an operator. - -Examples: -operator m,n; -antisymmetric m,n; -m(x,n(1,2)); - M( - N(2,1),X) -operator p; -antisymmetric p; -p(a,b,c); P(A,B,C) -p(b,a,c); - P(A,B,C) - -If identifier has not been declared an operator, the flag -ANTISYMMETRIC is still attached to it. When identifier is -subsequently used as an operator, the message - Declare identifier operator? (Y or N) -is printed. If the user replies Y, the antisymmetric property of the -operator is used. - -Note in the first example, identifiers are customarily ordered -alphabetically, while numbers are ordered from largest to smallest. -The operators may have any desired number of arguments (less than 128). - -\endsection -\item[APPEND] -APPEND (page 50) - -The APPEND operator constructs a new list from the elements of its two -arguments (which must be lists). - - APPEND(lst,lst) - -lst must be a list, though it may be the empty list ({}). Any -arguments beyond the first two are ignored. - -Examples: -alist := {1,2,{a,b}}; ALIST := {1,2,{A,B}} -blist := {3,4,5,sin(y)}; BLIST := {3,4,5,SIN(Y)} -append(alist,blist); {1,2,{A,B},3,4,5,SIN(Y)} -append(alist,{}); {1,2,{A,B}} -append(list z,blist); {Z,3,4,5,SIN(Y)} - -Comment The new list consists of the elements of the second list -appended to the elements of the first list. You can append new -elements to the beginning or end of an existing list by putting the -new element in a list (use curly braces or the operator list). This -is particularly helpful in an iterative loop. - -\endsection -\item[ARBCONST] -ARBCONST operator (page 350) - -See the ODESOLVE package - -\endsection -\xitem[arbitrary ordering] -arbitrary ordering, 316 - -\endsection -\item[ARGLENGTH] -ARGLENGTH (page 117) -The operator ARGLENGTH returns the number of arguments of the top-level -operator in its argument. - - ARGLENGTH(expression) - -expression can be any valid REDUCE algebraic expression. - -Examples: -arglength(a + b + c + d); 4 -arglength(a/b/c); 2 -arglength(log(sin(df(r**3*x,x)))); 1 - -In the first example, + is an n-ary operator, so the number of terms -is returned. In the second example, since / is a binary operator, the -argument is actually (a/b)/c, so there are two terms at the top level. -In the last example, no matter how deeply the operators are nested, -there is still only one argument at the top level. - -\endsection -\item[ARNUM] -ARNUM (pages 179, 223) -Author: Eberhard Schruefer - -This package provides facilities for handling algebraic numbers as polynomial -coefficients in REDUCE calculations. It includes facilities for introducing -indeterminates to represent algebraic numbers, for calculating splitting -fields, and for factoring and finding greatest common divisors in such -domains. - -\endsection -\item[ARRAY] -ARRAY (page 67) - -The ARRAY declaration declares a list of identifiers to be of type -ARRAY, and sets all their entries to 0. - - ARRAY identifier(dimensions){,identifier(dimensions)} - -identifier may be any valid REDUCE identifier. If the identifier -was already an array, a warning message is given that the array has been -redefined. dimensions are of form - integer{,integer}. - -array a(2,5),b(3,3,3),c(200); -array a(3,5); *** ARRAY A REDEFINED -a(3,4); 0 -length a; {4,6} - -Arrays are always global, even if defined inside a procedure or block -statement. Their status as an array remains until the variable is -reset by CLEAR. Arrays may not have the same names as operators, -procedures or scalar variables. - -Array elements are referred to by the usual notation: A(I,J) returns -the jth element of the ith row. The ASSIGNment operator := is used to -put values into the array. Arrays as a whole cannot be subject to -assignment by LET or := ; the assignment operator := is only valid for -individual elements. - -When you use LET on an array element, the contents of that element -become the argument to LET. Thus, if the element contains a number or -some other expression that is not a valid argument for this command, -you get an error message. If the element contains an identifier, the -identifier has the substitution rule attached to it globally. The -same behaviour occurs with CLEAR. If the array element contains an -identifier or simple_expression, it is cleared. Do NOT use CLEAR to -try to set an array element to 0. Because of the side effects of -either LET or CLEAR, it is unwise to apply either of these to array -elements. - -Array indices always start with 0, so that the declaration ARRAY A(5) -sets aside 6 units of space, indexed from 0 through 5, and initialises -them to 0. The LENGTH command returns a list of the true number of -elements in each dimension. - -\endsection -\item[ASEC] -ASEC (pages 76, 78) - -The ASEC operator returns the arccosecant of its argument. - - ASEC(expression) or ASEC simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -asec(ab); ASEC(AB) -asec 15; ASEC(15) - 2 2 - SQRT(X *Y - 1) -df(asec(x*y),x); ----------------- - 2 2 - X*(X *Y - 1) -on rounded; -res := asec sqrt(2); RES := 0.785398163397 -res-pi/4; 0 - -An explicit numeric value is not given unless the switch ROUNDED is -on and the argument has an absolute numeric value greater or equal to 1. - -\endsection -\item[ASECH] -ASECH (pages 76, 78) - -ASECH represents the hyperbolic arccosecant of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of -ASECH is known to the system. Numerical values may also be found by -turning on the switch ROUNDED. - - ASECH(expression) or ASECH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -asech a; ASECH(A) -asech(1); 0 - 4 - 2*SQRT(A - 1)*A -df(acosh(a**2),a); ------------------ - 4 - A - 1 -int(asech(x),x); INT(ASECH(X),X) - -You may attach functionality by defining ASECH to be the inverse of -SECH. This is done by the commands - put('sech,'inverse,'asech); - put('asech,'inverse,'sech); -You can write a procedure to attach integrals or other functions to -ASECH. You may wish to add a check to see that its argument is -properly restricted. - -\endsection -\item[ASIN] -ASIN (pages 76, 78) - -The ASIN operator returns the arcsine of its argument. - - ASIN(expression) or ASIN simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -asin(givenangle); ASIN(GIVENANGLE) -asin(5); ASIN(5) - 2 - - 2*SQRT( - 4*X + 1) -df(asin(2*x),x); ------------------------ - 2 - 4*X - 1 -on rounded; -asin .5; 0.523598775598 -asin(sqrt(3)); ASIN(1.73205080757) -asin(sqrt(3)/2); 1.04719755120 - -A numeric value is not returned by ASIN unless the switch -ROUNDED is on and its argument has an absolute value less than or -equal to 1. - -\endsection -\item[ASINH] -ASINH (pages 76, 78) - -The ASINH operator returns the hyperbolic arcsine of its argument. -The derivative of ASINH and some simple transformations are known -to the system. - - ASINH(expression) or ASINH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -asinh d; ASINH(D) -asinh(1); ASINH(1) - 2 - 2*SQRT(4*X + 1) -df(asinh(2*x),x); ------------------ - 2 - 4*X + 1 - -You may attach further functionality by defining ASINH to be the -inverse of SINH. This is done by the commands - put('sinh,'inverse,'asinh); - put('asinh,'inverse,'sinh); - -A numeric value is not returned by ASINH unless the switch ROUNDED is -on and its argument evaluates to a number. - -\endsection -\xitem[Assignment] -Assignment, 54, 55, 57, 63, 195, 198 - -\endsection -\item[Asymptotic command] -Asymptotic command (pages 139, 151) - -See WEIGHT and WTLEVEL - -\endsection -\item[ATAN] -ATAN (pages 76, 78, 81) - -The ATAN operator returns the arctangent of its argument. - - ATAN(expression) or ATAN simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -atan(middle); ATAN(MIDDLE) -on rounded; -atan 45; 1.54857776147 -off rounded; - 2 - 2*ATAN(X)*X - LOG(X + 1) -int(atan(x),x); --------------------------- - 2 - 2*Y -df(atan(y**2),y); -------- - 4 - Y + 1 - -A numeric value is not returned by ATAN unless the switch ROUNDED is -on and its argument evaluates to a number. - -\endsection -\item[ATAN2] -ATAN2 (pages 76, 78) - - ATAN2(expression,expression) - -expression is any valid scalar REDUCE expression. In ROUNDED mode, if -a numerical value exists, ATAN2 returns the principal value of the arc -tangent of the second argument divided by the first in the range -[-pi,+pi] radians, using the signs of both arguments to determine the -quadrant of the return value. An expression in terms of ATAN2 is -returned in other cases. - -Examples: -atan2(3,2); ATAN2(3,2); -on rounded; -atan2(3,2); 0.982793723247 -atan2(a,b); ATAN2(a,b); -atan2(1,0); 1.57079632679 - -ATAN2 returns a numeric value only if ROUNDED is on. Then the -arctangent is calculated to the current degree of floating point precision. - -\endsection -\item[ATANH] -ATANH (pages 76, 78) - -The ATANH operator returns the hyperbolic arctangent of its argument. -The derivative of ASINH and some simple transformations are known -to the system. - - ATANH(expression) or ATANH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -atanh aa; ATANH(AA) -atanh(1); ATANH(1) - - X -df(atanh(x*y),y); ----------- - 2 2 - X *Y - 1 - -A numeric value is not returned by ASINH unless the switch ROUNDED is -on and its argument evaluates to a number. You may attach additional -functionality by defining ATANH to be the inverse of TANH. This is -done by the commands - put('tanh,'inverse,'atanh); - put('atanh,'inverse,'tanh); - -\endsection -\xitem[AVEC function] -AVEC function, 232 - -\endsection -\item[AVECTOR] -AVECTOR (pages 179, 231) - -Author: David Harper - -A Vector Algebra and Calculus Package. - -This package provides REDUCE with the ability to perform vector -algebra using the same notation as scalar algebra. The basic -algebraic operations are supported, as are differentiation and -integration of vectors with respect to scalar variables, cross -product and dot product, component manipulation and application of -scalar functions (e.g. cosine) to a vector to yield a vector -result. - -\endsection -\item[BALANCED_MOD] -BALANCED_MOD - -MODULAR numbers are normally produced in the range [0,...n), where -n is the current modulus. With BALANCED_MOD on, the range -[-n/2,n/2] is used instead. - -Examples: - setmod 7; 1 - on modular; - 4; 4 - on balanced_mod; - 4; -3 - -\endsection -\item[BEGIN...END] -BEGIN ... END (pages 61, 62, 64) - -BEGIN is used to start a BLOCK statement, which is closed with END. - - BEGIN statement{; statement} END - -statement is any valid REDUCE statement. - -Examples: - begin for i := 1:3 do write i end; 1 - 2 - -begin scalar n;n:=1;b:=for i:=1:4 product(x-i);return n end; - 1 - 4 3 2 -b; X - 10*X + 35*X - 50*X + 24 - -A BEGIN...END block can do actions (such as WRITE), but does not -return a value unless instructed to by a RETURN statement, which must -be the last statement executed in the block. It is unnecessary to -insert a semicolon before the END. - -Local variables, if any, are declared in the first statement -immediately after BEGIN, and may be defined as SCALAR, INTEGER, or -REAL. ARRAY variables declared within a BEGIN...END block are global -in every case, and LET statements have global effects. A LET -statement involving a formal parameter affects the calling parameter -that corresponds to it. LET statements involving local variables make -global assignments, overwriting outside variables by the same name or -creating them if they do not exist. You can use this feature to -affect global variables from procedures, but be careful that you do -not do it inadvertently. - -\endsection -\item[BERNOULLI] -BERNOULLI (pages 185, 393) -[Part of SPECFN package] - -The BERNOULLI operator returns the nth Bernoulli number. - - BERNOULLI(integer) - -Examples: -load_package specfn; (SPECFN) - - - 174611 -bernoulli 20; ----------- - 330 - -bernoulli 17; 0 - -All Bernoulli numbers with odd indices except for 1 are zero. - -The BERNOULLIP operator returns the nth Bernoulli Polynomial evaluated -at x. - - BERNOULLIP(integer,expression) - -Examples: -load_package specfn; (SPECFN) - - 2 - Z*(2*Z - 3*Z + 1) -BernoulliP(3,z); -------------------- - 2 - - 338585 -BernoulliP(10,3); -------- - 66 - -The value of the nth Bernoulli Polynomial at 0 is the nth Bernoulli number. - -\endsection -\item[BESSELI] -BESSELI (pages 185, 396) -[Part of SPECFN package] - -The BESSELI operator returns the modified Bessel function I. - - BESSELI(order,argument) - -Examples: -load_package specfn; (SPECFN) -on rounded; -Besseli (1,1); 0.565159103992 - -The knowledge about the operator BESSELI is currently fairly limited. - -\endsection -\item[BESSELJ] -BESSELJ (pages 185, 396) -[Part of SPECFN package] - -The BESSELJ operator returns the Bessel function of the first kind. - - BESSELJ(order,argument) - -Examples: -load_package specfn; (SPECFN) -BesselJ(1/2,pi); 0 -on rounded; -BesselJ(0,1); 0.765197686558 - -\endsection -\item[BESSELK] -BESSELK (pages 185, 396) -[Part of SPECFN package] - -The BESSELK operator returns the modified Bessel function K. - - BESSELK(order,argument) - -Examples: -load_package specfn; (SPECFN) -df(besselk(0,x),x); - BESSELK(1,X) - -There is currently no numeric support for the operator BesselK. - -\endsection -\item[BESSELY] -BESSELY (pages 185, 396) -[Part of SPECFN package] - -The BESSELY operator returns the Bessel function of the second kind. - BESSELY(order,argument) - -Examples: -load_package specfn; (SPECFN) -Bessely (1/2,pi); - SQRT(2) / PI -on rounded; -Bessely (1,3); 0.324674424792 - -The operator BESSELY is also called Weber's function. - -\endsection -\item[BETA] -BETA (pages 185, 397) -[Part of SPECFN package] - -The BETA operator returns the Beta function defined by - - Beta (z,w) := defint(t**(z-1)* (1 - t)**(w-1),t,0,1) . - - - BETA(expression,expression) - - -Examples: -load_package specfn; (SPECFN) -Beta(2,2); 1 / 6 -Beta(x,y); GAMMA(X)*GAMMA(Y) / GAMMA(X + Y) - -The operator BETA is simplified towards the GAMMA operator. - -\endsection -\item[BFSPACE] -BFSPACE (page 133) - -Floating point numbers are normally printed in a compact notation -(either fixed point or in scientific notation if SCIENTIFIC_NOTATION -has been used). In some (but not all) cases, it helps -comprehensibility if spaces are inserted in the number at regular -intervals. The switch BFSPACE, if on, will cause a blank to be -inserted in the number after every five characters. - -Examples: - on rounded; - 1.2345678; 1.2345678 - on bfspace; - 1.2345678; 1.234 5678 - -BFSPACE is normally off. - -\endsection -\item[BINOMIAL] -BINOMIAL (page 185) - -The BINOMIAL operator returns the Binomial coefficient if both -parameter are integer and expressions involving the Gamma function otherwise. - - BINOMIAL(integer,integer) - -Examples: -Binomial(49,6); 13983816 - - GAMMA(N + 1) -Binomial(n,3); ---------------- - 6*GAMMA(N - 2) - -The operator BINOMIAL evaluates the Binomial coefficients from the -explicit form and therefore it is not the best algorithm if you want -to compute many binomial coefficients with big indices in which case a -recursive algorithm is preferable. - -\endsection -\xitem[Block] -Block, 61, 64 - -\endsection -\xitem[BNDEQ!*] -BNDEQ!*, 257 - -\endsection -\xitem[Boolean] -Boolean, 45 - -\endsection -\item[BOUNDS] -BOUNDS (page 182) - -Upper and lower bounds of a real valued function over an INTERVAL or a -rectangular multivariate domain are computed by the operator -BOUNDS. The algorithmic basis is the computation with inequalities: -starting from the interval(s) of the variables, the bounds are -propagated in the expression using the rules for inequality -computation. Some knowledge about the behavior of special functions -like ABS, SIN, COS, EXP, LOG, fractional exponentials etc. is -integrated and can be evaluated if the operator bounds is called with -rounded mode on (otherwise only algebraic evaluation rules are -available). - -If BOUNDS finds a singularity within an interval, the evaluation is -stopped with an error message indicating the problem part of the -expression. - - BOUNDS(exp,var=(l .. u) [,var=(l .. u) ...]) - BOUNDS(exp,{var=(l .. u) [,var=(l .. u) ...]}) - -where exp is the function to be investigated, var are the variables of -exp, l and u specify the area as set of INTERVAL s. - -BOUNDS computes upper and lower bounds for the expression in the given -area. An INTERVAL is returned. - -Examples: - bounds(sin x,x=(1 .. 2)); - 1 .. 1 - on rounded; - bounds(sin x,x=(1 .. 2)); 0.84147098481 .. 1 - bounds(x**2+x,x=(-0.5 .. 0.5)); - 0.25 .. 0.75 - -\endsection -\xitem[BROEBFULLREDUCTION] -BROEBFULLREDUCTION, 303 - -\endsection -\xitem[Buchberger's Algorithm] -Buchberger's Algorithm, 292, 295 - -\endsection -\item[BYE] -BYE (page 70) - -The BYE command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are -at the top level, the BYE command exits REDUCE. QUIT is a synonym for -BYE. - -\endsection -\xitem[C(I)] -C(I), 379 - -\endsection -\xitem[Call by value] -Call by value, 171, 173 - -\endsection -\xitem[Canonical form] -Canonical form, 97 - -\endsection -\item[CARD_NO] -CARD_NO (page 108) - -CARD_NO sets the total number of cards allowed in a Fortran -output statement when FORT is on. Default is 20. - -Examples: -on fort; -card_no := 4; CARD_NO=4. -z := (x + y)**15; - ANS1=5005.*X**6*Y**9+3003.*X**5*Y**10+1365.*X**4*Y** - . 11+455.*X**3*Y**12+105.*X**2*Y**13+15.*X*Y**14+Y**15 - Z=X**15+15.*X**14*Y+105.*X**13*Y**2+455.*X**12*Y**3+ - . 1365.*X**11*Y**4+3003.*X**10*Y**5+5005.*X**9*Y**6+ - . 6435.*X**8*Y**7+6435.*X**7*Y**8+ANS1 - -Twenty total cards means 19 continuation cards. You may set it for -more if your Fortran system allows more. Expressions are broken apart -in a Fortran-compatible way if they extend for more than CARD_NO -continuation cards. - -\endsection -\xitem[cartesian coordinates] -cartesian coordinates, 354 - -\endsection -\item[CEILING] -CEILING (page 72) - - CEILING(expression) - -This operator returns the ceiling (i.e., the least integer greater -than or equal to its argument) if its argument has a numerical value. -For negative numbers, this is equivalent to FIX. For non-numeric -arguments, the value is an expression in the original operator. - -Examples: -ceiling 3.4; 4 -fix 3.4; 3 -ceiling(-5.2); -5 -fix(-5.2); -5 -ceiling a; CEILING(A) - -\endsection -\item[CENTERED_MOD] -CENTERED_MOD (page 134) - -This is an error in the Reduce manual. It should be BALANCED_MOD. -For more information select that entry. - -\endsection -\xitem[chain rule] -chain rule, 254 - -\endsection -\xitem[Character set] -Character set, 33 - -\endsection -\item[Chebyshev_fit] -Chebyshev fit (page 182) - -The operator family CHEBYSHEV_... implements approximation and -evaluation of functions by the Chebyshev method. Let T(n,a,b,x) be -the Chebyshev polynomial of order n transformed to the interval (a,b). -Then a function f(x) can be approximated in (a,b) by a series - - for i := 0:n sum c(i)*T(i,a,b,x) - -The operator CHEBYSHEV_FIT computes this approximation and returns a -list, which has as first element the sum expressed as a polynomial and -as second element the sequence of Chebyshev coefficients. - -CHEBYSHEV_DF and CHEBYSHEV_INT transform a Chebyshev coefficient list -into the coefficients of the corresponding derivative or integral -respectively. For evaluating a Chebyshev approximation at a given -point in the basic interval the operator CHEBYSHEV_EVAL can be used. - -CHEBYSHEV_EVAL is based on a recurrence relation which is in general -more stable than a direct evaluation of the complete polynomial. - - CHEBYSHEV_FIT(fcn,var=(lo .. hi),n) - - CHEBYSHEV_EVAL(coeffs,var=(lo .. hi), var=pt) - - CHEBYSHEV_DF(coeffs,var=(lo .. hi)) - - CHEBYSHEV_INT(coeffs,var=(lo .. hi)) - - -where fcn is an algebraic expression (the target function), var is the -variable of fcn, lo and hi are numerical real values which describe an -INTERVAL lo < hi, the integer n is the approximation order (set to 20 -if missing), pt is a number in the interval and coeffs is a series of -Chebyshev coefficients. - -Examples: - -on rounded; -w:=chebyshev_fit(sin x/x,x=(1 .. 3),5); - 3 2 - W := {0.0382345446975*X - 0.239802588672*X + 0.0651206939005*X - - + 0.977836217464, - - {0.899091895826,-0.406599215895,-0.00519766024352,0.00946374143 - - 079,-0.0000948947435875}} - -chebyshev_eval(second w, x=(1 .. 3), x=2.1); - 0.411091086819 - -\xitem[Chebyshev Polynomials] -Chebyshev Polynomials, 185 - -\endsection -\item[CHEBYSHEVT] -CHEBYSHEVT (page 185) - -The CHEBYSHEVT operator computes the nth Chebyshev T Polynomial (of the -first kind). - -CHEBYSHEVT(integer,expression) - -Examples: -load_package specfn; (SPECFN) - 2 -ChebyshevT(3,xx); XX*(4*XX - 3) - -ChebyshevT(3,4); 244 - -Chebyshev's T polynomials are computed using the recurrence relation: - -ChebyshevT(n,x) := 2x*ChebyshevT(n-1,x) - ChebyshevT(n-2,x) with -ChebyshevT(0,x) := 0 and ChebyshevT(1,x) := x - -\endsection -\item[CHEBYSHEVU] -CHEBYSHEVU (page 185) - -The CHEBYSHEVU operator returns the nth Chebyshev U Polynomial (of the -second kind). - -CHEBYSHEVU(integer,expression) - -Examples: -load_package specfn; (SPECFN) - 2 -ChebyshevU(3,xx); 4*X*(2*X - 1) - -ChebyshevU(3,4); 496 - -Chebyshev's U polynomials are computed using the recurrence relation: - -ChebyshevU(n,x) := 2x*ChebyshevU(n-1,x) - ChebyshevU(n-2,x) with -ChebyshevU(0,x) := 0 and ChebyshevU(1,x) := 2x - -\endsection -\item[CLEAR] -CLEAR (pages 142, 146) - -The CLEAR command is used to remove assignments or remove substitution -rules from any expression. - -CLEAR identifier{,identifier} or - let-type statement CLEAR identifier - -identifier can be any SCALAR, MATRIX, or ARRAY variable or PROCEDURE -name. let-type statement can be any general or specific LET statement -(see below). - -Examples: -array a(2,3); -a(2,2) := 15; A(2,2) := 15 -clear a; -a(2,2); Declare A operator? (Y or N) -let x = y + z; -sin(x); SIN(Y + Z) -clear x; -sin(x); SIN(X) -let x**5 = 7; -clear x; -x**5; 7 -clear x**5; - 5 -x**5; X - -Although it is not a good idea, operators of the same name but taking -different numbers of arguments can be defined. Using a CLEAR -statement on any of these operators clears every one with the same -name, even if the number of arguments is different. - -The CLEAR command is used to ``forget'' matrices, arrays, operators -and scalar variables, returning their identifiers to the pristine -state to be used for other purposes. When CLEAR is applied to array -elements, the contents of the array element becomes the argument for -CLEAR. Thus, you get an error message if the element contains a -number, or some other expression that is not a legal argument to -CLEAR. If the element contains an identifier, it is cleared. When -clear is applied to matrix elements, an error message is returned if -the element evaluates to a number, otherwise there is no effect. Do -NOT try to use CLEAR to set array or matrix elements to 0. You will -not be pleased with the results. - -If you are trying to clear power or product substitution rules made -with either LET or FORALL...LET, you must reproduce the rule, exactly -as you typed it with the same arguments, up to but not including the -equal sign, using the word CLEAR instead of the word LET. This is -shown in the last example. Any other type of LET or FORALL...LET -substitution can be cleared with just the variable or operator name. -MATCH behaves the same as LET in this situation. There is a more -complicated example under FORALL. - -\endsection -\item[CLEARRULES] -CLEARRULES (page 148) - - CLEARRULES list{,list} - -The operator CLEARRULES is used to remove previously defined -RULE lists from the system. list can be an explicit rule -list, or evaluate to a rule list. - -Examples: -trig1 := {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2, - cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, - sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, - cos(~x)^2 => (1+cos(2*x))/2, - sin(~x)^2 => (1-cos(2*x))/2}$ -let trig1; - COS(A - B) + COS(A + B) -cos(a)*cos(b); ------------------------- - 2 -clearrules trig1; -cos(a)*cos(b); COS(A)*COS(B) - -\endsection -\item[COEFF] -COEFF (page 115) - -The COEFF operator returns the coefficients of the powers of the -specified variable in the given expression, in a list. - - COEFF(expression,variable) - -expression is expected to be a polynomial expression, not a rational -expression. Rational expressions are accepted when the switch RATARG -is on. variable must be a kernel. The results are returned in a -list. - -Examples: - 3 2 -coeff((x+y)**3,x); {Y ,3*Y ,3*Y,1} -coeff((x+2)**4 + sin(x),x); {SIN(X) + 16,32,24,8,1} -high_pow; 4 -low_pow; 0 - 7 9 -ab := x**9 + sin(x)*x**7 + sqrt(y); AB := SQRT(Y) + SIN(X)*X + X -coeff(ab,x); {SQRT(Y),0,0,0,0,0,0,SIN(X),0,1} - -The variables HIGH_POW and LOW_POW are set to the highest and lowest -powers of the variable, respectively, appearing in the expression. - -The coefficients are put into a list, with the coefficient of the -lowest (constant) term first. You can use the usual list access -methods (first, second, third, rest, length, and part) to extract -them. If a power does not appear in the expression, the corresponding -element of the list is zero. Terms involving functions of the -specified variable but not including powers of it (for example in the -expression x**4 + 3*x**2 + tan(x)) are placed in the constant term. - -Since the COEFF command deals with the expanded form of the -expression, you may get unexpected results when EXP is off, or when -FACTOR or IFACTOR are on. - -If you want only a specific coefficient rather than all of them, use the -COEFFN operator. - -\endsection -\item[Coefficient] -Coefficient (pages 132, 134) - -REDUCE allows for a variety of numerical domains for the numerical -coefficients of polynomials used in calculations. The default mode is -integer arithmetic, although the possibility of using real -coefficients has been discussed elsewhere. Rational coefficients have -also been available by using integer coefficients in both the -numerator and denominator of an expression, using the ON DIV option to -print the coefficients as rationals. However, REDUCE includes several -other coefficient options in its basic version. - -See ADJPREC, BFSPACE, COMPLEX, MODULAR, PRECISION, PRINT_PRECISION, -RATIONAL, RATIONALIZE, ROUNDALL, ROUNDBF, ROUNDED and SETMOD. - -\endsection -\item[COEFFN] -COEFFN (page 116) - -The COEFFN operator takes three arguments: an expression, a kernel, -and a non-negative integer. It returns the coefficient of the kernel -to that integer power, appearing in the expression. - - COEFFN(expression,kernel,integer) - -expression must be a polynomial, unless RATARG is on which allows -rational expressions. kernel must be a Kernel, and integer must be a -non-negative integer. - -Examples: - -ff := x**7 + sin(y)*x**5 + y**4 + x + 7$ -coeffn(ff,x,5); SIN(Y) -coeffn(ff,z,3); 0 - 5 7 -coeffn(ff,y,0); SIN(Y)*X + X + X + 7 - -rr := 1/y**2+y**3+sin(y); -on ratarg; - -coeffn(rr,y,-2); ***** -2 invalid as COEFFN index - -coeffn(rr,y,5); 1 - ---- - 2 - y - -If the given power of the kernel does not appear in the expression, -COEFFN returns 0. Negative powers are never detected, even if they -appear in the expression and RATARG are on. COEFFN with an integer -argument of 0 returns any terms in the expression that do not contain -the given kernel. - -\endsection -\item[COFACTOR] -COFACTOR (page 166) - -The operator COFACTOR returns the cofactor of the element in row -row and column column of a MATRIX. Errors occur -if row or column do not evaluate to integer expressions or if -the matrix is not square. - - COFACTOR(matrix_expression,row,column) - -Examples: -cofactor(mat((a,b,c),(d,e,f),(p,q,r)),2,2); A*R - C*P -cofactor(mat((a,b,c),(d,e,f)),1,1); ***** non-square matrix - -\endsection -\xitem[COFRAME] -COFRAME (pages 257, 262, 271) - WITH METRIC (page 263 - WITH SIGNATURE (page 263 - -\endsection -\item[COLLECT] -COLLECT (page 57) - -COLLECT is a key word of the FOR construction. Details are given there. - -\endsection -\item[COMBINEEXPT] -COMBINEEXPT (page 77) - -REDUCE is in general poor at surd simplification. However, when the -switch COMBINEEXPT is on, the system attempts to combine -exponentials whenever possible. - -Example: 1/3 1/6 -3^(1/2)*3^(1/3)*3^(1/6); SQRT(3)*3 *3 -on combineexpt; -ws; 1 - -\endsection -\item[COMBINELOGS] -COMBINELOGS (page 77) - -In many cases it is desirable to expand product arguments of logarithms, -or collect a sum of logarithms into a single logarithm. Since these are -inverse operations, it is not possible to provide rules for doing both at -the same time and preserve the REDUCE concept of idempotent evaluation. -As an alternative, REDUCE provides two switches EXPANDLOGS and -COMBINELOGS to carry out these operations. -Examples: - on expandlogs; - log(x*y); LOG(X) + LOG(Y) - on combinelogs; - ws; LOG(X*Y) - -At the present time, it is possible to have both switches on at once, -which could lead to infinite recursion. However, an expression is -switched from one form to the other in this case. Users should not -rely on this behaviour, since it may change in the next release. - -\endsection -\xitem[COMM] -COMM (page 378 - -\endsection -\xitem[Command] -Command (page 67 - -\endsection -\xitem[Command terminator] -Command terminator (page 153 - -\endsection -\item[COMMENT] -COMMENT (page 38) - -Beginning with the word COMMENT, all text until the next statement -terminator (; or $) is ignored. - -Examples: - 2 -x := a**2 comment--a is the velocity of the particle;; X := A - -Note that the first semicolon ends the comment and the second one -terminates the original REDUCE statement. - -Multiple-line comments are often needed in interactive files. The -COMMENT command allows a normal-looking text to accompany the REDUCE -statements in the file. - -\endsection -\item[COMP] -COMP (page 213) - -(Not available in Personal REDUCE} - -When COMP is on, any succeeding function definitions are compiled -into a faster-running form. Default is OFF. - -Examples: -The following procedure finds Fibonacci numbers recursively. -Create a new file ``refib'' in your current directory with the following -lines in it: - -procedure refib(n); - if fixp n and n >= 0 then - if n <= 1 then 1 - else refib(n-1) + refib(n-2) - else rederr "nonnegative integer only"; - -end; - -{Now load REDUCE and run the following:} - -on time; Time: 100 ms - -in "refib"$ Time: 0 ms - - REFIB - - Time: 260 ms - - Time: 20 ms - -refib(80); 37889062373143906 - - Time: 14840 ms - -on comp; Time: 80 ms - -in "refib"$ Time: 20 ms - - REFIB - - Time: 640 ms - -refib(80); 37889062373143906 - - Time: 10940 ms - -Note that the compiled procedure runs faster. Your time messages will -differ depending upon which system you have. Compiled functions -remain so for the duration of the REDUCE session, and are then lost. -They must be recompiled if wanted in another session. With the switch -TIME on as shown above, the CPU time used in executing the command is -returned in milliseconds. Be careful not to leave COMP on unless you -want it, as it makes the processing of procedures much slower. - -\endsection -\item[COMPACT] -COMPACT (pages 179, 241) - -Author: Anthony C. Hearn - -COMPACT is a package of functions for the reduction of a polynomial in -the presence of side relations. COMPACT applies the side relations to -the polynomial so that an equivalent expression results with as few -terms as possible. For example, the evaluation of - - compact(s*(1-sin x^2)+c*(1-cos x^2)+sin x^2+cos x^2, - {cos x^2+sin x^2=1}); - -yields the result - - 2 2 - SIN(X) *C + COS(X) *S + 1 - -\endsection -\item[Compiler] -Compiler (page 213) - -A compiler is available in the Professional REDUCE to convert -functions into a compiled form for faster execution. See the switch -COMP for more details. - -\endsection -\item[COMPLEX] -COMPLEX (pages 135, 372) - -When the COMPLEX switch is on, full complex arithmetic is used in -simplification, function evaluation, and factorisation. Default is OFF. - -Examples: - 2 2 -factorize(a**2 + b**2); {A + B } -on complex; -factorize(a**2 + b**2); {A - I*B,A + I*B} -(x**2 + y**2)/(x + i*y); X - I*Y -on rounded; *** Domain mode COMPLEX changed to COMPLEX_FLOAT -sqrt(-17); 4.12310562562*I -log(7*i); 1.94591014906 + 1.57079632679*I - -Complex floating-point can be done by turning on ROUNDED in addition -to COMPLEX. With COMPLEX off however, REDUCE knows that i is the -square root of -1 but will not carry out more complicated complex -operations. If you want complex denominators cleared by -multiplication by their conjugates, turn on the switch RATIONALIZE. - -\endsection -\xitem[Compound statement] -Compound statement (pages 61, 63 - -\endsection -\xitem[Conditional statement] -Conditional statement (page 56) - -\endsection -\item[CONJ] -CONJ (page 72) - - CONJ(expression) or CONJ simple_expression - -This operator returns the complex conjugate of an expression, if that -argument has an numerical value. A non-numerical argument is returned as -an expression in the operators REPART and IMPART. - -Examples: -conj(1+i); 1-I -conj(a+i*b); REPART(A) - REPART(B)*I - IMPART(A)*I - IMPART(B) - -\endsection -\xitem[Constructor] -Constructor (page 198) - -\endsection -\item[CONT] -CONT (page 160) - -The command CONT returns control to an interactive file after a -PAUSE command that has been answered with N. - -Examples: -Suppose you are in the middle of an interactive file. - factorize(x**2 + 17*x + 60); {X + 5,X + 12} - pause; Cont? (Y or N) - n - saveas results; - factor1 := first results; FACTOR1 := X + 5 - factor2 := second results; FACTOR2 := X + 12 - cont; -....the file resumes - -A PAUSE allows you to enter your own REDUCE commands, change switch -values, inquire about results, or other such activities. When you -wish to resume operation of the interactive file, use CONT. - -\endsection -\xitem[COORDINATES operator] -COORDINATES operator (page 234) - -\endsection -\xitem[COORDS vector] -COORDS vector (page 234) - -\endsection -\xitem[CORFACTOR] -CORFACTOR (page 350) - -\endsection -\item[COS] -COS (pages 76, 78) - -The COS operator returns the cosine of its argument. - - COS(expression) or COS simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -cos abc; COS(ABC) -cos(pi); -1 -cos 4; COS(4) -on rounded; -cos(4); - 0.653643620864 -cos log 5; - 0.0386319699339 - -COS returns a numeric value only if ROUNDED is on. Then the cosine is -calculated to the current degree of floating point precision. - -\endsection -\item[COSH] -COSH (pages 76, 78) - -The COSH operator returns the hyperbolic cosine of its argument. The -derivative of COSH and some simple transformations are known to the -system. - - COSH(expression) or COSH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -cosh b; COSH(B) -cosh(0); 1 -df(cosh(x*y),x); SINH(X*Y)*Y -int(cosh(x),x); SINH(X) - -You may attach further functionality by defining its inverse (see -ACOSH). A numeric value is not returned by COSH unless the switch -ROUNDED is on and its argument evaluates to a number. - -\endsection -\item[COT] -COT (pages 76, 78) - -COT represents the cotangent of its argument. It takes an arbitrary -scalar expression as its argument. The derivative of ACOT and some -simple properties are known to the system. - - COT(expression) or COT simple_expression - -expression may be any scalar REDUCE expression. simple_expression -must be a single identifier or begin with a prefix operator name. - -Examples: -cot(a)*tan(a); COT(A)*TAN(A)) -cot(1); COT(1) - 2 -df(cot(2*x),x); - 2*(COT(2*X) + 1) - -Numerical values of expressions involving COT may be found by -turning on the switch ROUNDED. - -\endsection -\item[COTH] -COTH (pages 76, 78) - -The COTH operator returns the hyperbolic cotangent of its argument. -The derivative of COTH and some simple transformations are known to -the system. - - COTH(expression) or COTH simple_expression - -expression may be any scalar REDUCE expression. simple_expression -must be a single identifier or begin with a prefix operator name. - -Examples: - 2 -df(coth(x*y),x); - Y*(COTH(X*Y) - 1) -coth acoth z; Z - -You can write LET statements and procedures to add further -functionality to COTH if you wish. Numerical values of expressions -involving COTH may also be found by turning on the switch ROUNDED. - -\endsection -\item[CRAMER] -CRAMER (pages 85, 163) - -When the CRAMER switch is on, MATRIX inversion and linear equation -solving (operator SOLVE) is done by Cramer's rule, through exterior -multiplication. Default is OFF. - -Examples: -on time; Time: 80 ms -off output; Time: 100 ms -mm := mat((a,b,c,d,f),(a,a,c,f,b), - (b,c,a,c,d), (c,c,a,b,f), - (d,a,d,e,f)); - Time: 300 ms -inverse := 1/mm; Time: 18460 -on cramer; Time: 80 ms -cramersinv := 1/mm; Time: 9260 MS - -Your time readings will vary depending on the REDUCE version you use. -After you invert the matrix, turn on OUTPUT and ask for one of the -elements of the inverse matrix, such as CRAMERSINV(3,2), so that you -can see the size of the expressions produced. - -Inversion of matrices and the solution of linear equations with dense -symbolic entries in many variables is generally considerably faster -with CRAMER on. However, inversion of numeric-valued matrices is -slower. Consider the matrices you're inverting before deciding -whether to turn CRAMER on or off. A substantial portion of the time -in matrix inversion is given to formatting the results for printing. -To save this time, turn OUTPUT off, as shown in this example or -terminate the expression with a dollar sign instead of a semicolon. -The results are still available to you in the workspace associated -with your prompt number, or you can assign them to an identifier for -further use. - -\endsection -\item[CREF] -CREF (pages 215, 216) - -The switch CREF invokes the CREF cross-reference program that -processes a set of procedure definitions to produce a summary of their -entry points, undefined procedures, non-local variables and so on. The -program will also check that procedures are called with a consistent -number of arguments, and print a diagnostic message otherwise. - -The output is alphabetised on the first seven characters of each function -name. - -To invoke the cross-reference program, CREF is first turned on. -This causes the program to load and the cross-referencing process to -begin. After all the required definitions are loaded, turning CREF -off will cause a cross-reference listing to be produced. - - - -Algebraic procedures in REDUCE are treated as if they were symbolic, so -that algebraic constructs will actually appear as calls to symbolic -functions, such as AEVAL. - -\endsection -\xitem[CRESYS] -CRESYS (pages 378, 380) - -\endsection -\xitem[CROSS] -CROSS - vector (page 233) - -\endsection -\xitem[cross product] -cross product (pages 233, 357) - -\endsection -\xitem[Cross reference] -Cross reference (page 215) - -\endsection -\item[CSC] -CSC (pages 76, 78) - -The CSC operator returns the cosecant of its argument. The derivative -of CSC and some simple transformations are known to the system. - - CSC(expression) or CSC simple_expression - -expression may be any scalar REDUCE expression. simple_expression -must be a single identifier or begin with a prefix operator name. - -Examples: -csc(q)*sin(q); CSC(Q)*SIN(Q) -df(csc(x*y),x); -COT(X*Y)*CSC(X*Y)*Y - - -You can write LET statements and procedures to add further -functionality to CSC if you wish. Numerical values of expressions -involving CSC may also be found by turning on the switch ROUNDED. - -\endsection -\item[CSCH] -CSCH (pages 76, 78) - -The COSH operator returns the hyperbolic cosecant of its argument. -The derivative of CSCH and some simple transformations are known to -the system. - - CSCH(expression) or CSCH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -csch b; CSCH(B) -csch(0); 0 -df(csch(x*y),x); - COTH(X*Y)*CSCH(X*Y)*Y -int(csch(x),x); INT(CSCH(X),X) - -A numeric value is not returned by CSCH unless the switch ROUNDED is -on and its argument evaluates to a number. - -\endsection -\xitem[CURL operator] -CURL operator (page 234) - -\endsection -\xitem[curl vector field] -curl vector field (page 234) - -\endsection -\xitem[curl operator] -curl operator (page 358) - -\endsection -\xitem[cylindrical coordinates] -cylindrical coordinates (page 355) - -\endsection -\xitem[d exterior differentiation] -d - exterior differentiation (page 271) - -\endsection -\xitem[Declaration] -Declaration (page 67) - -\endsection -\item[DECOMPOSE] -DECOMPOSE (page 127) - -The DECOMPOSE operator takes a multivariate polynomial as argument, -and returns an expression and a LIST of EQUATIONs from which the -original polynomial can be found by composition. - - DECOMPOSE(expression) or DECOMPOSE simple_expression - -Examples: -decompose(x^8-88*x^7+2924*x^6-43912*x^5+263431*x^4- - 218900*x^3+65690*x^2-7700*x+234) - 2 - {U + 35*U + 234, - - 2 - U=V + 10*V, - - 2 - V=X - 22*X } - - 2 -decompose(u^2+v^2+2u*v+1); {W + 1,W=U + V} - -Unlike factorisation, this decomposition is not unique. Further -details can be found in V.S. Alagar, M.Tanh, Fast Polynomial -Decomposition, Proc. EUROCAL 1985, pp 150-153 (Springer) and J. von -zur Gathen, Functional Decomposition of Polynomials: the Tame Case, J. -Symbolic Computation (1990) 9, 281-299. - -\endsection -\item[DEFINE] -DEFINE (page 70) - -The command DEFINE allows you to supply a new name for an identifier -or replace it by any valid REDUCE expression. - - DEFINE identifier = substitution {,identifier = substitution} - -identifier is any valid REDUCE identifier, substitution can be a -number, an identifier, an operator, a reserved word, or an expression. - -Examples: - -define is= :=, xx=y+z; -a is 10; A := 10 - 2 2 -xx**2; Y + 2*Y*Z + Z - -xx := 10; Y + Z := 10 - -The renaming is done at the input level, and therefore takes precedence -over any other replacement or substitution declared for the same identifier. -It remains in effect until the end of the REDUCE session. Be careful with -it, since you cannot easily undo it without ending the session. - -\endsection -\xitem[definite integration (simple)] -definite integration (simple) (page 236) - -\endsection -\xitem[DEFINT function] -DEFINT function (page 236) - -\endsection -\xitem[DEFLINEINT function] -DEFLINEINT function (page 238) - -\endsection -\item[DEFN] -DEFN (pages 197, 218) - -When the switch DEFN is on, the Standard Lisp equivalent of the -input statement or procedure is printed, but not evaluated. Default is -OFF. - -Examples: - -on defn; -17/3; (AEVAL (LIST 'QUOTIENT 17 3)) - -df(sin(x),x,2); (AEVAL (LIST 'DF (LIST 'SIN 'X) 'X 2)) -procedure coshval(a); - begin scalar g; - g := (exp(a) + exp(-a))/2; - return g - end; (AEVAL - (PROGN - (FLAG '(COSHVAL) 'OPFN) - (DE COSHVAL (A) - (PROG (G) - (SETQ G - (AEVAL - (LIST - 'QUOTIENT - (LIST - 'PLUS - (LIST 'EXP A) - (LIST 'EXP (LIST 'MINUS A))) - 2))) - (RETURN G)))) ) -coshval(1); (AEVAL (LIST 'COSHVAL 1)) -off defn; -coshval(1); Declare COSHVAL operator? (Y or N) -n -procedure coshval(a); - begin scalar g; - g := (exp(a) + exp(-a))/2; - return g - end; COSHVAL -on rounded; -coshval(1); 1.54308063482 - -The above function COSHVAL finds the hyperbolic cosine (cosh) of its -argument. When DEFN is on, you can see the Standard Lisp equivalent -of the function, but it is not entered into the system as shown by the -message DECLARE COSHVAL OPERATOR?. It must be reentered with DEFN off -to be recognised. This procedure is used as an example; a more -efficient procedure would eliminate the unnecessary local variable -with - procedure coshval(a); - (exp(a) + exp(-a))/2; - -\endsection -\item[DEFPOLY] -DEFPOLY statement (page 225) - -DEFPOLY is used to introduce a defining polynoimial for an algebraic -number. For example, to define an atom to stand for teh square root -of 2 one would say - - load arnum; - defpoly sqrt2^2 -2; - -This associates a simplification function for the variable and also -generates a power reduction rule used by the operations * and / for -the reduction of their result modulo the defining polynomial. A basis -for the representation of an algebraic number is also set up by the -statement. If the defining polynomial is not monic, it will be made -so by an appropriate substitution. - -\endsection -\item[DEG] -DEG (page 128) - -The operator DEG returns the highest degree of its variable argument -found in its expression argument. - - DEG(expression,kernel) - -expression is expected to be a polynomial expression, not a rational -expression. Rational expressions are accepted when the switch RATARG -is on. variable must be a Kernel. The results are returned in a -list. - -Examples: -deg((x+y)**5,x); 5 -deg((a+b)*(c+2*d)**2,d); 2 -deg(x**2 + cos(y),sin(x)); -deg((x**2 + sin(x))**5,sin(x)); 5 - -\endsection -\xitem[Degree] -Degree (page 128) - -\endsection -\xitem[DELSQ operator] -DELSQ - operator (page 234) - -\endsection -\xitem[delsq operator] -delsq operator (page 358) - -\endsection -\item[DEMO] -DEMO (page 69) - -The DEMO switch is used for interactive files, causing the system -to pause after each command in the file until you type a Return. -Default is OFF. - -The switch DEMO has no effect on top level interactive statements. -Use it when you want to slow down operations in a file so you can see -what is happening. - -You can either include the ON DEMO command in the file, or enter it -from the top level before bringing in any file. Unlike the PAUSE -command, ON DEMO does not permit you to interrupt the file for -questions of your own. - -\endsection -\item[DEN] -DEN (pages 120, 129) - -The DEN operator returns the denominator of its argument. - - DEN(expression) - -expression is ordinarily a rational expression, but may be any valid -scalar REDUCE expression. - -Examples: - 2 -a := x**3 + 3*x**2 + 12*x; A := X*(X + 3*X + 12) -b := 4*x*y + x*sin(x); B := X*(SIN(X) + 4*Y) -den(a/b); SIN(X) + 4*Y -den(aa/4 + bb/5); 20 -den(100/6); 3 -den(sin(x)); 1 - -DEN returns the denominator of the expression after it has been -simplified by REDUCE. As seen in the examples, this includes putting -sums of rational expressions over a common denominator, and reducing -common factors where possible. If the expression does not have any -other denominator, 1 is returned. - -Switch settings, such as MCD or RATIONAL, have an effect on the -denominator of an expression. - -\endsection -\item[DEPEND] -DEPEND (page 95) - -DEPEND declares that its first argument depends on the rest of its -arguments. - - DEPEND kernel{,kernel} - -kernel must be a legal variable name or a prefix operator (see -Kernel). - -Examples: - -depend y,x; -df(y**2,x); 2*DF(Y,X)*Y -depend z,cos(x),y; -df(sin(z),cos(x)); COS(Z)*DF(Z,COS(X)) -df(z**2,x); 2*DF(Z,X)*Z -nodepend z,y; -df(z**2,x); 2*DF(Z,X)*Z -cc := df(y**2,x); CC := 2*DF(Y,X)*Y -y := tan x; Y := TAN(X); - 2 -cc; 2*TAN(X)*(TAN(X) + 1) - -Dependencies can be removed by using the declaration NODEPEND. The -differentiation operator uses this information, as shown in the -examples above. Linear operators also use knowledge of dependencies -(see LINEAR). Note that dependencies can be nested: Having declared y -to depend on x, and z to depend on y, we see that the chain rule was -applied to the derivative of a function of z with respect to x. If -the explicit function of the dependency is later entered into the -system, terms with DF(Y,X), for example, are expanded when they are -displayed again, as shown in the last example. - -\endsection -\xitem[DEPEND statement] -DEPEND statement (page 359) - -\endsection -\xitem[DEQ(I)] -DEQ(I) (page 379) - -\endsection -\xitem[derivative variational] -derivative - variational (page 257) - -\endsection -\item[DET] -DET (pages 97, 163) - -The operator COFACTOR returns the cofactor of the element in row -row and column column of a MATRIX. Errors occur -if row or column do not evaluate to integer expressions or if -the matrix is not square. - - COFACTOR(matrix_expression,row,column) - -Examples: -cofactor(mat((a,b,c),(d,e,f),(p,q,r)),2,2); A*R - C*P -cofactor(mat((a,b,c),(d,e,f)),1,1); ***** non-square matrix - -\endsection -\xitem[determinant] -determinant - in DETM!* (page 263) - -\endsection -\xitem[DETM!*] -DETM!* (page 263) - -\endsection -\item[DF] -DF (pages 79, 80) - -The DF operator finds partial derivatives with respect to one or -more variables. - - DF(expression,var - [,number] - {,var [ ,number] } ) - -expression can be any valid REDUCE algebraic expression. var must be -a Kernel, and is the differentiation variable. number must be a -non-negative integer. - -Examples: -df(x**2,x); 2*X - 2 -df(x**2*y + sin(y),y); COS(Y) + X - -df((x+y)**10,z); 0 - 6 -df(1/x**2,x,2); ---- - 4 - X -df(x**4*y + sin(y),y,x,3); 24*X - -for all x let df(tan(x),x) = sec(x)**2; - 2 -df(tan(3*x),x); 3*SEC(3*X) - -An error message results if a non-kernel is entered as a -differentiation operator. If the optional number is omitted, it is -assumed to be 1. See the declaration DEPEND to establish dependencies -for implicit differentiation. - -You can define your own differentiation rules, expanding REDUCE's -capabilities, using the LET command as shown in the last example -above. Note that once you add your own rule for differentiating a -function, it supersedes REDUCE's normal handling of that function for -the duration of the REDUCE session. If you clear the rule -(CLEARRULES), you don't get back to the previous rule. - -\endsection -\item[DFPRINT] -DFPRINT - -When DFPRINT is on, expressions in the differentiation operator -DF are printed in a more ``natural'' notation, with the -differentiation variables appearing as subscripts. In addition, if the -switch NOARG is on (the default), the arguments of the -differentiated operator are suppressed. - -Examples: -operator f; -df(f x,x); DF(F(X),X); -on dfprint; -ws; F - X -df(f(x,y),x,y); F - X,Y -off noarg; -ws; F(X,Y) - X - -\endsection -\xitem[differential geometry] -differential geometry (page 248) - -\endsection -\xitem[Differentiation] -Differentiation (pages 79, 80, 95) - -\endsection -\xitem[differentiation] -differentiation - partial (page 251) - vector (page 233) - -\endsection -\item[DIGAMMA] -DIGAMMA (page 185, 395) - -See PSI -\endsection -\item[DILOG] -DILOG (pages 76, 81, 185) - -The DILOG operator is known to the differentiation and integration -operators, but has numeric value attached only at DILOG(0). DILOG is -defined by - log(x) - dilog(x) = -int ------ dx - x-1 - - dilog(x) = -int(log(x),x)/(x-1) - -Examples: 2 2 -df(dilog(x**2),x); - (2*LOG(X )*X)/(X - 1) - -int(dilog(x),x); DILOG(X)*X - DILOG(X) + LOG(X)*X - X - 2 -dilog(0); PI /6 - -\endsection -\xitem[dimension] -dimension (page 251) - -\endsection -\xitem[Dirac gamma matrix] -Dirac gamma matrix (page 206) - -\endsection -\item[DISPLAY] -DISPLAY (page 158)) - -When given a numeric argument n, DISPLAY prints the n most recent -input statements, identified by prompt numbers. If an empty pair of -parentheses is given, or if n is greater than the current number of -statements, all the input statements since the beginning of the -session are printed. - - DISPLAY(n) or DISPLAY() - -n should be a positive integer. However, if it is a real number, the -truncated integer value is used, and if a non-numeric argument is -used, all the input statements are printed. - -The statements are displayed in upper case, with lines split at -semicolons or dollar signs, as they are in editing. If long files -have been input during the session, the DISPLAY command is slow to -format these for printing. - -\endsection -\xitem[Display] -Display (page 97) - -\endsection -\xitem[DISPLAYFRAME command] -DISPLAYFRAME command (pages 266, 271) - -\endsection -\xitem[Displaying structure] -Displaying structure (page 112) - -\endsection -\item[DIV] -DIV (pages 103, 132) - -When DIV is on, the system divides any simple factors found in the -denominator of an expression into the numerator. Default is OFF. - -Examples: - -on div; - 2 -2 -a := x**2/y**2; A := X *Y - 1 2 -2 -1 -b := a/(3*z); B := ---*X *Y *Z - 3 -off div; - 2 - X -a; ---- - 2 - Y - - 2 - X -b; -------- - 2 - 3*Y *Z - -The DIV switch only has effect when the PRI switch is on. When PRI is -off, regardless of the setting of DIV, the printing behaviour is as if -DIV were off. - -\endsection -\xitem[DIV operator] -DIV - operator (page 234) - -\endsection -\xitem[div operator] -div operator (page 358) - -\endsection -\xitem[divergence vector field] -divergence - vector field (page 234) - -\endsection -\xitem[DLINEINT] -DLINEINT (page 360) - -\endsection -\xitem[DO] -DO (pages 57--59) - -\endsection -\xitem[Dollar sign] -Dollar sign (page 53) - -\endsection -\item[DOT] -DOT product of vectors (pages 205, 233, 357) - -The . operator is used to denote the scalar product of two Lorentz -four-vectors. - vector . vector - -vector must be an identifier declared to be of type VECTOR to have -the scalar product definition. When applied to arguments that are not -vectors, the CONS operator is used, -whose symbol is also ``dot.'' - -Examples: -vector aa,bb,cc; -let aa.bb = 0; -aa.bb; 0 -aa.cc; AA.CC -q := aa.cc; Q := AA.CC -q; AA.CC - -Since vectors are special high-energy physics entities that do not -contain values, the . product will not return a true scalar product. -You can assign a scalar identifier to the result of a . operation, or -assign a . operation to have the value of the scalar you supply, as -shown above. Note that the result of a . operation is a scalar, not a -vector. - -The metric tensor g(u,v) can be represented by U.V. If contraction -over the indices is required, U and V should be declared to be of type -INDEX. - -The dot operator has the highest precedence of the infix operators, so -expressions involving . and other operators have the scalar product -evaluated first before other operations are done. - -\endsection -\xitem[Dot product] -Dot product (pages 205, 233, 357) - -\endsection -\xitem[DOTGRAD operator] -DOTGRAD operator (page 358) - -\endsection -\xitem[DVINT] -DVINT (page 360) - -\endsection -\xitem[DVOLINT] -DVOLINT (page 360) - -\endsection -\item[E] -E (page 36) - -The constant E is reserved for use as the base of the natural -logarithm. Its value is approximately 2.71828284590, which REDUCE gives -to the current decimal precision when the switch ROUNDED is on. - - -E may be used as an iterative variable in a FOR statement, -or as a local variable or a PROCEDURE. If E is defined as a local -variable inside the procedure, the normal definition as the base of the -natural logarithm would be suspended inside the procedure. - -\endsection -\item[ECHO] -ECHO (page 153) - -The ECHO switch is normally off for top-level entry, and on when files -are brought in. If ECHO is turned on at the top level, your input -statements are echoed to the screen (thus appearing twice). Default -OFF (but note default ON for files). - - -If you want to display certain portions of a file and not others, use the -commands OFF ECHO and ON ECHO inside the file. If you want -no display of the file, use the input command - - IN filename$ - -rather than using the semicolon delimiter. - -Be careful when you use commands within a file to generate another file. -Since ECHO is on for files, the output file echoes input statements -(unlike its behaviour from the top level). You should explicitly turn off -ECHO when writing output, and turn it back on when you're done. - -\endsection -\item[ED] -ED (pages 157, 158) - -The ED command invokes a simple line editor for REDUCE input -statements. - - ED integer or ED - -ED called with no argument edits the last input statement. If integer -is greater than or equal to the current line number, an error message -is printed. Reenter a proper ED command or return to the top level -with a semicolon. - -The editor formats REDUCE's version of the desired input statement, -dividing it into lines at semicolons and dollar signs. The statement -is printed at the beginning of the edit session. The editor works on -one line at a time, and has a pointer (shown by ^) to the current -character of that line. When the session begins, the pointer is at -the left hand side of the first line. The editing prompt is >. - -The following commands are available. They may be entered in either -upper or lower case. All commands are activated by the carriage -return, which also prints out the current line after changes. Several -commands can be placed on a single line, except that commands -terminated by an Cntrl-G must be the last command before the carriage -return. - -b -Move pointer to beginning of current line. - -ddigit -Delete current character and next (digit-1) characters. An error -message is printed if anything other than a single digit follows d. -If there are fewer than digit characters left on the line, all but the -final dollar sign or semicolon is removed. To delete a line -completely, use the k command. - -e -End the current session, causing the edited expression to be reparsed by -REDUCE. - -fchar -Find the next occurrence of the character char to the right of the -pointer on the current line and move the pointer to it. If the -character is not found, an error message is printed and the pointer -remains in its original position. Other lines are not searched. The -f command is not case-sensitive. - -istring{Cntrl-G} -Insert string in front of pointer. The Cntrl-G key is your delimiter for -the input string. No other command may follow this one on the same -line. - -k -Kill rest of the current line, including the semicolon or dollar sign -terminator. If there are characters remaining on the current line, and it -is the last line of the input statement, a semicolon is added to the line -as a terminator for REDUCE. If the current line is now empty, one of the -following actions is performed: If there is a following line, it becomes -the current line and the pointer is placed at its first character. If the -current line was the final line of the statement, and there is a previous -line, the previous line becomes the current line. If the current line was -the only line of the statement, and it is empty, a single semicolon is -inserted for REDUCE to parse. - -l -Finish editing this line and move to the last previous line. An error message -is printed if there is no previous line. - -n -Finish editing this line and move to the next line. An error message is -printed if there is no next line. - -p -Print out all the lines of the statement. Then a dotted line is printed, and -the current line is reprinted, with the pointer under it. - -q -Quit the editing session without saving the changes. If a semicolon is -entered after q, a new line prompt is given, otherwise REDUCE prompts you -for another command. Whatever you type in to the prompt appearing after -the q is entered is stored as the input for the line number in which you -called the edit. Thus if you enter a semicolon, neither INPUT -ED will find anything under the current number. - -rchar -Replace the character at the pointer by char. - -sstring{Cntrl-G} -Search for the first occurrence of string to the right of the -pointer on the current line and move the pointer to its first character. -The Cntrl-G key is your delimiter for the input string. The s function -does not search other lines of the statement. If the string is not found, -an error message is printed and the pointer remains in its original -position. The s command is not case-sensitive. No other command may -follow this one on the same line. - -x or space -Move the pointer one character to the right. If the pointer is already at -the end of the line, an error message is printed. - -- (minus) -Move the pointer one character to the left. If the pointer is already at the -beginning of the line, an error message is printed. - -? -Display the Help menu, showing the commands and their actions. - -Examples: -(Line numbers are shown in the following examples) - 2 -2: x**2 + y; X + Y -3: ed 2; - X**2 + Y; - ^ -For help, type '?' -?- {(Enter three spaces and Return})} - X**2 + Y; - ^ -?- r5 - X**5 + Y; - ^ -?- fY - X**5 + Y; - ^ -?- iabc{(Terminate with Cntrl-G and Return)} - X**5 + abcY; - ^ -?- ---- - X**5 + abcY; - ^ -?- fbd2 - X**5 + aY; - ^ -?- b - X**5 + aY; - ^ 5 -?- e AY + X -4: procedure dumb(a); - write a; -DUMB -5: dumb(17); 17 -6: ed 4; - PROCEDURE DUMB (A); - ^ -WRITE A; -?- fArBn - WRITE A; - ^ -?- ibegin scalar a; a := b + 10;{space Cntrl-G and Return} - begin scalar a; a := b + 10; WRITE A; -?- f;i end {Cntrl-G Return} - begin scalar b; b := a + 10; WRITE A end; - ^ -?- p - PROCEDURE DUMB (B); - begin scalar b; b := a + 10; WRITE A end; - - - - - - - - - - - - begin scalar b; b := a + 10; WRITE A end; - ^ -?- e DUMB -7: dumb(17); 27 -8: - -Note that REDUCE reparsed the procedure DUMB and updated the -definition. - -Since REDUCE divides the expression to be edited into lines at -semicolons or dollar sign terminators, some lines may occupy more than -one line of screen space. If the pointer is directly beneath the last -line of text, it refers to the top line of text. If there is a blank -line between the last line of text and the pointer, it refers to the -second line of text, and likewise for cases of greater than two lines -of text. In other words, the entire REDUCE statement up to the next -terminator is printed, even if it runs to several lines, then the -pointer line is printed. - -You can insert new statements which contain semicolons of their own -into the current line. They are run into the current line where you -placed them until you edit the statement again. REDUCE will -understand the set of statements if the syntax is correct. - -If you leave out needed closing brackets when you exit the editor, a -message is printed allowing you to redo the edit (you can edit the -previous line number and return to where you were). If you leave out -a closing double-quotation mark, an error message is printed, and the -editing must be redone from the original version; the edited version -has been destroyed. Most syntax errors which you inadvertently leave -in an edited statement are caught as usual by the REDUCE parser, and -you will be able to re-edit the statement. - -When the editor processes a previous statement for your editing, -escape characters are removed. Most special characters that you may -use in identifiers are printed in legal fashion, prefixed by the -exclamation point. Be sure to treat the special character and its -escape as a pair in your editing. The characters ( ) # ; ' ` are -different. Since they have special meaning in Lisp, they are -double-escaped in the editor. It is unwise to use these characters -inside identifiers anyway, due to the probability of confusion. - -If you see a Lisp error message during editing, the edit has been -aborted. Enter a semicolon and you will see a new line prompt. - -Since the editor has no dependence on any window system, it can be -used if you are running REDUCE without windows. - -\endsection -\item[EDITDEF] -EDITDEF (page 159) - -The interactive editor ED may be used to edit a user-defined -procedure that has not been compiled. - - EDITDEF(identifier) - -where identifier is the name of the procedure. When EDITDEF is -invoked, the procedure definition will be displayed in editing mode, -and may then be edited and redefined on exiting from the editor using -standard ED commands. - -\endsection -\item[END] -END (page 69) - -The command END has two main uses: - -(i) as the ending of a BEGIN...END BLOCK; and -(ii) to end input from a file. - -In a BEGIN...END BLOCK, there need not be a delimiter (; or $) before -the END, though there must be one after it, or a right bracket -matching an earlier left bracket. - -Files to be read into REDUCE should end with END;, which must be -preceded by a semicolon (usually the last character of the previous -line). The additional semicolon avoids problems with mistakes in the -files. If you have suspended file operation by answering N to a PAUSE -command, you are still, technically speaking, ``in'' the file. Use END -to exit the file. - -An END at the top level of a program is ignored. - -\endsection -\item[EPS] -EPS (pages 207, 267) - -The EPS operator denotes the completely antisymmetric tensor of -order 4 and its contraction with Lorentz four-vectors, as used in -high-energy physics calculations. - - EPS(vector-expr,vector-expr,vector-expr,vector-expr) - -vector-expr must be a valid vector expression, and may be an index. - -Examples: -vector g0,g1,g2,g3; -eps(g1,g0,g2,g3); - EPS(G0,G1,G2,G3); -eps(g1,g2,g0,g3); EPS(G0,G1,G2,G3); -eps(g1,g2,g3,g1); 0 - - -Vector identifiers are ordered alphabetically by REDUCE. When an odd -number of transpositions is required to restore the canonical order to -the four arguments of EPS, the term is ordered and carries a minus -sign. When an even number of transpositions is required, the term is -returned ordered and positive. When one of the arguments is repeated, -the value 0 is returned. A contraction of the form eps(_i j mu nu -p_mu q_nu) is represented by EPS(I,J,P,Q) when I and J have been -declared to be of type INDEX. - -\endsection -\xitem[EPS Levi-Civita tensor] -EPS - Levi-Civita tensor (page 271) - -\endsection -\item[Equation] -Equation (page 47) - -An Equation is an expression where two algebraic expressions -are connected by the (infix) operator EQUAL or by =. -For access to the components of an EQUATION the operators -LHS, RHS or PART can be used. The -evaluation of the left-hand side of an EQUATION is controlled -by the switch EVALLHSEQP, while the right-hand side is -evaluated unconditionally. When an EQUATION is part of a -logical expression, e.g. in a IF or WHILE statement, -the equation is evaluated by subtracting both sides and comparing -the result with zero. - -\endsection -\item[ERF] -ERF (page 81) - -The ERF operator represents the error function, defined by - erf(x) = (2/sqrt(pi))*int(e^(-x^2),x) - -A limited number of its properties are known to the system, including -the fact that it is an odd function. Its derivative is known, and -from this, some integrals may be computed. However, a complete -integration procedure for this operator is not currently included. - -Examples: -erf(0); 0 -erf(-a); - ERF(A) - 4*SQRT(PI)*X -df(erf(x**2),x); -------------- - 4 - X - - 2 - X - E *ERF(X)*PI*X + SQRT(PI) -int(erf(x),x); ---------------------------- - 2 - X - E *PI - -\endsection -\item[ERRCONT] -ERRCONT (page 157) - -When the ERRCONT switch is on, error conditions do not stop file -execution. Error messages will be printed whether ERRCONT is on or off. -Default is OFF. - -The table below shows REDUCE behaviour under the settings of ERRCONT and -INT : - -Behaviour in Case of Error in Files - -errcont int Behaviour when errors in files are encountered - off off Message is printed and parsing continues, but - no further statements are executed; no commands - from keyboard accepted except bye or end - off on Message is printed, and you are asked if you - wish to continue. (This is the default behaviour) - on off Message is printed, and file continues to execute - without pause - on on Message is printed, and file continues to execute - without pause - - -\endsection -\xitem[ETA(ALFA)] -ETA(ALFA) (page 379) - -\endsection -\xitem[euclidean metric] -euclidean metric (page 263) - -\endsection -\item[EULER] -EULER (pages 185, 393) - -The EULER operator returns the nth Euler number. - -EULER(integer) - -Examples: -load_package specfn; (SPECFN) -Euler 20; 370371188237525 -Euler 0; 1 - -The EULER numbers are evaluated by a recursive algorithm which makes -it hard to compute Euler numbers above say 200. - -Euler numbers appear in the coefficients of the power series -representation of 1/cos(z). - -\endsection -\item[EULERP] -Euler Polynomials (page 185) - -The EULERP operator returns the nth Euler Polynomial. - -EULERP(integer,expression) - -Examples: - load_package specfn; (SPECFN) - EulerP(2,xx); XX*(XX - 1) - EulerP(10,3); 2046 - -The Euler numbers are the values of the Euler Polynomials at 1/2 -multiplied by 2**n. - -\endsection -\item[Euler Numbers] -Euler Numbers (pages 185, 393) - -See EULERP. -The Euler numbers are the values of the Euler Polynomials at 1/2 -multiplied by 2**n. - -\endsection -\item[EVAL_MODE] -EVAL_MODE (page 191) - -The constant E is reserved for use as the base of the natural -logarithm. Its value is approximately 2.71828284590, which REDUCE gives -to the current decimal precision when the switch ROUNDED is on. - - -E may be used as an iterative variable in a FOR statement, -or as a local variable or a PROCEDURE. If E is defined as a local -variable inside the procedure, the normal definition as the base of the -natural logarithm would be suspended inside the procedure. - -\endsection -\item[EVALLHSEQP] -EVALLHSEQP (page 47) - -Under normal circumstances, the right-hand-side of an EQUATION is evaluated -but not the left-hand-side. If both sides are to be evaluated, the switch -EVALLHSEQP should be turned on. - -\endsection -\item[EVEN] -EVEN (page 90) - - EVEN identifier{,identifier} - -This declaration is used to declare an operator even in its first -argument. Expressions involving an operator declared in this manner -are transformed if the first argument contains a minus sign. Any -other arguments are not affected. - -Examples: - even f; - f(-a) F(A) - f(-a,-b) F(A,-B) - -\endsection -\xitem[Even operator] -Even operator (page 90) - -\endsection -\item[EVENP] -EVENP (page 46) - -The EVENP logical operator returns TRUE if its argument is an even -integer, and NIL if its argument is an odd integer. An error message -is returned if its argument is not an integer. - - EVENP(integer) or EVENP integer - -integer must evaluate to an integer. - -Examples: -aa := 1782; AA := 1782 -if evenp aa then yes else no; YES -if evenp(-3) then yes else no; NO - -Although you would not ordinarily enter an expression such as the last -example above, note that the negative term must be enclosed in -parentheses to be correctly parsed. The EVENP operator can only be -used in conditional statements such as IF...THEN...ELSE or WHILE...DO. - -\endsection -\item[EXCALC] -EXCALC (pages 180, 247) - -Author: Eberhard Schruefer - -The EXCALC package is designed for easy use by all who are familiar -with the calculus of Modern Differential Geometry. The program is currently -able to handle scalar-valued exterior forms, vectors and operations between -them, as well as non-scalar valued forms (indexed forms). It is thus an ideal -tool for studying differential equations, doing calculations in general -relativity and field theories, or doing simple things such as calculating the -Laplacian of a tensor field for an arbitrary given frame. - -\endsection -\xitem[Exclamation mark] -Exclamation mark (page 33) - -\endsection -\xitem[EXCLUDE] -EXCLUDE (page 368) - -\endsection -\xitem[EXDEGREE] -EXDEGREE (page 271) - -\endsection -\xitem[EXDEGREE command] -EXDEGREE command (page 249) - -\endsection -\item[EXP] -EXP (operator and switch) (pages 76, 78, 81, 120, 124) - -The EXP operator returns E raised to the power of its argument. - - EXP(expression) or EXP simple_expression - -expression can be any valid REDUCE scalar expression. -simple_expression must be a single identifier or begin with a -prefix operator. - -Examples: - SIN X -exp(sin(x)); E - 11 -exp(11); E -on rounded; -exp sin(pi/3); 2.37744267524 - -Numeric values are returned only when ROUNDED is on. The single -letter E with the exponential operator ^ or ** may be substituted for -EXP without change of function. - -EXP switch - -When the EXP switch is on, powers and products of expressions are -expanded. Default is ON. - -Examples: 3 2 -(x+1)**3; X + 3*X + 3*X + 1 -(a + b*i)*(c + d*i); A*C + A*D*I + B*C*I - B*D -off exp; 3 -(x+1)**3; (X + 1) -(a + b*i)*(c + d*i); (A + B*I)*(C + D*I) -length((x+1)**2/(y+1)); 2 - - -Note that REDUCE knows that i^2 = -1. When EXP is off, equivalent -expressions may not simplify to the same form, although zero -expressions still simplify to zero. Several operators that expect a -polynomial argument behave differently when EXP is off, such as -LENGTH. Be cautious about leaving EXP off. - - -\endsection -\item[EXPAND_CASES] -EXPAND_CASES (page 86) - -When a ROOT_OF form in a result of SOLVE has been converted to a -ONE_OF form, EXPAND_CASES can be used to convert this into form -corresponding to the normal explicit results of SOLVE. See ROOT_OF. - -\endsection -\item[EXPANDLOGS] -EXPANDLOGS (page 77) - -In many cases it is desirable to expand product arguments of -logarithms, or collect a sum of logarithms into a single logarithm. -Since these are inverse operations, it is not possible to provide -rules for doing both at the same time and preserve the REDUCE concept -of idempotent evaluation. As an alternative, REDUCE provides two -switches EXPANDLOGS and COMBINELOGS to carry out these operations. -Both are off by default. - -Examples: - on expandlogs; - log(x*y); LOG(X) + LOG(Y) - on combinelogs; - ws; LOG(X*Y) - -At the present time, it is possible to have both switches on at once, -which could lead to infinite recursion. However, an expression is -switched from one form to the other in this case. Users should not rely -on this behaviour, since it may change in the next release. - -\endsection -\item[EXPINT] -EXPINT (page 76) - -The EXPINT operator represents the exponential integral defined by: - - expint(x) = int(e^x,x)/x - -A limited number of its properties are known to the system, including -its derivative. From this, some integrals may be computed. However, -a complete integration procedure for this operator is not currently -included. - -Examples: -expint(0); EXPINT(0) - 2 - X - 2*E -df(expint(x**2),x); ------- - X - X -int(expint(x),x); EXPINT(X)*X - E - -\endsection -\xitem[EXPR] -EXPR (page 196) - -\endsection -\xitem[Expression] -Expression (page 43) - -\endsection -\item[exterior calc] -exterior calculus (page 248) - -See the EXCALC package - -\endsection -\item[exterior df] -exterior differentiation (page 252) - -See the EXCALC package - -\endsection -\xitem[exterior form] -exterior form - declaration (page 249) - vector (page 249) - with indices (pages 249, 259) - -\endsection -\xitem[exterior product] -exterior product (pages 250, 269) - -\endsection -\item[EZGCD] -EZGCD (page 124) - -When EZGCD and GCD are on, greatest common divisors are -computed using the EZ GCD algorithm that uses modular arithmetic (and is -usually faster). Default is OFF. - - -As a side effect of the gcd calculation, the expressions involved are -factored, though not the heavy-duty factoring of FACTORIZE. The -EZ GCD algorithm was introduced in a paper by J. Moses and D.Y.Y. Yun in -Proceedings of the ACM, 1973, pp. 159-166. - -Note that the GCD switch must also be on for EZGCD to have -effect. - -\endsection -\item[FACTOR] -FACTOR (Declaration and Switch) (pages 101, 121, 122) - -When a kernel is declared by FACTOR, all terms involving fixed powers -of that kernel are printed as a product of the fixed powers and the -rest of the terms. - - FACTOR kernel {,kernel} - -kernel must be a Kernel. - -Examples: 2 2 2 -a := (x + y + z)**2; A := X + 2*X*Y + 2*X*Z + Y + 2*Y*Z + Z -factor y; 2 2 2 -a; Y + 2*Y*(X + Z) + X + 2*X*Z + Z -factor sin(x); 4 3 2 -c := df(sin(x)**4*x**2*z,x); C := 2*SIN(X) *X*Z + 4*SIN(X) *COS(X)*X *Z -remfac sin(x); 3 -c; 2*SIN(X) *X*Z*(2*COS(X)*X + SIN(X)) - -Use the FACTOR declaration to display variables of interest so that -you can see their powers more clearly, as shown in the example. -Remove this special treatment with the declaration REMFAC. The FACTOR -declaration is only effective when the switch PRI is on. - -The FACTOR declaration is not a factoring command; to factor -expressions use the FACTOR switch or the FACTORIZE command. - -FACTOR (switch) - -When the FACTOR switch is on, input expressions and results are -automatically factored. - -Examples: - -on factor; -aa := 3*x**3*a + 6*x**2*y*a + 3*x**3*b + 6*x**2*y*b -+ x*y*a + 2*y**2*a + x*y*b + 2*y**2*b; - 2 - AA := (A + B)*(3*X + Y)*(X + 2*Y) -off factor; -aa; - 3 2 2 3 2 2 - 3*A*X + 6*A*X *Y + A*X*Y + 2*A*Y + 3*B*X + 6*B*X *Y + B*X*Y + 2*B*Y -on factor; - 2 -ab := x**2 - 2; AB := X - 2 - -REDUCE factors univariate and multivariate polynomials with integer -coefficients, finding any factors that also have integer coefficients. -The factoring is done by reducing multivariate problems to univariate -ones with symbolic coefficients, and then solving the univariate ones -modulo small primes. The results of these calculations are merged to -determine the factors of the original polynomial. The factoriser -normally selects evaluation points and primes using a random number -generator. Thus, the detailed factoring behaviour may be different -each time any particular problem is tackled. - -When the FACTOR switch is turned on, the EXP switch is turned off, and -when the FACTOR switch is turned off, the EXP switch is turned on, -whether it was on previously or not. - -When the switch TRFAC is on, informative messages are generated at -each call to the factoriser. The TRALLFAC switch causes the -production of a more verbose trace message. It takes precedence over -TRFAC if they are both on. - -To factor a polynomial explicitly and store the results, use the operator -FACTORIZE. - -\endsection -\item[FACTORIAL] -FACTORIAL (pages 72, 174) - -FACTORIAL(expression) - -If the argument of FACTORIAL is a positive integer or zero, its -factorial is returned. Otherwise the result is expressed in terms of -the original operator. For more general operations, the GAMMA -operator is available in the SPECFN package. - -Examples: -factorial 4; 24 -factorial 30 ; 265252859812191058636308480000000 -factorial(a) ; FACTORIAL(A) - -\endsection -\item[Factorization] -Factorization (page 121) - -Operations for factorising expressions exist in REDUCE. See the -operator FACTORIZE and the switch FACTOR. - -The command FACTOR controls output format. - -\endsection -\item[FACTORIZE] -FACTORIZE (pages 121, 122) - -The FACTORIZE operator factors a given expression. - - FACTORIZE(expression) - -expression should be a polynomial, otherwise an error will result. - -Examples: - 2 2 -fff := factorize(x^3 - y^3); FFF := {X - Y,X + X*Y + Y } -fac1 := first fff; FAC1 := X - Y -factorize(x^15 - 1); {X - 1, - - 2 - X + X + 1, - - 4 3 2 - X + X + X + X + 1, - - 8 7 5 4 3 - X - X + X - X + X - X + 1} - - 8 7 5 4 3 -lastone := part(ws,length ws); lastone := x - x + x - x + x - x + 1 -setmod 2; 1 -on modular; -factorize(x^15 - 1); {X + 1, - - 2 - X + X + 1, - - 4 - X + X + 1, - - 4 3 - X + X + 1, - - 4 3 2 - X + X + X + X + 1} - -The FACTORIZE command returns the factors it finds as a LIST. You can -therefore use the usual list access methods (FIRST, SECOND, THIRD, -REST, LENGTH and PART) to extract the factors. - -If the expression given to FACTORIZE is an integer, it will be -factored into its prime components. To factor any integer factor of a -non-numerical expression, the switch IFACTOR should be turned on. Its -default is off. IFACTOR has effect only when factoring is explicitly -done by FACTORIZE, not when factoring is automatically done with the -FACTOR switch. If full factorisation is not needed the switch -LIMITEDFACTORS allows you to reduce the computing time of calls to -FACTORIZE. - -Factoring can be done in a modular domain by calling FACTORIZE when -MODULAR is on. You can set the modulus with the SETMOD command. The -last example above shows factoring modulo 2. - -For general comments on factoring, see comments under the switch -FACTOR. - -\endsection -\item[FAILHARD] -FAILHARD - -When the FAILHARD switch is on, the integration operator INT terminates -with an error message if the integral cannot be done in closed terms. -Default is off. - -Use the FAILHARD switch when you are dealing with complicated integrals -and want to know immediately if REDUCE was unable to handle them. The -integration operator sometimes returns a formal integration form that is -more complicated than the original expression, when it is unable to -complete the integration. - -\endsection -\xitem[Fast loading of code] -Fast loading of code (page 214) - -\endsection -\xitem[FDOMAIN command] -FDOMAIN command (pages 251, 271) - -\endsection -\xitem[FEXPR] -FEXPR (page 196) - -\endsection -\xitem[File handling] -File handling (page 153) - -\endsection -\item[FIRST] -FIRST (page 50) - -The FIRST operator returns the first element of a LIST. - FIRST(list) or FIRST list - -list must be a non-empty list to avoid an error message. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D} -first alist; A -blist := {x,y,{ww,aa,qq},z}; BLIST := {X,Y,{WW,AA,QQ},Z} -first third blist; WW - -\endsection -\item[FIRSTROOT] -FIRSTROOT (page 370) - - FIRSTROOT(expression) - FIRSTROOT simple_exprerssion - -FIRSTROOT is like ROOTS but only the first root determined by ROOTS is -computed. Note that this is not in general the first root that would -be listed in ROOTS output, since the ROOTS outputs are sorted into a -canonical order. Also, in some difficult root finding cases, the -first root computed might be incorrect. - -\endsection -\item[FIX] -FIX (page 73) - FIX(expression) - -The operator FIX returns the integer part of its argument, if that -argument has a numerical value. For positive numbers, this is equivalent -to FLOOR, and, for negative numbers, CEILING. For -non-numeric arguments, the value is an expression in the original operator. - -Examples: -fix 3.4; 3 -floor 3.4; 3 -ceiling 3.4; 4 -fix(-5.2); -5 -floor(-5.2); -6 -ceiling(-5.2); -5 -fix(a); FIX(A) - -\endsection -\item[FIXP] -FIXP (page 46) - -The FIXP logical operator returns true if its argument is an integer. - - FIXP(expression) or FIXP simple_expression - -expression can be any valid REDUCE expression, simple_expression -must be a single identifier or begin with a prefix operator. - -Examples: -if fixp 1.5 then write "ok" else write "not"; not -if fixp(a) then write "ok" else write "not"; not -a := 15; A := 15 -if fixp(a) then write "ok" else write "not"; ok - -Logical operators can only be used inside conditional expressions such as -IF...THEN or WHILE...DO. - -\endsection -\item[FLOOR] -FLOOR (page 73) - - FLOOR(expression) - -This operator returns the floor (i.e., the greatest integer less than -or equal to its argument) if its argument has a numerical value. For -positive numbers, this is equivalent to FIX. For non-numeric -arguments, the value is an expression in the original operator. - -Examples: -floor 3.4; 3 -fix 3.4; 3 -floor(-5.2); -6 -fix(-5.2); -5 -floor a; FLOOR(A) - -\endsection -\item[FOR] -FOR (page 65) - -The FOR command is used for iterative loops. There are many -possible forms it can take. - - / \ - / |STEP UNTIL| \ - |:=| || -FOR| | : | | - | \ / | - |EACH IN | - \ / - - where ::= DO|PRODUCT|SUM|COLLECT|JOIN. - -var can be any valid REDUCE identifier except T or NIL, inc, start and -stop can be any expression that evaluates to a positive or negative -integer. list must be a valid LIST structure. The action taken must -be one of the actions shown above, each of which is followed by a -single REDUCE expression, statement or a GROUP (<<...>>) or BLOCK -(BEGIN...END) statement. - -Examples: -for i := 1:10 sum i; 55 -for a := -2 step 3 until 6 product a; - -8 -a := 3; A := 3 -for iter := 4:a do write iter; -m := 0; M := 0 -for s := 10 step -1 until 3 do - <>; -m; 520 - 2 2 2 -for each x in {q,r,s} sum x**2; Q + R + S - 1 1 1 -for i := 1:4 collect 1/i; {1,---,---,---} - 2 3 4 - -for i := 1:3 join list solve(x**2 + i*x + 1,x); - SQRT(3)*I - 1 - {{X=---------------, - 2 - - - (SQRT(3)*I + 1) - X=--------------------}, - 2 - - {X=-1}, - - SQRT(5) - 3 - SQRT(5) - 3 - {X=-------------,X=----------------}} - 2 2 - -The behaviour of each of the five action words follows: - - Action Word Behaviour -Keyword Argument Type Action - do statement, command, group Evaluates its argument once - or block for each iteration of the loop, - not saving results -collect expression, statement, Evaluates its argument once for - command, group, block, list each iteration of the loop, - storing the results in a list - which is returned by the for - statement when done - join list or an operator which Evaluates its argument once for - produces a list each iteration of the loop, - appending the elements in each - individual result list onto the - overall result list -product expression, statement, Evaluates its argument once for - command, group or block each iteration of the loop, - multiplying the results together - and returning the overall product - sum expression, statement, Evaluates its argument once for - command, group or block each iteration of the loop, - adding the results together and - returning the overall sum - -For number-driven FOR statements, if the ending limit is smaller than -the beginning limit (larger in the case of negative steps) the action -statement is not executed at all. The iterative variable is local to -the FOR statement, and does not affect the value of an identifier with -the same name. For list-driven FOR statements, if the list is empty, -the action statement is not executed, but no error occurs. - -You can use nested FOR statements, with the inner FOR statement after -the action keyword. You must make sure that your inner statement -returns an expression that the outer statement can handle. - -\endsection -\item[FORALL] -FORALL (pages 141, 142) - -See the LET construction. - -\endsection -\item[FOREACH] -FOREACH (page 57--59, 195) - -FOREACH is a synonym for the FOR EACH variant of the -FOR construct. It is designed to iterate down a list, and an -error will occur if a list is not used. The use of FOR EACH is -preferred to FOREACH. - - FOREACH variable in list action expression - where action ::= DO|PRODUCT|SUM|COLLECT|JOIN - -Example: - 2 2 2 -foreach x in {q,r,s} sum x**2; Q + R + S - -\endsection -\xitem[FORDER command] -FORDER command (pages 268, 271) - -\endsection -\item[FORT] -FORT (page 108) - -When FORT is on, output is given Fortran-compatible syntax. Default -is OFF. - -Examples: -on fort; -df(sin(7*x + y),x); ANS=7.*COS(7*X+Y) -on rounded; -b := log(sin(pi/5 + n*pi)); B=LOG(SIN(3.14159265359*N+0.628318530718)) - -REDUCE results can be written to a file (using OUT) and used as data -by Fortran programs when FORT is in effect. FORT knows about correct -statement length, continuation characters, defining a symbol when it -is first used, and other Fortran details. - -The GENTRAN package offers many more possibilities than the FORT -switch. It produces Fortran (or C or Ratfor) code from REDUCE -procedures or structured specifications, including facilities for -producing double precision output. - -\endsection -\item[FORT_WIDTH] -FORT_WIDTH (page 111) - -The FORT_WIDTH variable sets the number of characters in a line of -Fortran-compatible output produced when the FORT switch is on. -Default is 70. - -Examples: -fort_width := 30; FORT_WIDTH := 30 -on fort; -df(sin(x**3*y),x); ANS=3.*COS(X - . **3*Y)*X**2* - . Y - -FORT_WIDTH includes the usually blank characters at the beginning -of the card. As you may notice above, it is conservative and makes the -lines even shorter than it was told. - -\endsection -\item[FORTRAN] -FORTRAN (pages 108, 110) - -REDUCE can produce FORTRAN syntax printed expressions with the switch - ON FORT - -There are also two major packages concerned with generating FORTRAN, -GENTRAN and SCOPE. The first of these is an embedded language for -defining FORTRAN program fragments or program units, with parts -substituted from algebraic calculation. SCOPE is a system for -optimising the form of FORTRAN expressions, usually used in -conjunction with GENTRAN. - -\endsection -\xitem[FRAME command] -FRAME command (pages 265, 271) - -\endsection -\item[FREEOF] -FREEOF (page 46) -The FREEOF logical operator returns TRUE if its first argument does -not contain its second argument anywhere in its structure. - - FREEOF(expression,kernel) or expression FREEOF kernel - -expression can be any valid scalar REDUCE expression, kernel must -be a kernel expression (see Kernel). - -Examples: 2 -a := x + sin(y)**2 + log sin z; A := LOG(SIN(Z)) + SIN(Y) + X -if freeof(a,sin(y)) - then write "free" else write "not free"; - not free -if freeof(a,sin(x)) - then write "free" else write "not free"; - free -if a freeof sin z - then write "free" else write "not free"; - not free - -Logical operators can only be used in conditional expressions such as -IF...THEN or WHILE...DO. - -\endsection -\item[FULLPREC] -FULLPREC - -Trailing zeroes of rounded numbers to the full system precision are -normally not printed. If this information is needed, for example to get a -more understandable indication of the accuracy of certain data, the switch -FULLPREC can be turned on. - -Examples: - on rounded; - 1/2; 0.5 - on fullprec; - ws; 0.500000000000 - -This is just an output options which neither influences the accuracy -of the computation nor does it give additional information about the -precision of the results. See also SCIENTIFIC_NOTATION. - -\endsection -\item[FULLROOTS] -FULLROOTS (page 87) - -Since roots of cubic and quartic polynomials can often be very -messy, a switch FULLROOTS controls the production -of results in closed form. SOLVE will apply the -formulas for explicit forms for degrees 3 and 4 only if -FULLROOTS is ON. Otherwise the result forms -are built using ROOT_OF. Default is OFF. - -\endsection -\xitem[Function] -Function (page 175) - -\endsection -\item[G] -G (page 206) - -G is an n-ary operator used to denote a product of gamma matrices -contracted with Lorentz four-vectors, in high-energy physics. - G(identifier,vector-expr -{,vector-expr}) - -identifier is a scalar identifier representing a fermion line -identifier, vector-expr can be any valid vector expression, -representing a vector or a gamma matrix. - -Examples: -vector aa,bb,cc; -vector a; -g(line1,aa,bb); AA.BB -g(line2,aa,a); 0 -g(id,aa,bb,cc); 0 -g(li1,aa,bb) + k; AA.BB + K -let aa.bb = m*k; -g(ln1,aa)*g(ln1,bb); K*M -g(ln1,aa)*g(ln2,bb); 0 - -The vector A is reserved in arguments of G to denote the special gamma -matrix gamma_5. It must be declared to be a vector before you use it. - -Gamma matrix expressions are associated with fermion lines in a -Feynman diagram. If more than one line occurs in an expression, the -gamma matrices involved are separate (operating in independent spin -space), as shown in the last two example lines above. A product of -gamma matrices associated with a single line can be entered either as -a single G command with several vector arguments, or as products of -separate G commands each with a single argument. - -While the product of vectors is not defined, the product, sum and -difference of several gamma expressions are defined, as is the product -of a gamma expression with a scalar. If an expression involving gamma -matrices includes a scalar, the scalar is treated as if it were the -product of itself with a unit 4 x 4 matrix. - -Dirac expressions are evaluated by computing the trace of the -expression using the commutation algebra of gamma matrices. The -algorithms used are described in articles by J. S. R. Chisholm in Il -Nuovo Cimento X, Vol. 30, p. 426, 1963, and J. Kahane, Journal of -Mathematical Physics, Vol. 9, p. 1732, 1968. The trace is then -divided by 4 to distinguish between the trace of a scalar and the -trace of an expression that is the product of a scalar with a unit 4 x -4 matrix. - -Trace calculations may be prevented over any line identifier by -declaring it to be NOSPUR. If it is later desired to evaluate these -traces, the declaration can be undone with the SPUR declaration. - -The notation of Bjorken and Drell, Relativistic Quantum Mechanics, -1964, is assumed in all operations involving gamma matrices. For an -example of the use of G in a calculation, see the REDUCE -User's Manual. - -\endsection -\item[GAMMA] -GAMMA (pages 185, 394) - -The GAMMA operator returns the Gamma function. - - GAMMA(expression) - -Examples: - load_package specfn; (SPECFN) - gamma(10); 362880 - gamma(1/2); SQRT(PI) - -\endsection -\item[Gamma Function] -Gamma Function (pages 185, 394) - -See GAMMA. - -\endsection -\item[GC] -GC - -With the GC switch, you can turn the garbage collection messages on -or off. The form of the message depends on the particular Lisp used for -the REDUCE implementation. - -See RECLAIM for an explanation of garbage collection. REDUCE does -garbage collection when needed even if you have turned the notices off. - -\endsection -\item[GCD] -GCD (operator and switch) (pages 123, 124) - -The GCD operator returns the greatest common divisor of two -polynomials. - - GCD(expression,expression) - -expression must be a polynomial (or integer), otherwise an error -occurs. - -Examples: -gcd(2*x**2 - 2*y**2,4*x + 4*y); 2*(X + Y) -gcd(sin(x),x**2 + 1); 1 -gcd(765,68); 17 - -The operator GCD described here provides an explicit means to find the -gcd of two expressions. The switch GCD described below simplifies -expressions by finding and cancelling gcd's at every opportunity. When -the switch EZGCD is also on, gcd's are figured using the EZ GCD -algorithm, which is usually faster. - -GCD switch - -With the GC switch, you can turn the garbage collection messages on -or off. The form of the message depends on the particular Lisp used for -the REDUCE implementation. - - -See RECLAIM for an explanation of garbage collection. REDUCE does -garbage collection when needed even if you have turned the notices off. - -\endsection -\item[GDIMENSION] -GDIMENSION (page 300) - - GDIMENSION(bas[,vars]) - -where bas is a GROEBNER basis in the current term order which must be -LEX term order (see IDEAL parameters). GDIMENSION computes the -dimension of the ideal spanned by the given basis. - -GDIMENSION cannot be called with other TERM orders. - -\endsection -\item[GEGENBAUERP] -GEGENBAUERP (page 185) - -The GEGENBAUERP operator computes Gegenbauer's (ultraspherical) -polynomials. - - GEGENBAUERP(integer,expression,expression) - -Examples: - load_package specfn; (SPECFN) - 2 - GegenbauerP(3,2,xx); 4*XX*(8*XX - 3) - - GegenbauerP(3,2,4); 2000 - -\endsection -\xitem[GEN(I)] -GEN(I) (page 379) - -\endsection -\xitem[Generalized Hypergeometric functions] -Generalized Hypergeometric functions (page 187) - -\endsection -\item[GENTRAN] -GENTRAN (page 180) - -Author: Barbara L. Gates - -This package is an automatic code GENerator and TRANslator. It constructs -complete numerical programs based on sets of algorithmic specifications and -symbolic expressions. Formatted FORTRAN, RATFOR or C code can be generated -through a series of interactive commands or under the control of a template -processing routine. Large expressions can be automatically segmented into -subexpressions of manageable size, and a special file-handling mechanism -maintains stacks of open I/O channels to allow output to be sent to any -number of files simultaneously and to facilitate recursive invocation of the -whole code generation process. - -\endsection -\xitem[GETCSYSTEM command] -GETCSYSTEM command (page 235) - -\endsection -\xitem[GETROOT] -GETROOT (page 370) - -\endsection -\xitem[GFNEWT] -GFNEWT (page 371) - -\endsection -\xitem[GFROOT] -GFROOT (page 371) - -\endsection -\item[GINDEPENDENT_SETS] -GINDEPENDENT_SETS (page 300) - - GINDEPENDENT_SETS(bas[,vars]) - -where bas is a GROEBNER basis in LEX term order (which must be the -current TERM order) with the specified variables (see IDEAL -parameters). - -GINDEPENDENT_SETS computes the maximal left independent variable sets -of the ideal, that are the variable sets which play the role of free -parameters in the current ideal basis. Each set is a list which is a -subset of the variable list. The result is a list of these sets. For -an ideal with dimension zero the list is empty. The -Kredel-Weispfenning algorithm is used. - -The operator cannot be called under another TERM order. - -\endsection -\xitem[GL(I)] -GL(I) (page 379) - -\endsection -\item[GLEXCONVERT] -GLEXCONVERT (page 300) - - GLEXCONVERT(bas[,vars][,MAXDEG=mx][,NEWVARS=nv]) - -where bas is a GROEBNER basis in the current term order, mx (optional) -is a positive integer and nvl (optional) is a list of variables (see -IDEAL parameters). - -The operator GLEXCONVERT converts the basis of a zero-dimensional -ideal (finite number of isolated solutions) from arbitrary ordering -into a basis under LEX term order. - -The parameter newvars defines the new variable sequence. If omitted, -the original variable sequence is used. If only a subset of variables -is specified here, the partial ideal basis is evaluated. - -If newvars is a list with one element, the minimal UNIVARIATE -polynomial is computed. - -maxdeg is an upper limit for the degrees. The algorithm stops with an -error message, if this limit is reached. - -A warning occurs, if the ideal is not zero dimensional. - -During the call the TERM order of the input basis must be active. - -\endsection -\item[GLTBASIS] -GLTBASIS (pages 299, 303) - -If GLTBASIS set on, the leading terms of the result basis of a -GROEBNER or GROEBNERF calculation are extracted. They are collected as -a basis of monomials, which is available as value of the global -variable GLTB. - -\endsection -\xitem[GNUPLOT] -GNUPLOT (page 181) - -\endsection -\xitem[GO TO] -GO TO (page 63) - -\endsection -\item[GosperAlg] -Gosper's Algorithm (page 403) - -See SUM and PROD. - -\endsection -\xitem[GRAD operator] -GRAD - operator (page 234) - -\endsection -\xitem[grad operator] -grad operator (page 358) - -\endsection -\xitem[gradient vector field] -gradient - vector field (page 234) - -\endsection -\item[GRADLEX] -GRADLEX (page 293) - -The terms are ordered first with their total degree, and if the total -degree is identical the comparison is LEX term order. With Groebner -basis calculations this term order produces polynomials of lowest -degree. - -\endsection -\item[GRADLEXGRADLEX] -GRADLEXGRADLEX - -The terms are separated into two groups where the second parameter of -the TORDER call determines the length of the first group. For a -comparison first the total degrees of both variable groups are -compared. If both are equal GRADLEX term order comparison is applied -to the first group, and if that does not decide GRADLEX term order is -applied for the second group. This order has the elimination property -for the variable groups. It can be used e.g. for separating variables -from parameters. The terms are ordered first with their total degree, -and if the total degree is identical the comparison is LEX term order. -With Groebner basis calculations this term order produces polynomials -of lowest degree. - -\endsection -\item[GREDUCE] -GREDUCE (page 307) - - GREDUCE(exp, exp1, exp2, ... , expm}[,vars]) - -where exp is an expression, and {exp1, exp2, ... , expm} is a list of -expressions or equations and vars is an optional list of variables -(see IDEAL parameters). - -GREDUCE is functionally equivalent with a call to GROEBNER and then a -call to PREDUCE. - -\endsection -\item[GROEBFULLREDUCTION] -GROEBFULLREDUCTION (page 298) - -If GROEBFULLREDUCTION set off, the polynomial reduction steps during -GROEBNER and GROEBNERF are limited to the pure head term reduction; -subsequent terms are reduced otherwise. - -By default GROEBFULLREDUCTION is on. - -\endsection -\item[GROEBMONFAC] -GROEBMONFAC (page 304) - -The variable GROEBMONFAC is connected to the handling of monomial -factors. A monomial factor is a product of variable powers as a -factor, e.g. x**2*y in x**3*y - 2*x**2*y**2. A monomial factor -represents a solution of the type x = 0 or y = 0 with a certain -multiplicity. With GROEBNERF the multiplicity of monomial factors is -lowered to the value of the shared variable GROEBMONFAC which by -default is 1 (= monomial factors remain present, but their -multiplicity is brought down). With GROEBMONFAC:= 0 the monomial -factors are suppressed completely. - -\endsection -\item[GROEBNER] -GROEBNER (pages 181, 296) - - GROEBNER({exp, ...}[,{var, ...}]) - -where {exp, ... } is a list of expressions or equations, {var, ... } -is an optional list of variables (see IDEAL PARAMETERS). - -The operator GROEBNER implements the Buchberger algorithm for -computing Groebner bases for a given set of expressions with respect -to the given set of variables in the order given. As a side effect, -the sequence of variables is stored as a REDUCE list in the shared -variable GVARSLAST - this is important in cases where the algorithm -rearranges the variable sequence because GROEBOPT is ON. - -Example: - - groebner({x**2+y**2-1,x-y}) {X - Y,2*Y**2 -1} - -See also GROEBNERF, GVARSLAST, GROEBOPT, GROEBPREREDUCE, -GROEBFULLREDUCTION, GLTBASIS, GLTB, GLTERMS, GROEBSTAT, TRGROEB, -TRGROEBS, GROEBPROT, GROEBPROTFILE, GROEBNERT. - -\endsection -\item[Groebner_Bases] -Groebner Bases (page 291) - -The GROEBNER package calculates Groebner bases using the Buchberger -algorithm and provides related algorithms for arithmetic with ideal -bases, such as ideal quotients, Hilbert polynomials, basis conversion, -independent variable set. - -Some routines of the Groebner package are used by SOLVE -- in -that context the package is loaded automatically. However, if you -want to use the package by explict calls you must load it by - - load_package groebner; - -For the common parameter setting of most operators in this package -see IDEAL PARAMETERS. - -\endsection -\item[GROEBNERF] -GROEBNERF (pages 302, 304, 318) - - GROEBNERF({exp, ...}[,{var, ...}] [,{nz, ... }]); - -where {exp, ... } is a list of expressions or equations, {var, ...} is -an optional list of variables (see IDEAL parameters) and {nz,... } is -an optional list of polynomials to be considered as non zero for this -calculation. - -GROEBNERF tries to separate polynomials into individual factors and to -branch the computation in a recursive manner (factorization tree). -The result is a list of partial Groebner bases. Multiplicities (one -factor with a higher power, the same partial basis twice) are deleted -as early as possible in order to speed up the calculation. - -The third parameter of GROEBNERF declares some polynomials -nonzero. If any of these is found in a branch of the calculation -the branch is canceled. - -Example: - -groebnerf({ 3*x**2*y+2*x*y+y+9*x**2+5*x = 3, - 2*x**3*y-x*y-y+6*x**3-2*x**2-3*x = -3, - x**3*y+x**2*y+3*x**3+2*x**2 }, {y,x}); - - {{Y - 3,X}, - - 2 - {2*Y + 2*X - 1,2*X - 5*X - 5}} - -See also GROEBRESMAX, GROEBMONFAC, GROEBRESTRICTION, GROEBNER, -GVARSLAST, GROEBOPT, GROEBPREREDUCE, GROEBFULLREDUCTION, GLTBASIS, -GLTB, GLTERMS, GROEBSTAT, TRGROEB, TRGROEBS, GROEBNERT. - -\endsection -\item[GROEBNERT] -GROEBNERT (page 311) - - GROEBNERT(v}=exp,...}[,vars]) - -where v are KERNELS (simple or indexed variables), exp are polynomials -and optional vars are variables (see IDEAL parameters). - -GROEBNERT is functionally equivalent to a GROEBNER call for {exp,...}, -but the result is a set of equations where the left-hand sides are the -basis elements while the right-hand sides are the same values -expressed as combinations of the input formulas, expressed in terms of -the names v. - -Example: - - groebnert({p1=2*x**2+4*y**2-100,p2=2*x-y+1}); - - GB1 := {2*X - Y + 1=P2, - - 2 - 9*Y - 2*Y - 199= - 2*X*P2 - Y*P2 + 2*P1 + P2} - - -\endsection -\item[GROEBOPT] -GROEBOPT (pages 298, 303) - -If GROEBOPT is set ON, the sequence of variables is optimized with -respect to execution speed of GROEBNER calculations; note that the -final list of variables is available in GVARSLAST. By default -GROEBOPT is off, conserving the original variable sequence. - -An explicitly declared dependency using the DEPEND declaration -superseeds the variable optimization. - -Example: - - depend a, x, y; - -guarantees that a will be placed in front of x and y. - -\endsection -\item[GROEBPREREDUCE] -GROEBPREREDUCE (pages 298, 303) - -If GROEBPREREDUCE set ON, GROEBNER and GROEBNERF try to simplify the -input expressions: if the head term of an input expression is a -multiple of the head term of another expression, it can be reduced; -these reductions are done cyclicly as long as possible in order to -shorten the main part of the algorithm. - -By default GROEBPREREDUCE is off. - -\endsection -\item[GROEBPROT] -GROEBPROT (page 309) - -If GROEBPROT is ON the computation steps during PREDUCE, GREDUCE and -GROEBNER are collected in a list which is assigned to the variable -GROEBPROTFILE. - -\endsection -\item[GROEBPROTFILE] -GROEBPROTFILE (page 309) - -If GROEBPROT is ON the computation steps during PREDUCE, GREDUCE and -GROEBNER are collected in a list which is assigned to the variable -GROEBPROTFILE. - -\endsection -\xitem[GROEBRES] -GROEBRES (page 304) - -\endsection -\item[GROEBRESMAX] -GROEBRESMAX (page 305) - -The variable GROEBRESMAX controls during GROEBNERF calculations the -number of partial results. Its default value is 300. If more partial -results are calculated, the calculation is terminated. - -\endsection -\item[GROEBRESTRICTION] -GROEBRESTRICTION (page 306) - -During GROEBNERF calculations irrelevant branches can be excluded by -setting the variable GROEBRESTRICTION. The following restrictions are -implemented: - - GROEBRESTRICTION := NONNEGATIVE - GROEBRESTRICTION := POSITIVE - -With NONNEGATIVE branches are excluded where one polynomial has no -nonnegative real zeros; with POSITIVE the restriction is sharpened to -positive zeros only. - -\endsection -\item[GROEBSTAT] -GROEBSTAT (pages 299, 303) - -If GROEBSTAT is on, a summary of the GROEBNER or GROEBNERF computation -is printed at the end including the computing time, the number of -intermediate H polynomials and the counters for the criteria hits. - -\endsection -\xitem[GROEPOSTPROC] -GROEPOSTPROC (page 319) - -\endsection -\xitem[GROESOLVE] -GROESOLVE (page 318) - -\endsection -\xitem[Group statement] -Group statement (pages 55, 56, 61) - -\endsection -\xitem[grouped ordering] -grouped ordering (page 315) - -\endsection -\item[GSORT] -GSORT (page 322) - - GSORT(p[,vars]) - -where p is a polynomial or a list of polynomials, vars in an optional -list of variables (see IDEAL parameters). - -The polynomials are reordered and sorted corresponding to the current -TERM order. - -Example: - - torder lex; - 2 2 - gsort(x**2+2x*y+y**2,{y,x}); {Y + 2 * Y * X + X } - -\endsection -\item[GSPLIT] -GSPLIT (page 323) - - GSPLIT(p[,vars]); - -where p is a polynomial or a list of polynomials, vars in an optional -list of variables (see IDEAL parameters). - -The polynomial is reordered corresponding to the the current TERM -order and then separated into leading term and reductum. Result is a -list with the leading term as first and the reductum as second -element. - -Example: - - torder lex; - 2 2 - gsplit(x**2+2x*y+y**2,{y,x}); {Y , 2*Y*X + X } - -\endsection -\item[GSPOLY] -GSPOLY (page 324) - - GSPOLY(p1,p2[,vars]); - -where p1 and p2 are polynomials, vars in an optional list of variables -(see IDEAL parameters). - -The SUBTRACTION polynomial of p1 and p2 is computed corresponding to -the method of the Buchberger algorithm for computing GROEBNER bases: -p1 and p2 are multiplied with terms such that when subtracting them -the leading terms cancel each other. - -\endsection -\item[GVARS] -GVARS (page 296) - - GVARS({exp,exp,... }) - - where exp are expressions or equations. - -GVARS extracts from the expressions the KERNELs which can -play the role of variables for a GROEBNER or GROEBNERF calculation. - -\endsection -\item[GVARSLAST] -GVARSLAST (page 298) - -After a GROEBNER or GROEBNERF calculation the actual variable sequence -is stored in the variable GVARSLAST. If GROEBOPT is ON, GVARSLAST -shows the variable sequence after reordering. - -\endsection -\item[GZERODIM?] -GZERODIM? (page 299) - - GZERODIM!?(basis[,vars]) - -where basis is a Groebner basis in the current -TERM order with the specified variables (see IDEAL parameters). - -GZERODIM!? tests whether the ideal spanned by the given basis -has dimension zero. If yes, the number of zeros is returned, -NIL otherwise. - -\endsection -\item[Hankel Functions] -Hankel Functions (pages 185, 396) - -Part of the SPECFN package. See HANKEL1 and HANKEL2. - -\endsection -\item[HANKEL1] -HANKEL1 (pages 185, 396) - -The HANKEL1 operator returns the Hankel function of the first kind. - -HANKEL1(order,argument) - -Examples: - load_package specfn; (SPECFN) - Hankel1 (1/2,pi); - SQRT(2) / PI - on rounded; - Hankel1 (1,3); 0.324674424792 - -The operator HANKEL1 is also called Bessel function of the third kind. -There is currently no numeric evaluation of Hankel functions. - -\endsection -\item[HANKEL2] -HANKEL2 (pages 185, 396) - -The HANKEL2 operator returns the Hankel function of the second kind. - - HANKEL2(order,argument) - -Examples: - load_package specfn; (SPECFN) - Hankel2 (1/2,pi); - SQRT(2) / PI - on rounded; - Hankel2 (1,3); 0.324674424792 - -The operator HANKEL2 is also called Bessel function of the third kind. -There is currently no numeric evaluation of Hankel functions. - -\endsection -\item[HERMITEP] -HERMITEP (page 185) - -The HERMITEP operator returns the nth Hermite Polynomial. - - HERMITEP(integer,expression) - -Examples: - load_package specfn; (SPECFN) - 2 - HermiteP(3,xx); 4*XX*(2*XX - 3) - HermiteP(3,4); 464 - -Hermite polynomials are computed using the recurrence relation: - -HermiteP(n,x) := 2x*HermiteP(n-1,x) - 2*(n-1)*HermiteP(n-2,x) with -HermiteP(0,x) := 1 and HermiteP(1,x) := 2x - -\endsection -\xitem[HFACTORS scale factors] -HFACTORS scale factors (page 234) - -\endsection -\xitem[High energy trace] -High energy trace (page 209) - -\endsection -\item[HIGH_POW] -HIGH_POW (page 115) - -The variable HIGH_POW is set by COEFF to the highest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. - -Examples: -coeff((x+1)^5*(x*(y+3)^2)^2,x); {0, - - 0, - - 4 3 2 - Y + 12*Y + 54*Y + 108*Y + 81, - - 4 3 2 - 5*(Y + 12*Y + 54*Y + 108*Y + 81), - - 4 3 2 - 10*(Y + 12*Y + 54*Y + 108*Y + 81), - - 4 3 2 - 10*(Y + 12*Y + 54*Y + 108*Y + 81), - - 4 3 2 - 5*(Y + 12*Y + 54*Y + 108*Y + 81), - - 4 3 2 - Y + 12*Y + 54*Y + 108*Y + 81} - -high_pow; 7 - -\endsection -\xitem[HIGHESTDERIV] -HIGHESTDERIV (page 350) - -\endsection -\item[HILBERTPOLYNOMIAL] -HILBERTPOLYNOMIAL (page 321) - - HILBERTPOLYNOMIAL(bas,[vars]) - -where bas is a GROEBNER basis in the current TERM order and vars is an -optional variable list(see IDEAL parameters). - -The degree of the HILBERT polynomial is the dimension of the ideal -spanned by the basis. For an ideal of dimension zero the Hilbert -polynomial is a constant which is the number of common zeros of the -ideal (including eventual multiplicities). The HOLLMANN algorithm is -used. - -\endsection -\xitem[History] -History (page 158) - -\endsection -\xitem[Hodge-* duality operator] -Hodge-* duality operator (pages 256, 266) - -\endsection -\item[HORNER] -HORNER - -When the HORNER switch is on, polynomial expressions are printed -in Horner's form for faster and safer numerical evaluation. Default -is OFF. The leading variable of the expression is selected as -Horner variable. To select the Horner variable explicitly use the -KORDER declaration. - -Examples: -on horner; 3 2 -(13p-4q)^3; ( - 64)*Q + P*(624*Q + P*(( - 2028)*Q + P*2197)) -korder q; - 3 2 -ws; 2197*P + Q*(( - 2028)*P + Q*(624*P + Q*(-64))) - -\endsection -\xitem[HYPERGEOMETRIC] -Hypergeometric Functions (page 397) - -The HYPERGEOMETRIC operator provides simplifications for the -generalised hypergeometric functions. -The HYPERGEOMETRIC operator is included in the package specfn2. - - HYPERGEOMETRIC(list_of_parameters,list_of_parameters,argument) - -Examples: -load_package specfn; (SPECFN) -hypergeometric ({1/2,1},{3/2},-x^2); \rfrac{atan(x)}{x} -hypergeometric ({},{},z); e^z - -The special case with length of the first list equals 2 and -length of the second list equals 1 is often called "hypergeometric function". - -\endsection -\item[HYPOT] -HYPOT (pages 76, 78) - - HYPOT(expression,expression) - -If ROUNDED is on, and the two arguments evaluate to numbers, this -operator returns the square root of the sums of the squares of the -arguments in a manner that avoids intermediate overflow. In other cases, -an expression in the original operator is returned. - -Examples: -hypot(3,4); HYPOT(3,4) -on rounded; -ws; 5.0 -hypot(a,b); HYPOT(A,B) - -\endsection -\item[I] -I (page 36) - -REDUCE knows I is the square root of -1, and that i^2 = -1. - -Examples: -(a + b*i)*(c + d*i); A*C + A*D*I + B*C*I - B*D -i**2; -1 - -I cannot be used as an identifier. It is all right to use I as an -index variable in a FOR loop, or as a local (SCALAR) variable inside a -BEGIN...END block, but it loses its definition as the square root of --1 inside the block in that case. - -Only the simplest properties of i are known by REDUCE unless the -switch COMPLEX is turned on, which implements full complex arithmetic -in factoring, simplification, and functional values. COMPLEX is -ordinarily off. - -\endsection -\xitem[i] -i (page 223) - -\endsection -\xitem[ideal dimension] -ideal dimension (page 300) - -\endsection -\item[IDEAL PARAMETERS] -IDEAL PARAMETERS - -Most operators of the Groebner package compute expressions in a -polynomial ring which given as R[var,var,...] where R is the current -REDUCE coefficient domain. All algebraically exact domains of REDUCE -are supported. The package can operate over rings and fields. The -operation mode is distinguished automatically. In general the ring -mode is a bit faster than the field mode. The factoring variant can -be applied only over domains which allow you factoring of multivariate -polynomials. - -The variable sequence var is either given explicitly as argument in -form of a list, or it is extracted automatically from the -expressions. In the second case the current REDUCE system order is -used (see KORDER) for arranging the variables. If some kernels should -play the role of formal parameters (the ground domain R then is the -polynomial ring over these), the variable sequences must be given -explicitly. - -All REDUCE kernels can be used as variables. But please note, that -all variables are considered as independent; e.g. when using SIN(A) -and COS(A) as variables, the basic relation SIN(A)^2+COS(A)^2-1=0 must -be explicitly added to an equation set because the Groebner operators -do not include such knowledge automatically. - -The terms (monomials) in polynomials are arranged according to the -current TERM ORDER. Note that the algebraic properties of the -computed results only are valid as long as neither the ordering nor -the variable sequence changes. - -The input expressions exp can be polynomials P, rational functions N/D -or equations LH=RH built from polynomials or rational functions. -Apart from the tracing algorithms GROEBNERT and PREDUCET, where the -equations have a specific meaning, equations are converted to simple -expressions by taking the difference of the left-hand and right-hand -sides LH-RH=>P. Rational functions are converted to polynomials by -converting the expression to a common denominator form first, and then -using the numerator only N=>P. So eventual zeros of the denominators -are ignored. - -A basis on input or output of an algorithm is coded as a list -of expressions {exp,exp,...}. - -\endsection -\item[IDEALQUOTIENT] -IDEALQUOTIENT (page 320) - - IDEALQUOTIENT({exp, ...}, d [,{var, ...}]) - -where {exp,...} is a list of expressions or equations, d is a single -expression or equation and {var,...} is an optional list of variables -(see IDEAL parameters). - -IDEALQUOTIENT computes the ideal quotient: ideal spanned by the -expressions {exp,...} divided by the single polynomial/expression -f. The result is the GROEBNER basis of the quotient ideal. - -\endsection -\item[Identifier] -Identifier (page 35) - -Identifiers in REDUCE consist of one or more alphanumeric characters, -of which the first must be alphabetical. The maximum number of -characters allowed is system dependent, but is usually over 100. -However, printing is simplified if they are kept under 25 characters. - -You can also use special characters in your identifiers, but each must be -preceded by an exclamation point ! as an escape character. Useful -special characters are # $ % ^ & * - + = ? < > ~ | / ! and -the space. Note that the use of the exclamation point as a special -character requires a second exclamation point as an escape character. -The underscore _ is special in this regard. It must be preceded -by an escape character in the first position in an identifier, but is -treated like a normal letter within an identifier. - -Other characters, such as ( ) # ; ` ' " can also be used if preceded -by a !, but as they have special meanings to the Lisp reader it is -best to avoid them to avoid confusion. - -Many system identifiers have * before or after their names, or - -between words. If you accidentally pick one of these names for your -own identifier, it could have disastrous effects. For this reason it -is wise not to include * or - anywhere in your identifiers. - -You will notice that REDUCE does not use the escape characters when it -prints identifiers containing special characters; however, you still -must use them when you refer to these identifiers. Be careful when -editing statements containing escaped special characters to treat the -character and its escape as an inseparable pair. - -Identifiers are used for variable names, labels for GO TO statements, -and names of arrays, matrices, operators, and procedures. Once an -identifier is used as a matrix, array, scalar or operator identifier, -it may not be used again as a matrix, array or operator. An operator -or array identifier may later be used as a scalar without problems, -but a matrix identifier cannot be used as a scalar. All procedures -are entered into the system as operators, so the name of a procedure -may not be used as a matrix, array, or operator identifier either. - -\endsection -\item[IF] -IF (pages 55, 56) - -The IF command is a conditional statement that executes a statement -if a condition is true, and optionally another statement if it is not. - - IF condition THEN statement {ELSE statement} - -condition must be a logical or comparison operator that evaluates to -true or false. statement must be a single REDUCE statement or a GROUP -(<<...>>) or BLOCK (BEGIN...END) statement. - -Examples: -if x = 5 then a := b+c else a := d+f; D + F -x := 9; X := 9 -if numberp x and x<20 then y := sqrt(x) else write "illegal"; 3 -clear x; -if numberp x and x<20 then y := sqrt(x) else write "illegal"; illegal -x := 12; X := 12 -a := if x < 5 then 100 else 150; A := 150 -b := u**(if x < 10 then 2); B := 1 - 2 -bb := u**(if x > 10 then 2); BB := U - -An IF statement may be used inside an assignment statement and sets -its value depending on the conditions, or used anywhere else an -expression would be valid, as shown in the last example. If there is -no ELSE clause, the value is 0 if a number is expected, and nothing -otherwise. - -The ELSE clause may be left out if no action is to be taken if the -condition is false. - -The condition may be a compound conditional statement using AND or -OR. If a non-conditional statement, such as a constant, is used by -accident, it is assumed to have value true. - -Be sure to use GROUP or BLOCK statements after THEN or ELSE. - -The IF operator is right associative. The following constructions are -examples: - -(1) - IF condition THEN IF condition THEN action ELSE action - -which is equivalent to - IF condition THEN (IF condition THEN action ELSE action); - -(2) IF condition THEN action ELSE IF condition THEN action ELSE action -which is equivalent to - IF condition THEN action ELSE - (IF condition THEN action ELSE action). - -\endsection -\item[IFACTOR] -IFACTOR (page 121) - -When the IFACTOR switch is on, any integer terms appearing as a result -of the FACTORIZE command are factored themselves into primes. Default -is OFF. If the argument of FACTORIZE is an integer, -IFACTOR has no effect, since the integer is always factored. - -Examples: -factorize(4*x**2 + 28*x + 48); {4,X + 3,X + 4} -factorize(22587); {3,7529} -on ifactor; -factorize(4*x**2 + 28*x + 48); {2,2,X + 4,X + 3} -factorize(22587); {3,7529} - -Constant terms that appear within nonconstant polynomial factors are -not factored. - -The IFACTOR switch affects only factoring done specifically with -FACTORIZE, not on factoring done automatically when the FACTOR switch -is on. - -\endsection -\xitem[imaginary unit] -imaginary unit (page 223) - -\endsection -\item[IMPART] -IMPART (pages 72, 73, 75) - - IMPART(expression) or IMPART simple_expression - -This operator returns the imaginary part of an expression, if that -argument has an numerical value. A non-numerical argument is returned -as an expression in the operators REPART and IMPART. - -Examples: -impart(1+i); 1 -impart(a+i*b); REPART(B) + IMPART(A) - -\endsection -\item[IN] -IN (page 153) - -The IN command takes a list of file names and inputs each file into -the system. - IN filename{,filename} - -filename must be in the current directory, or be a valid pathname. -If the file name is not an identifier, double quote marks (") are -needed around the file name. - - -A message is given if the file cannot be found, or has a mistake -in it. - -Ending the command with a semicolon causes the file to be echoed to the -screen; ending it with a dollar sign does not echo the file. If you want -some but not all of a file echoed, turn the switch ECHO on or off -in the file. - -An efficient way to develop procedures in REDUCE is to write them into a file -using a system editor of your choice, and then input the -files into an active REDUCE session. REDUCE reparses the procedure as -it takes information from the file, overwriting the previous procedure -definition. When it accepts the procedure, it echoes its name to the screen. -Data can also be input to the system from files. - -Files to be read in should always end in END; to avoid -end-of-file problems. Note that this is an additional END; to any -ending procedures in the file. - -\endsection -\item[Indefinite integration] -Indefinite integration (page 80) - -See the INT operator. - -\endsection -\xitem[independent sets] -independent sets (page 300) - -\endsection -\item[INDEX] -INDEX (page 206) - -The declaration INDEX flags a four-vector as an index for subsequent -high-energy physics calculations. - INDEX vector-id{,vector-id} - -vector-id must have been declared of type VECTOR. - -Examples: -vector aa,bb,cc; -index uu; -let aa.bb = 0; -(aa.uu)*(bb.uu); 0 -(aa.uu)*(cc.uu); AA.CC - -Index variables are used to represent contraction over components of -vectors when scalar products are taken by the . operator, as well as -indicating contraction for the EPS operator or metric tensor. - -The special status of a vector as an index can be revoked with the -declaration REMIND. The object remains a vector, however. - -\endsection -\xitem[INDEX_SYMMETRIES command] -INDEX_SYMMETRIES command (page 271) - -\endsection -\xitem[INDEXRANGE command] -INDEXRANGE command (page 271) - -\endsection -\xitem[INDEXSYMMETRIES command] -INDEXSYMMETRIES command (page 262) - -\endsection -\item[INFINITY] -INFINITY (pages 37, 368) - -The name INFINITY is used to represent the infinite positive number. -However, at the present time, arithmetic in terms of this operator reflects -finite arithmetic, rather than true operations on infinity. - -\endsection -\item[INFIX] -INFIX (page 94) - -INFIX declares identifiers to be infix operators. - - INFIX identifier {,identifier} - -identifier can be any valid REDUCE identifier, which has not already -been declared an operator, array or matrix, and is not reserved by the -system. - -Examples: -infix aa; -for all x,y let aa(x,y) = cos(x)*cos(y) - sin(x)*sin(y); -x aa y; COS(X)*COS(Y) - SIN(X)*SIN(Y) - - SQRT(3) -pi/3 aa pi/2; ------------ - 2 -aa(pi,pi); 1 - -A LET statement must be used to attach functionality to the operator. -Note that the operator is defined in prefix form in the LET statement. -After its definition, the operator may be used in either prefix or infix -mode. The above operator aa finds the cosine of the sum of two angles by -the formula - cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y). -Precedence may be attached to infix operators with the PRECEDENCE declaration. - -User-defined infix operators may be used in prefix form. If they are used -in infix form, a space must be left on each side of the operator to avoid -ambiguity. Infix operators are always binary. - -\endsection -\xitem[Infix operator] -Infix operator (pages 38--41) - -\endsection -\xitem[inner product] -inner product (page 357) - -\endsection -\xitem[inner product exterior form] -inner product - exterior form (page 254) - -\endsection -\item[INPUT] -INPUT (page 158) - -The INPUT command returns the input expression to the REDUCE numbered -prompt that is its argument. - INPUT(number) or INPUT number - - -number must be between 1 and the current REDUCE prompt number. - -An expression brought back by INPUT can be re-executed with new -values or switch settings, or used as an argument in another expression. -The command WS brings back the results of a numbered REDUCE -statement. Two lists contain every input and every output statement since -the beginning of the session. If your session is very long, storage space -begins to fill up with these expressions, so it is a good idea to end the -session once in a while, saving needed expressions to files with the -SAVEAS and OUT commands. - -Switch settings and LET statements can also be re-executed by using -INPUT. - -An error message is given if a number is called for that has not yet been used. - -\endsection -\xitem[Input] -Input (page 153) - -\endsection -\xitem[Instant evaluation] -Instant evaluation (pages 68, 117, 140, 162, 164) - -\endsection -\item[INT] -INT (operator and switch) (pages 80, 160) - -The INT operator performs analytic integration on a variety of -functions. - - INT(expression,kernel) - -expression can be any scalar expression. involving polynomials, log -functions, exponential functions, or tangent or arctangent -expressions. INT attempts expressions involving error functions, -dilogarithms and other trigonometric expressions. Integrals involving -algebraic extensions (such as square roots) may not succeed. kernel -must be a REDUCE KERNEL. - -Examples: - 3 - X*(X + 12) -int(x**3 + 3,x); ------------- - 4 - - 2*X - E *( - COS(X) + 2*SIN(x)) -int(sin(x)*exp(2*x),x); ----------------------------- - 5 - - SQRT(2)*(LOG( - SQRT(2) + X) - LOG(SQRT(2) + X)) -int(1/(x^2-2),x); -------------------------------------------------- - 4 - - COS(X) - - ATAN(--------) - 2 -int(sin(x)/(4 + cos(x)**2),x); ------------------- - 2 - - SQRT(x - 1) -int(1/sqrt(x^2-x),x); INT(---------------------,X) - SQRT(X)*X - SQRT(X) - -Note that REDUCE could not handle the last integral with its default -integrator, since the integrand involves a square root. However, the -integral can be found using the ALGINT package. Alternatively, you -could add a rule using the LET statement to evaluate this integral. - -The arbitrary constant of integration is not shown. Definite -integrals can be found by evaluating the result at the limits of -integration (use ROUNDED) and subtracting the lower from the higher. -Evaluation can be easily done by the SUB operator. - -When INT cannot find an integral it returns an expression involving -formal INT expressions unless the switch FAILHARD has been set. If not -all of the expression can be integrated, the switch NOLNR controls -whether a partially integrated result should be returned or not. - -INT switch - -The INT switch specifies an interactive mode of operation. Default -ON. - - -There is no reason to turn INT off during interactive calculations, -since there are no benefits to be gained. If you do have INT off -while inputting a file, and REDUCE finds an error, it prints the message -``Continuing with parsing only''. In this state, REDUCE accepts only -END; or BYE; from the keyboard; -everything else is ignored, even the command ON INT. - -\endsection -\item[INTEGER] -INTEGER (page 61) - -The INTEGER declaration must be made immediately after a BEGIN (or -other variable declaration such as REAL and SCALAR) and declares local -integer variables. They are initialised to 0. - - INTEGER identifier{,identifier} - -identifier may be any valid REDUCE identifier, except T or NIL. - -Integer variables remain local, and do not share values with variables -of the same name outside the BEGIN...END block. When the block is -finished, the variables are removed. You may use the words REAL or -SCALAR in the place of INTEGER. INTEGER does not indicate -type-checking by the current REDUCE; it is only for your own -information. Declaration statements must immediately follow the -BEGIN, without a semicolon between BEGIN and the first variable -declaration. - -Any variables used inside BEGIN...END blocks that were not declared -SCALAR, REAL or INTEGER are global, and any change made to them inside -the block affects their global value. Any ARRAY or MATRIX declared -inside a block is always global. - -\endsection -\xitem[Integer] -Integer (page 44) - -\endsection -\item[Integration] -Integration (pages 80, 92) - -See INT, ALGINT or NUM_INT. -\endsection -\xitem[integration definite (simple)] -integration definite (simple) (page 236) - line (page 238) - volume (page 237) - -\endsection -\xitem[Interactive use] -Interactive use (pages 157, 160) - -\endsection -\item[INTERPOL] -INTERPOL (page 127) - -INTERPOL generates an interpolation polynomial. - - INTERPOL(values,variable,points) - -values and points are LISTs of equal length and variable is an -algebraic expression (preferably a KERNEL). The interpolation -polynomial is generated in the given variable of degree -length(values)-1. The unique polynomial F is defined by the property -that for corresponding elements V of values and P of points the -relation F(P)=V holds. - -Examples: -f := for i:=1:4 collect(i**3-1); F := {0,7,26,63} -p := {1,2,3,4}; P := {1,2,3,4} - 3 -interpol(f,x,p); X - 1 - -The Aitken-Neville interpolation algorithm is used which guarantees a -stable result even with rounded numbers and an ill-conditioned problem. - -\endsection -\item[INTSTR] -INTSTR (page 98) - -If INTSTR (for ``internal structure'') is on, arguments of an -operator are printed in a more structured form. - -Examples: - operator f; - f(2x+2y); F(2*X + 2*Y) - on intstr; - ws; F(2*(X + Y)) - -\endsection -\item[ISOLATER] -ISOLATER (page 369) - - ISOLATER(expression) - ISOLATER simple_expresion - ISOLATER(expression, POSITIVE) - ISOLATER(expression, NEGATIVE) - ISOLATER(expression, lo, hi) - -The ISOLATER function produces a list of rational intervals, each -containing a single real root of the univariate polynomial p, within -the specified region, but does not find the roots. If arg2 and arg3 -are not present, all real roots are found. If the additional -arguments are present, they restrict the region of consideration. - -If arg2=NEGATIVE then only negative roots of p are included; if -arg2=POSITIVE then only positive roots of p are included. Zero roots -are excluded. - -If arguments are (p,arg2,arg3) then Arg2 and Arg3 must be r (a real -number) or EXCLUDE r, or a member of the list POSITIVE, NEGATIVE, -INFINITY, -INFINITY. EXCLUDE r causes the value r to be excluded from -the region. The order of the sequence arg2, arg3 is unimportant. -Assuming that arg2 <= arg3 when both are numeric, then - - {-INFINITY,INFINITY} is equivalent to {} represents all roots; - {arg2,NEGATIVE} represents -1 (cos(x+y)+cos(x-y))/2, - cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, - sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, - cos(~x)^2 => (1+cos(2*x))/2, - sin(~x)^2 => (1-cos(2*x))/2}$ -let trig1; - COS(A - B) + COS(A + B) -cos(a)*cos(b); ------------------------- - 2 - -A LET command returns no value, though the substitution rule is -entered. Assignment rules made by ASSIGN and LET rules are at the -same level, and cancel each other. There is a difference in their -operation, however, as shown in the first example: a LET assignment -tracks the changes in what it is assigned to, while a := assignment is -fixed at the value it originally had. - -The use of expressions as left-hand sides of LET statements is a -little complicated. The rules of operation are: - -(i) Expressions of the form A*B = C do not change A, B or C, but set -A*B to C. - -(ii) Expressions of the form A+B = C substitute C - B for A, but do -not change B or C. - -(iii) Expressions of the form A-B = C substitute B + C for A, but do -not change B or C. - -(iv) Expressions of the form A/B = C substitute B*C for A, but do not -change B or C. - -(v) Expressions of the form A**N = C substitute C for A**N in every -expression of a power of A to N or greater. An asymptotic command -such as A**N = 0 sets all terms involving A to powers greater than or -equal to N to 0. Finite fields may be generated by requiring modular -arithmetic (the MODULAR switch) and defining the primitive polynomial -via a LET statement. - -LET substitutions involving expressions are cleared by using the CLEAR -command with exactly the same expression. - -Note when a simple LET statement is used to assign functionality to an -operator, it is valid only for the exact identifiers used. For the -use of the LET command to attach more general functionality to an -operator, see FORALL. - -Arrays as a whole cannot be arguments to LET statements, but matrices -as a whole can be legal arguments, provided both arguments are -matrices. However, it is important to note that the two matrices are -then linked. Any change to an element of one matrix changes the -corresponding value in the other. Unless you want this behaviour, you -should not use LET for matrices. The assignment operator ASSIGN can -be used for non-tracking assignments, avoiding the side effects. -Matrices are redimensioned as needed in LET statements. - -When array or matrix elements are used as the left-hand side of LET -statements, the contents of that element is used as the argument. -When the contents is a number or some other expression that is not a -valid left-hand side for LET, you get an error message. If the -contents is an identifier or simple expression, the LET rule is -globally attached to that identifier, and is in effect not only inside -the array or matrix, but everywhere. Because of such unwanted side -effects, you should not use LET with array or matrix elements. The -assignment operator := can be used to put values into array or matrix -elements without the side effects. - -Local variables declared inside BEGIN...END blocks cannot be used as -the left-hand side of LET statements. However, BEGIN...END blocks -themselves can be used as the right-hand side of LET statements. The -construction: - FOR ALL vars - LET operator(vars) = block -is an alternative to the - PROCEDURE name(vars); block -construction. One important difference between the two constructions -is that the vars as formal parameters to a procedure have their global -values protected against change by the procedure, while the vars of a -LET statement are changed globally by its actions. - -Be careful in using a construction such as LET x = x + 1 except inside -a controlled loop statement. The process of resubstitution continues -until a stack overflow message is given. - -The LET statement may be used to make global changes to variables from -inside procedures. If X is a formal parameter to a procedure, the -command LET x = ... makes the change to the calling variable. For -example, if a procedure was defined by - procedure f(x,y); - let x = 15; -and the procedure was called as - f(a,b); -A would have its value changed to 15. Be careful when using LET -statements inside procedures to avoid unwanted side effects. - -It is also important to be careful when replacing LET statements with -other LET statements. The overlapping of these substitutions can be -unpredictable. Ordinarily the latest-entered rule is the first to be -applied. Sometimes the previous rule is superseded completely; other -times it stays around as a special case. The order of entering a set -of related LET expressions is very important to their eventual -behaviour. The best approach is to assume that the rules will be -applied in an arbitrary order. - -\endsection -\xitem[Levi-Cevita tensor] -Levi-Cevita tensor (page 267) - -\endsection -\item[LEX] - -The terms are ordered lexicographically: two terms t1 t2 are compared -for their degrees along the fixed variable sequence: t1 is higher than -t2 if the first different degree is higher in t1. This order has the -elimination property for GROEBNER BASIS calculations. If the ideal -has a univariate polynomial in the last variable the groebner basis -will contain such polynomial. LEX is best suited for solving of -polynomial equation systems. - -\endsection -\item[LHS] -LHS (page 47) - -The LHS operator returns the left-hand side of an EQUATION, such as -those returned in a list by SOLVE. - - LHS(equation) or LHS equation - -equation must be an equation of the form - LEFT-HAND SIDE = RIGHT-HAND SIDE. - -Examples: -polly := (x+3)*(x^4+2x+1); - 5 4 2 - POLLY := X + 3*X + 2*X + 7*X + 3 - -pollyroots := solve(polly,x); - 3 2 - POLLYROOTS := {X=ROOT_OF(X_ - X_ + X_ + 1,X_),X=-1,X=-3} - -variable := lhs first pollyroots; - VARIABLE := X - -\endsection -\xitem[LIE Derivative] -Lie Derivative (page 255) - -\endsection -\item[LIMIT] -LIMIT (pages 329, 360) - -LIMITS is a fast limit package for REDUCE for functions which are -continuous except for computable poles and singularities, based on -some earlier work by Ian Cohen and John P. Fitch. The Truncated -Power Series package is used for non-critical points, at which -the value of the function is the constant term in the expansion -around that point. l'Hopital's rule is used in critical cases, -with preprocessing of 1-1 forms and reformatting of product forms -in order to apply l'Hopital's rule. A limited amount of bounded -arithmetic is also employed where applicable. - - LIMIT(expr,var,limpoint) or - LIMIT!+(expr,var,limpoint) or - LIMIT!-(expr,var,limpoint) - -where expr is an expression depending of the variable var (a KERNEL) -and limpoint is the limit point. If the limit depends upon the -direction of approach to the limpoint, the operators LIMIT!+ and -LIMIT!- may be used. - -Examples: - limit(x*cot(x),x,0); 0 - 2 - limit((2x+5)/(3x-2),x,infinity); --- - 3 - -\endsection -\xitem[LIMIT0] -LIMIT0 (page 330) - -\endsection -\xitem[LIMIT1] -LIMIT1 (page 330) - -\endsection -\xitem[LIMIT2] - -\endsection -\xitem[LIMIT2] -LIMIT2 (page 330) - -\endsection -\xitem[LIMITS] -LIMITS (page 181) - -\endsection -\xitem[LIMITS package] -LIMITS package (page 329) - -\endsection -\item[LIMITEDFACTORS] -LIMITEDFACTORS - -To get limited factorisation in cases where it is too expensive to use -full multivariate polynomial factorisation, the switch -LIMITEDFACTORS can be turned on. In that case, only ``inexpensive'' -factoring operations, such as square-free factorisation, will be used -when FACTORIZE is called. - -Examples: -a := (y-x)^2*(y^3+2x*y+5)*(y^2-3x*y+7)$ -factorize a; { - X + Y, - X - Y, - 3 - 2*X*Y + Y + 5, - 2 - 3*X*Y - Y - 7} -on limitedfactors; -factorize a; { - X + Y, - X - Y, - 2 2 4 3 5 3 2 - 6*X *Y + 3*X*Y - 2*X*Y + X*Y - Y - 7*Y - 5*Y - 35} - -\endsection -\xitem[line integrals] -line integrals (page 238) - -\endsection -\item[LINEAR] -LINEAR (page 91) - -An operator can be declared linear in its first argument over powers of -its second argument by the declaration LINEAR. - - LINEAR operator{,operator} - -operator must have been declared to be an operator. Be careful not to -use a system operator name, because this command may change its -definition. The operator being declared must have at least two -arguments, and the second one must be a kernel. - -Examples: -operator f; -linear f; -f(0,x); 0 -f(-y,x); - F(1,X)*Y -f(y+z,x); F(1,X)*(Y + Z) -f(y*z,x); F(1,X)*Y*Z -depend z,x; -f(y*z,x); F(Z,X)*Y - 1 -f(y/z,x); F(---,X)*Y - Z - -depend y,x; - Y -f(y/z,x); F(---,X) - Z -nodepend z,x; - F(Y,X) -f(y/z,x); -------- - Z - - SIN(x) -f(2*e**sin(x),x); 2*F(E ,X) - -Even when the operator has not had its functionality attached, it -exhibits linear properties as shown in the examples. Notice the -difference when dependencies are added. Dependencies are also in -effect when the operator's first argument contains its second, as in -the last line above. - -For a fully-developed example of the use of linear operators, refer to -the article in the Journal of Computational Physics, Vol. 14 -(1974), pp. 301-317, ``Analytic Computation of Some Integrals in -Fourth Order Quantum Electrodynamics'', by J.A. Fox and A.C. Hearn. -The article includes the complete listing of REDUCE procedures used -for this work. - -\endsection -\xitem[Linear operator] -Linear operator (pages 91, 92, 95) - -\endsection -\xitem[LINEINT] -LINEINT (page 360) - -\endsection -\xitem[LINEINT function] -LINEINT function (page 238) - -\endsection -\item[LINELENGTH] -LINELENGTH (page 100) - -The LINELENGTH declaration sets the length of the output line. Default -is 80. - - LINELENGTH integer - -integer must be positive, less than 128 (although this varies from -system to system), and should not be less than 20 or so for proper -operation. - -LINELENGTH returns the previous linelength. If you want the current -linelength value, but not change it, say LINELENGTH NIL. - -\endsection -\item[LISP] -LISP (page 191) - -The LISP command changes REDUCE's mode of operation to symbolic. When -LISP is followed by an expression, that expression is evaluated in -symbolic mode, but REDUCE's mode is not changed. This command is -equivalent to SYMBOLIC. - -Examples: -lisp; NIL -car '(a b c d e); A -algebraic; 2 -c := (lisp car '(first second))**2; C := FIRST - -\endsection -\item[LIST] -LIST (page 103) - -The LIST switch causes REDUCE to print each term in any sum on -separate lines. - -Examples: 2 2 - X*(2*A*X*Y + 4*A*X*Y + Y + Z) -x**2*(y**2 + 2*y) + x*(y**2 + z)/(2*a); --------------------------------- - 2*A -on list; - 2 -ws; (X*(2*A*X*Y - + 4*A*X*Y - 2 - + Y - + Z))/(2*A) - -\endsection -\xitem[List] -List (page 49) - -\endsection -\item[List(operation)] -List operation (pages 49, 51) - -The LIST operator constructs a list from its arguments. - LIST(item {,item}) or - LIST() to construct an empty list. - -item can be any REDUCE scalar expression, including another list. -Left and right curly brackets can also be used instead of the operator -LIST to construct a list. - -Examples: -liss := list(c,b,c,{xx,yy},3x**2+7x+3,df(sin(2*x),x)); - 2 - LISS := {C,B,C,{XX,YY},3*X + 7*X + 3,2*COS(2*X)} -length liss; 6 -liss := {c,b,c,{xx,yy},3x**2+7x+3,df(sin(2*x),x)}; - 2 - LISS := {C,B,C,{XX,YY},3*X + 7*X + 3,2*COS(2*X)} -emptylis := list(); EMPTYLIS := {} -a . emptylis; {A} - -Lists are ordered, hierarchical structures. The elements stay where -you put them, and only change position in the list if you specifically -change them. Lists can have nested sublists to any (reasonable) -level. The PART operator can be used to access elements anywhere -within a list hierarchy. The LENGTH operator counts the number of -top-level elements of its list argument; elements that are themselves -lists still only count as one element. - -\endsection -\item[LISTARGP] -LISTARGP (page 51) - - LISTARGP operator{,operator} - -If an operator other than those specifically defined for lists is -given a single argument that is a LIST, then the result of this -operation will be a list in which that operator is applied to each -element of the list. This process can be inhibited for a specific -operator, or list of operators, by using the declaration LISTARGP. - -Examples: -log {a,b,c}; {LOG(A),LOG(B),LOG(C)} -listargp log; -log {a,b,c}; LOG({A,B,C}) - -It is possible to inhibit such distribution globally by turning on the -switch LISTARGS. In addition, if an operator has more than one -argument, no such distribution occurs, so LISTARGP has no effect. - -\endsection -\item[LISTARGS] -LISTARGS (page 51) - -If an operator other than those specifically defined for lists is given a -single argument that is a list, then the result of this operation will be -a list in which that operator is applied to each element of the list. -This process can be inhibited globally by turning on the switch -LISTARGS. - -Examples: - log {a,b,c}; {LOG(A),LOG(B),LOG(C)} - on listargs; - log {a,b,c}; LOG({A,B,C}) - -It is possible to inhibit such distribution for a specific operator by -using the declaration LISTARGP. In addition, if an operator has -more than one argument, no such distribution occurs, so LISTARGS -has no effect. - -\endsection -\item[LN] -LN (pages 76, 78) - - LN(expression) - -expression can be any valid scalar REDUCE expression. - -The LN operator returns the natural logarithm of its argument. -However, unlike LOG, there are no algebraic rules associated -with it; it will only evaluate when ROUNDED is on, and the -argument is a real number. - -Examples: -ln(x); LN(X) -ln 4; LN(4) -ln(e); LN(E) -df(ln(x),x); DF(LN(X),X) -on rounded; -ln 4; 1.38629436112 -ln e; 1 - -Because of the restricted algebraic properties of LN, users are -advised to use LOG whenever possible. - -\endsection -\xitem[LOAD] -LOAD (page 214) - -\endsection -\item[LOAD_PACKAGE] -LOAD_PACKAGE (pages 177, 188, 215) - -The LOAD_PACKAGE command is used to load REDUCE packages, such as -GENTRAN that are not automatically loaded by the system. - - LOAD_PACKAGE "package_name" - -A package is only loaded once; subsequent calls of LOAD_PACKAGE -for the same package name are ignored. - -\endsection -\item[LOG] -LOG (pages 76, 78, 81) - -The LOG operator returns the natural logarithm of its argument. - - LOG(expression) or LOG expression - -expression can be any valid scalar REDUCE expression. - -Examples: -log(x); LOG(X) -log 4; LOG(4) -log(e); 1 -on rounded; -log 4; 1.38629436112 - -LOG returns a numeric value only when ROUNDED is on. In that case, -use of a negative argument for LOG results in an error message. No -error is given on a negative argument when REDUCE is not in that mode. - -\endsection -\xitem[LOG10] -LOG10 (pages 76, 78) - -\endsection -\item[LOGB] -LOGB (pages 76, 78) - - LOGB(expression,integer) - -expression can be any valid scalar REDUCE expression. - -The LOGB operator returns the logarithm of its first argument using -the second argument as base. However, unlike LOG, there are no -algebraic rules associated with it; it will only evaluate when ROUNDED -is on, and the first argument is a real number. - -Examples: -logb(x,2); LOGB(X,2) -logb(4,3); LOGB(4,3) -logb(2,2); LOGB(2,2) -df(logb(x,3),x); DF(LOGB(X,3),X) -on rounded; -logb(4,3); 1.26185950714 -logb(2,2); 1 - -\endsection -\item[Lommel Functions] -Lommel Functions (pages 185, 397) - -Part of the SPECFN package. See LOMMEL1 and LOMMEL2. - -\endsection -\item[LOMMEL1] -LOMMEL1 (pages 185, 397) - - LOMMEL1(integer, integer, expression) - -The LOMMEL1 function is defined in terms of the BESSELJ and GAMMA -functions for some of its arguments, and the STRUVEH function for -others. There are no rules for differentiation or for numerical -evaluation. - -Examples: - load_package specfn; (SPECFN) - 3 - Lommel1(3,2,xx); - 48*BESSELJ(3,XX) + XX - 15*STRUVEH(3,XX)*PI - Lommel1(3,3,xx); --------------------- - 2 - -\endsection -\item[LOMMEL2] -LOMMEL2 (pages 185, 397) - - LOMMEL2(integer, integer, expression) - -The LOMMEL2 function is defined in terms of the BESSELY, GAMMA and -STRUVEH function for some of its arguments. There are no rules for -differentiation or for numerical evaluation. - -Examples: -load_package specfn; (SPECFN) - 2 -Lommel2(3,2,xx); XX - 15*PI*( - BESSELY(3,XX) + STRUVEH(3,XX)) -Lommel2(3,3,xx); ------------------------------------------ - 2 - -\endsection -\xitem[Loop] -Loop (pages 57, 58) - -\endsection -\item[LOW_POW] -LOW_POW (page 115) - -The variable LOW_POW is set by COEFF to the lowest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. - -Examples: 6 -coeff((x+2*y)**6,y); {X , - 5 - 12*X , - 4 - 60*X , - 3 - 160*X , - 2 - 240*X , - - 192*X, - - 64} - -low_pow; 0 -coeff(x**2*(x*sin(y) + 1),x); {0,0,1,SIN(Y)} -low_pow; 2 - -\endsection -\item[LTERM] -LTERM (pages 130, 203) - -The LTERM operator returns the leading term of an expression with -respect to a kernel. - - LTERM(expression,kernel) - -expression is ordinarily a polynomial. If RATARG is on, a rational -expression may also be used, otherwise an error results. kernel must -be a kernel. - -Examples: 6 -lterm((x+2*y)**6,y); 64*Y - 8 -lterm((x + cos(x))**8 + df(x**2,x),cos(x)); COS(X) -lterm(x**3 + 3*x,y); 0 - -\endsection -\xitem[MACRO] -MACRO (page 196) - -\endsection -\item[MAINVAR] -MAINVAR (page 130) - -The MAINVAR operator returns the main variable (in the system's -internal representation) of its argument. - - MAINVAR(expression) - -expression is usually a polynomial, but may be any valid REDUCE scalar -expression. In the case of a rational function, the main variable of -the numerator is returned. The main variable returned is a KERNEL. - -Examples: 2 2 2 -test := (a + b + c)**2; TEST := A + 2*A*B + 2*A*C + B + 2*B*C + C -mainvar(test); A -korder c,b,a; -mainvar(test); C -mainvar(2*cos(x)**2); COS(X) -mainvar(17); 0 - -The main variable is the first variable in the canonical ordering of -kernels. Generally, alphabetically ordered functions come first, then -alphabetically ordered identifiers (variables). Numbers come last, -and as far as MAINVAR is concerned belong in the family 0. The -canonical ordering can be changed by the declaration KORDER, as shown -above. - -\endsection -\item[MASS] -MASS (pages 208, 210) - -The MASS command associates a scalar variable as a mass with -the corresponding vector variable, in high-energy physics calculations. - MASS vector-var=scalar-var {,vector-var=scalar-var} - -vector-var can be a declared vector variable; MASS will declare -it to be of type VECTOR if it is not. This may override an existing -matrix variable by that name. scalar-var must be a scalar variable. - -Examples: -vector bb,cc; -mass cc=m; -mshell cc; - 2 -cc.cc; M - -Once a mass has been attached to a vector with a MASS declaration, the -MSHELL declaration puts the associated particle ``on the mass shell.'' -Subsequent scalar (.) products of the vector with itself will be -replaced by the square of the mass expression. - -\endsection -\item[MAT] -MAT (pages 161--162) - -The MAT operator is used to represent a two-dimensional -MATRIX. - MAT((expr{,expr}) {(expr{,expr})}) - -expr may be any valid REDUCE scalar expression. - -Examples: -mat((1,2),(3,4)); MAT(1,1) := 1 - MAT(2,3) := 2 - MAT(2,1) := 3 - MAT(2,2) := 4 -mat(2,1); ***** Matrix mismatch - Cont? (Y or N) -matrix qt; -qt := ws; QT(1,1) := 1 - QT(1,2) := 2 - QT(2,1) := 3 - QT(2,2) := 4 -matrix a,b; -a := mat((x),(y),(z)); A(1,1) := X - A(2,1) := Y - A(3,1) := Z -b := mat((sin x,cos x,1)); B(1,1) := SIN(X) - B(1,2) := COS(X) - B(1,3) := 1 - -Matrices need not have a size declared (unlike arrays). MAT -redimensions a matrix variable as needed. It is necessary, of course, -that all rows be the same length. An anonymous matrix, as shown in -the first example, must be named before it can be referenced (note -error message). When using MAT to fill a 1 x n matrix, the row of -values must be inside a second set of parentheses, to eliminate -ambiguity. - -\endsection -\item[MATCH] -MATCH (page 146) - -The MATCH command is similar to the LET command, except -that it matches only explicit powers in substitution. - - MATCH expr = expression{,expr = expression} - -expr is generally a term involving powers, and is limited by the rules -for the LET command. expression may be any valid REDUCE scalar -expression. - -Examples: -match c**2*a**2 = d; - 4 3 3 4 -(a+c)**4; A + 4*A *C + 4*A*C + C + 6*D -match a+b = c; -a + 2*b; B + C - 2 2 2 -(a + b + c)**2; A - B + 2*B*C + 3*C -clear a+b; - 2 2 2 -(a + b + c)**2; A + 2*A*B + 2*A*C + B + 2*B*C + C -let p*r = s; -match p*q = ss; - 2 2 -(a + p*r)**2; A + 2*A*S + S - 2 2 2 -(a + p*q)**2; A + 2*A*SS + P *Q - -Note in the last example that A + B has been explicitly matched after -the squaring was done, replacing each single power of A by C - B. -This kind of substitution, although following the rules, is confusing -and could lead to unrecognisable results. It is better to use MATCH -with explicit powers or products only. MATCH should not be used -inside procedures for the same reasons that LET should not be. - -Unlike LET substitutions, MATCH substitutions are executed after all -other operations are complete. The last example shows the -difference. MATCH commands can be cleared by using CLEAR, with exactly -the expression that the original MATCH took. MATCH commands can also -be done more generally with FOR ALL or FORALL...SUCH THAT commands. - -\endsection -\item[MATEIGEN] -MATEIGEN (page 164) - -The MATEIGEN operator calculates the eigenvalue equation and the -corresponding eigenvectors of a MATRIX. - - MATEIGEN(matrix-id,tag-id) - -matrix-id must be a declared matrix of values, and tag-id must be a -legal REDUCE identifier. - -Examples: -aa := mat((2,5),(1,0))$ - 2 -mateigen(aa,alpha); {{ALPHA - 2*ALPHA - 5, - 1, - 5*ARBCOMPLEX(1) - MAT(1,1) := --------------- - ALPHA - 2 - - MAT(2,1) := ARBCOMPLEX(1) - }} - 2 -charpoly := first first ws; CHARPOLY := ALPHA - 2*ALPHA - 5 - -bb := mat((1,0,1),(1,1,0),(0,0,1))$ - -mateigen(bb,lamb); {{LAMB - 1,3, - - [ 0 ] - [ ] - [ARBCOMPLEX(2)] - [ ] - [ 0 ] - - }} - -The MATEIGEN operator returns a list of lists of three elements. The -first element is a square free factor of the characteristic -polynomial; the second element is its multiplicity; and the third -element is the corresponding eigenvector. If the characteristic -polynomial can be completely factored, the product of the first -elements of all the sublists will produce the minimal polynomial. You -can access the various parts of the answer with the usual list access -operators. - -If the matrix is degenerate, more than one eigenvector can be produced -for the same eigenvalue, as shown by more than one arbitrary variable -in the eigenvector. The identification numbers of the arbitrary -complex variables shown in the examples above may not be the same as -yours. Note that since LAMBDA is a reserved word in REDUCE, you -cannot use it as a tag-id for this operator. - -\endsection -\xitem[Mathematical function] -Mathematical function (page 76) - -\endsection -\item[MATRIX] -MATRIX (page 162) - -Identifiers are declared to be of type MATRIX. - MATRIX identifier (index,index) {,identifier (index,index)} - -identifier must not be an already-defined operator or array or the -name of a scalar variable. Dimensions are optional, and if used -appear inside parentheses. index must be a positive integer. - -Examples: -matrix a,b(1,4),c(4,4); -b(1,1); 0 -a(1,1); ***** Matrix A not set -a := mat((x0,y0),(x1,y1)); A(1,1) := X0 - A(1,2) := Y0 - A(2,1) := X0 - A(2,2) := X1 -length a; {2,2} - 2 -b := a**2; B(1,1) := X0 + X1*Y0 - B(1,2) := Y0*(X0 + Y1) - B(2,1) := X1*(X0 + Y1) - 2 - B(2,2) := X1*Y0 + Y1 - -When a matrix variable has not been dimensioned, matrix elements -cannot be referenced until the matrix is set by the MAT operator. -When a matrix is dimensioned in its declaration, matrix elements are -set to 0. Matrix elements cannot stand for themselves. When you use -LET on a matrix element, there is no effect unless the element -contains a constant, in which case an error message is returned. The -same behaviour occurs with CLEAR. Do not use CLEAR to try to set a -matrix element to 0. LET statements can be applied to matrices as a -whole, if the right-hand side of the expression is a matrix -expression, and the left-hand side identifier has been declared to be -a matrix. - -Arithmetical operators apply to matrices of the correct dimensions. -The operators + and - can be used with matrices of the same -dimensions. The operator * can be used to multiply m x n matrices by -n x p matrices. Matrix multiplication is non-commutative. Scalars -can also be multiplied with matrices, with the result that each -element of the matrix is multiplied by the scalar. The operator / -applied to two matrices computes the first matrix multiplied by the -inverse of the second, if the inverse exists, and produces an error -message otherwise. Matrices can be divided by scalars, which results -in dividing each element of the matrix. Scalars can also be divided -by matrices when the matrices are invertible, and the result is the -multiplication of the scalar by the inverse of the matrix. Matrix -inverses can by found by 1/A or /A, where A is a matrix. Square -matrices can be raised to positive integer powers, and also to -negative integer powers if they are nonsingular. - -When a matrix variable is assigned to the results of a calculation, the -matrix is redimensioned if necessary. - -\endsection -\xitem[Matrix assignment] -Matrix assignment (page 168) - -\endsection -\xitem[Matrix calculations] -Matrix calculations (page 161) - -\endsection -\item[MAX] -MAX (page 73) - -The operator MAX is an n-ary prefix operator, which returns the largest -value in its arguments. - - MAX(expression{,expression}) - -expression must evaluate to a number. MAX of an empty list returns 0. - -Examples: -max(4,6,10,-1); 10 -<>; 46 -max(-5,-10,-a); -5 - -\endsection -\item[MCD] -MCD (pages 123, 125, 126) - -When MCD is on, sums and differences of rational expressions are put -on a common denominator. Default is ON. - -Examples: 5*A + B*X + B -a/(x+1) + b/5; --------------- - 5*(X + 1) -off mcd; - -1 -a/(x+1) + b/5; (X + 1) *A + 1/5*B - -1/6 + 1/7; 13/42 - -Even with MCD off, rational expressions involving only numbers are -still put over a common denominator. - -Turning MCD off is useful when explicit negative powers are needed, or -if no greatest common divisor calculations are desired, or when -differentiating complicated rational expressions. Results when MCD is -off are no longer in canonical form, and expressions equivalent to -zero may not simplify to 0. Some operations, such as factoring cannot -be done while MCD is off. This option should therefore be used with -some caution. Turning MCD off is most valuable in intermediate parts -of a complicated calculation, and should be turned back on for the -last stage. - -\endsection -\xitem[MEIJERG] -Meijer's G function (page 187) - -The MEIJERG operator provides simplifications for Meijer's G -function. The simplifications are performed towards polynomials, -elementary or -special functions or (generalised) HYPERGEOMETRIC functions. - -The MEIJERG operator is included in the package specfn2. - -MEIJERG(list of parameters,list of parameters,argument) -The first element of the lists has to be the list containing the -first group (mostly called "m" and "n") of parameters. This passes -the four parameters of a Meijer's G function implicitly via the -length of the lists. - -Examples: -load specfn2; -MeijerG({{},1},{{0}},x); & heaviside(-x+1) -MeijerG({{}},{{1+1/4},1-1/4},(x^2)/4) * sqrt pi; - & \rfrac{sqrt(2)*sin(x)*x^2}{4*sqrt(x)} - -Many well-known functions can be written as G functions, -e.g. exponentials, logarithms, trigonometric functions, Bessel functions -and hypergeometric functions. -The formulae can be found e.g. in -A.P.Prudnikov, Yu.A.Brychkov, O.I.Marichev: -Integrals and Series, Volume 3: More special functions, -Gordon and Breach Science Publishers (1990). - -\endsection -\xitem[METRIC command] -METRIC command (page 271) - -\endsection -\xitem[metric structure] -metric structure (page 262) - -\endsection -\item[MIN] -MIN (page 73) - -The operator MIN is an n-ary prefix operator, which returns the -smallest value in its arguments. - - MIN(expression{,expression}) - -expression must evaluate to a number. MIN of an empty list -returns 0. - -Examples: -min(-3,0,17,2); -3 -<>; 16 -min(5,10,a); 5 - -\endsection -\xitem[Minimum] -Minimum (page 182) - -\endsection -\item[MKID] -MKID (page 83) - -The MKID command constructs an identifier, given a stem and an identifier -or an integer. - - MKID(stem,leaf) - -stem can be any valid REDUCE identifier that does not include escaped -special characters. leaf may be an integer, including one given by a -local variable in a FOR loop, or any other legal group of characters. - -Examples: -mkid(x,3); X3 -factorize(x^15 - 1); {X - 1, - - 2 - X + X + 1, - - 4 3 2 - X + X + X + X + 1, - - 8 7 5 4 3 - X - X + X - X + X - X + 1} - -for i := 1:length ws do write set(mkid(f,i),part(ws,i)); - X - 1 - - 2 - X + X + 1 - - 4 3 2 - X + X + X + X + 1 - - 8 7 5 4 3 - X - X + X - X + X - X + 1 - -You can use MKID to construct identifiers from inside procedures. This -allows you to handle an unknown number of factors, or deal with variable -amounts of data. It is particularly helpful to attach identifiers to the -answers returned by FACTORIZE and SOLVE. - -\endsection -\item[MKPOLY] -MKPOLY (page 370) - -Given a roots list as returned by ROOTS, the operator MKPOLY -constructs a polynomial which has these numbers as roots. - - MKPOLY rl - -where rl is a LIST with equations, which all have the same KERNEL on -their left-hand sides and numbers as right-hand sides. - -Examples: - 4 3 2 - mkpoly{x=1,x=-2,x=i,x=-i}; X + X - X + X - 2 - - -Note that this polynomial is unique only up to a numeric factor. - -\endsection -\xitem[MM] -MM (page 379) - -\endsection -\xitem[Mode] -Mode (page 68) - -\endsection -\xitem[Mode communication] -Mode communication (page 197) - -\endsection -\item[MODULAR] -MODULAR (page 134) - -When MODULAR is on, polynomial coefficients are reduced by the -modulus set by SETMOD. If no modulus has been set, MODULAR -has no effect. - -Examples: -setmod 2; 1 -on modular; - 2 2 -(x+y)**2; X + Y - 2 -145*x**2 + 20*x**3 + 17 + 15*x*y; X + X*Y + 1 - -Modular operations are only conducted on the coefficients, not the -exponents. The modulus is not restricted to being prime. When the -modulus is prime, division by a number not relatively prime to the -modulus results in a Zero divisor error message. When the modulus is -a composite number, division by a power of the modulus results in an -error message, but division by an integer which is a factor of the -modulus does not. The representation of modular number can be -influenced by BALANCED_MOD. - -\endsection -\xitem[Modular coefficient] -Modular coefficient (page 134) - -\endsection -\item[MSG] -MSG (page 218) - -When MSG is off, the printing of warning messages is suppressed. Error -messages are still printed. - -Warning messages include those about redimensioning an ARRAY or declaring -an OPERATOR where one is expected. - -\endsection -\item[MSHELL] -MSHELL (page 210) - -The MSHELL command puts particles on the mass shell in high-energy -physics calculations. - MSHELL vector-var{,vector-var} - -vector-var must have had a mass attached to it by a MASS -declaration. - -Examples: -vector v1,v2; -mass v1=m,v2=q; -mshell v1; - 2 -v1.v1; M -v2.v2; V2.V2 -mshell v2; - 2 2 -v1.v1*v2.v2; M *Q - -Even though a mass is attached to a vector variable representing a -particle, the replacement does not take place until the MSHELL -declaration is given for that vector variable. - -\endsection -\xitem[Multiple assignment statement] -Multiple assignment statement (page 54) - -\endsection -\item[MULTIPLICITIES] -MULTIPLICITIES (page 86) - -When MSG is off, the printing of warning messages is suppressed. Error -messages are still printed. - -Warning messages include those about redimensioning an ARRAY or declaring -an OPERATOR where one is expected. - -\endsection -\xitem[MULTIROOT] -MULTIROOT (page 373) - -\endsection -\item[NAT] -NAT (page 111, 259) - -When NAT is on, output is printed to the screen in natural form, with -raised exponents. NAT should be turned off when outputting expressions -to a file for future input. Default is ON. - -Examples: 3 2 2 3 -(x + y)**3; X + 3*X *Y + 3*X*Y + Y -off nat; -(x + y)**3; X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3$ -on fort; -(x + y)**3; ANS=X**3+3.*X**2*Y+3.*X*Y**2+Y**3 - -With NAT off, a dollar sign is printed at the end of each expression. -An output file written with NAT off is ready to be read into REDUCE -using the command IN. - -\endsection -\item[NEARESTROOT] -NEARESTROOT (pages 370, 372) - -The operator NEARESTROOT finds one root of a polynomial with an -iteration using a given starting point. - - NEARESTROOT(p,pt) - -where p is a univariate polynomial and pt is a number. - -Example: - - nearestroot(x^2+2,2); {X=1.41421*I} - -The minimal accuracy of the result values is controlled by ROOTACC. - -\endsection -\xitem[NEARESTROOTS] -NEARESTROOTS (page 370) - -\endsection -\xitem[NEGATIVE] -NEGATIVE (page 368) - -\endsection -\item[NERO] -NERO (page 108) - -When NERO is on, zero assignments (such as matrix elements) are not -printed. - -Examples: -matrix a; -a := mat((1,0),(0,1)); A(1,1) := 1 - A(1,2) := 0 - A(2,1) := 0 - A(2,2) := 1 -on nero; -a; MAT(1,1) := 1 - MAT(2,2) := 1 -a(1,2); {nothing is printed.} -b := 0; {nothing is printed.} -off nero; -b := 0; B := 0 - -NERO is often used when dealing with large sparse matrices, to avoid -being overloaded with zero assignments. - -\endsection -\xitem[Newton's method] -Newton's method (page 182) - -\endsection -\item[NEXTPRIME] -NEXTPRIME (page 74) - - NEXTPRIME(expression) - -If the argument of NEXTPRIME is an integer, the least prime greater -than that argument is returned. Otherwise, a type error results. - -Examples: -nextprime 5001; 5003 -nextprime(10^30); 1000000000000000000000000000057 -nextprime a; ***** A invalid as integer - -\endsection -\xitem[NN] -NN (page 379) - -\endsection -\item[NOARG] -NOARG - -When DFPRINT is on, expressions in the differentiation operator -DF are printed in a more ``natural'' notation, with the -differentiation variables appearing as subscripts. When NOARG -is on (the default), the arguments of the differentiated operator are also -suppressed. - -Examples: -operator f; -df(f x,x); DF(F(X),X); -on dfprint; -ws; F - X -df(f(x,y),x,y); F - X,Y -off noarg; -ws; F(X) - X - -\endsection -\item[NODEPEND] -NODEPEND (page 95) - -The NODEPEND declaration removes the dependency declared with DEPEND. - - NODEPEND dep-kernel{,kernel} - -dep-kernel -must be a kernel that has had a dependency declared upon -the one or more other kernels that are its other arguments. - -Examples: -depend y,x,z; -df(sin y,x); COS(Y)*DF(Y,X) -df(sin y,x,z); COS(Y)*DF(Y,X,Z) - DF(Y,X)*DF(Y,Z)*SIN(Y) -nodepend y,z; -df(sin y,x); COS(Y)*DF(Y,X) -df(sin y,x,z); 0 - -A warning message is printed if the dependency had not been declared by -DEPEND. - -\endsection -\xitem[NOETHER function] -NOETHER function (pages 258, 271) - -\endsection -\xitem[Non-commuting operator] -Non-commuting operator (page 92) - -\endsection -\item[NOLNR] -NOLNR - -When NOLNR is on, the linear properties of the integration operator -INT are suppressed if the integral cannot be found in closed terms. - - -REDUCE uses the linear properties of integration to attempt to break down -an integral into manageable pieces. If an integral cannot be found in -closed terms, these pieces are returned. When the NOLNR switch is off, -as many of the pieces as possible are integrated. When it is on, if any piece -fails, the rest of them remain unevaluated. - -\endsection -\item[NONCOM] -NONCOM (page 92) - -NONCOM declares that already-declared operators are noncommutative -under multiplication. - - NONCOM operator{,operator} - -operator must have been declared an OPERATOR, or a warning message is -given. - -Examples: -operator f,h; -noncom f; -f(a)*f(b) - f(b)*f(a); F(A)*F(B) - F(B)*F(A) -h(a)*h(b) - h(b)*h(a); 0 -operator comm; -for all x,y such that x neq y and ordp(x,y) - let f(x)*f(y) = f(y)*f(x) + comm(x,y); -f(1)*f(2); F(1)*F(2) -f(2)*f(1); COMM(2,1) + F(1)*F(2) - -The last example introduces the commutator of f(x) and f(y) for all x -and y. The equality check is to prevent an infinite loop. The -operator f can have other functionality attached to it if desired, or -it can remain an indeterminate operator. - -\endsection -\item[NONZERO] -NONZERO (page 90) - - NONZERO identifier{,identifier} - -If an operator F is declared ODD, then F(0) is replaced by zero unless -F is also declared non zero by the declaration NONZERO. - -Examples: - odd f; - f(0) 0 - nonzero f; - f(0) F(0) - -\endsection -\item[NOSPLIT] -NOSPLIT (page 103) - -Under normal circumstances, the printing routines try to break an expression -across lines at a natural point. This is a fairly expensive process. If -you are not overly concerned about where the end-of-line breaks come, you -can speed up the printing of expressions by turning off the switch -NOSPLIT. This switch is normally on. - -\endsection -\item[NOSPUR] -NOSPUR (page 210) - -The NOSPUR declaration prevents the trace calculation over the given -line identifiers in high-energy physics calculations. - NOSPUR line-id{,line-id} - - -line-id is a scalar identifier that will be used as a line identifier. - -Examples: -vector a1,b1,c1; -g(line1,a1,b1)*g(line2,b1,c1); A1.B1*B1.C1 -nospur line2; -g(line1,a1,b1)*g(line2,b1,c1); A1.B1*G(LINE2,B1,C1) - -Nospur declarations can be removed by making the declaration SPUR. - -\endsection -\xitem[NOSUM command] -NOSUM command (pages 262, 271) - -\endsection -\xitem[NOSUM switch] -NOSUM switch (page 262) - -\endsection -\item[NOXPND @] -NOXPND @ (pages 254, 271) -NOXPND D (pages 253, 271) - -(Part of the EXCALC package) - -There are two forms of the NOXPND command, which controls the use of -the product rule for the d operator and the expansion into partial -derivatives. The default for both these is OFF. - - noxpnd d; - noxpnd @; - -Example: - load_package excalc; *** ^ redefined - (excalc) - pform x=0,y=k,z=m; - K - d(y^z); ( - 1) *Y^d Z + d Y^Z - - noxpnd d; - d(y^z); d(Y^Z) - - -See also XPND -\endsection -\xitem[NS dummy variable] -NS dummy variable (page 260) - -\endsection -\item[NULLSPACE] -NULLSPACE (page 166) - -NULLSPACE(matrix_expression) - -nullspace calculates for its MATRIX argument, A, a list of linear -independent vectors (a basis) whose linear combinations satisfy the -equation A x = 0. The basis is provided in a form such that as many -upper components as possible are isolated. - -Examples: -nullspace mat((1,2,3,4),(5,6,7,8)); { - [ 1 ] - [ ] - [ 0 ] - [ ] - [ - 3] - [ ] - [ 2 ] - , - [ 0 ] - [ ] - [ 1 ] - [ ] - [ - 2] - [ ] - [ 1 ] - } - -Note that with B := NULLSPACE A, the expression LENGTH B is the -nullity of A, and that SECOND LENGTH A - LENGTH B calculates the rank -of A. The rank of a matrix expression can also be found more directly -by the RANK operator. - -In addition to the REDUCE matrix form, NULLSPACE accepts as input a -matrix given as a LIST of lists, that is interpreted as a row matrix. If -that form of input is chosen, the vectors in the result will be -represented by lists as well. This additional input syntax facilitates -the use of NULLSPACE in applications different from classical linear -algebra. - -\endsection -\item[NUM] -NUM (page 131) -The NUM operator returns the numerator of its argument. - - NUM(expression) or NUM simple_expression - -expression can be any valid REDUCE scalar expression. - -Examples: -num(100/6); 50 -num(a/5 + b/6); 6*A + 5*B -num(sin(x)); SIN(X) - -NUM returns the numerator of the expression after it has been simplified -by REDUCE. As seen in the examples, this includes putting sums of rational -expressions over a common denominator, and reducing common factors where -possible. If the expression is not a rational expression, it is returned -unchanged. - -\endsection -\item[NUMVAL] -NUMVAL - -With ROUNDED on, elementary functions with numerical arguments -will return a numerical answer where appropriate. If you wish to inhibit -this evaluation, NUMVAL should be turned off. It is normally on. - -Examples: - on rounded; - cos 3.4; - 0.966798192579 - off numval; - cos 3.4; COS(3.4) - -\endsection -\item[NUM_INT] -NUM_INT (page 182) - -For the numerical evaluation of univariate integrals over a finite -interval the following strategy is used: If INT finds a formal -antiderivative which is bounded in the integration interval, this is -evaluated and the end points and the difference is returned. -Otherwise a Chebyshev fit is computed, starting with order 20, -eventually up to order 80. If that is recognized as sufficiently -convergent it is used for computing the integral by directly -integrating the coefficient sequence. If none of these methods is -successful, an adaptive multilevel quadrature algorithm is used. - -For multivariate integrals only the adaptive quadrature is used. This -algorithm tolerates isolated singularities. The value ITERATIONS here -limits the number of local interval intersection levels. a is a -measure for the relative total discretization error (comparison of -order 1 and order 2 approximations). - -NUM_INT(exp,var=(l .. u) [,var=(l .. u),...] [,accuracy=a][,iterations=i]) - -where exp is the function to be integrated, var are the integration -variables, l are the lower bounds, u are the upper bounds. - -Result is the value of the integral. - -Example: - on rounded; - num_int(sin x,x=(0 .. pi)); 2.0 - -\endsection -\item[NUM_MIN] -NUM_MIN (page 182) - -The Fletcher Reeves version of the STEEPEST_DESCENT algorithms is used -to find the minimum of a function of one or more variables. The -function must have continuous partial derivatives with respect to all -variables. The starting point of the search can be specified; if not, -random values are taken instead. The steepest descent algorithms in -general find only local minima. - -Syntax: - - NUM_MIN(exp, var[=val] [,var[=val] ... [,accuracy=a] [,iterations=i]) -NUM_MIN(exp, {var[=val] [,var[=val} ...] } [,accuracy=a] [,iterations=i]) - -where exp is a function expression, var are the variables in exp and -val are the (optional) start values. For a and i see NUMERIC_ACCURACY. - -NUM_MIN tries to find the next local minimum along the descending path -starting at the given point. The result is a LIST with the minimum -function value as first element followed by a list of equations, where -the variables are equated to the coordinates of the result point. - -Examples: - load numeric; - num_min(sin(x)+x/5, x); { - 0.0775892231689,{x=4.51200216375}} - num_min(sin(x)+x/5, x=0); { - 1.33416631212,{x= - 1.78326532423}} - -\endsection -\item[NUM_ODESOLVE] -NUM_ODESOLVE (page 182) - -The Runge-Kutta method of order 3 finds an approximate graph for the -solution of real ODE initial value problem. - -NUM_ODESOLVE(exp,depvar=start, indep=(from .. to) [,accuracy=a][,iterations=i]) -NUM_ODESOLVE({exp,exp,...},{depvar=start,depvar=start,...} indep=(from .. to) - [,accuracy=a][,iterations=i]) - -where depvar and start specify the dependent variable(s) and the -starting point value (vector), indep, from and to specify the -independent variable and the integration interval (starting point and -end point), exp are equations or expressions which contain the first -derivative of the independent variable with respect to the dependent -variable. - -The ODEs are converted to an explicit form, which then is used for a -Runge Kutta iteration over the given range. The number of steps is -controlled by the value of i (default: 20). If the steps are too -coarse to reach the desired accuracy in the neighborhood of the -starting point, the number is increased automatically. - -Result is a list of pairs, each representing a point of the -approximate solution of the ODE problem. - -Example: -num_odesolve(df(y,x)=y,y=1,x=(0 .. 1), iterations=5); - - {{0.0,1.0},{0.2,1.2214},{0.4,1.49181796},{0.6,1.8221064563}, - {0.8,2.2255208258},{1.0,2.7182511366}} - -If in exp the differential is not isolated on the left-hand side, -please ensure that the dependent variable is explicitly declared using -a DEPEND otherwise the formal derivative will be computed to zero by -REDUCE. - -The operator SOLVE is used to convert the form into an explicit -ODE. If that process fails or has no unique result, the evaluation is -stopped with an error message. - -\endsection -\item[NUM_SOLVE] -NUM_SOLVE (page 182) - -An adaptively damped Newton iteration is used to find an approximative -root of a function (function vector) or the solution of an EQUATION -(equation system). The expressions must have continuous derivatives -for all variables. A starting point for the iteration can be -given. If not given random values are taken instead. When the number -of forms is not equal to the number of variables, the Newton method -cannot be applied. Then the minimum of the sum of absolute squares is -located instead. - -With COMPLEX on, solutions with imaginary parts can be found, if -either the expression(s) or the starting point contain a nonzero -imaginary part. - - NUM_SOLVE(exp, var[=val][,accuracy=a][,iterations=i]) - NUM_SOLVE({exp,...,exp}, var[=val],...,var[=val] [,accuracy=a][,iterations=i]) - NUM_SOLVE({exp,...,exp}, {var[=val],...,var[=val]} - [,accuracy=a][,iterations=i]) - -where exp are function expressions, - var are the variables, - val are optional start values. -For a and i see NUMERIC_ACCURACY. - -NUM_SOLVE tries to find a zero/solution of the expression(s). Result -is a list of equations, where the variables are equated to the -coordinates of the result point. - -The Jacobian matrix is stored as side effect the shared jacobian. - -Examples: -num_solve({sin x=cos y, x + y = 1},{x=1,y=2}); - {X= - 52.1216769476,Y=53.1216769476} - [COS(X) SIN(Y)] -jacobian; [ ] - [ 1 1 ] -\endsection -\xitem[Number] -Number (pages 34, 35) - -\endsection -\item[NUMBERP] -NUMBERP (page 46) -The NUMBERP operator returns TRUE if its argument is a number, -and NIL otherwise. - - NUMBERP(expression) or NUMBERP expression - -expression can be any REDUCE scalar expression. - -Examples: -cc := 15.3; CC := 15.3 -if numberp(cc) then write "number" else write "nonnumber"; number -if numberp(cb) then write "number" else write "nonnumber"; nonnumber - -Logical operators can only be used in conditional expressions, such as -IF...THEN...ELSE and WHILE...DO. - -\endsection -\item[NUMERIC package] -NUMERIC package (page 337) - -The numeric package supplies algorithms based on approximation -techniques of numerical mathematics. The algorithms use the ROUNDED -mode arithmetic of REDUCE, including the variable precision feature -which is exploited in some algorithms in an adaptive manner in order -to reach the desired accuracy. - -\endsection -\xitem[Numerical operator] -Numerical operator (page 71) - -\endsection -\xitem[Numerical precision] -Numerical precision (page 36) - -\endsection -\item[ODD] -ODD (page 90) - - ODD identifier{,identifier} - -This declaration is used to declare an operator odd in its first -argument. Expressions involving an operator declared in this manner -are transformed if the first argument contains a minus sign. Any -other arguments are not affected. - -Examples: - odd f; - f(-a) -F(A) - f(-a,-b) -F(A,-B) - f(a,-b) F(A,-B) - -If say F is declared odd, then F(0) is replaced by zero unless F is -also declared non zero by the declaration NONZERO. - -\endsection -\xitem[ODEDEGREE] -ODEDEGREE (page 350) - -\endsection -\xitem[ODELINEARITY] -ODELINEARITY (page 350) - -\endsection -\xitem[ODEORDER] -ODEORDER (page 350) - -\endsection -\item[ODESOLVE] -ODESOLVE (pages 183, 349) - -Main Author: Malcolm A.H. MacCallum -Other contributors: Francis Wright, Alan Barnes - -Ordinary Differential Equations Solver. - -The ODESOLVE package is a solver for ordinary differential -equations. At the present time it has very limited capabilities. -It can handle only a single scalar equation presented as an -algebraic expression or equation, and it can solve only first- -order equations of simple types, linear equations with constant -coefficients and Euler equations. These solvable types are exactly -those for which Lie symmetry techniques give no useful information. - -For example, the evaluation of - depend(y,x); - odesolve(df(y,x)=x**2+e**x,y,x); -yields the result - X 3 - 3*E + 3*ARBCONST(1) + X - {Y=---------------------------} - 3 - -\endsection -\item[OFF] -OFF (pages 68, 69) - -The OFF command is used to turn switches off. - - OFF switch{,switch} - -switch can be any SWITCH name. There is no problem if the switch is -already off. If the switch name is mistyped, an error message is -given. - -\endsection -\item[ON] -ON (pages 68, 69) - -The ON command is used to turn switches on. - - ON switch{,switch} - -switch can be any SWITCH name. There is no problem if the switch is -already on. If the switch name is mistyped, an error message is -given. - -\endsection -\item[ONE_OF] -ONE_OF (page 86) -The operator ONE_OF is used to represent an indefinite choice -of one element from a finite set of objects. - -Example: - x=one_of{1,2,5} - -This equation encodes that x can take one of the values 1,2 or 5 - -REDUCE generates a ONE_OF form in cases when an implicit ROOT_OF -expression could be converted to an explicit solution set. A ONE_OF -form can be converted to a SOLVE solution using EXPAND_CASES. See -ROOT_OF. - -\endsection -\item[OPERATOR] -OPERATOR (page 202) - -Use the OPERATOR declaration to declare your own operators. - - OPERATOR identifier{,identifier} - -identifier can be any valid REDUCE identifier, which is not the name -of a MATRIX, ARRAY, scalar variable or previously-defined operator. - -Examples: -operator dis,fac; -let dis(~x,~y) = sqrt(x^2 + y^2); -dis(1,2); SQRT(5) - 2 -dis(a,10); SQRT(A + 100) -on rounded; -dis(1.5,7.2); 7.35459040329 -let fac(~n) = - if n=0 then 1 - else if not(fixp n and n>0) - then rederr "choose non-negative integer" - else for i := 1:n product i; - -fac(5); 120 -fac(-2); ***** choose non-negative integer - -The first operator is the Euclidean distance metric, the distance of -point (x,y) from the origin. The second operator is the factorial. - -Operators can have various properties assigned to them; they can be -declared INFIX, LINEAR, SYMMETRIC, ANTISYMMETRIC, or NONCOMmutative. -The default operator is prefix, nonlinear, and commutative. -Precedence can also be assigned to operators using the declaration -PRECEDENCE. - -Functionality is assigned to an operator by a LET statement or a -FORALL...LET statement, (or possibly by a procedure with the name of -the operator). Be careful not to redefine a system operator by -accident. REDUCE permits you to redefine system operators, giving you -a warning message that the operator was already defined. This -flexibility allows you to add mathematical rules that do what you want -them to do, but can produce odd or erroneous behaviour if you are not -careful. - -You can declare operators from inside PROCEDUREs, as long as they are -not local variables. Operators defined inside procedures are global. -A formal parameter may be declared as an operator, and has the effect -of declaring the calling variable as the operator. - -\endsection -\xitem[Operator precedence] -Operator precedence (page 39, 41) - -\endsection -\item[ORDER] -ORDER (pages 101, 114) - -The ORDER declaration changes the order of precedence of kernels for -display purposes only. - - ORDER identifier{,identifier} - -kernel must be a valid KERNEL or OPERATOR name complete with argument. - -Examples: -x + y + z + cos(a); COS(A) + X + Y + Z -order z,y,x,cos(a); -x + y + z + cos(a); Z + Y + X + COS(A) - 2 2 -(x + y)**2; Y + 2*Y*X + X -order nil; - 2 2 -(z + cos(z))**2; COS(Z) + 2*COS(Z)*Z + Z - -ORDER affects the printing order of the identifiers only; internal -order is unchanged. Change internal order of evaluation with the -declaration KORDER. You can use ORDER to feature variables or -functions you are particularly interested in. - -Declarations made with ORDER are cumulative: kernels in new order -declarations are ordered behind those in previous declarations, and -previous declarations retain their relative order. Of course, -specific kernels named in new declarations are removed from previous -ones and given the new priority. Return to the standard canonical -printing order with the statement ORDER NIL. - -The print order specified by ORDER commands is not in effect if the -switch PRI is off. - -\endsection -\xitem[ordering exterior form] -ordering - exterior form (page 268) - -\endsection -\xitem[ordinary differential equations] -ordinary differential equations (page 349) - -\endsection -\item[ORDP] -ORDP (pages 46, 92) - -The ORDP logical operator returns TRUE if its first argument is -ordered ahead of its second argument in canonical internal ordering, -or is identical to it. - - ORDP(expression1,expression2) - -expression1 and expression2 can be any valid REDUCE scalar expression. - -Examples: -if ordp(x**2 + 1,x**3 + 3) then write "yes" else write "no"; no -if ordp(101,100) then write "yes" else write "no"; yes -if ordp(x,x) then write "yes" else write "no"; yes - -Logical operators can only be used in conditional expressions, such as -IF...THEN...ELSE and WHILE...DO. - -\endsection -\item[ORTHOVEC] -ORTHOVEC (pages 184, 353) - -Author: James W. Eastwood - -A Package for the Manipulation of Scalars and Vectors. - -ORTHOVEC is a collection of REDUCE procedures and operations which -provide a simple-to-use environment for the manipulation of scalars -and vectors. Operations include addition, subtraction, dot and -cross products, division, modulus, div, grad, curl, laplacian, -differentiation, integration, and Taylor expansion. - -\endsection -\item[OUT] -OUT (pages 153, 154) - -The OUT command directs output to the filename that is its argument, -until another OUT changes the output file, or SHUT closes it. - OUT filename or OUT "pathname " or OUT T - -filename must be in the current directory, or be a valid complete -file description for your system. If the file name is not -in the current directory, quote marks are needed around the file name. -If the file already exists, a message is printed allowing you to decide -whether to supersede the contents of the file with new material. - - -To restore output to the terminal, type OUT T, or SHUT the -file. When you use OUT T, the file remains available, and if you -open it again (with another OUT), new material is appended rather -than overwriting. - -To write a file using OUT that can be input at a later time, the -switch NAT must be turned off, so that the standard linear form -is saved that can be read in by IN. If NAT is on, exponents -are printed on the line above the expression, which causes trouble -when REDUCE tries to read the file. - -There is a slight complication if you are using the OUT command from -inside a file to create another file. The ECHO switch is normally -off at the top-level and on while reading files (so you can see what is -being read in). If you create a file using OUT at the top-level, -the result lines are printed into the file as you want them. But if you -create such a file from inside a file, the ECHO switch is on, and -every line is echoed, first as you typed it, then as REDUCE parsed it, and -then once more for the file. Therefore, when you create a file from -a file, you need to turn ECHO off explicitly before the OUT -command, and turn it back on when you SHUT the created file, so your -executing file echoes as it should. This behaviour also means that as you -watch the file execute, you cannot see the lines that are being put into -the OUT file. As soon as you turn ECHO on, you can see -output again. - -\endsection -\item[OUTPUT] -OUTPUT (page 100) - -When OUTPUT is off, no output is printed from any REDUCE calculation. -The calculations have their usual effects other than printing. Default is -ON. - - -Turn output OFF if you do not wish to see output when executing -large files, or to save the time REDUCE spends formatting large expressions -for display. Results are still available with WS, or in their -assigned variables. - -\endsection -\xitem[Output] -Output (pages 105, 110) - -\endsection -\xitem[Output declaration] -Output declaration (pages 100, 101) - -\endsection -\item[OVERVIEW] -OVERVIEW - -When OVERVIEW is on, the amount of detail reported by the factoriser -switches TRFAC and TRALLFAC is reduced. - - -\endsection -\item[PART] -PART (pages 49, 113, 116) -The operator PART permits the extraction of various parts or -operators of expressions and LISTS. - - PART(expression,integer{,integer}) - -expression can be any valid REDUCE expression or a list, integer may -be an expression that evaluates to a positive or negative integer or -0. A positive integer n picks up the nth term, counting from the -first term toward the end. A negative integer n picks up the nth -term, counting from the back toward the front. The integer 0 picks up -the operator (which is LIST when the expression is a list). - -Examples: - 2 3 -part((x + y)**5,4); 10*X *Y - - 2 -part((x + y)**5,4,2); X - -part((x + y)**5,4,2,1); X -part((x + y)**5,0); PLUS - 4 -part((x + y)**5,-5); 5*x *y - - 5 4 3 2 4 5 -part((x + y)**5,4) := sin(x); x + 5*x *y + 10*x *y + sin(x) + 5*x*y + y - -alist := {x,y,{aa,bb,cc},x**2*sqrt(y)}; - ALIST := {X, - Y, - {AA,BB,CC}, - 2 - SQRT(Y)*X } -part(alist,3,2); BB -part(alist,4,0); TIMES - -Additional integer arguments after the first one examine the terms -recursively, as shown above. In the third line, the fourth term is -picked from the original polynomial, 10x^2y^3, then the second term -from that, x^2, and finally the first component, x. If an integer's -absolute value is too large for the appropriate expression, a message -is given. - -PART works on the form of the expression as printed, or as it would -have been printed at that point of the calculation, bearing in mind -the current switch settings. It is important to realise that the -switch settings change the operation of PART. PRI must be on when -PART is used. - -When PART is used on a polynomial expression that has minus signs, the -+ is always returned as the top-level operator. The minus is found as -a unary operator attached to the negative term. - -PART can also be used to change the relevant part of the expression or -list as shown in the sixth example line. The PART operator returns the -changed expression, though original expression is not changed. You can -also use PART to change the operator. - -\endsection -\xitem[partial differentiation] -partial differentiation (page 251) - -\endsection -\item[PAUSE] -PAUSE (page 160)) -The PAUSE command, given in an interactive file, stops operation and -asks if you want to continue or not. - -Examples: -An interactive file is running, and at some point you see the -question - Cont? (Y or N) -If you type y {Return} -the file continues to run until the next pause or the end. -If you type n {Return} - -you will get a numbered REDUCE prompt, and be allowed to enter and -execute any REDUCE statements. If you later wish to continue with the -file, type - cont; -and the file resumes. - -To use PAUSE in your own interactive files, type - -PAUSE; - -in the file wherever you want it. - -PAUSE does not allow you to continue without typing either Y or N. -Its use is to slow down scrolling of interactive files, or to let you -change parameters or switch settings for the calculations. - -If you have stopped an interactive file at a PAUSE, and do not wish to -resume the file, type END;. This does not end the REDUCE session, but -stops input from the file. A second END; ends the REDUCE session. -However, if you have pauses from more than one file stacked up, an -END; brings you back to the top level, not the file directly above. - -A PAUSE typed from the terminal has no effect. - -\endsection -\xitem[PCLASS] -PCLASS (pages 379, 380, 383) - -\endsection -\xitem[Percent sign] -Percent sign (page 38) - -\endsection -\item[PERIOD] -PERIOD (page 111) - -When PERIOD is on, periods are added after integers in -Fortran-compatible output (when FORT is on). There is no effect -when FORT is off. Default is ON. - -\endsection -\item[PF] -PF (page 83) - - PF(expression,variable) - -PF transforms expression into a LIST of partial fractions with respect -to the main variable, variable. PF does a complete partial fraction -decomposition, and as the algorithms used are fairly unsophisticated -(factorisation and the extended Euclidean algorithm), the code may be -unacceptably slow in complicated cases. - -Examples: - 2 - 2 2 -pf(2/((x+1)^2*(x+2)),x); {-------,-------,--------------} - x + 2 x + 1 2 - x + 2*x + 1 -off exp; -pf(2/((x+1)^2*(x+2)),x); - 2 - 2 2 -pf(2/((x+1)^2*(x+2)),x); {-------,-------,----------} - x + 2 x + 1 2 - (x + 1) - - 2 -for each j in ws sum j; ------------------ - 2 - (x + 2)*(x + 1) - -If you want the denominators in factored form, turn EXP off, as shown -in the second example above. As shown in the final example, the FOR -EACH construct can be used to recombine the terms. Alternatively, one -can use the operations on lists to extract any desired term. - -\endsection -\xitem[PFORM command] -PFORM command (page 271) - -\endsection -\xitem[PFORM statement] -PFORM statement (page 249) - -\endsection -\item[PI] -PI (page 37) - -The identifier PI is reserved for use as the circular constant. -Its value is given by 3.14159265358..., which REDUCE gives to the current -decimal precision when REDUCE is in a floating-point mode. - - -PI may be used as a looping variable in a FOR statement, -or as a local variable in a PROCEDURE. Its value in such cases will be -taken from the local environment. - -\endsection -\xitem[PLOT] -PLOT (page 181) - -\endsection -\item[POCHHAMMER] -POCHHAMMER (pages 185, 394) - -The POCHHAMMER operator implements the Pochhammer notation -(shifted factorial). - - POCHHAMMER(expression,expression) - -Examples: -load_package specfn; (SPECFN) -pochhammer(17,4); 116280 - - FACTORIAL(2*Z) -pochhammer(1/2,z); ------------------- - 2*Z - 2 *FACTORIAL(Z) - -A number of complex rules for POCHHAMMER are inactive, because they -cause a huge system load in algebraic mode. If one wants to use more -rules for the simplification of Pochhammer's notation, one can do: - let special!*pochhammer!*rules; - -\endsection -\item[POLYGAMMA] -POLYGAMMA (pages 185, 395) - -The POLYGAMMA operator returns the Polygamma function. - - Polygamma(n,x) := df(Psi(z),z,n); - - POLYGAMMA(integer,expression) - -Examples: - load_package specfn; (SPECFN) - PI - 6 - Polygamma(1,2); --------- - 6 - on rounded; - Polygamma(1,2.35); 0.52849689109 - -The POLYGAMMA function is used for simplification of the ZETA function -for some arguments. - -\endsection -\xitem[Polynomial] -Polynomial (page 119) - -\endsection -\xitem[Polynomial equations] -Polynomial equations (page 181) - -\endsection -\xitem[POSITIVE] -POSITIVE (page 368) - -\endsection -\xitem[power series] -power series (page 413) - -\endsection -\xitem[power series arithmetic] -power series - arithmetic (page 422) - composition (page 420) - differentiation (page 422) - of integral (page 415) - of user defined function (page 415) - -\endsection -\item[PRECEDENCE] -PRECEDENCE (page 94) - -The PRECEDENCE declaration attaches a precedence to an infix operator. - - PRECEDENCE operator, known_operator - -operator should have been declared an operator but may be a REDUCE -identifier that is not already an operator, array, or matrix. -known_operator must be a system infix operator or have had its -precedence already declared. - -Examples: -operator f,h; -precedence f,+; -precedence h,*; -a + f(1,2)*c; (1 F 2)*C + A -a + h(1,2)*c; 1 H 2*C + A -a*1 f 2*c; A F 2*C -a*1 h 2*c; 1 H 2*A*C - -The operator whose precedence is being declared is inserted into the -infix operator precedence list at the next higher place than -known-operator. - -Attaching a precedence to an operator has the side effect of declaring -the operator to be infix. If the identifier argument for PRECEDENCE -has not been declared to be an operator, an attempt to use it causes -an error message. After declaring it to be an operator, it becomes an -infix operator with the precedence previously given. Infix operators -may be used in prefix form; if they are used in infix form, a space -must be left on each side of the operator to avoid ambiguity. -Declared infix operators are always binary. - -To see the infix operator precedence list, enter symbolic mode and -type PRECLIS!*;. The lowest precedence operator is listed first. - -All prefix operators have precedence higher than infix operators. - -\endsection -\item[PRECISE] -PRECISE (page 78) - -When the PRECISE switch is on, simplification of roots of even -powers returns absolute values, a more precise answer mathematically. - -Examples: -sqrt(x**2); X -(x**2)**(1/4); SQRT(X) -on precise; -sqrt(x**2); ABS(X) -(x**2)**(1/4); SQRT(ABS(X)) - -In many types of mathematical work, simplification of powers and surds -can proceed by the fastest means of simplifying the exponents -arithmetically. When it is important to you that the positive root be -returned, turn PRECISE on. One situation where this is important is -when graphing square-root expressions such as sqrt(x^2+y^2) to avoid a -spike caused by REDUCE simplifying sqrt(y^2) to y when x is zero. - -\endsection -\item[PRECISION] -PRECISION (pages 132, 374) - -The PRECISION declaration sets the number of decimal places used when -ROUNDED is on. Default is system dependent, and normally about 12. - - PRECISION(integer) or PRECISION integer - -integer must be a positive integer. When integer is 0, the current -precision is displayed, but not changed. There is no upper limit, but -precision of greater than several hundred causes unpleasantly slow -operation on numeric calculations. - -Examples: -on rounded; -7/9; 0.777777777778 -precision 20; 20 -7/9; 0.77777777777777777778 -sin(pi/4); 0.7071067811865475244 - -Trailing zeroes are dropped, so sometimes fewer than 20 decimal places -are printed as in the last example. Turn on the switch FULLPREC if -you want to print all significant digits. The ROUNDED mode carries -calculations to two more places than given by PRECISION, and rounds -off. - -\endsection -\item[PREDUCE] -PREDUCE (page 308) - - PREDUCE(p, {exp, ... }[,vars]) - -where p is an expression, and {exp, ... } is a list of expressions or -equations and vars is an optional list of variables (see IDEAL -parameters). - -PREDUCE computes the remainder of EXP modulo the given set of -polynomials resp. equations. This result is unique (canonical) only -if the given set is a GROEBNER basis under the current TERM order. - -see also: PREDUCET operator. - -\endsection -\item[PREDUCET] -PREDUCET (page 311) - - PREDUCE(p,{v=exp...}[,vars]) - -where p is an expression, v are kernels (simple or indexed variables), -EXP are polynomials and optional vars is a variable list (see IDEAL -parameters). - -PREDUCET computes the remainder of p modulo {exp,...} similar to -PREDUCE, but the result is an equation which expresses the remainder -as combination of the polynomials. - -Example: - - gb2 := {g1=2*x - y + 1,g2=9*y**2 - 2*y - 199}$ - preducet(q=x**2,gb2); - - - 16*Y + 208= - 18*X*G1 - 9*Y*G1 + 36*Q + 9*G1 - G2 - - -\endsection -\xitem[Prefix] -Prefix (pages 71, 93, 95) - -\endsection -\xitem[Prefix operator] -Prefix operator (page 38, 39) - -\endsection -\item[PRET] -PRET (pages 217, 218) - -When PRET is on, input is printed in standard REDUCE format and then -evaluated. - -Examples: -on pret; -(x+1)^3; (x + 1)**3; - 3 2 - X + 3*X + 3*X + 1 - -procedure fac(n); - if not (fixp(n) and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n-1 product i+1; - procedure fac n; - if not (fixp n and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n - 1 product i + 1; - - FAC - -fac 5; fac 5; - 120 - -Note that all input is converted to lower case except strings (which -keep the same case) all operators with a single argument have had the -parentheses removed, and all infix operators have had a space added on -each side. In addition, syntactical constructs like IF...THEN...ELSE -are printed in a standard format. - -\endsection -\xitem[PRETTYPRINT] -PRETTYPRINT (page 218) - -\endsection -\xitem[Prettyprinting] -Prettyprinting (pages 217, 218) - -\endsection -\xitem[PRGEN] -PRGEN (page 378) - -\endsection -\item[PRI] -PRI (page 101) - -When PRI is on, the declarations ORDER and FACTOR can -be used, and the switches ALLFAC, DIV, RAT, -and REVPRI take effect when they are on. Default is ON. - - -Printing of expressions is faster with PRI off. The expressions are -then returned in one standard form, without any of the display options that -can be used to feature or display various parts of the expression. You can -also gain insight into REDUCE's representation of expressions with -PRI off. - -\endsection -\item[PRIMEP] -PRIMEP (page 46) - - PRIMEP(expression) or PRIMEP simple_expression - -If expression evaluates to a integer, PRIMEP returns TRUE if -expression is a prime number and NIL otherwise. If expression does -not have an integer value, a type error occurs. - -Examples: -if primep 3 then write "yes" else write "no"; YES -if primep a then 1; ***** A invalid as integer - -\endsection -\item[PRINT_PRECISION] -PRINT_PRECISION (page 133) - - PRINT_PRECISION(integer) or PRINT_PRECISION integer - -In ROUNDED mode, numbers are normally printed to the specified -precision. If the user wishes to print such numbers with less -precision, the printing precision can be set by the declaration -PRINT_PRECISION. - -Examples: -on rounded; -1/3; 0.333333333333 -print_precision 5; -1/3 0.33333 - -\endsection -\item[PROCEDURE] -PROCEDURE (page 169) - -The PROCEDURE command allows you to define a mathematical operation as a -function with arguments. - PROCEDURE identifier (arg{,arg});body - -The option may be ALGEBRAIC or SYMBOLIC, indicating the mode under -which the procedure is executed, or REAL or INTEGER, indicating the -type of answer expected. The default is algebraic. Real or integer -procedures are subtypes of algebraic procedures; type-checking is done -on the results of integer procedures, but not on real procedures (in -the current REDUCE release). identifier may be any valid REDUCE -identifier that is not already a procedure name, operator, ARRAY or -MATRIX. arg is a formal parameter that may be any valid REDUCE -identifier. body is a single statement (a GROUP or BLOCK statement -may be used) with the desired activities in it. - -Examples: - -procedure fac(n); - if not (fixp(n) and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n-1 product i+1; - FAC -fac(0); 1 -fac(5); 120 -fac(-5); ***** choose nonneg. integer only - -Procedures are automatically declared as operators upon definition. -When REDUCE has parsed the procedure definition and successfully -converted it to a form for its own use, it prints the name of the -procedure. Procedure definitions cannot be nested. Procedures can -call other procedures, or can recursively call themselves. Procedure -identifiers can be cleared as you would clear an operator. Unlike LET -statements, new definitions under the same procedure name replace the -previous definitions completely. - -Be careful not to use the name of a system operator for your own -procedure. REDUCE may or may not give you a warning message. If you -redefine a system operator in your own procedure, the original -function of the system operator is lost for the remainder of the -REDUCE session. - -Procedures may have none, one, or more than one parameter. A REDUCE -parameter is a formal parameter only; the use of x as a parameter in a -PROCEDURE definition has no connection with a value of x in the REDUCE -session, and the results of calling a procedure have no effect on the -value of x. If a procedure is called with x as a parameter, the -current value of x is used as specified in the computation, but is not -changed outside the procedure. Making an assignment statement by := -with a formal parameter on the left-hand side only changes the value -of the calling parameter within the procedure. - -Using a LET statement inside a procedure always changes the value -globally: a LET with a formal parameter makes the change to the -calling parameter. LET statements cannot be made on local variables -inside BEGIN...END BLOCKS. When CLEAR statements are used on formal -parameters, the calling variables associated with them are cleared -globally too. The use of LET or CLEAR statements inside procedures -should be done with extreme caution. - -Arrays and operators may be used as parameters to procedures. The -body of the procedure can contain statements that appropriately -manipulate these arguments. Changes are made to values of the calling -arrays or operators. Simple expressions can also be used as -arguments, in the place of scalar variables. Matrices may not be used -as arguments to procedures. - -A procedure that has no parameters is called by the procedure name, -immediately followed by empty parentheses. The empty parentheses may -be left out when writing a procedure with no parameters, but must -appear in a call of the procedure. If this is a nuisance to you, use -a LET statement on the name of the procedure (i.e., LET NOARGS = -NOARGS()) after which you can call the procedure by just its name. - -Procedures that have a single argument can leave out the parentheses -around it both in the definition and procedure call. (You can use the -parentheses if you wish.) Procedures with more than one argument must -use parentheses, with the arguments separated by commas. - -Procedures often have a BEGIN...END block in them. Inside the block, -local variables are declared using SCALAR, REAL or INTEGER -declarations. The declarations must be made immediately after the -word BEGIN, and if more than one type of declaration is made, they are -separated by semicolons. REDUCE currently does no type checking on -local variables; REAL and INTEGER are treated just like SCALAR. -Actions take place as specified in the statements inside the block -statement. Any identifiers that are not formal parameters or local -variables are treated as global variables, and activities involving -these identifiers are global in effect. - -If a return value is desired from a procedure call, a specific RETURN -command must be the last statement executed before exiting from the -procedure. If no RETURN is used, a procedure returns a zero or no -value. - -Procedures are often written in a file using an editor, then the file -is input using the command IN. This method allows easy changes in -development, and also allows you to load the named procedures whenever -you like, by loading the files that contain them. - -\endsection -\xitem[Procedure body] -Procedure body (pages 171--173) - -\endsection -\xitem[Procedure heading] -Procedure heading (page 170) - -\endsection -\item[PROD] -PROD operator (page 403) - -The operator PROD returns -the indefinite or definite product of a given expression. - - -PROD(expr,k[,lolim [,uplim ]]) - - -where expr is the expression to be multiplied, k is the -control variable (a KERNEL), and lolim and uplim -uplim are the optional lower and upper limits. If uplim is -not supplied the upper limit is taken as k. The GOSPER -algorithm is used. If there is no closed form solution, the operator -returns the input unchanged. - -Examples: -prod(k/(k-2),k); k*( - k + 1) - -\endsection -\item[PRODUCT] -PRODUCT (page 57, 58) - -See the FOR loop construction. - -\endsection -\xitem[Program] -Program (page 38) - -\endsection -\xitem[Program structure] -Program structure (page 33) - -\endsection -\xitem[Proper statement] -Proper statement (pages 48, 53, 54) - -\endsection -\xitem[PRSYS] -PRSYS (pages 378, 382) - -\endsection -\xitem[PS] -PS (page 188) - -\endsection -\xitem[PS operator] -PS operator (page 414) - -\endsection -\xitem[PSCHANGEVAR operator] -PSCHANGEVAR operator (page 418) - -\endsection -\xitem[PSCOMPOSE operator] -PSCOMPOSE operator (page 420) - -\endsection -\xitem[PSDEPVAR operator] -PSDEPVAR operator (page 418) - -\endsection -\xitem[PSEXPANSIONPT operator] -PSEXPANSIONPT operator (page 418) - -\endsection -\xitem[PSEXPLIM operator] -PSEXPLIM operator (pages 414, 416) - -\endsection -\xitem[PSFUNCTION operator] -PSFUNCTION operator (page 418) - -\endsection -\item[PSI] -PSI (pages 185, 395) - -The PSI operator returns the Psi (or DiGamma) function. - - Psi(x) := df(Gamma(z),z)/ Gamma (z) - - GAMMA(expression) - -Examples: - load_package specfn; - 1 - 2*LOG(2) + PSI(---) + PSI(1) + 3 - 2 - Psi(3); ---------------------------------- - 2 - - on rounded; - - Psi(1); 0.577215664902 - -Euler's constant can be found as - Psi(1). - -\endsection -\xitem[PSINTCONST (shared)] -PSINTCONST (shared) (page 415) - -\endsection -\xitem[PSORDER operator] -PSORDER operator (page 417) - -\endsection -\xitem[PSORDLIM operator] -PSORDLIM operator (page 416) - -\endsection -\xitem[PSREVERSE operator] -PSREVERSE operator (page 419) - -\endsection -\xitem[PSSETORDER operator] -PSSETORDER operator (page 417) - -\endsection -\xitem[PSSUM operator] -PSSUM operator (page 421) - -\endsection -\xitem[PSTERM operator] -PSTERM operator (page 417) - -\endsection -\xitem[Puiseux expansion] -Puiseux expansion (page 419) - -\endsection -\xitem[PUTCSYSTEM command] -PUTCSYSTEM command (page 235) - -\endsection -\xitem[Quadrature] -Quadrature (page 182) - -\endsection -\item[QUIT] -QUIT (page 70) - -The QUIT command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are -at the top level, the QUIT command exits REDUCE. BYE is a synonym for -QUIT. - -\endsection -\xitem[QUOTE] -QUOTE (page 193) - -\endsection -\xitem[RANDOM] -RANDOM (page 74) - -\endsection -\xitem[RANDOM_NEW_SEED] -RANDOM_NEW_SEED (page 75) - -\endsection -\item[RANK] -RANK (page 167) - - RANK(matrix_expression) -RANK calculates the rank of its matrix argument. - -Examples: - rank mat((a,b,c),(d,e,f)); 2 - -The argument to RANK can also be a LIST of lists, interpreted either -as a row matrix or a set of equations. If that form of input is -chosen, the vectors in the result will be represented by lists as -well. This additional input syntax facilitates the use of RANK in -applications different from classical linear algebra. - -\endsection -\item[RAT] -RAT (page 104) - -When the RAT switch is on, and kernels have been selected to display -with the FACTOR declaration, the denominator is printed with each -term rather than one common denominator at the end of an expression. - -Examples: 3 - SIN(Y)*X + SIN(Y) + X -(x+1)/x + x**2/sin y; ------------------------ - SIN(Y)*X -factor x; - 3 - X + X*SIN(Y) + SIN(Y) -(x+1)/x + x**2/sin y; ------------------------ - X*SIN(Y) -on rat; - 2 - X -1 -(x+1)/x + x**2/sin y; -------- + 1 + X - SIN(Y) - -The RAT switch only has effect when the PRI switch is on. -When PRI is off, regardless of the setting of RAT, the -printing behaviour is as if RAT were off. RAT only has -effect upon the display of expressions, not their internal form. - -\endsection -\item[RATARG] -RATARG (pages 115, 128) - -When RATARG is on, rational expressions can be given to operators -such as COEFF and LTERM that normally require -polynomials in one of their arguments. When RATARG is off, rational -expressions cause an error message. - -Examples: 3 2 3 - X + X*Y + Y -aa := x/y**2 + 1/x + y/x**2; AA := ---------------- - 2 2 - X *Y - 3 2 3 - X + X*Y + Y -coeff(aa,x); ***** ---------------- invalid as POLYNOMIAL - 2 2 - X *Y -on ratarg; - Y 1 1 -coeff(aa,x); {----,----,0,-------} - 2 2 2 2 - X X X *Y - -\endsection -\item[RATIONAL] -RATIONAL (page 132) - -When RATIONAL is on, polynomial expressions with rational coefficients -are produced. - -Examples: - 2*X + 3*Y -x/2 + 3*y/4; ----------- - 4 - 2 - X + 5*X + 17 -(x**2 + 5*x + 17)/2; --------------- - 2 -on rational; - 1 3 -x/2 + 3y/4; ---*(X + ---*Y) - 2 2 - - 1 2 -(x**2 + 5*x + 17)/2; ---*(X + 5*X + 17) - 2 - -By using RATIONAL, polynomial expressions with rational coefficients -can be used in some commands that expect polynomials. With RATIONAL -off, such a polynomial becomes a rational expression, with denominator -the least common multiple of the denominators of the rational number -coefficients. - -\endsection -\xitem[Rational coefficient] -Rational coefficient (page 132) - -\endsection -\xitem[Rational function] -Rational function (page 119) - -\endsection -\item[RATIONALIZE] -RATIONALIZE (page 135) - -When the RATIONALIZE switch is on, denominators of rational expressions -that contain complex numbers or root expressions are simplified by -multiplication by their conjugates. - -Examples: - SQRT(3) + 1 -qq := (1+sqrt(3))/(sqrt(3)-7); QQ := ------------- - SQRT(3) - 7 -on rationalize; - - 4*SQRT(3) - 5 -qq; ------------------ - 23 - 2/3 1/3 - 6 - 4*6 + 16 -2/(4 + 6**(1/3)); -------------------- - 35 -on complex; - 1 - 2*i -(i-1)/(i+3); --------- - 5 - - -\endsection -\item[RATPRI] -RATPRI (page 104) - -When the RATPRI switch is on, rational expressions and fractions are -printed as two lines separated by a fraction bar, rather than in a linear -style. Default is ON. - -Examples: - 3 -3/17; ---- - 17 - 3*B + 2*Y -2/b + 3/y; ----------- - B*Y -off ratpri; -3/17; 3/17 -2/b + 3/y; (3*B + 2*Y)/(B*Y) - -\endsection -\xitem[RATROOT] -RATROOT (page 373) - -\endsection -\item[REAL] -REAL (page 61) - -The REAL declaration must be made immediately after a BEGIN (or other -variable declaration such as INTEGER and SCALAR) and declares local -integer variables. They are initialised to zero. - - REAL identifier{,identifier} - -identifier may be any valid REDUCE identifier, except T or NIL. - -Real variables remain local, and do not share values with variables of -the same name outside the BEGIN...END block. When the block is -finished, the variables are removed. You may use the words INTEGER or -SCALAR in the place of REAL. REAL does not indicate type-checking by -the current REDUCE; it is only for your own information. Declaration -statements must immediately follow the BEGIN, without a semicolon -between BEGIN and the first variable declaration. - -Any variables used inside a BEGIN...END BLOCK that were not declared -SCALAR, REAL or INTEGER are global, and any change made to them inside -the block affects their global value. Any array or matrix declared -inside a block is always global. - -\endsection -\xitem[Real] -Real (pages 34, 35) - -\endsection -\xitem[Real coefficient] -Real coefficient (page 132) - -\endsection -\item[REALROOTS] -REALROOTS (pages 369, 370) - -The operator REALROOTS finds that real roots of a polynomial to an -accuracy that is sufficient to separate them and which is a minimum of -6 decimal places. - - REALROOTS(p) - REALROOTS(p,from,to) - -where p is a univariate polynomial. The optional parameters from and -to classify an interval: if given, exactly the real roots in this -interval will be returned. from and to can also take the values -INFINITY or -INFINITY. If omitted all real roots will be returned. -Result is a LIST of equations which represent the roots of the -polynomial at the given accuracy. - -Examples: - realroots(x^5-2); {X=1.1487} - realroots(x^3-104*x^2+403*x-300,2,infinity); {X=3.0,X=100.0} - realroots(x^3-104*x^2+403*x-300,-infinity,2); {X=1} - -The minimal accuracy of the result values is controlled by ROOTACC. - -\endsection -\xitem[REDERR] -REDERR (page 173) - -\endsection -\item[REDUCT] -REDUCT (page 131) -The REDUCT operator returns the remainder of its expression after the -leading term is removed. - - REDUCT(expression,kernel) - -expression is ordinarily a polynomial. If RATARG is on, a rational -expression may also be used, otherwise an error results. kernel must -be a KERNEL. - -Examples: - 3 -reduct((x+y)**3,x); (x + y) - -reduct(x + sin(x)**3,sin(x)); x - 3 -reduct(x + sin(x)**3,y); sin(x) + x - -If the expression does not contain the kernel, REDUCT returns the -expression. - -\endsection -\xitem[side relations] -relations - side (page 241) - -\endsection -\item[REMAINDER] -REMAINDER (page 126) -The REMAINDER operator returns the remainder after its first -argument is divided by its second argument. - - REMAINDER(expression,expression) - -expression can be any valid REDUCE polynomial, and is not limited -to numeric values. - -Examples: -remainder(13,6); 1 -remainder(x**2 + 3*x + 2,x+1); 0 -remainder(x**3 + 12*x + 4,x**2 + 1); 11*X + 4 -remainder(sin(2*x),x*y); SIN(2*X) - -If the first argument to REMAINDER contains a denominator not equal to -1, an error occurs. - -\endsection -\item[REMFAC] -REMFAC (page 102) - -The REMFAC declaration removes the special factoring treatment of its -arguments that was declared with FACTOR. - -REMFAC kernel{,kernel} - -kernel must be a KERNEL or OPERATOR name that was declared as special -with the FACTOR declaration. - -\endsection -\xitem[REMFORDER command] -REMFORDER command (pages 268, 271) - -\endsection -\item[REMIND] -REMIND (page 206) - -The REMIND declaration removes the special status of its arguments -as indices, which was set in the INDEX declaration, in -high-energy physics calculations. - REMIND identifier{,identifier} - -identifier must have been declared to be of type INDEX. - -\endsection -\xitem[RENOSUM command] -RENOSUM command (pages 262, 271) - -\endsection -\item[REPART] -REPART (pages 72, 73, 75) - - REPART(expression) or REPART simple_expression - -This operator returns the real part of an expression, if that argument -has an numerical value. A non-numerical argument is returned as an -expression in the operators REPART and IMPART. - -Examples: -repart(1+i); 1 -repart(a+i*b); REPART(A) - IMPART(B) - -\endsection -\item[REPEAT] -REPEAT (pages 60, 61, 63, 65) - -The REPEAT command causes repeated execution of a statement UNTIL -the given condition is found to be true. The statement is always executed -at least once. - REPEAT statement UNTIL condition - -statement can be a single statement, GROUP statement, or -a BEGIN...END BLOCK. condition must be a logical -operator that evaluates to rue or nil. - -Examples: -<> until m = 0>>; - 400*X - 300*X - 200*X - 100*X - -<> until m <= 0>>; - -1 - -REPEAT must always be followed by an UNTIL with a condition. Be -careful not to generate an infinite loop with a condition that is -never true. In the second example, if the condition had been M = 0, -it would never have been true since M already had value -2 when the -condition was first evaluated. - -\endsection -\xitem[Reserved variable] -Reserved variable (pages 36, 37) - -\endsection -\item[REST] -REST (page 50) - -The REST operator returns a LIST containing all but the first element of -the list it is given. - REST(list) or REST list - - -list must be a non-empty list, but need not have more than one element. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D}; -rest alist; {B,C,D} -blist := {x,y,{aa,bb,cc},z}; BLIST := {X,Y,{AA,BB,CC},Z} -second rest blist; {AA,BB,CC} -clist := {c}; CLIST := C -rest clist; {} - -\endsection -\xitem[RESULT] -RESULT (page 378) - -\endsection -\item[RESULTANT] -RESULTANT (page 126) -The RESULTANT operator computes the resultant of two polynomials with -respect to a given variable. If the resultant is 0, the polynomials -have a root in common. - RESULTANT(expression,expression,kernel) - -expression must be a polynomial containing kernel ; -kernel must be a KERNEL. - -Examples: -resultant(x**2 + 2*x + 1,x+1,x); 0 -resultant(x**2 + 2*x + 1,x-3,x); 16 -resultant(z**3 + z**2 + 5*z + 5, - z**4 - 6*z**3 + 16*z**2 - 30*z + 55, - z); 0 -resultant(x**3*y + 4*x*y + 10,y**2 + 6*y + 4,y); - 6 4 3 2 - 4*(x + 8*x - 15*x + 16*x - 60*x + 25) - -The resultant is the determinant of the Sylvester matrix, formed from the -coefficients of the two polynomials in the following way: - -Given two polynomials: - - n n-1 - a x + a1 x + ... + an - -and - m m-1 - b x + b1 x + ... + bm - -form the (m+n)x(m+n-1) Sylvester matrix by the following means: - - 0.......0 a a1 .......... an - 0....0 a a1 .......... an 0 - . . . . - a0 a1 .......... an 0.......0 - 0.......0 b b1 .......... bm - 0....0 b b1 .......... bm 0 - . . . . - b b1 .......... bm 0.......0 - -If the determinant of this matrix is 0, the two polynomials have a -common root. Finding the resultant of large expressions is -time-consuming, due to the time needed to find a large determinant. - -The sign conventions RESULTANT uses are those given in the article, -``Computing in Algebraic Extensions,'' by R. Loos, appearing in -Computer Algebra--Symbolic and Algebraic Computation, 2nd ed., edited -by B. Buchberger, G.E. Collins and R. Loos, and published by - -Springer-Verlag, 1983. - -These are: - resultant(p(x),q(x),x) = (-1)^{deg p(x)*deg q(x)} * resultant(q(x),p(x),x), - resultant(a,p(x),x) = a^{deg p(x)}, - resultant(a,b,x) = 1 - -where p(x) and q(x) are polynomials which have x as a variable, and -a and b are free of x. - -Error messages are given if RESULTANT is given a non-polynomial -expression, or a non-kernel variable. - -\endsection -\item[RETRY] -RETRY (page 157) -The RETRY command allows you to retry the latest statement that resulted -in an error message. - -Examples: -matrix a; -det a; ***** Matrix A not set -a := mat((1,2),(3,4)); A(1,1) := 1 - A(1,2) := 2 - A(2,1) := 3 - A(2,2) := 4 -retry; -2 - -RETRY remembers only the most recent statement that resulted in an -error message. It allows you to stop and fix something obvious, then -continue on your way without retyping the original command. - -\endsection -\item[RETURN] -RETURN (pages 62--64) - -The RETURN command causes a value to be returned from inside a -BEGIN...END BLOCK. - BEGIN statements RETURN (expression) - END - -statements can be any valid REDUCE statements. The value of -expression is returned. - -Examples: -begin write "yes"; return a end; yes - A -procedure dumb(a); - begin if numberp(a) then return a - else return 10 end; - DUMB -dumb(x); 10 -dumb(-5); -5 -procedure dumb2(a); - begin c := a**2 + 2*a + 1; - d := 17; c*d; return end; - DUMB2 -dumb2(4); -c; 25 -d; 17 - -Note in DUMB2 above that the assignments were made as requested, but -the product C*D cannot be accessed. Changing the procedure to read -RETURN C*D would remedy this problem. - -The RETURN statement is always the last statement executed before -leaving the block. If RETURN has no argument, the block is exited but -no value is returned. A block statement does not need a RETURN ; the -statements inside terminate in their normal fashion without one. In -that case no value is returned, although the specified actions inside -the block take place. - -The RETURN command can be used inside <<...>> GROUP statements and -IF...THEN...ELSE commands that are inside BEGIN...END BLOCKs. It is -not valid in these constructions that are not inside a BEGIN...END -block. It is not valid inside FOR, REPEAT...UNTIL or WHILE...DO loops -in any construction. To force early termination from loops, the GO -TO(GOTO) command must be used. When you use nested block statements, -a RETURN from an inner block exits returning a value to the -next-outermost block, rather than all the way to the outside. - -\endsection -\item[REVERSE] -REVERSE (page 51) - -The REVERSE operator returns a LIST that is the reverse of the list it -is given. - REVERSE(list) or REVERSE list - -list must be a LIST. - -Examples: - 2 3 -aa := {c,b,a,{x**2,z**3},y}; AA := {C,B,A,{X ,Z },Y} - 2 3 -reverse aa; {Y,{X ,Z},A,B,C} - 2 3 -reverse(q . reverse aa); {C,B,A,{X ,Z },Y,Q} - -REVERSE and CONS can be used together to add a new element to the end -of a list (. adds its new element to the beginning). The REVERSE -operator uses a noticeable amount of system resources, especially if -the list is long. If you are doing much heavy-duty list manipulation, -you should probably design your algorithms to avoid much reversing of -lists. A moderate amount of list reversing is no problem. - -\endsection -\item[REVGRADLEX] -REVGRADLEX (page 293) - -The terms are ordered first with their total degree (degree sum), and -if the total degree is identical the comparison is the inverse of LEX -term order. With GROEBNER and GROEBNERF calculations this term order -is similar to GRADLEX term order; it is known as most efficient -ordering with respect to computing time. - -\endsection -\item[REVPRI] -REVPRI (page 105) - -When the REVPRI switch is on, terms are printed in reverse order from -the normal printing order. - -Examples: - 5 2 -x**5 + x**2 + 18 + sqrt(y); SQRT(Y) + X + X + 18 - -a + b + c + w; A + B + C + W - -on revpri; - 2 5 -x**5 + x**2 + 18 + sqrt(y); 17 + X + X + SQRT(Y) - -a + b + c + w; W + C + B + A - -Turn REVPRI on when you want to display a polynomial in ascending -rather than descending order. - -\endsection -\item[RHS] -RHS (page 47) -The RHS operator returns the right-hand side of an EQUATION, such as -those returned in a LIST by SOLVE. - - RHS(equation) or RHS equation - -equation must be an equation of the form left-hand side = right-hand side. - -Examples: - roots := solve(x**2 + 6*x*y + 5x + 3y**2,x); - - 2 - - (SQRT(24*Y + 60*Y + 25) + 6*Y + 5) - ROOTS := {X=----------------------------------------, - 2 - - 2 - SQRT(24*Y + 60*Y + 25) - 6*Y - 5 - X=-----------------------------------} - 2 - -root1 := rhs first roots; - - (SQRT(24*Y + 60*Y + 25) + 6*Y + 5) - ROOT1 := ---------------------------------------- - 2 -root2 := rhs second roots; - 2 - SQRT(24*Y + 60*Y + 25) - 6*Y - 5 - ROOT2 := ----------------------------------- - 2 - -An error message is given if RHS is applied to something other than an -equation. - -\endsection -\xitem[Riemann Zeta Function] -Riemann Zeta Function (pages 185, 395) - -\endsection -\xitem[RIEMANNCONX command] -RIEMANNCONX command (pages 267, 271) - -\endsection -\xitem[Riemannian Connections] -Riemannian Connections (page 267) - -\endsection -\xitem[Rlisp] -Rlisp (page 213) - -\endsection -\item[RLISP88] -RLISP88 (page 204) - -Rlisp '88 is a superset of the Rlisp that has been traditionally used -for the support of REDUCE. It is fully documented in the book Marti, -J.B., ``RLISP '88: An Evolutionary Approach to Program Design and -Reuse'', World Scientific, Singapore (1993). It supports different -looping constructs from the traditional Rlisp, and treats ``-'' as a -letter unless separated by spaces. Turning on the switch RLISP88 -converts to Rlisp '88 parsing conventions in symbolic mode, and -enables the use of Rlisp '88 extensions. Turning off the switch -reverts to the traditional Rlisp and the previous mode (SYMBOLIC or -ALGEBRAIC) in force before RLISP88 was turned on. - -\endsection -\item[RLROOTNO] -RLROOTNO (page 369) - -The function RLROOTNO computes the number of real roots of p in the -specified region, but does not find the roots. - - RLROOTNO(expression) - RLROOTNO(expression, POSITIVE) - RLROOTNO(expression, NEGATIVE) - RLROOTNO(expression, lo, hi) - -For more details on the specification of an interval, see ISOLATER. - -Examples: - load_package roots; - rlrootno (x^3-3x^2+2x+10); 1 - rlrootno(x^3-3x^2+2x+10,positive); 0 -\endsection -\xitem[root finding] -root finding (page 367) - -\endsection -\item[ROOT_OF] -ROOT_OF (pages 85, 86) - -When the operator SOLVE is unable to find an explicit solution or if -that solution would be too complicated, the result is presented as -formal root expression using the internal operator ROOT_OF and a new -local variable. An expression with a top level ROOT_OF is implicitly a -list with an unknown number of elements since we can't always know how -many solutions an equation has. If a substitution is made into such an -expression, closed form solutions can emerge. If this occurs, the -ROOT_OF construct is replaced by an operator ONE_OF. At this point it -is of course possible to transform the result if the original SOLVE -operator expression into a standard SOLVE solution. To effect this, -the operator EXPAND_CASES can be used. - -Examples: 7 2 -solve(a*x^7-x^2+1,x); {x=root_of(a*x_ - x_ + 1,x_)} -sub(a=0,ws); {x=one_of(1,-1)} -expand_cases ws; {x=1,x=-1} - -The components of ROOT_OF and ONE_OF expressions can be processed as -usual with operators ARGLENGTH and PART. - -\endsection -\item[ROOT_MULTIPLICITES] -ROOT_MULTIPLICITES - -The ROOT_MULTIPLICITIES variable is set to the list of the -multiplicities of the roots of an equation by the SOLVE operator. - - -SOLVE returns its solutions in a list. The multiplicities of -each solution are put in the corresponding locations of the list -ROOT_MULTIPLICITIES. - -\endsection -\xitem[ROOT_VAL] -ROOT_VAL (page 370) - -\endsection -\item[ROOTACC] -ROOTACC (page 373) - -The operator ROOTACC allows you to set the accuracy up to which the -roots package computes its results. - - ROOTACC(n) - -Here n is an integer value. The internal accuracy of the ROOTS package -is adjusted to a value of MAX(6,N). The default value is 6. - -\endsection -\xitem[ROOTMSG] -ROOTMSG (page 373) - -\endsection -\xitem[ROOTPREC] -ROOTPREC (page 374) - -\endsection -\item[ROOTS] -ROOTS (pages 184, 369, 370) - -The operator ROOTS is the main top level function of the roots -package. It will find all roots, real and complex, of the polynomial -p to an accuracy that is sufficient to separate them and which is a -minimum of 6 decimal places. - - ROOTS(p) - -where p is a univariate polynomial. Result is a LIST of equations -which represent the roots of the polynomial at the given accuracy. In -addition, ROOTS stores separate lists of real roots and complex roots -in the global variables ROOTSREAL and ROOTSCOMPLEX. - -Examples: - - roots(x^5-2); {X=-0.929316 + 0.675188*I, - X=-0.929316 - 0.675188*I, - X=0.354967 + 1.09248*I, - X=0.354967 - 1.09248*I, - X=1.1487} - -The minimal accuracy of the result values is controlled by -ROOTACC. - -\endsection -\xitem[ROOTS package] -ROOTS package (page 367) - -\endsection -\xitem[ROOTS_AT_PREC] -ROOTS_AT_PREC (page 370) - -\endsection -\item[ROOTSCOMPLEX] -ROOTSCOMPLEX (page 369) - -When the operator ROOTS is called the complex roots are collected in -the global variable ROOTSCOMPLEX as LIST. - -\endsection -\item[ROOTSREAL] -ROOTSREAL (page 369) - -When the operator ROOTS is called the real roots are collected in the -global variable ROOTREAL as LIST. - -\endsection -\item[ROUND] -ROUND (page 75) - - ROUND(expression) - -If its argument has a numerical value, ROUND rounds it to the nearest -integer. For non-numeric arguments, the value is an expression in the -original operator. - -Examples: -round 3.4; 3 -round 3.5; 4 -round a; ROUND(A) - -\endsection -\item[ROUNDALL] -ROUNDALL (page 133) - -In ROUNDED mode, rational numbers are normally converted to a -floating point representation. If ROUNDALL is off, this conversion -does not occur. ROUNDALL is normally ON. - -Examples: -on rounded; -1/2; 0.5 -off roundall; - 1 -1/2; --- - 2 - -\endsection -\item[ROUNDBF] -ROUNDBF (page 133) - -When ROUNDED is on, the normal defaults cause underflows to be -converted to zero. If you really want the small number that results -in such cases, ROUNDBF can be turned on. - -Examples: -on rounded; -exp(-100000.1^2); 0 -on roundbf; -exp(-100000.1^2); 1.18441281937E-4342953505 - -If a polynomial is input in ROUNDED mode at the default precision into -any ROOTS function, and it is not possible to represent any of the -coefficients of the polynomial precisely in the system floating point -representation, the switch ROUNDBF will be automatically turned on. -All rounded computation will use the internal bigfloat representation -until the user subsequently turns ROUNDBF off. (A message is output to -indicate that this condition is in effect.) - -\endsection -\item[ROUNDED] -ROUNDED (pages 36, 44, 78, 108, 132, 372) - -When ROUNDED is on, floating-point arithmetic is enabled, with -precision initially at a system default value, which is usually 12 -digits. The precise number can be found by the command PRECISION(0). - -Examples: -pi; PI - - 5 -35/217; ---- - 31 -on rounded; -pi; 3.14159265359 - -35/217; 0.161 - -sqrt(3); 1.73205080756 - -If more than the default number of decimal places are required, use the -PRECISION command to set the required number. - -\endsection -\item[Rule lists] -Rule lists (page 147) - -A RULE is an instruction to replace an algebraic expression -or a part of an expression by another one. - lhs => rhs or - lhs => rhs WHEN cond -lhs is an algebraic expression used as search pattern and -rhs is an algebraic expression which replaces matches of -rhs. => is the operator REPLACE. - -lsh can contain free variables which are preceded by a tilde ~ in -their leftmost position in lhs. If a rule has a WHEN cond part it -will fire only if the evaluation of cond has a result TRUE. cond may -contain references to free variables of lhs. - -Rules can be collected in a LIST which then forms a RULE LIST. RULE -LISTS can be used to collect algebraic knowledge for a specific -evaluation context. - -RULES and RULE LISTS are globally activated and deactivated by LET, -FORALL, CLEARRULES. For a single evaluation they can be locally -activate by WHERE. The active rules for an operator can be visualised -by SHOWRULES. - -Examples: -operator f,g,h; -let f(x) => x^2; - 2 -f(x); X -g_rules:={g(~n,~x)=>h(n/2,x) when evenp n, -g(~n,~x)=>h((1-n)/2,x) when not evenp n}$ -let g_rules; -g(3,x); H(-1,X) - -\endsection -\item[SAVEAS] -SAVEAS (page 99)) -The SAVEAS command saves the current workspace under the name of its -argument. - - SAVEAS identifier - -identifier can be any valid REDUCE identifier. - -Examples: - -(The numbered prompts are shown below, unlike in most examples) -1: solve(x^2-3); {x=sqrt(3),x= - sqrt(3)} -2: saveas rts(0)$ -3: rts(0); {x=sqrt(3),x= - sqrt(3)} - -SAVEAS works only for the current workspace, the last algebraic -expression produced by REDUCE. This allows you to save a result that -you did not assign to an identifier when you originally typed the -input. For access to previous output use WS. - -\endsection -\xitem[savesfs] -savesfs (page 393) - -\endsection -\item[SAVESTRUCTR] -SAVESTRUCTR (page 113) - -When SAVESTRUCTR is on, results of the STRUCTR command are returned as -a list whose first element is the representation for the expression -and the remaining elements are equations showing the relationships of -the generated variables. - -Examples: -off exp; - -structr((x+y)^3 + sin(x)^2); ANS3 - where - 3 2 - ANS3 := ANS1 + ANS2 - - ANS2 := SIN(X) - - ANS1 := X + Y - -ans3; ANS3 -on savestructr; - 3 2 -structr((x+y)^3 + sin(x)^2); {ANS3,ANS3=ANS1 + ANS2 ,ANS2=SIN(X),ANS1=X + Y} - 3 2 -ans3 where rest ws; (X + Y) + SIN(X) - -In normal operation, STRUCTR is only a display command. With -SAVESTRUCTR on, you can access the various parts of the expression -produced by STRUCTR. - -The generic system names use the stem ANS. You can change this to your -own stem by the command VARNAME. REDUCE adds integers to this stem -to make unique identifiers. - -\endsection -\xitem[Saving an expression] -Saving an expression (page 111) - -\endsection -\item[SCALAR] -SCALAR (pages 61, 62) - -The SCALAR declaration must be made immediately after a BEGIN (or -other variable declaration such as INTEGER and REAL) and declares -local scalar variables. They are initialised to 0. - - SCALAR identifier{,identifier} - -identifier may be any valid REDUCE identifier, except T or NIL. - -Scalar variables remain local, and do not share values with variables -of the same name outside the BEGIN...END BLOCK. When the block is -finished, the variables are removed. You may use the words REAL or -INTEGER in the place of SCALAR. REAL and INTEGER do not indicate -type-checking by the current REDUCE; they are only for your own -information. Declaration statements must immediately follow the -BEGIN, without a semicolon between BEGIN and the first variable -declaration. - -Any variables used inside BEGIN...END blocks that were not declared -SCALAR, REAL or INTEGER are global, and any change made to them inside -the block affects their global value. Arrays declared inside a block -are always global. - -\endsection -\xitem[Scalar] -Scalar (page 43) - -\endsection -\xitem[SCALEFACTORS operator] -SCALEFACTORS operator (page 234) - -\endsection -\item[SCIENTIFIC_NOTATION] -SCIENTIFIC_NOTATION (page 34) - - SCIENTIFIC_NOTATION(m) or SCIENTIFIC_NOTATION(m,n) - -m and n are positive integers. SCIENTIFIC_NOTATION controls the -output format of floating point numbers. At the default settings, any -number with five or less digits before the decimal point is printed in -a fixed-point notation, e.g., 12345.6. Numbers with more than five -digits are printed in scientific notation, e.g., 1.234567E+5. -Similarly, by default, any number with eleven or more zeros after the -decimal point is printed in scientific notation. - -When SCIENTIFIC_NOTATION is called with the numerical argument m a -number with more than m digits before the decimal point, or m or more -zeros after the decimal point, is printed in scientific notation. -When SCIENTIFIC_NOTATION is called with a list {m, n}, a number with -more than m digits before the decimal point, or n or more zeros after -the decimal point is printed in scientific notation. - -Examples: - -on rounded; -12345.6; 12345.6 - -123456.5; 1.234565e+5 - -0.00000000000000012; 1.2e-16 - -scientific_notation 20; {5,11} - -5: 123456.7; 123456.7 - -0.00000000000000012; 0.00000000000000012 - -\endsection -\item[SCOPE] -SCOPE (page 185) - -Author: J.A. van Hulzen - -REDUCE Source Code Optimization Package. - -SCOPE is a package for the production of an optimised form of a -set of expressions. It applies an heuristic search for common -(sub)expressions to almost any set of proper REDUCE assignment -statements. The output is obtained as a sequence of assignment -statements. GENTRAN is used to facilitate expression output. - -\endsection -\xitem[SDER(I)] -SDER(I) (page 379) - -\endsection -\item[SEC] -SEC (pages 76, 78) - -The SEC operator returns the secant of its argument. - - SEC(expression) or SEC simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -sec abc; SEC(ABC) -sec(pi); -1 -sec 4; SEC(4) -on rounded; -sec(4); - 1.52988565647 -sec log 5; - 25.8852966005 - -SEC returns a numeric value only if ROUNDED is on. Then the secant is -calculated to the current degree of floating point precision. - -\endsection -\item[SECH] -SECH (pages 76, 78) - -The SECH operator returns the hyperbolic secant of its argument. - - SECH(expression) or SECH simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -sech abc; SECH(ABC) -sech(0); 1 -sech 4; SECH(4) -on rounded; -sech(4); 0.0366189934737 -sech log 5; 0.384615384615 - -SECH returns a numeric value only if ROUNDED is on. Then the -expression is calculated to the current degree of floating point -precision. - -\endsection -\item[SECOND] -SECOND (page 50) - -The SECOND operator returns the second element of a list. - SECOND(list) or SECOND list - -list must be a list with at least two elements, to avoid an error -message. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D} -second alist; B -blist := {x,{aa,bb,cc},z}; BLIST := {X,{AA,BB,CC},Z} -second second blist; BB - -\endsection -\xitem[Selector] -Selector (page 198) - -\endsection -\xitem[Semicolon] -Semicolon (page 53) - -\endsection -\item[SET] -SET (pages 55, 83) - -The SET operator is used for assignments when you want both sides of -the assignment statement to be evaluated. - - SET(restricted_expression,expression) - -expression can be any REDUCE expression; restricted_expression -must be an identifier or an expression that evaluates to an identifier. - -Examples: -a := y; A := Y - 2 -set(a,sin(x^2)); SIN(X ) - 2 -a; SIN(X ) - 2 -y; SIN(X ) - -a := b + c; A := B + C - -set(a-c,z); Z - -b; Z - -Using an ARRAY or MATRIX reference as the first argument to SET has -the result of setting the contents of the designated element to SET's -second argument. You should be careful to avoid unwanted side effects -when you use this facility. - -\endsection -\item[SETMOD] -SETMOD (page 134) - -The SETMOD command sets the modulus value for subsequent MODULAR -arithmetic. - - SETMOD integer - -integer must be positive, and greater than 1. It need not be a prime -number. - -Examples: -setmod 6; 1 -on modular; -16; 4 - 2 -x^2 + 5x + 7; X + 5*X + 1 - X -x/3; --- - 3 -setmod 2; 6 - 4 -(x+1)^4; X + 1 -x/3; X - -SETMOD returns the previous modulus, or 1 if none has been set before. -SETMOD only has effect when MODULAR is on. - -Modular operations are done only on numbers such as coefficients of -polynomials, not on the exponents. The modulus need not be prime. -Attempts to divide by a power of the modulus produces an error -message, since the operation is equivalent to dividing by 0. However, -dividing by a factor of a non-prime modulus does not produce an error -message. - -\endsection -\xitem[SGN indeterminate sign] -SGN - indeterminate sign (page 257) - -\endsection -\item[SHARE] -SHARE (page 197) - -The SHARE declaration allows access to its arguments by both -algebraic and symbolic modes. - - SHARE identifier{,identifier} - -identifier can be any valid REDUCE identifier. - -Programming in SYMBOLIC as well as algebraic mode allows you a wider -range of techniques than just algebraic mode alone. Expressions do -not cross the boundary since they have different representations, -unless the SHARE declaration is used. For more information on using -symbolic mode, see the REDUCE User's Manual, and the Standard Lisp -Report. - -You should be aware that a previously-declared array is destroyed by -the SHARE declaration. Scalar variables retain their values. You can -share a declared MATRIX that has not yet been dimensioned so that it -can be used by both modes. Values that are later put into the matrix -are accessible from symbolic mode too, but not by the usual matrix -reference mechanism. In symbolic mode, a matrix is stored as a list -whose first element is MAT, and whose next elements are the rows of -the matrix stored as lists of the individual elements. Access in -symbolic mode is by the operators FIRST, SECOND, THIRD and REST. - -\endsection -\item[SHOWRULES] -SHOWRULES (page 150) - - SHOWRULES(expression) or SHOWRULES simple_expression - -SHOWRULES returns in RULE-LIST form any OPERATOR rules associated with -its argument. - -Examples: -showrules log; {log(e) => 1, - - log(1) => 0, - - ~x - log(e ) => ~x, - - 1 - df(log(~x),~x) => ----} - ~x - -Such rules can then be manipulated further as with any LIST. For example -RHS FIRST WS; has the value 1. - -An operator may have properties that cannot be displayed in such a form, -such as the fact it is an odd function, or has a definition defined as a -procedure. - -\endsection -\item[SHOWTIME] -SHOWTIME (page 70) - -The SHOWTIME command prints the elapsed system time since the last -call of this command or since the beginning of the session, if it has -not been called before. - -Examples: -showtime; Time: 1020 ms - 2 -factorize(x^4 - 8x^4 + 8x^2 - 136x - 153); {X - 9,X + 17,X + 1} -showtime; Time: 920 ms - -The time printed is either the elapsed cpu time or the elapsed wall -clock time, depending on your system. SHOWTIME allows you to see the -system time resources REDUCE uses in its calculations. Your time -readings will of course vary from this example according to the system -you use. - -\endsection -\item[SHUT] -SHUT (pages 153--155) - -The SHUT command closes output files. - SHUT filename{,filename} - -filename must have been a file opened by OUT. - - -A file that has been opened by OUT must be SHUT before it is -brought in by IN. Files that have been opened by OUT should -always be SHUT before the end of the REDUCE session, to avoid either -loss of information or the printing of extraneous information into the file. -In most systems, terminating a session by BYE closes all open -output files. - -\endsection -\xitem[Side effect] -Side effect (page 48) - -\endsection -\xitem[side relations] -side relations (page 241) - -\endsection -\item[SIGN] -SIGN (page 75) - - SIGN expression - -SIGN tries to evaluate the sign of its argument. If this is possible -SIGN returns one of 1, 0 or -1. Otherwise, the result is the original -form or a simplified variant. - -Examples: - sign(-5) -1 - sign(-a^2*b) -SIGN(B) - -Even powers of formal expressions are assumed to be positive only as long -as the switch COMPLEX is off. - -\endsection -\xitem[SIGNATURE command] -SIGNATURE command (page 271) - -\endsection -\xitem[Simplification] -Simplification (pages 44, 97) - -\endsection -\xitem[SIMPSYS] -SIMPSYS (pages 378, 380, 383) - -\endsection -\item[SIN] -SIN (pages 76, 78) - -The SIN operator returns the sine of its argument. - - SIN(expression) or SIN simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -sin aa; SIN(AA) -sin(pi/2); 1 -on rounded; -sin 3; 0.14112000806 -sin(pi/2); 1.0 - -SIN returns a numeric value only if ROUNDED is on. Then the sine is -calculated to the current degree of floating point precision. The -argument in this case is assumed to be in radians. - -\endsection -\item[SINH] -SINH (pages 76, 78) - -The SINH operator returns the hyperbolic sine of its argument. The -derivative of SINH and some simple transformations are known to the -system. - - SINH(expression) or SINH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -sinh b; SINH(B) -sinh(0); 0 - 2 -df(sinh(x**2),x); 2*COSH(X )*X - COSH(4*X) -int(sinh(4*x),x); ----------- - 4 -on rounded; -sinh 4; 27.2899171971 - - -You may attach further functionality by defining its inverse (see -ASINH). A numeric value is not returned by SINH unless the switch -ROUNDED is on and its argument evaluates to a number. - -\endsection -\xitem[SMACRO] -SMACRO (page 196) - -\endsection -\item[SOLVE] -SOLVE (pages 84, 85, 90, 181) - -The SOLVE operator solves a single algebraic EQUATION or a system of -simultaneous equations. - - SOLVE(expression [ , kernel]) or - - SOLVE({expression,...} [ ,{ kernel ,...}] ) - -If the number of equations equals the number of distinct kernels, the -optional kernel argument(s) may be omitted. expression is either a -scalar expression or an EQUATION. When more than one expression is -given, the LIST of expressions is surrounded by curly braces. The -optional list of KERNELs follows, also in curly braces. - -Examples: -sss := solve(x^2 + 7); Unknown: X - SSS := {X= - SQRT(7)*I, - X=SQRT(7)*I} -rhs first sss; - SQRT(7)*I -solve(sin(x^2*y),y); - PI*(2*ARBINT(1) + 1) - {Y=----------------------, - 2 - X - - 2*ARBINT(1)*PI - Y=----------------} - 2 - X - -off allbranch; -solve(sin(x**2*y),y); {Y=0} -solve({3x + 5y = -4,2*x + y = -10},{x,y}); - 46 22 - {{x=-------,y=----}} - 7 7 -solve({x + a*y + z,2x + 5},{x,y}); - 5 - 2*z + 5 - {{x=------,y=------------}} - 2 2*a -ab := (x+2)^2*(x^6 + 17x + 1); - 8 7 6 3 2 - ab := x + 4*x + 4*x + 17*x + 69*x + 72*x + 4 - - 6 -www := solve(ab,x); {X=ROOT_OF(X_ + 17*X_ + 1),X=-2} -root_multiplicities; {1,2} - -Results of the SOLVE operator are returned as EQUATIONS in a LIST. -You can use the usual list access methods (FIRST, SECOND, THIRD, REST -and PART) to extract the desired equation, and then use the operators -RHS and LHS to access the right-hand or left-hand expression of the -equation. When SOLVE is unable to solve an equation, it returns the -unsolved part as the argument of ROOT_OF, with the variable renamed to -avoid confusion, as shown in the last example above. - -For one equation, SOLVE uses square-free factorisation, roots of -unity, and the known inverses of the LOG, SIN, COS, ACOS, ASIN, and -exponentiation operators. The quadratic, cubic and quartic formulas -are used if necessary, but these are applied only when the switch -FULLROOTS is set on; otherwise or when no closed form is available the -result is returned as ROOT_OF expression. The switch TRIGFORM -determines which type of cubic and quartic formula is used. The -multiplicity of each solution is given in a list as the system -variable ROOT_MULTIPLICITIES. For systems of simultaneous linear -equations, matrix inversion is used. For nonlinear systems, the -Groebner basis method is used. - -Linear equation system solving is influenced by the switch CRAMER. - -Singular systems can be solved when the switch SOLVESINGULAR is on, -which is the default setting. A message is given if the system of -equations is inconsistent. - -Related: ALLBRANCH switch, FULLROOTS switch, ROOTS operator, ROOT_OF -operator, TRIGFORM switch. - -\endsection -\item[SOLVESINGULAR] -SOLVESINGULAR (page 89) - -When SOLVESINGULAR is on, singular or under determined systems of -linear equations are solved, using arbitrary real, complex or integer -variables in the answer. Default is ON. - -Examples: - - ARBCOMPLEX(1) -solve({2x + y,4x + 2y},{x,y}); {{X=------------------,Y=ARBCOMPLEX(1)}} - 2 - - 8*arbcomplex(2) -solve({7x + 15y - z,x - y - z},{x,y,z});{{x=-----------------, - 11 - - - 3*ARBCOMPLEX(2) - Y=--------------------, - 11 - - Z=ARBCOMPLEX(2)}} - -off solvesingular; -solve({2x + y,4x + 2y},{x,y}); ***** SOLVE given singular equations -solve({7x + 15y - z,x - y - z},{x,y,z});***** SOLVE given singular equations - -The integer following the identifier ARBCOMPLEX above is assigned by -the system, and serves to identify the variable uniquely. It has no other -significance. - -\endsection -\xitem[SORTOUTODE] -SORTOUTODE (page 350) - -\endsection -\xitem[SPACEDIM command] -SPACEDIM command (pages 251, 271) - -\endsection -\item[SPDE] -SPDE (page 185) - -Author: Fritz Schwartz - -The package SPDE provides a set of functions which may be used to -determine the symmetry group of Lie- or point-symmetries of a given system of -partial differential equations. In many cases the determining system is -solved completely automatically. In other cases the user has to provide -additional input information for the solution algorithm to terminate. - - -\endsection -\xitem[SPECFN] -SPECFN (page 185) - -\endsection -\xitem[SPECFN package] -SPECFN package (page 391) - -\endsection -\xitem[SPECFN2] -SPECFN2 (page 187) - -\endsection -\xitem[spherical coordinates] -spherical coordinates (pages 265, 355) - -\endsection -\item[SPLIT_FIELD] -SPLIT_FIELD function (page 227) - -SPLIT_FIELD is part of the ARNUM package for algebraic numbers. It -calculates a primitive element of minimal degree for which a given -polynomial splits into linear factors. The algorithm as described by -Trager. - -Example: - load arnum; - split!_field(x**3-3*x+7); - - *** Splitting field is generated by: - - 6 4 2 - A5 - 18*A5 + 81*A5 + 1215 - - - - 4 2 - {1/126*A5 - 5/42*A5 - 1/2*A5 + 2/7, - - - 4 2 - - (1/63*A5 - 5/21*A5 + 4/7), - - - 4 2 - 1/126*A5 - 5/42*A5 + 1/2*A5 + 2/7} - - - for each j in ws product (x-j); - - 3 - X - 3*X + 7 - - -\endsection -\item[SPUR] -SPUR (page 210) - -The SPUR declaration removes the special exemption from trace -calculations that was declared by NOSPUR, in high-energy physics -calculations. - SPUR line-id{,line-id} - -line-id must be a line-identifier that has previously been declared -NOSPUR. - -\endsection -\xitem[SQFRF] -SQFRF (page 373) - -\endsection -\item[SQRT] -SQRT (pages 76, 78) - -The SQRT operator returns the square root of its argument. - - SQRT(expression) - -expression can be any REDUCE scalar expression. - -Examples: -sqrt(16*a^3); 4*SQRT(A)*A -sqrt(17); SQRT(17) -on rounded; -sqrt(17); 4.12310562562 -off rounded; 2 -sqrt(a*b*c^5*d^3*27); 3*SQRT(D)*SQRT(C)*SQRT(B)*SQRT(A)*SQRT(3)*C *D - -SQRT checks its argument for squared factors and removes them. - -Numeric values for square roots that are not exact integers are given -only when ROUNDED is on. - -Please note that SQRT(A**2) is given as A, which may be incorrect if A -eventually has a negative value. If you are programming a calculation -in which this is a concern, you can turn on the PRECISE switch, which -causes the absolute value of the square root to be returned. - -\endsection -\xitem[Standard form] -Standard form (page 198) - -\endsection -\xitem[Standard quotient] -Standard quotient (page 198) - -\endsection -\xitem[Statement] -Statement (page 53) - -\endsection -\xitem[Stirling Numbers] -Stirling Numbers (page 185, 394) - -\endsection -\item[STIRLING1] -STIRLING1 (pages 185, 394) - -The STIRLING1 operator returns the Stirling Numbers S(n,m) of the -first kind, i.e. the number of permutations of n symbols which have -exactly m cycles (divided by (-1)**(n-m)). - - STIRLING1(integer,integer) - -Examples: - load_package specfn; (SPECFN) - Stirling1 (17,4); -87077748875904 - - GAMMA(N + 1) - Stirling1 (n,n-1); ----------------- - 2*GAMMA(N - 1) - -The operator STIRLING1 evaluates the Stirling numbers of the first -kind by rulesets for special cases or by a computing the closed form, -which is a series involving the operators BINOMIAL and STIRLING2. - -\endsection -\item[STIRLING2] -STIRLING2 (pages 185, 394) - -The STIRLING1 operator returns the Stirling Numbers S(n,m) of the -second kind, i.e. the number of ways of partitioning a set of n elements -into m non-empty subsets. - - STIRLING2(integer,integer) - -Examples: - load_package specfn; (SPECFN) - Stirling2 (17,4); 694337290 - GAMMA(N + 1) - Stirling2 (n,n-1); ---------------- - 2*GAMMA(N - 1) - -The operator STIRLING2 evaluates the Stirling numbers of the second -kind by rulesets for special cases or by a computing the closed form. - -\endsection -\item[String] -String (page 37)) -A STRING is any collection of characters enclosed in double quotation -marks ("). It may be used as an argument for a variety of commands -and operators, such as IN, REDERR and WRITE. -Examples: -write "this is a string"; this is a string -write a, " ", b, " ",c,"!"; A B C! - -\endsection -\item[STRUCTR] -STRUCTR (pages 112, 113) - -The STRUCTR operator breaks its argument expression into named -subexpressions. - - STRUCTR(expression [,identifier[,identifier ...]]) - -expression may be any valid REDUCE scalar expression. identifier may -be any valid REDUCE IDENTIFIER. The first identifier is the stem for -subexpression names, the second is the name to be assigned to the -structured expression. - -Examples: -structr(sqrt(x**2 + 2*x) + sin(x**2*z)); ANS1*ANS3 + ANS2 - - WHERE - - 1/2 - ANS3 := X - - 2 - ANS2 := SIN(X *Z) - - 1/2 - ANS1 := (X + 2) - -ans3; ANS3 -on fort; -structr((x+1)**5 + tan(x*y*z),var,aa); - VAR1=TAN(X*Y*Z) - AA=VAR1+X**5+5.*X**4+10.*X**3+10.*X**2+5.*X+1. - -The second argument to STRUCTR is optional. If it is not given, the -default stem ANS is used by REDUCE to construct names for the -subexpression. The names are only for display purposes: REDUCE does -not store the names and their values unless the switch SAVESTRUCTR is -on. - -If a third argument is given, the structured expression as a whole is -named by this argument, when FORT is on. The expression is not stored -under this name. You can send these structured Fortran expressions to -a file with the OUT command. - -\endsection -\xitem[Structuring] -Structuring (page 97) - -\endsection -\xitem[Struve Functions] -Struve Functions (pages 185, 397) - -\endsection -\item[STRUVEH] -STRUVEH (pages 185, 397) - -The STRUVEH operator returns Struve's H function. - - STRUVEH(order,argument) - -Examples: -load_package specfn; (SPECFN) - - 3 - - BESSELJ(---,X) - 2 -struveh(-3/2,x); ------------------- - I - - -There is currently no numeric support for the operator STRUVEH. - -\endsection -\item[STRUVEL] -STRUVEL (pages 185, 397) - -The STRUVEL operator returns the modified Struve L function . - - STRUVEL(order,argument) - -Examples: - load_package specfn; (SPECFN); - 3 - struvel(-3/2,x); BESSELI(---,X) - 2 - -There is currently no numeric support for the operator STRUVEL. - -\endsection -\xitem[Sturm Sequences] -Sturm Sequences (page 369) - -\endsection -\item[SUB] -SUB (page 137) - -The SUB operator substitutes a new expression for a kernel in an -expression. - - SUB(kernel=expression {,kernel=expression} expression) - or - SUB({kernel=expression, kernel=EXPRESSION},expression}) - -kernel must be a KERNEL, expression can be any REDUCE scalar -expression. - -Examples: -sub(x=3,y=4,(x+y)**3); 343 -x; X -sub({cos=sin,sin=cos},cos a+sin b} COS(B) + SIN(A) - -Note in the second example that operators can be replaced using the -SUB operator. - -\endsection -\xitem[SUCH THAT] -SUCH THAT (page 142) - -\endsection -\item[SUM] -SUM (pages 57, 58, 187) - -The operator SUM returns -the indefinite or definite summation of a given expression. - - -SUM(expr,k[,lolim [,uplim ]]) - - -where expr is the expression to be added, k is the -control variable (a KERNEL), and lolim and uplim -uplim are the optional lower and upper limits. If uplim is -not supplied the upper limit is taken as k. The GOSPER -algorithm is used. If there is no closed form solution, the operator -returns the input unchanged. - -Examples: - 2 2 -sum(4n**3,n); N *(N + 2*N + 1) - -sum(2a+2k*r,k,0,n-1); N*(2*A + N*R - R) - -\endsection -\xitem[SUM-SQ] -SUM-SQ (page 404) - -\endsection -\xitem[SVEC] -SVEC (page 355) - -\endsection -\xitem[Switch] -Switch (pages 68, 69) - -\endsection -\item[SYMBOLIC] -SYMBOLIC (page 191) - -The SYMBOLIC command changes REDUCE's mode of operation to symbolic. -When SYMBOLIC is followed by an expression, that expression is -evaluated in symbolic mode, but REDUCE's mode is not changed. It is -equivalent to the LISP command. - -Examples: -symbolic; NIL -cdr '(a b c); (B C) -algebraic; -x + symbolic car '(y z); X + Y - -\endsection -\xitem[Symbolic mode] -Symbolic mode (pages 191, 193, 197, 198) - -\endsection -\xitem[Symbolic procedure] -Symbolic procedure (page 196) - -\endsection -\item[SYMMETRIC] -SYMMETRIC (page 93) - -When an operator is declared SYMMETRIC, its arguments are reordered -to conform to the internal ordering of the system. - - SYMMETRIC identifier{,identifier} - -identifier is an identifier that has been declared an operator. - -Examples: -operator m,n; -symmetric m,n; -m(y,a,sin(x)); M(SIN(X),A,Y) -n(z,m(b,a,q)); N(M(A,B,Q),Z) - -If identifier has not been declared to be an operator, the flag -SYMMETRIC is still attached to it. When identifier is subsequently -used as an operator, the message - DECLARE identifier OPERATOR ? (Y OR N) -is printed. If the user replies Y, the symmetric property of the -operator is used. - -\endsection -\xitem[system precision] -system precision (page 374) - -\endsection -\item[T] -T (page 37) - -The constant T stands for the truth value true. It cannot be used as -a scalar variable in a BLOCK, as a looping variable in a FOR statement -or as an OPERATOR name. - -\endsection -\item[TAN] -TAN (pages 76, 78, 81) - -The TAN operator returns the tangent of its argument. - - TAN(expression) or TAN simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -tan a; TAN(A) -tan(pi/3); SQRT(3) -on rounded; -tan(pi/3); 1.73205080757 - -TAN returns a numeric value only if ROUNDED is on. Then the tangent -is calculated to the current degree of floating point accuracy. - -When ON ROUNDED is in force, no check is made to see if the argument -to TAN is a multiple of pi/2, for which the tangent goes to positive -or negative infinity. (Of course, since REDUCE uses a fixed-point -representation of pi/2, it produces a large but not infinite number). -You need to make a check for multiples of pi/ in any program you use -that might possibly ask for the tangent of such a quantity. - -\endsection -\xitem[tangent vector] -tangent vector (page 252) - -\endsection -\item[TANH] -TANH (pages 76, 78) - -The TANH operator returns the hyperbolic tangent of its argument. The -derivative of TANH and some simple transformations are known to the -system. - - TANH(expression) or TANH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -tanh b; TANH(B) -tanh(0); 0 - 2 -df(tanh(x*y),x); Y*( - TANH(X*Y) + 1) - 2*X -int(tanh(x),x); LOG(E + 1) - X -on rounded; -tanh 2; 0.964027580076 - -You may attach further functionality by defining its inverse (see -ATANH). A numeric value is not returned by TANH unless the switch -ROUNDED is on and its argument evaluates to a number. - -\endsection -\item[TAYLOR] -TAYLOR (page 188, 406) - -The TAYLOR operator is used for expanding an expression into a Taylor -series. - -TAYLOR(expression, var, expression, number) -TAYLOR(expression, var, expression, number {,var, expression, number}) - -expression can be any valid REDUCE algebraic expression. var must be -a KERNEL, and is the expansion variable. The expression following it -denotes the point about which the expansion is to take place. number -must be a non-negative integer and denotes the maximum expansion -order. If more than one triple is specified TAYLOR will expand its -first argument independently with respect to all the variables. Note -that once the expansion has been done it is not possible to calculate -higher orders. - -Instead of a KERNEL, var may also be a list of kernels. In this case -expansion will take place in a way so that the sum of the degrees of -the kernels does not exceed the maximum expansion order. If the -expansion point evaluates to the special identifier INFINITY, TAYLOR -tries to expand in a series in 1/var. - -The expansion is performed variable per variable, i.e. in the example -below by first expanding exp(x^2+y^2) with respect to x and then -expanding every coefficient with respect to y. - -Examples: - 2 2 2 2 3 3 -taylor(e^(x^2+y^2),x,0,2,y,0,2); 1 + Y + X + Y *X + O(X ,Y ) - - 2 2 3 -taylor(e^(x^2+y^2),{x,y},0,2); 1 + Y + X + O({X,Y} ) - -taylor(x*y/(x+y),x,0,2,y,0,2); ***** Not a unit in argument to quottaylor - -Note that it is not generally possible to apply the standard REDUCE -operators to a Taylor kernel. For example, PART, COEFF, or COEFFN -cannot be used. Instead, the expression at hand has to be converted -to standard form first using the TAYLORTOSTANDARD operator. - -Differentiation of a Taylor expression is possible. If you -differentiate with respect to one of the Taylor variables the order -will decrease by one. - -Substitution is a bit restricted: Taylor variables can only be -replaced by other kernels. There is one exception to this rule: you -can always substitute a Taylor variable by an expression that -evaluates to a constant. Note that REDUCE will not always be able to -determine that an expression is constant: an example is sin(acos(4)). - -Only simple taylor kernels can be integrated. More complicated -expressions that contain Taylor kernels as parts of themselves are -automatically converted into a standard representation by means of the -TAYLORTOSTANDARD operator. In this case a suitable warning is -printed. - -\endsection -\xitem[TAYLOR package] -TAYLOR package (page 405) - -\endsection -\xitem[Taylor series arithmetic] -Taylor series - arithmetic (page 407) - differentiation (page 408) - integration (page 408) - reversion (page 408) - substitution (page 408) - -\endsection -\item[TAYLORAUTOCOMBINE] -TAYLORAUTOCOMBINE switch (page 408) - -If you set TAYLORAUTOCOMBINE to ON, REDUCE automatically combines -Taylor expressions during the simplification process. This is -equivalent to applying TAYLORCOMBINE to every expression that contains -Taylor kernels. Default is ON. - -\endsection -\item[TAYLORAUTOEXPAND] -TAYLORAUTOEXPAND switch (pages 408, 409) - -TAYLORAUTOEXPAND makes Taylor expressions ``contagious'' in the sense -that TAYLORCOMBINE tries to Taylor expand all non-Taylor -subexpressions and to combine the result with the rest. Default is -OFF. - -\endsection -\item[TAYLORCOMBINE] -TAYLORCOMBINE (page 407) - -This operator tries to combine all Taylor kernels found in its -argument into one. Operations currently possible are: - -Addition, subtraction, multiplication, and division. -Roots, exponentials, and logarithms. -Trigonometric and hyperbolic functions and their inverses. - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 - 3 -taylorcombine log hugo; X + O(X ) - - 1 2 3 -taylorcombine(hugo + x); (1 + X + ---*X + O(X )) + X - 2 -on taylorautoexpand; - 1 2 3 -taylorcombine(hugo + x); 1 + 2*X + ---*X + O(X ) - 2 - -Application of unary operators like LOG and ATAN will nearly always -succeed. For binary operations their arguments have to be Taylor -kernels with the same template. This means that the expansion -variable and the expansion point must match. Expansion order is not -so important, different order usually means that one of them is -truncated before doing the operation. - -If TAYLORKEEPORIGINAL is set to ON and if all Taylor kernels in its -argument have their original expressions kept TAYLORCOMBINE will also -combine these and store the result as the original expression of the -resulting Taylor kernel. There is also the switch TAYLORAUTOEXPAND. - -There are a few restrictions to avoid mathematically undefined -expressions: it is not possible to take the logarithm of a Taylor -kernel which has no terms (i.e. is zero), or to divide by such a -beast. There are some provisions made to detect singularities during -expansion: poles that arise because the denominator has zeros at the -expansion point are detected and properly treated, i.e. the Taylor -kernel will start with a negative power. (This is accomplished by -expanding numerator and denominator separately and combining the -results.) Essential singularities of the known functions (see above) -are handled correctly. - -\endsection -\item[TAYLORKEEPORIGINAL] -TAYLORKEEPORIGINAL (pages 406, 407, 409, 411) - -TAYLORKEEPORIGINAL, if set to ON, forces the TAYLOR and all Taylor -kernel manipulation operators to keep the original expression, -i.e. the expression that was Taylor expanded. All operations -performed on the Taylor kernels are also applied to this expression -which can be recovered using the operator TAYLORORIGINAL. Default is -OFF. - -\endsection -\item[TAYLORORIGINAL] -TAYLORORIGINAL (pages 411, 412) - -TAYLORORINAL can recover the original expression (the one that was -expanded) from the Taylor kernel that is given as its argument. - - TAYLORORIGINAL(expression) - TAYLORORIGINAL simple_expression - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 -taylororiginal hugo; - ***** Taylor kernel doesn't have an original part in taylororiginal - -on taylorkeeporiginal; - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 - X -taylororiginal hugo; E - -An error is signalled if the argument is not a Taylor kernel or if the -original expression was not kept, i.e. if TAYLORKEEPORIGINAL was set -OFF during expansion. - -\endsection -\item[TAYLORPRINTORDER] -TAYLORPRINTORDER switch (page 409) - -TAYLORPRINTORDER, if set to ON, causes the remainder to be printed in -big-O notation. Otherwise, three dots are printed. Default is -ON. - -\endsection -\item[TAYLORPRINTTERMS] -TAYLORPRINTTERMS (pages 406, 412) - -Only a certain number of (non-zero) coefficients are printed. If there -are more, an expression of the form N TERMS is printed to indicate how -many non-zero terms have been suppressed. The number of terms printed -is given by the value of the shared algebraic variable -TAYLORPRINTTERMS. Allowed values are integers and the special -identifier ALL. The latter setting specifies that all terms are to be -printed. The default setting is 5. - -Examples: -taylor(e^(x^2+y^2),x,0,4,y,0,4); - 2 1 4 2 2 2 5 5 - 1 + Y + ---*Y + X + Y *X + (4 TERMS) + O(X ,Y ) - 2 -taylorprintterms := all; - ALL -taylor(e^(x^2+y^2),x,0,4,y,0,4); - 2 1 4 2 2 2 1 4 2 1 4 1 2 4 - 1 + y + ---*y + x + y *x + ---*y *x + ---*x + ---*y *x - 2 2 2 2 - - 1 4 4 5 5 - + ---*y *x + O(x ,y ) - 4 - -\endsection -\item[TAYLORREVERT] -TAYLORREVERT (page 411) - -TAYLORREVERT allows reversion of a Taylor series of a function f, -i.e., to compute the first terms of the expansion of the inverse of f -from the expansion of f. - - TAYLORREVERT(expression, var, var) - -The first argument must evaluate to a Taylor kernel with the second -argument being one of its expansion variables. - -Examples: - 2 6 -taylor(u - u**2,u,0,5); U - U + O(U ) - 2 3 4 5 6 -taylorrevert(ws,u,x); X + X + 2*X + 5*X + 14*X + O(X ) - -\endsection -\item[TAYLORSERIESP] -TAYLORSERIESP (page 407) - -The TAYLORSERIESP operator may be used to determine if its argument is -a Taylor kernel. - - TAYLORSERIESP(expression) - TAYLORSERIESP simple_expression - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 -if taylorseriesp hugo then OK; OK -if taylorseriesp(hugo + y) then OK else NO; NO - -Note that this operator is subject to the same restrictions as, e.g., -ORDP or NUMBERP, i.e. it may only be used in boolean expressions in IF -or LET statements. -\endsection -\item[TAYLORTEMPLATE] -TAYLORTEMPLATE (pages 407, 412) - -The template of a Taylor kernel, i.e. the list of all variables with -respect to which expansion took place together with expansion point -and order can be extracted using - - TAYLORTEMPLATE(expression) - TAYLORTEMPLATE simple_expression - -The operator returns a list of lists with the three elements -(VAR,VAR0,ORDER). An error is signalled if the argument is not a -Taylor kernel. - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 -taylortemplate hugo; {{X,0,2}} - -\endsection -\item[TAYLORTOSTANDARD] -TAYLORTOSTANDARD (page 407) - -The TAYLORTOSTANDARD operator converts all Taylor kernels in its -argument into standard form and resimplifies the result. - - TAYLORTOSTANDARD(expression) - TAYLORTOSTANDARD simple_expression - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 - 2 - X + 2*X + 2 -taylortostandard hugo; -------------- - 2 -\endsection -\xitem[Terminator] -Terminator (page 53) - -\endsection -\item[THIRD] -THIRD (page 50) - -The THIRD operator returns the third item of a LIST. - THIRD(list) or THIRD list - - - -list must be a list containing at least three items to avoid an error -message. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D} -third alist; C -blist := {x,{aa,bb,cc},y,z}; BLIST := {X,{AA,BB,CC},Y,Z}; -third second blist; CC -third blist; Y - -\endsection -\item[TIME] -TIME (page 68) - -When TIME is on, the system time used in executing each REDUCE -statement is printed after the answer is printed. - -Examples: -on time; Time: 4940 ms - 2 -df(sin(x**2 + y),y); COS(X + Y ) - Time: 180 ms -solve(x**2 - 6*y,x); {X= - SQRT(Y)*SQRT(6), - X=SQRT(Y)*SQRT(6)} - Time: 320 ms - -When TIME is first turned on, the time since the beginning of the -REDUCE session is printed. After that, the time used in computation, -(usually in milliseconds, though this is system dependent) is printed -after the results of each command. Idle time or time spent typing in -commands is not counted. If TIME is turned off, the first reading -after it is turned on again gives the time elapsed since it was turned -off. The time printed is CPU or wall clock time, depending on the -system. - -\endsection -\item[TORDER] -TORDER (pages 296, 315, 316) - -The operator TORDER sets the actual term order. - -1. simple term order: - TORDER m - -where m is the name of a term order mode LEX term order, GRADLEX term -order, REVGRADLEX term order or another implemented parameterless -mode. - -2. stepped term order: - TORDER m,n - TORDER {m,n} - -where m is the name of a two step term order, one of GRADLEXGRADLEX -term order, GRADLEXREVGRADLEX term order, LEXGRADLEX term order or -LEXREVGRADLEX term order, and n is a positive integer. - -3. weighted term order - TORDER WEIGHTED, n,n,... - TORDER WEIGHTED, {n,n,...} - -where the n are positive integers, see weighted term order. - -TORDER sets the term order mode. The default mode is LEX. The -previous order mode is returned. - -\endsection -\item[TP] -TP (page 165) - -The TP operator returns the transpose of its MATRIX - argument. - TP identifier or TP(identifier) - -identifier must be a matrix, which either has had its dimensions set -in its declaration, or has had values put into it by MAT. - -Examples: -matrix m,n; -m := mat((1,2,3),(4,5,6))$ -n := tp m; N(1,1) := 1 - N(1,2) := 4 - N(2,1) := 2 - N(2,2) := 5 - N(3,1) := 3 - N(3,2) := 6 - -In an assignment statement involving TP, the matrix identifier on the -left-hand side is redimensioned to the correct size for the transpose. - -\endsection -\item[TPS] -TPS (pages 188, 330) - -Authors: Alan Barnes and Julian Padget - -A Truncated Power Series Package. - -This package implements formal Laurent series expansions in one -variable using the domain mechanism of REDUCE. This means that power -series objects can be added, multiplied, differentiated etc., like -other first class objects in the system. A lazy evaluation scheme -is used and thus terms of the series are not evaluated until they -are required for printing or for use in calculating terms in other -power series. The series are extendible giving the user the -impression that the full infinite series is being manipulated. The -errors that can sometimes occur using series that are truncated at -some fixed depth (for example when a term in the required series -depends on terms of an intermediate series beyond the truncation -depth) are thus avoided. - -\endsection -\xitem[TRA] -TRA (page 178) - -\endsection -\item[TRACE] -TRACE (page 166) - -The TRACE operator finds the trace of its MATRIX argument. - TRACE(expression) or TRACE simple_expression - -expression or simple_expression must evaluate to a square -matrix. - -Examples: -matrix a; -a := mat((x1,y1),(x2,y2))$ -trace a; X1 + Y2 - -The trace is the sum of the entries along the diagonal of a square matrix. -Given a non-matrix expression, or a non-square matrix, TRACE returns -an error message. - -\endsection -\xitem[tracing EXCALC] -tracing - EXCALC (page 266) - ODESOLVE (page 351) - ROOTS package (page 373) - SPDE package (page 380) - SUM package (page 404) - -\endsection -\item[TRALLFAC] -TRALLFAC - -When TRALLFAC is on, a more detailed trace of factoriser calls is -generated. - - -The TRALLFAC switch takes precedence over TRFAC if they are -both on. TRFAC gives a factorisation trace with less detail in it. -When the FACTOR switch is on also, all input polynomials are sent to -the factoriser automatically and trace information is generated. The -OUT command saves the results of the factoring, but not the trace. - - -\endsection -\item[TRFAC] -TRFAC (page 122) - -When TRFAC is on, a narrative trace of any calls to the factoriser is -generated. Default is OFF. - - -When the switch FACTOR is on, and TRFAC is on, every input -polynomial is sent to the factoriser, and a trace generated. With -FACTOR off, only polynomials that are explicitly factored with the -command FACTORIZE generate trace information. - -The OUT command saves the results of the factoring, but not -the trace. The TRALLFAC switch gives trace information to a -greater level of detail. - -\endsection -\item[TRGROEB] -TRGROEB (pages 299, 303) - -If TRGROEB is on, intermediate H polynomials are printed during a -GROEBNER or GROEBNERF calculation. - -\endsection -\xitem[TRGROEB1] -TRGROEB1 (pages 299, 303) - -\endsection -\xitem[TRGROEBR] -TRGROEBR (page 304) - -\endsection -\item[TRGROEBS] -TRGROEBS (pages 299, 303) - -If TRGROEBS is on, intermediate H and S polynomials are printed during -a GROEBNER or GROEBNERF calculation. - -\endsection -\item[TRIGFORM] -TRIGFORM (page 87) - -When FULLROOTS is on, SOLVE will compute the -roots of a cubic or quartic polynomial is closed form. When -TRIGFORM is on, the roots will be expressed by trigonometric -forms. Otherwise nested surds are used. Default is ON. - -\endsection -\item[TRINT] -TRINT (page 178) - -When TRINT is on, a narrative tracing various steps in the -integration process is produced. - -The OUT command saves the results of the integration, but not the -trace. - -\endsection -\item[TRNONLNR] -TRNONLNR - -When TRNONLNR is on, a narrative tracing various steps in -the process for solving non-linear equations is produced. - - -TRNONLNR can only be used after the solve package has been loaded -(e.g., by an explicit call of LOAD_PACKAGE). The OUT -command saves the results of the equation solving, but not the trace. - -\endsection -\xitem[TRODE] -TRODE (page 351) - -\endsection -\xitem[TRROOT] -TRROOT (page 373) - -\endsection -\xitem[TRSUM] -TRSUM (page 404) - -\endsection -\xitem[truncated power series] -truncated power series (page 413) - -\endsection -\xitem[TVECTOR command] -TVECTOR command (pages 249, 271) - -\endsection -\xitem[U(ALFA)] -U(ALFA) (page 379) - -\endsection -\xitem[U(ALFA] -U(ALFA,I) (page 379) - -\endsection -\item[UNTIL] -UNTIL (page 57) - -See the FOR loop construction. -\endsection -\xitem[User packages] -User packages (page 177) - -\endsection -\xitem[VARDF] -VARDF (pages 257, 271) - -\endsection -\xitem[Variable] -Variable (page 36) - -\endsection -\xitem[Variable elimination] -Variable elimination (page 181) - -\endsection -\xitem[variational derivative] -variational derivative (page 257) - -\endsection -\item[VARNAME] -VARNAME (pages 111, 112) - -The declaration VARNAME instructs REDUCE to use its argument as the -default Fortran (when FORT is on) or STRUCTR identifier and identifier -stem, rather than using ANS. - - VARNAME identifier - -identifier can be any combination of one or more alphanumeric -characters. Try to avoid REDUCE reserved words. - -Examples: -varname ident; IDENT -on fort; -x**2 + 1; IDENT=X**2+1. - -off fort,exp; 3 -structr(((x+y)**2 + z)**3); IDENT2 - where - 2 - IDENT2 := IDENT1 + Z - IDENT1 := X + Y - -EXP was turned off so that STRUCTR could show the structure. If EXP -had been on, the expression would have been expanded into a -polynomial. - -\endsection -\xitem[VDF] -VDF (page 359) - -\endsection -\xitem[VEC command] -VEC command (page 232) - -\endsection -\item[VECDIM] -VECDIM (page 212) - -The SPUR declaration removes the special exemption from trace -calculations that was declared by NOSPUR, in high-energy physics -calculations. - SPUR line-id{,line-id} - -line-id must be a line-identifier that has previously been declared -NOSPUR. - -\endsection -\item[VECTOR] -VECTOR (High Energy Physics) (page 208) - -The VECTOR declaration declares that its arguments are of type VECTOR. - VECTOR identifier{,identifier} - -identifier must be a valid REDUCE identifier. It may have already -been used for a matrix, array, operator or scalar variable. After an -identifier has been declared to be a vector, it may not be used as a -scalar variable. - -Vectors are special entities for high-energy physics calculations. -You cannot put values into their coordinates; they do not have -coordinates. They are legal arguments for the high-energy physics -operators EPS, G and . (dot). Vector variables are used to represent -gamma matrices and gamma matrices contracted with Lorentz 4-vectors, -since there are no Dirac variables per se in the system. Vectors do -follow the usual vector rules for arithmetic operations: + and - -operate upon two or more vectors, producing a vector; * and / cannot -be used between vectors; the scalar product is represented by the -. operator; and the product of a scalar and vector expression is well -defined, and is a vector. - -You can represent components of vectors by including representations -of unit vectors in your system. For instance, letting E0 represent -the unit vector (1,0,0,0), the command - -V1.E0 := 0; - -would set up the substitution of zero for the first component of the -vector V1. - -Identifiers that are declared by the INDEX and MASS declarations are -automatically declared to be vectors. - -The following errors can occur in calculations using the high energy -physics package: - -A REPRESENTS ONLY GAMMA5 IN VECTOR EXPRESSIONS -You have tried to use A in some way other than gamma5 in a high-energy -physics expression. - -GAMMA5 NOT ALLOWED UNLESS VECDIM IS 4 -You have used gamma_5 in a high-energy physics computation involving a -vector dimension other than 4. - -ID HAS NO MASS -One of the arguments to MSHELL has had no mass assigned to it, in -high-energy physics calculations. - -MISSING ARGUMENTS FOR G OPERATOR -A line symbol is missing in a gamma matrix expression in high-energy physics -calculations. - -UNMATCHED INDEX list -The parser has found unmatched indices during the evaluation of a -gamma matrix expression in high-energy physics calculations. - -\endsection -\xitem[vector] -vector , integration 233 - addition (page 356) - cross product (page 357) - differentiation (page 233) - division (page 357) - dot product (page 357) - exponentiation (page 357) - inner product (page 357) - modulus (page 357) - multiplication (page 357) - subtraction (page 356) - -\endsection -\xitem[vector algebra] -vector algebra (page 231) - -\endsection -\xitem[VECTORADD] -VECTORADD (page 356) - -\endsection -\xitem[VECTORCROSS] -VECTORCROSS (page 357) - -\endsection -\xitem[VECTORDIFFERENCE] -VECTORDIFFERENCE (page 356) - -\endsection -\xitem[VECTOREXPT] -VECTOREXPT (page 357) - -\endsection -\xitem[VECTORMINUS] -VECTORMINUS (page 356) - -\endsection -\xitem[VECTORPLUS] -VECTORPLUS (page 356) - -\endsection -\xitem[VECTORQUOTIENT] -VECTORQUOTIENT (page 357) - -\endsection -\xitem[VECTORRECIP] -VECTORRECIP (page 357) - -\endsection -\xitem[VECTORTIMES] -VECTORTIMES (page 357) - -\endsection -\xitem[VERBOSELOAD switch] -VERBOSELOAD switch (page 409) - -\endsection -\xitem[VINT] -VINT (page 360) - -\endsection -\xitem[VMOD] -VMOD (page 357) - -\endsection -\xitem[VMOD operator] -VMOD operator (page 233) - -\endsection -\xitem[VOLINT] -VOLINT (page 360) - -\endsection -\xitem[VOLINTEGRAL function] -VOLINTEGRAL function (page 237) - -\endsection -\xitem[VOLINTORDER vector] -VOLINTORDER vector (page 237) - -\endsection -\xitem[VORDER] -VORDER (page 359) - -\endsection -\xitem[VOUT] -VOUT (page 355) - -\endsection -\xitem[VSTART] -VSTART (page 354) - -\endsection -\xitem[VTAYLOR] -VTAYLOR (page 359) - -\endsection -\xitem[wedge] -wedge (page 271) - -\endsection -\item[WEIGHT] -WEIGHT (page 152) - -The WEIGHT command is used to attach weights to kernels for asymptotic -constraints. - - WEIGHT kernel = number - -kernel must be a REDUCE KERNEL, number must be a positive integer, not -0. - -Examples: 4 3 2 2 3 4 -a := (x+y)**4; A := X + 4*X *Y + 6*X *Y + 4*X*Y + Y -weight x=2,y=3; -wtlevel 8; - 4 -a; X -wtlevel 10; - 2 2 2 -a; X *(6*Y + 4*X*Y + X ) -int(x**2,x); ***** X invalid as KERNEL - -Weights and WTLEVEL are used for asymptotic constraints, where -higher-order terms are considered insignificant. - -Weights are originally equivalent to 0 until set by a WEIGHT command. -To remove a weight from a kernel, use the CLEAR command. Weights once -assigned cannot be changed without clearing the identifier. Once a -weight is assigned to a kernel, it is no longer a kernel and cannot be -used in any REDUCE commands or operators that require kernels, until -the weight is cleared. Note that terms are ordered by greatest -weight. - -The weight level of the system is set by WTLEVEL, initially at 2. -Since no kernels have weights, no effect from WTLEVEL can be seen. -Once you assign weights to kernels, you must set WTLEVEL correctly for -the desired operation. When weighted variables appear in a term, -their weights are summed for the total weight of the term (powers of -variables multiply their weights). When a term exceeds the weight -level of the system, it is discarded from the result expression. - -\endsection -\xitem[weighted ordering] -weighted ordering (page 316) - -\endsection -\item[WHEN] -WHEN (page 147) - -The WHEN operator is used inside a RULE to make the -execution of the rule depend on a boolean condition which is -evaluated at execution time. For the use see RULE. - -\endsection -\item[WHERE] -WHERE (page 148) - -The WHERE operator provides an infix notation for one-time -substitutions for kernels in expressions. - - expression WHERE kernel = expression{,kernel = expression} - -expression can be any REDUCE scalar expression, kernel must be a -KERNEL. Alternatively a RULE or a RULE LIST can be a member of the -right-hand part of a WHERE expression. - -Examples: -x**2 + 17*x*y + 4*y**2 where x=1,y=2; - 51 -for i := 1:5 collect x**i*q where q= for j := 1:i product j; - 2 3 4 5 - {X,2*X ,6*X ,24*X ,120*X } - 2 3 -x**2 + y + z where z=y**3,y=3; X + Y + 3 - -Substitution inside a WHERE expression has no effect upon the values -of the kernels outside the expression. The WHERE operator has the -lowest precedence of all the infix operators, which are lower than -prefix operators, so that the substitutions apply to the entire -expression preceding the WHERE operator. However, WHERE is applied -before command keywords such as THEN, REPEAT, or DO. - -A RULE or a RULE SET in the right-hand part of the WHERE expression -act as if the rules were activated by LET immediately before the -evaluation of the expression and deactivated by CLEARRULES immediately -afterwards. - -WHERE gives you a natural notation for auxiliary variables in -expressions. As the second example shows, the substitute expression -can be a command to be evaluated. The substitute assignments are made -in parallel, rather than sequentially, as the last example shows. The -expression resulting from the first round of substitutions is not -reexamined to see if any further such substitutions can be made. -WHERE can also be used to define auxiliary variables in PROCEDURE -definitions. - -\endsection -\item[WHILE] -WHILE (pages 59, 61, 63, 65) - -The WHILE command causes a statement to be repeatedly executed until a -given condition is true. If the condition is initially false, the -statement is not executed at all. - - WHILE condition DO statement - -condition is given by a logical operator, statement must be a single -REDUCE statement, or a GROUP (<<...>>) or BEGIN...END block. - -Examples: -a := 10; A := 10 -while a <= 12 do <>; 10 - 11 - 12 -while a < 5 do <>; .... nothing is printed - -\endsection -\xitem[WHITTAKERM] -WHITTAKERM (pages 185, 397) - -\endsection -\item[WHITTAKERW] -WHITTAKERW (pages 185, 397) - -The WHITTAKERW operator returns Whittaker's W function. - - WHITTAKERW(parameter,parameter,argument) - -Examples: -load_package specfn; (SPECFN) - 1 - 4*SQRT(2)*KUMMERU(---,5,2) - 2 -WhittakerW(2,2,2); ---------------------------- - E - -Whittaker's W function is one of the Confluent Hypergeometric functions. -For reference see the HYPERGEOMETRIC operator. - -\endsection -\xitem[Workspace] -Workspace (page 99) - -\endsection -\item[WRITE] -WRITE (page 105)) - -The WRITE command explicitly writes its arguments to the output device -(terminal or file). - - WRITE item{,item} - -item can be an expression, an assignment or a STRING enclosed in -double quotation marks ("). - -Examples: -write a, sin x, "this is a string"; ASIN(X)this is a string -write a," ",sin x," this is a string"; A SIN(X) this is a string -if not numberp(a) then write "the symbol ",a; the symbol A -array m(10); -for i := 1:5 do write m(i) := 2*i; - M(1) := 2 - M(2) := 4 - M(3) := 6 - M(4) := 8 - M(5) := 10 -m(4); 8 - -The items specified by a single WRITE statement print on a single line -unless they are too long. A printed line is always ended with a carriage -return, so the next item printed starts a new line. - -When an assignment statement is printed, the assignment is also made. -This allows you to get feedback on filling slots in an array with a -FOR statement, as shown in the last example above. - -\endsection -\item[WS] -WS (pages 29, 158) - -The WS operator alone returns the last result; WS with a number -argument returns the results of the REDUCE statement executed after -that numbered prompt. - - WS or WS(number) - -number must be an integer between 1 and the current REDUCE prompt number. - -Examples: -(In the following examples, unlike most others, the numbered -prompt is shown.) -1: df(sin y,y); COS(Y) - 2 -2: ws^2; COS(Y) - -3: df(ws 1,y); -SIN(Y) - -WS and WS(number) can be used anywhere the expression they stand for -can be used. Calling a number for which no result was produced, such -as a switch setting, will give an error message. - -The current workspace always contains the results of the last REDUCE -command that produced an expression, even if several input statements -that do not produce expressions have intervened. For example, if you -do a differentiation, producing a result expression, then change -several switches, the operator WS; returns the results of the -differentiation. The current workspace (WS) can also be used inside -files, though the numbered workspace contains only the IN command that -input the file. - -There are three history lists kept in your REDUCE session. The first -stores raw input, suitable for the statement editor. The second -stores parsed input, ready to execute and accessible by INPUT. The -third stores results, when they are produced by statements, which are -accessible by the WS n operator. If your session is very long, -storage space begins to fill up with these expressions, so it is a -good idea to end the session once in a while, saving needed -expressions to files with the SAVEAS and OUT commands. - -An error message is given if a reference number has not yet been used. - -\endsection -\item[WTLEVEL] -WTLEVEL (page 152) - -In conjunction with WEIGHT, WTLEVEL is used to implement asymptotic -constraints. Default value is 2. - - WTLEVEL integer - -integer is a positive integer that is the greatest weight term to be -retained in expressions involving kernels with weight assignments. - -Examples: 4 3 2 2 3 4 -(x+y)**4; X + 4*X *Y + 6*X *Y + 4*X*Y + Y -weight x=2,y=3; -wtlevel 8; - 4 -(x+y)**4; X -wtlevel 10; - 2 2 2 -(x+y)**4; X *(6*Y + 4*X*Y + X ) -int(x**2,x); ***** X invalid as KERNEL - -WTLEVEL is used in conjunction with the command WEIGHT to enable -asymptotic constraints. Weight of a term is computed by multiplying -the weights of each variable in it by the power to which it has been -raised, and adding the resulting weights for each variable. If the -weight of the term is greater than WTLEVEL, the term is dropped from -the expression, and not used in any further computation involving the -expression. - -Once a weight has been attached to a KERNEL, it is no longer -recognised by the system as a kernel, though still a variable. It -cannot be used in REDUCE commands and operators that need kernels. -The weight attachment can be undone with a CLEAR command. WTLEVEL can -be changed as desired. - -\endsection -\xitem[X(I)] -X(I) (page 379) - -\endsection -\xitem[XI(I)] -XI(I) (page 379) - -\endsection -\item[XPND command] -XPND command (pages 253, 254, 271) - -(Part of the EXCALC package) - -There are two forms of the XPND command, which controls the use of the -product rule for the d operator and the expansion into partial -derivatives. The default for both these is ON. - - xpnd d; - xpnd @; - -Example: - load_package excalc; *** ^ redefined - (excalc) - pform x=0,y=k,z=m; - K - d(y^z); ( - 1) *Y^d Z + d Y^Z - - noxpnd d; - d(y^z); d(Y^Z) - - -See also NOXPND - -\endsection -\item[ZETA] -ZETA (pages 185, 395) - -The ZETA operator returns Riemann's Zeta function, - - Zeta (z) := sum(1/(k**z),k,1,infinity) - - ZETA(expression) - -Examples: - load_package specfn; (SPECFN) - 2 - PI - Zeta(2); ----- - 6 - on rounded; - Zeta 1.01; 100.577943338 - -Numerical computation for the Zeta function for arguments close to 1 -are tedious, because the series is converging very slowly. In this -case a formula (e.g. found in Bender/Orzag: Advanced Mathematical -Methods for Scientists and Engineers, McGraw-Hill) is used. - -No numerical approximation for complex arguments is done. - -\endsection -\xitem[ZETA(ALFA,I)] -ZETA(ALFA,I) (page 379) - -\endsection DELETED r36/announce.ps Index: r36/announce.ps ================================================================== --- r36/announce.ps +++ r36/announce.ps @@ -1,365 +0,0 @@ -%!PS-Adobe-2.0 -%%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software -%%Title: announce.dvi -%%CreationDate: Sun Sep 17 10:21:20 1995 -%%Pages: 2 -%%PageOrder: Ascend -%%BoundingBox: 0 0 596 842 -%%EndComments -%DVIPSCommandLine: F:\EMTEX\DVIPS32.EXE announce -%DVIPSParameters: dpi=300, compressed, comments removed -%DVIPSSource: TeX output 1995.09.17:1021 -%%BeginProcSet: texc.pro -/TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N -/X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 -mul N 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y(notionally)f -(inserted)g(and)f(then)g(the)g(resulting)h(\014rst)e(line)j(is)e(then)g -(thro)o(wn)f(a)o(w)o(a)o(y)l(.)24 b(I)191 1685 y(guess)14 -b(in)g(this)h(case)f(it)g(means)f(that)h(if)g(the)g(caret)f(is)i(righ)o -(t)e(at)g(the)h(start)f(of)g(the)h(bu\013er)191 1742 -y(all)i(the)f(inserted)h(stu\013)f(gets)g(abandoned.)262 -1798 y(Characters)i(inserted)h(at)g(\(or)f(in)i(some)e(cases)h(near\))g -(the)g(end)g(of)g(the)g(text)f(can)191 1855 y(b)q(e)i(used)h(as)e -(program)f(input.)32 b(If)19 b(the)f(caret)h(is)g(at)f(the)g(end)i(of)e -(the)h(text)f(t)o(yp)q(ed-in)191 1911 y(c)o(haarcters)g(are)g(placed)i -(in)g(a)f(t)o(yp)q(e-ahead)g(bu\013er)f(un)o(til)i(the)f(program)f -(requests)h(a)191 1968 y(line)h(of)e(input.)30 b(When)19 -b(that)f(happ)q(ens)h(c)o(haracters)f(are)g(accepted)h(from)f(the)g(t)o -(yp)q(e-)191 2024 y(ahead)f(bu\013er)h(\(and/or)e(the)h(P)l(ASTE)h -(source\))f(and)g(ec)o(ho)q(ed)h(to)f(the)g(screen)h(un)o(til)h(a)191 -2081 y(newline)j(is)f(seen.)35 b(If)20 b(the)h(user)f(t)o(yp)q(es)g(a)g -(newline)i(at)d(the)i(end)g(of)e(the)h(input)i(line)191 -2137 y(the)14 b(c)o(haracters)f(in)i(it)f(are)g(mo)o(v)o(ed)f(to)h(a)f -(program-input-bu\013er)i(whic)o(h)f(is)h(where)f(the)191 -2193 y(program)j(reads)h(them)g(from.)27 b(If)19 b(the)f(user)g(re-p)q -(ositions)h(the)f(caret)g(and)g(inserts)h(a)191 2250 -y(newline)h(in)o(to)e(the)f(middle)j(of)d(the)h(input)h(line)h(then)e -(pre-t)o(yp)q(ed)g(c)o(haracters)g(in)g(the)191 2306 -y(line)h(but)f(after)f(where)i(the)e(newline)j(w)o(as)d(get)h(pushed)h -(bac)o(k)e(in)o(to)h(the)g(t)o(yp)q(e-ahead)191 2363 -y(bu\013er)f(\(and)g(if)h(that)e(o)o(v)o(er\015o)o(ws)g(they)i(are)e -(lost)i(with)f(a)g(b)q(eep\).)27 b(The)17 b(e\013ect)g(is)h(that)191 -2419 y(the)e(program)f(gets)g(one)h(line)i(at)d(once)h(and)h(when)f -(that)f(line)j(is)e(placed)h(in)g(its)f(input)191 2476 -y(bu\013er)d(it)h(will)h(just)e(ha)o(v)o(e)g(b)q(een)i(ec)o(ho)q(ed)f -(to)f(the)h(screen.)19 b(The)14 b(program-input-bu\013er)191 -2532 y(will)22 b(ha)o(v)o(e)e(limited)i(length)f(and)g(truly)f -(ridiculously)k(long)c(input)i(will)g(b)q(e)f(silen)o(tly)191 -2589 y(truncated)e(when)g(mo)o(v)o(ed)g(in)o(to)g(it.)32 -b(I)19 b(will)i(feel)f(en)o(titled)g(to)e(reject)h(input)h(activit)o(y) -191 2645 y(that)14 b(I)i(notice)g(creating)f(an)g(input)i(line)f(that)f -(is)h(longer)f(then)h(that)e(limit.)262 2702 y(An)d(elab)q(oration)g -(on)g(this)h(explanation)g(is)g(that)e(part)g(of)h(the)g(\014nal)h -(line)h(in)f(the)f(text)927 2826 y(5)p eop -%%Page: 6 6 -6 5 bop 191 274 a Fb(bu\013er)17 b(can)g(b)q(e)h(an)f(incomplete)i -(input)g(line.)27 b(This)18 b(can)f(start)f(part)h(w)o(a)o(y)f(along)h -(the)191 330 y(line)e(\(eg)e(it)h(will)h(tend)e(to)g(start)f(after)h -(the)h(displa)o(y)o(ed)g(prompt\).)19 b(After)13 b(v)m(arious)h(CUT)191 -387 y(and)g(DELETE)f(op)q(erations)h(or)f(when)h(the)f(program)g -(requests)g(input)i(after)d(prin)o(ting)191 443 y(a)20 -b(line)h(that)f(w)o(as)f(not)h(terminated)g(it)g(can)h(start)e(w)o(ell) -i(along)f(the)g(\014nal)h(line.)36 b(An)191 500 y(incomplete)16 -b(line)f(is)g(created)f(when)g(the)h(program)d(requests)i(a)g(line)i -(of)d(input.)21 b(When)191 556 y(the)16 b(user)g(inserts)g(a)g(newline) -h(in)o(to)f(the)g(incomplete)i(line)f(it)f(b)q(ecomes)h(complete,)f -(its)191 613 y(con)o(ten)o(ts)e(are)h(mo)o(v)o(ed)g(elsewhere)h(and)g -(there)f(is)h(no)f(longer)g(an)h(incomplete)g(line.)262 -669 y(When)c(a)g(P)l(ASTE)h(op)q(eration)f(copies)h(material)g(in)o(to) -f(the)g(middle)i(of)e(a)g(do)q(cumen)o(t)191 726 y(an)o(y)i(prompts)g -(are)g(inserted.)21 b(But)14 b(if)h(then)g(some)f(of)g(that)g(line)i -(is)f(mo)o(v)o(ed)f(out)g(to)g(the)191 782 y(program-input-bu\013er)20 -b(prompts)g(are)g(discarded)h(during)g(the)g(mo)o(v)o(e.)34 -b(If)20 b(P)l(ASTE)191 839 y(puts)c(stu\013)e(righ)o(t)i(at)f(the)g -(end)i(of)e(the)g(bu\013er)h(it)g(omits)f(an)o(y)g(prompts)g(in)i(the)e -(pasted)191 895 y(stu\013.)36 b(But)21 b(the)g(start)f(of)g(eac)o(h)h -(line)h(of)f(input)h(that)e(is)h(ec)o(ho)q(ed)h(will)g(get)f(a)f(fresh) -191 951 y(prompt)15 b(displa)o(y)o(ed)h(on)f(it.)262 -1008 y(When)g(the)f(program)g(that)g(is)h(b)q(eing)h(run)f(is)g(halted) -h(w)o(aiting)f(for)f(input)h(and)g(the)191 1064 y(screen)h(has)g(b)q -(een)h(scrolled)g(suc)o(h)f(that)g(the)f(end)i(of)e(the)h(bu\013er)g -(the)g(windo)o(w)g(title)h(is)191 1121 y(c)o(hanged)e(to)g(\\w)o -(aiting)g(for)g(input".)262 1177 y(Pressing)k(an)o(y)f(k)o(ey)g(or)h(p) -q(erforming)g(a)f(P)l(ASTE)h(op)q(eration)g(alw)o(a)o(ys)f(scrolls)h -(the)191 1234 y(windo)o(w)14 b(to)f(mak)o(e)h(the)g(caret)f(visible.)22 -b(The)14 b(caret)f(can)h(only)g(ha)o(v)o(e)g(b)q(ecome)g(in)o(visible) -191 1290 y(as)j(a)g(result)g(of)g(a)g(user-initiated)i(scroll)f -(request)f(\(or)f(HOME\))h(since)h(except)g(when)191 -1347 y(suc)o(h)j(a)f(request)g(has)h(hidden)h(it)f(the)f(windo)o(w)h -(scrolls)g(automatically)g(to)f(k)o(eep)h(it)191 1403 -y(visible.)262 1460 y(Note)14 b(that)f(the)i(rules)g(giv)o(en)g(here)g -(indicate)h(that)e(c)o(haracters)f(are)h(only)h(inserted)191 -1516 y(in)o(to)i(the)f(bu\013er)h(at)f(t)o(w)o(o)f(distinct)j(places:) -23 b(where)17 b(the)g(caret)f(is)h(and)g(at)f(the)g(end)i(of)191 -1572 y(the)e(bu\013er.)24 b(So)16 b(the)g(implemen)o(tation)i(can)e -(surviv)o(e)h(if)g(it)g(just)f(cac)o(hes)g(information)191 -1629 y(ab)q(out)f(those)g(t)o(w)o(o)f(p)q(ositions.)191 -1772 y Fc(6)67 b(UNDO)21 b(|)h(a)g(summary)191 1874 y -Fb(There)14 b(is)g(an)g(undo)g(bu\013er)g(that)f(can)h(store)f(a)h -(limited)h(n)o(um)o(b)q(er)f(of)g(c)o(haracters)f(and)h(a)191 -1930 y(limited)k(n)o(um)o(b)q(er)e(of)f(transactions.)21 -b(A)16 b(transaction)f(iden)o(ti\014es)i(a)f(caret)f(p)q(osition)i(or) -191 1986 y(a)e(range)g(within)h(the)f(text,)g(an)g(p)q(ossibly)i(a)d -(sequence)j(of)e(asso)q(ciated)g(c)o(haracters:)247 2080 -y(1.)22 b(After)c(a)h(P)l(ASTE)g(that)f(happ)q(ened)i(within)g(the)f(b) -q(o)q(dy)h(of)e(the)h(text)f(and)h(did)305 2137 y(not)14 -b(terminate)i(an)f(input)h(line)h(an)e(UNDO)g(discards)h(the)f -(inserted)i(material;)247 2231 y(2.)22 b(After)15 b(a)g(P)l(ASTE)h -(that)e(put)i(one)g(or)f(more)g(newlines)i(in)o(to)e(the)h(input)g -(area)f(no)305 2287 y(UNDO)k(will)h(b)q(e)f(p)q(ossible)i(\(b)q(ecause) -e(some)g(of)f(the)h(inserted)g(text)g(has)f(b)q(een)305 -2343 y(passed)d(on)g(to)g(the)g(appication)h(co)q(de)g(to)f(pro)q -(cess\);)247 2437 y(3.)22 b(After)h(a)g(CUT)g(follo)o(w)o(ed)g(p)q -(ossibly)i(b)o(y)f(op)q(erations)f(that)g(mo)o(v)o(e)f(the)i(caret)305 -2494 y(an)d(UNDO)h(re-p)q(ositions)h(the)f(caret)f(and)h(do)q(es)g -(inserts)h(c)o(haracters)e(as)g(for)305 2550 y(a)14 b(P)l(ASTE)i(\(but) -f(that)f(paste)h(is)h(not)f(itself)h(undoable\);)247 -2644 y(4.)22 b(After)16 b(a)h(sequence)h(of)f(DELETE)g(k)o(eys)g(ha)o -(v)o(e)f(b)q(een)j(pressed)e(an)g(UNDO)h(will)305 2700 -y(re-insert)d(the)h(deleted)g(c)o(haracters.)j(It)d(can)f(re-instate)g -(deleted)i(prompts.)927 2826 y(6)p eop -%%Page: 7 7 -7 6 bop 247 274 a Fb(5.)22 b(Sequences)c(of)f(non-delete)i(c)o -(haracters)e(are)g(collected)i(up)f(to)e(the)i(p)q(oin)o(t)g(of)f(a)305 -330 y(newline.)32 b(If)19 b(the)g(newline)i(causes)e(transmission)g(of) -f(the)h(c)o(haracters)f(to)g(the)305 387 y(program)c(no)h(UNDO)h(is)g -(p)q(ossible.)22 b(Otherwise)16 b(eac)o(h)g(blo)q(c)o(k)g(up)g(to)e(a)h -(newline)305 443 y(is)g(an)g(UNDO)h(unit.)247 537 y(6.)22 -b(Previously)12 b(stored)f(UNDO)h(op)q(erations)f(can)h(b)q(ecome)g(in) -o(v)m(alid)i(if)e(they)g(o)o(v)o(erlap)305 594 y(with)k(a)f -(non-undo-able)j(op)q(eration)e(or)f(if)h(the)g(text)f(that)g(they)h -(relate)g(to)f(gets)305 650 y(abandoned)g(as)f(the)h(main)h(text)e -(bu\013er)h(rolls,)g(or)f(if)h(the)g(undo)h(stac)o(k)e(b)q(ecomes)305 -707 y(o)o(v)o(er-full.)20 b([Is)15 b(this)h(hard)f(to)g(implemen)o(t)h -(reliably?])191 850 y Fc(7)67 b(Auto-scrolling)24 b(|)f(a)f(summary)191 -951 y Fb(If)16 b(the)f(user)h(nev)o(er)g(re-p)q(ositions)g(the)g(caret) 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-(just)h(go)q(es)g(in)h(a)f(t)o(yp)q(e-ahead)h(bu\013er)f(and)g(nothing) -h(sp)q(ecial)h(happ)q(ens\).)262 1571 y(If)h(the)g(user)g(has)g(mo)o(v) -o(ed)g(the)g(caret)f(to)h(other)f(than)h(at)g(the)g(end)h(of)e(the)h -(bu\013er)191 1628 y(then)g(the)h(windo)o(w)f(is)h(nev)o(er)f(scrolled) -i(b)o(y)e(cwin,)h(but)f(in)h(cases)g(when)f(it)h(migh)o(t)f(b)q(e)191 -1684 y(in)o(teresting)h(to)g(scroll)g(it)g(the)g(title)h(text)e(of)g -(the)h(windo)o(w)g(is)g(up)q(dated)h(to)e(giv)o(e)h(the)191 -1741 y(user)c(a)g(clue)i(to)d(that)h(fact.)191 1884 y -Fc(8)67 b(Prin)n(t)24 b(and)f(other)f(op)r(erations)191 -1985 y Fb(The)d(regular)g(PRINT)h(item)f(on)g(the)g(men)o(u)g(should)h -(just)e(prin)o(t)h(the)g(whole)h(of)e(the)191 2042 y(con)o(ten)o(ts)h -(of)g(the)h(text)f(bu\013er.)33 b(It)20 b(will)h(apply)g(a)e(\014xed)i -(with)f(limit)h(and)f(truncate)191 2098 y(an)o(y)d(material)h(that)f -(spills)j(o\013)d(to)g(the)h(righ)o(t.)27 b(It)18 b(will)i(pac)o(k)d -(lines)j(on)o(to)c(pages)i(in)h(a)191 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y(relativ)o(e)15 b(to)g(the)g(curren)o(t)g -(windo)o(w)h(size)g(is)f(unclear.)214 1439 y(Other)g(op)q(erations:)20 -b(READ)15 b(is)h(not)f(done)h(y)o(et,)e(but)h(is)h(probably)g(easy)l(.) -927 2826 y(8)p eop -%%Trailer -end -userdict /end-hook known{end-hook}if -%%EOF DELETED r36/cslbase/toacn.car Index: r36/cslbase/toacn.car ================================================================== --- r36/cslbase/toacn.car +++ r36/cslbase/toacn.car cannot compute difference between binary files DELETED r36/cslsrc/helpdata Index: r36/cslsrc/helpdata ================================================================== --- r36/cslsrc/helpdata +++ r36/cslsrc/helpdata @@ -1,11555 +0,0 @@ -\item[Contents] -Help is available on the following - -Algebra Arithmetic Booleans -Commands Declarations Functions -InputOutput Library Matrix -Operators Specfns Switches -Syntax Variables - -There are help windows for each of these topics - -To select a help page double click on the word in the help window or -use the Help Selection option on the menu. A backspace/delete will -return to this Index window. - -An alphabetical list of all topics follows - -. -# -ABS ACOS ACOSH -ACOT ACOTH ACSC -ACSCH ADJPREC ALGEBRAIC -Algebraic mode ALGINT ALGINT(Package) -ALLBRANCH ALLFAC ANTISYMMETRIC -APPEND ARBCONST ARGLENGTH -ARNUM ARRAY ASEC -ASECH ASIN ASINH -ATAN ATAN2 ATANH -AVECTOR -BALANCED_MOD BEGIN...END BERNOULLI -BESSELI BESSELJ BESSELK -BESSELY BETA BFSPACE -BINOMIAL BOUNDS BYE -CARD_NO CEILING CENTERED_MOD -CHEBYSHEV_FIT CHEBYSHEVT CHEBYSHEVU -CLEAR CLEARRULES COEFF -COEFFICIENT COEFFN COFACTOR -COLLECT COMBINEEXPT COMBINELOGS -COMMENT COMP COMPACT -Compiler COMPLEX CONJ -CONT COS COSH -COT COTH CRAMER -CREF CSC CSCH -DECOMPOSE DEFINE DEFN -DEFPOLY DEG DEMO -DEN DEPEND DET -DF DFPRINT DILOG -DISPLAY DIV DOT -E ECHO ED -EDITDEF END EPS -Equation ERF ERRCONT -EULER EULERP Euler Numbers -EVAL_MODE EVALLHSEQP EVEN -EVENP EXCALC EXP -EXPAND_CASES EXPANDLOGS EXPINT -exterior calc exterior df EZGCD -FACTOR FACTORIAL Factorization -FACTORIZE FAILHARD FIRST -FIRSTROOT FIX FIXP -FLOOR FOR FORALL -FOREACH FORT FORT_WIDTH -FORTRAN FREEOF FULLPREC -FULLROOTS -G GAMMA Gamma Function -GC GCD GEGENBAUERP -GENTRAN GosperAlg -Hankel Functions HANKEL1 HANKEL2 -HERMITEP HIGH_POW HORNER -HYPOT -I Identifier IF -IFACTOR IMPART IN -Indefinite integration INDEX INFINITY -INFIX INPUT INT -INTEGER INTERPOL INTSTR -ISOLATER -JACOBIP -KERNEL KORDER Kummer Functions -KUMMERM KUMMERU -LAGUERREP LCM LCOF -LEGENDREP LENGTH LESSSPACE -LET LHS LIMIT -LIMITEDFACTORS LINEAR LINELENGTH -LISP LIST List(operation) -LISTARGP LISTARGS LN -LOAD_PACKAGE LOG LOGB -Lommel Functions LOMMEL1 LOMMEL2 -LOW_POW LTERM -MAINVAR MASS MAT -MATCH MATEIGEN MATRIX -MAX MCD MIN -MKID MODULAR MSG -MSHELL MULTIPLICITIES -NAT NERO NEXTPRIME -NOARG NODEPEND NOLNR -NONCOM NONZERO NOSPLIT -NOSPUR NOXPND NULLSPACE -NUM NUMVAL NUMBERP -NUM_INT NUM_MIN NUM_ODESOLVE -NUM_SOLVE -ODD ODESOLVE OFF -ON ONE_OF OPERATOR -ORDER ORDP ORTHOVEC -OUT OUTPUT OVERVIEW -PART PAUSE PERIOD -PF PI POCHHAMMER -POLYGAMMA PRECEDENCE PRECISE -PRECISION PRET PRI -PRIMEP PRINT_PRECISION PROCEDURE -PROD PRODUCT PSI -QUIT -RANK RAT RATARG -RATIONAL RATIONALIZE RATPRI -REAL REDUCT REMAINDER -REMFAC REMIND REPART -REPEAT REST RESULTANT -RETRY RETURN REVERSE -REVPRI RHS RLISP88 -RLROOTNO ROOT_OF ROOT_MULTIPLICITES -ROUND ROUNDALL ROUNDBF -ROUNDED Rule_lists -SAVEAS SAVESTRUCTR SCALAR -SCIENTIFIC_NOTATION SCOPE SEC -SECH SECOND SET -SETMOD SHARE SHOWRULES -SHOWTIME SHUT SIGN -SIN SINH SOLVE -SOLVESINGULAR SPDE SPLIT_FIELD -SPUR SQRT STIRLING1 -STIRLING2 String STRUCTR -STRUVEH STRUVEL SUB -SUM SYMBOLIC SYMMETRIC -T TAN TANH -TAYLOR TAYLORAUTOCOMBINE TAYLORAUTOEXPAND -TAYLORCOMBINE TAYLORKEEPORIGINAL TAYLORORIGINAL -TAYLORPRINTORDER TAYLORPRINTTERMS TAYLORREVERT -TAYLORSERIESP TAYLORTEMPLATE TAYLORTOSTANDARD -THIRD TIME TP -TPS TRACE TRALLFAC -TRFAC TRIGFORM TRINT -TRNONLNR -VARNAME VECDIM VECTOR -WEIGHT WHEN WHERE -WHILE WHITTAKERW WRITE -WS WTLEVEL -XPND -ZETA - -\endsection -\item[Algebra] -Algebra Index - -Algebraic operators about which there is help are: - -APPEND ARBINT ARBCOMPLEX -ARGLENGTH COEFF COEFFN -CONJ DECOMPOSE DEG -DEN DF EXPAND_CASES -EXPREAD FACTORIZE HYPOT -IMPART INT INTERPOL -LCOF LENGTH LHS -LTERM MAINVAR NPRIMITIVE -NUM PART PF -REDUCT REPART RESULTANT -RHS ROOT_OF SHOWRULES -SOLVE STRUCTR SUB -WS - -\endsection -\item[Arithmetic] -Arithmetic Index - -This section considers operations defined in REDUCE that concern numbers, -or operators that can operate on numbers in addition, in most cases, to -more general expressions. - -Arithmetic operations about which there is help are: - -ABS ADJPREC CEILING -DILOG FACTORIAL FIX -FIXP FLOOR GCD -LN LOG LOGB -MAX MIN NEXTPRIME -REMAINDER ROUND SIGN -SQRT - -\endsection -\item[Booleans] -Booleans Index - -Boolean operations about which there is help are: - -EVENP FREEOF NUMBERP -ORDP PRIMEP - -\endsection -\item[Commands] -Commands Index - -Commands about which there is help are: - -BYE CONT DISPLAY -LOAD_PACKAGE PAUSE QUIT -RETRY SAVEAS SHOWTIME -WRITE - -\endsection -\item[Concepts] -Concepts Index - -There is help on the following basic concepts: - -Identifier Kernel String - -Also there is a simple editor, described by - -ED EDITDEF - -\endsection -\item[Declarations] -Declarations Index - -Declarations about which there is help are: - -ALGEBRAIC ANTISYMMETRIC ARRAY -CLEAR CLEARRULES DEFINE -DEPEND EVEN FACTOR -FORALL INFIX INTEGER -KORDER LET LINEAR -LINELENGTH LISP LISTARGP -MATCH NODEPEND NONCOM -NONZERO ODD OFF -ON OPERATOR ORDER -PRECEDENCE PRECISION PRINT_PRECISION -REAL REMFAC SCALAR -SCIENTIFIC_NOTATION SHARE SYMBOLIC -SYMMETRIC VARNAME WEIGHT -WHILE WTLEVEL - -\endsection -\item[Functions] -Functions Index - -Elementary functions about which there is help are: - -ACOS ACOSH ACOT -ACOTH ACSC ACSCH -ASEC ASECH ASIN -ASINH ATAN ATANH -ATAN2 COS COSH -COT COTH CSC -CSCH ERF EXP -EXPINT SEC SECH -SIN SINH TAN -TANH - -\endsection -\item[HighEnergy] -High Energy Physics Index - -The High-energy Physics package is historic for REDUCE, since REDUCE -originated as a program to aid in computations with Dirac expressions. -The commutation algebra of the gamma matrices is independent of their -representation, and is a natural subject for symbolic mathematics. -Dirac theory is applied to beta decay and the computation of -cross-sections and scattering. The high-energy physics operators are -available in the REDUCE main program, rather than as a module which -must be loaded. - -Arithmetic operations about which there is help are: - -DOT EPS G -INDEX MASS MSHELL -NOSPUR REMIND SPUR -VECDIM VECTOR - -\endsection -\item[InputOutput] -Input and Output Index - -Input and Output actions about which there is help are: - -IN INPUT OUT -SHUT - -\endsection -\item[Library] -Library Index - -The external modules that are included in your REDUCE system are the -first members of the REDUCE User's Library. They have been -contributed by REDUCE users from various fields for the convenience -and pleasure of the REDUCE user community. Future releases of REDUCE -will include other packages as they are developed. The packages in -the User's Library are unsupported; any questions or problems should -be directed to their authors. - -Each package comes with its own documentation, which you can find, -along with the source code, in the subdirectories lib of you REDUCE -directory (with suffix .txt, .tex and .red). The LOAD_PACKAGE command -is used to load the files you wish into your system. There will be a -short delay while the module is loaded. A module cannot be unloaded. -Once it is in your system, it stays there until you end the session. -Each package also has a test file, which you will find under its name -in the lib directory with suffix .tst. - -The following paragraphs, provided by the authors of each module, -briefly introduce packages which have not yet been described in more -detail in other sections of this document. Please refer to the -documentation for each module for detailed information on its use. -Each of them have their own switches, commands, and operators, and -some redefine special characters to aid in their notation. - -Libraries about which there is help are: - -ALGINT ARNUM AVECTOR -COMPACT EXCALC GENTRAN -NUMERIC ODESOLVE ORTHOVEC -SCOPE SPDE TPS - -\endsection -\item[Matrix] -Matrix Index - -Matrix operations about which there is help are: - -COFACTOR DET MAT -MATEIGEN MATRIX NULLSPACE -RANK TP TRACE - -\endsection -\item[Operators] -Operators Index - -Operations about which there is help are: - -LIMIT SUM PROD - -\endsection -\item[Specfns] -Special Functions - -The REDUCE Special Function Package supplies extended algebraic and -numeric support for a wide class of objects. This package is released -together with REDUCE 3.5 (October 1993) for the first time, therefore -it is far from being complete. - -The functions included in this package are in most cases (unless -otherwise stated) defined and named like in the book by Abramowitz and -Stegun: Handbook of Mathematical Functions, Dover Publications. - -The aim is to collect as much information on the special functions and -simplification capabilities as possible, i.e. algebraic -simplifications and numeric (rounded mode) code, limits of the -functions together with the definitions of the functions, which are in -most cases a power series, a (definite) integral and/or a differential -equation. - -What can be found: - A variety of Bessel functions, special polynomials, the Gamma -function, the Zeta function and integral functions. - -What is missing: - Airy functions, Mathieu functions, LerchPhi, etc.. The information -about the special functions which solve certain differential equation -is very limited. In several cases numerical approximation is -restricted to real arguments or is missing completely. - -The implementation of this package uses REDUCE rule sets to a large -extent, which guarantees a high 'readability' of the functions -definitions in the source file directory. It makes extensions to the -special functions code easy in most cases too. To look at these rules -it may be convenient to use the showrules operator e.g. - - showrules Besseli; - -Note: The special function package has to be loaded explicitly by calling - load_package specfn; - -Help is available on: - -BERNOUILLI BESSELI BESSELJ -BESSELK BESSELY BETA -CHEBYSHEVT CHEBYSHEVU EULER -EULERP GAMMA GEGENBAUERP -HANKEL1 HANKEL2 HERMITEP -JACOBIP KUMMERM KUMMERU -LAGUERREP LEGENDREP LOMMEL1 -LOMMEL2 POCHHAMMER POLYGAMMA -PSI STIRLING1 STIRLING2 -STRUVEH STRUVEL WHITTAKERW -ZETA - -\endsection -\item[Switches] -Switches Index - -Switches are set on or off using the commands ON or OFF, respectively. -The default setting of the switches described in this section is -OFF unless stated otherwise. - -Switches about which there is help are: - -ALGINT ALLBRANCH ALLFAC -BALANCED_MOD BFSPACE COMBINEEXPT -COMBINELOGS COMP COMPLEX -CREF CRAMER DEFN -DEMO DFPRINT DIV -ECHO ERRCONT EVALLHSEQP -EXP EXPANDLOGS EZGCD -FACTOR FAILHARD FORT -FULLPREC FULLROOTS GC -GCD HORNER IFACTOR -INT INTSTR LCM -LESSSPACE LIMITEDFACTORS LIST -LISTARGS MCD MODULAR -MSG MULTIPLICITIES NAT -NERO NOARG NOLNR -NOSPLIT NUMVAL OUTPUT -OVERVIEW PERIOD PRECISE -PRET PRI RAT -RATARG RATIONAL RATIONALIZE -RATPRI REVPRI RLISP88 -ROUNDALL ROUNDBF ROUNDED -SAVESTRUCTR SOLVESINGULAR TIME -TRALLFAC TRFAC TRIGFORM -TRINT TRNONLNR - -\endsection -\item[Syntax] -Syntax Index - -Syntax about which there is help are: - -BEGIN...END COMMENT CONS -END EQUATION FIRST -FOR FOREACH GOTO -IF List PROCEDURE -REPEAT REST RETURN -REVERSE RuleSet SECOND -SET THIRD WHEN - -\endsection -\item[Variables] -Variables Index - -Variables about which there is help are: - -CARD_NO E EVAL_MODE -FORT_WIDTH HIGH_POW I -INFINITY LOW_POW NIL -PI ROOT_MULTIPLICITIES T - -\endsection -\xitem[><] ->< 3-D vector and diphthong (page 356) - -\endsection -\xitem[*] -* - 3-D vector, 356 - algebraic numbers, 224 - power series, 422 - vector, 232 - -\endsection -\xitem[**] -** - power series, 422 - -\endsection -\xitem[+] -+ - 3-D vector, 356 - algebraic numbers, 223 - power series, 422 - vector, 232 - -\endsection -\xitem[-] -- - 3-D vector, 356 - power series, 422 - vector, 232 - -\endsection -\item[.] -. (CONS) (page 50) - -The CONS operator adds a new element to the beginning of a LIST. Its -operation is identical to the symbol DOT (dot). It can be used -infix or prefix. - - CONS(item,list) or item CONS list - -item can be any REDUCE scalar expression, including a list; list -must be a list. - -Examples: - -liss := cons(a,{b}); {A,B} - -liss := c cons liss; {C,A,B} - -newliss := for each y in liss collect cons(y,list x); - NEWLISS := {{C,X},{A,X},{B,X}} - -for each y in newliss sum (first y)*(second y); - X*(A + B + C) - -If you want to use CONS to put together two elements into a new list, -you must make the second one into a list with curly brackets or the -LIST command. You can also start with an empty list created by {}. - -The CONS operator is right associative: A CONS B CONS C is valid if C -is a list; B need not be a list. The list produced is {A,B,C}. - -\endsection -\xitem[/] -/ - 3-D vector, 356 - algebraic numbers, 224 - power series, 422 - vector, 232 - -\endsection -\xitem[@] -@ - partial differentiation, 271 - tangent vector, 271 - -\endsection -\xitem[@ operator] -@ operator, 251 - -\endsection -\item[#] -# (pages 256, 271) - -# is the syntax for the Hodge-* operator in the EXCALC package. - -\endsection -\xitem[^] -^ - 3-D vector, 356 - exterior multiplication, 250, 271 - -\endsection -\item[ABS] -ABS (page 72) -The ABS operator returns the absolute value of its argument. - -ABS(expression) - -expression can be any REDUCE scalar expression. - -Examples: -abs(-a); ABS(A) -abs(-5); 5 -a := -10; A := -10 -abs(a); 10 -abs(-a); 10 - -If the argument has had no numeric value assigned to it, such as an -identifier or polynomial, ABS returns an expression involving -ABS of its argument, doing as much simplification of the argument -as it can, such as dropping any preceding minus sign. - -\endsection -\item[ACOS] -ACOS (pages 76, 78) - -The ACOS operator returns the arccosine of its argument. - - ACOS(expression) or ACOS simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -acos(ab); ACOS(AB) -acos 15; ACOS(15) - 2 2 - SQRT( - X *Y + 1)*Y -df(acos(x*y),x); ---------------------- - 2 2 - X *Y - 1 -on rounded; -res := acos(sqrt(2)/2); RES := 0.785398163397 -res-pi/4; 0 - -An explicit numeric value is not given unless the switch ROUNDED is -on and the argument has an absolute numeric value less than or equal to 1. - -\endsection -\item[ACOSH] -ACOSH (pages 76, 78) - -ACOSH represents the hyperbolic arccosine of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of -ACOSH is known to the system. Numerical values may also be found by -turning on the switch ROUNDED. - - ACOSH(expression) or ACOSH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix or -vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -acosh a; ACOSH(A) -acosh(0); ACOSH(0) - 4 - 2*SQRT(A - 1)*A -df(acosh(a**2),a); ------------------ - 4 - A - 1 - -int(acosh(x),x); INT(ACOSH(X),X) - -You may attach functionality by defining ACOSH to be the inverse of -COSH. This is done by the commands - put('cosh,'inverse,'acosh); - put('acosh,'inverse,'cosh); -You can write a procedure to attach integrals or other functions to -ACOSH. You may wish to add a check to see that its argument is -properly restricted. - -\endsection -\item[ACOT] -ACOT (pages 76, 78) - -ACOT represents the arccotangent of its argument. It takes an -arbitrary scalar expression as its argument. The derivative of ACOT is -known to the system. Numerical values may also be found by turning on -the switch ROUNDED. - - ACOT(expression) or ACOT simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. You can add functionality -yourself with LET and procedures. - -Examples: -acot a; ACOT(A) - PI -acot(0); ---- - 2 - - - 2*A -df(acot(a**2),a); -------- - 4 - A + 1 - - 2 - 2*ACOT(X)*X + LOG(X + 1) -int(acot(x),x); --------------------------- - 2 -on rounded; -acot(1); 0.785398163397 - -\endsection -\item[ACOTH] -ACOTH (pages 76, 78) - -ACOTH represents the inverse hyperbolic cotangent of its argument. It -takes an arbitrary scalar expression as its argument. The derivative -of ACOTH is known to the system. Numerical values may also be found -by turning on the switch ROUNDED. - - ACOTH(expression) or ACOTH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. You can add functionality yourself -with LET and procedures. - -Examples: -acoth(0); 0 - - - 2*X -df(acoth(x^2),x); -------- - 4 - X - 1 - -int(acoth(x),x); ACOTH(X)*X + ACOTH(X) + LOG(X - 1) - -\endsection -\item[ACSC] -ACSC (pages 76, 78) - -The ACSC operator returns the arccosecant of its argument. - - ACSC(expression) or ACSC simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -acsc(ab); ACSC(AB) -acsc 15; ACSC(15) - 2 2 - - SQRT(X *Y - 1) -df(acsc(x*y),x); -------------------- - 2 2 - X*(X *Y - 1) -on rounded; -res := acsc(2/sqrt(3)); RES := 1.0471975512 -res-pi/3; 0 - -An explicit numeric value is not given unless the switch ROUNDED is on -and the argument has an absolute numeric value less than or equal to -1. - -\endsection -\item[ACSCH] -ACSCH (pages 76, 78) - -The ACSCH operator returns the hyperbolic arccosecant of its argument. - - ACSCH(expression) or ACSCH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -acsch(ab); ACSCH(AB) -acsch 15; ACSCH(15) - 2 2 - - SQRT(X *Y + 1) -df(acsch(x*y),x); -------------------- - 2 2 - X*(X *Y + 1) -on rounded; -res := acsch(3); RES := 0.327450150237 - -An explicit numeric value is not given unless the switch ROUNDED is on -and the argument has an absolute numeric value less than or equal to -1. - -\endsection -\item[ADJPREC] -ADJPREC (page 133) - -When a real number is input, it is normally truncated to the PRECISION -in effect at the time the number is read. If it is desired to keep -the full precision of all numbers input, the switch ADJPREC (for -adjust precision) can be turned on. While on, ADJPREC will -automatically increase the precision, when necessary, to match that of -any integer or real input, and a message printed to inform the user of -the precision increase. - -Examples: -on rounded; -1.23456789012345; 1.23456789012 -on adjprec; -1.23456789012345; *** precision increased to 15 - 1.23456789012345 - -\endsection -\item[ALGEBRAIC] -ALGEBRAIC (page 191) - -The ALGEBRAIC command changes REDUCE's mode of operation to -algebraic. When ALGEBRAIC is used as an operator (with an argument -inside parentheses) that argument is evaluated in algebraic mode, but -REDUCE's mode is not changed. - -Examples: -algebraic; -symbolic; NIL - 2 -algebraic(x**2); X -x**2; ***** The symbol X has no value. - -REDUCE's symbolic mode does not know about most algebraic commands. -Error messages in this mode may also depend on the particular Lisp -used for the REDUCE implementation. - -\endsection -\item[Algebraic mode] -Algebraic mode (pages 191, 197, 198) - -Most REDUCE calculatuons take place in Algebraic mode. The -alternative is Symbolic mode, which is a syntactic form of LISP. See -the commands ALGEBRAIC and SYMBOLIC for mor details. - -\endsection -\item[ALGINT] -ALGINT - -When the ALGINT switch is on, the algebraic integration module (which -must be loaded from the REDUCE library) is used for integration. - -Loading ALGINT from the library automatically turns on the -ALGINT switch. An error message will be given if ALGINT is -turned on when the ALGINT has not been loaded from the library. - -\endsection -\item[ALGINT(Package)] -ALGINT(Package) (page 178) - -Author: James H. Davenport - -The ALGINT package provides indefinite integration of square roots. -This package, which is an extension of the basic integration package -distributed with REDUCE, will analytically integrate a wide range of -expressions involving square roots. The ALGINT switch provides for -the use of the facilities given by the module, and is automatically -turned on when the package is loaded. If you want to return to the -standard integration algorithms, turn ALGINT off. An error message is -given if you try to turn the ALGINT switch on when its module is not -loaded. - -\endsection -\item[ALLBRANCH] -ALLBRANCH (page 89) - -When ALLBRANCH is on, the operator SOLVE selects all branches of -solutions. When ALLBRANCH is off, it selects only the principal -branches. Default is ON. - -Examples: - -solve(log(sin(x+3)),x); {X=2*ARBINT(1)*PI - ASIN(1) - 3, - X=2*ARBINT(1)*PI + ASIN(1) + PI - 3} -off allbranch; -solve(log(sin(x+3)),x); {X=ASIN(1) - 3} - -ARBINT(1) indicates an arbitrary integer, which is given a unique -identifier by REDUCE, showing that there are infinitely many solutions -of this type. When ALLBRANCH is off, the single canonical solution is -given. - -\endsection -\item[ALLFAC] -ALLFAC (pages 102, 104) - -The ALLFAC switch, when on, causes REDUCE to factor out automatically -common products in the output of expressions. Default is ON. - -Examples: 3 -x + x*y**3 + x**2*cos(z); X*(COS(Z)*X + Y + 1) -off allfac; - 2 3 -x + x*y**3 + x**2*cos(z); COS(Z)*X + X*Y + X - -The ALLFAC switch has no effect when PRI is off. Although the switch -setting stays as it was, printing behaviour is as if it were off. - -\endsection -\xitem[ansatz of symmetry generator] -ansatz of symmetry generator, 386 - -\endsection -\item[ANTISYMMETRIC] -ANTISYMMETRIC (page 93) - -When an operator is declared ANTISYMMETRIC, its arguments are -reordered to conform to the internal ordering of the system. If an -odd number of argument interchanges are required to do this ordering, -the sign of the expression is changed. - - ANTISYMMETRIC identifier {,identifier} - -identifier is an identifier that has been declared as an operator. - -Examples: -operator m,n; -antisymmetric m,n; -m(x,n(1,2)); - M( - N(2,1),X) -operator p; -antisymmetric p; -p(a,b,c); P(A,B,C) -p(b,a,c); - P(A,B,C) - -If identifier has not been declared an operator, the flag -ANTISYMMETRIC is still attached to it. When identifier is -subsequently used as an operator, the message - Declare identifier operator? (Y or N) -is printed. If the user replies Y, the antisymmetric property of the -operator is used. - -Note in the first example, identifiers are customarily ordered -alphabetically, while numbers are ordered from largest to smallest. -The operators may have any desired number of arguments (less than 128). - -\endsection -\item[APPEND] -APPEND (page 50) - -The APPEND operator constructs a new list from the elements of its two -arguments (which must be lists). - - APPEND(lst,lst) - -lst must be a list, though it may be the empty list ({}). Any -arguments beyond the first two are ignored. - -Examples: -alist := {1,2,{a,b}}; ALIST := {1,2,{A,B}} -blist := {3,4,5,sin(y)}; BLIST := {3,4,5,SIN(Y)} -append(alist,blist); {1,2,{A,B},3,4,5,SIN(Y)} -append(alist,{}); {1,2,{A,B}} -append(list z,blist); {Z,3,4,5,SIN(Y)} - -Comment The new list consists of the elements of the second list -appended to the elements of the first list. You can append new -elements to the beginning or end of an existing list by putting the -new element in a list (use curly braces or the operator list). This -is particularly helpful in an iterative loop. - -\endsection -\item[ARBCONST] -ARBCONST operator (page 350) - -See the ODESOLVE package - -\endsection -\xitem[arbitrary ordering] -arbitrary ordering, 316 - -\endsection -\item[ARGLENGTH] -ARGLENGTH (page 117) -The operator ARGLENGTH returns the number of arguments of the top-level -operator in its argument. - - ARGLENGTH(expression) - -expression can be any valid REDUCE algebraic expression. - -Examples: -arglength(a + b + c + d); 4 -arglength(a/b/c); 2 -arglength(log(sin(df(r**3*x,x)))); 1 - -In the first example, + is an n-ary operator, so the number of terms -is returned. In the second example, since / is a binary operator, the -argument is actually (a/b)/c, so there are two terms at the top level. -In the last example, no matter how deeply the operators are nested, -there is still only one argument at the top level. - -\endsection -\item[ARNUM] -ARNUM (pages 179, 223) -Author: Eberhard Schruefer - -This package provides facilities for handling algebraic numbers as polynomial -coefficients in REDUCE calculations. It includes facilities for introducing -indeterminates to represent algebraic numbers, for calculating splitting -fields, and for factoring and finding greatest common divisors in such -domains. - -\endsection -\item[ARRAY] -ARRAY (page 67) - -The ARRAY declaration declares a list of identifiers to be of type -ARRAY, and sets all their entries to 0. - - ARRAY identifier(dimensions){,identifier(dimensions)} - -identifier may be any valid REDUCE identifier. If the identifier -was already an array, a warning message is given that the array has been -redefined. dimensions are of form - integer{,integer}. - -array a(2,5),b(3,3,3),c(200); -array a(3,5); *** ARRAY A REDEFINED -a(3,4); 0 -length a; {4,6} - -Arrays are always global, even if defined inside a procedure or block -statement. Their status as an array remains until the variable is -reset by CLEAR. Arrays may not have the same names as operators, -procedures or scalar variables. - -Array elements are referred to by the usual notation: A(I,J) returns -the jth element of the ith row. The ASSIGNment operator := is used to -put values into the array. Arrays as a whole cannot be subject to -assignment by LET or := ; the assignment operator := is only valid for -individual elements. - -When you use LET on an array element, the contents of that element -become the argument to LET. Thus, if the element contains a number or -some other expression that is not a valid argument for this command, -you get an error message. If the element contains an identifier, the -identifier has the substitution rule attached to it globally. The -same behaviour occurs with CLEAR. If the array element contains an -identifier or simple_expression, it is cleared. Do NOT use CLEAR to -try to set an array element to 0. Because of the side effects of -either LET or CLEAR, it is unwise to apply either of these to array -elements. - -Array indices always start with 0, so that the declaration ARRAY A(5) -sets aside 6 units of space, indexed from 0 through 5, and initialises -them to 0. The LENGTH command returns a list of the true number of -elements in each dimension. - -\endsection -\item[ASEC] -ASEC (pages 76, 78) - -The ASEC operator returns the arccosecant of its argument. - - ASEC(expression) or ASEC simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -asec(ab); ASEC(AB) -asec 15; ASEC(15) - 2 2 - SQRT(X *Y - 1) -df(asec(x*y),x); ----------------- - 2 2 - X*(X *Y - 1) -on rounded; -res := asec sqrt(2); RES := 0.785398163397 -res-pi/4; 0 - -An explicit numeric value is not given unless the switch ROUNDED is -on and the argument has an absolute numeric value greater or equal to 1. - -\endsection -\item[ASECH] -ASECH (pages 76, 78) - -ASECH represents the hyperbolic arccosecant of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of -ASECH is known to the system. Numerical values may also be found by -turning on the switch ROUNDED. - - ASECH(expression) or ASECH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -asech a; ASECH(A) -asech(1); 0 - 4 - 2*SQRT(A - 1)*A -df(acosh(a**2),a); ------------------ - 4 - A - 1 -int(asech(x),x); INT(ASECH(X),X) - -You may attach functionality by defining ASECH to be the inverse of -SECH. This is done by the commands - put('sech,'inverse,'asech); - put('asech,'inverse,'sech); -You can write a procedure to attach integrals or other functions to -ASECH. You may wish to add a check to see that its argument is -properly restricted. - -\endsection -\item[ASIN] -ASIN (pages 76, 78) - -The ASIN operator returns the arcsine of its argument. - - ASIN(expression) or ASIN simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -asin(givenangle); ASIN(GIVENANGLE) -asin(5); ASIN(5) - 2 - - 2*SQRT( - 4*X + 1) -df(asin(2*x),x); ------------------------ - 2 - 4*X - 1 -on rounded; -asin .5; 0.523598775598 -asin(sqrt(3)); ASIN(1.73205080757) -asin(sqrt(3)/2); 1.04719755120 - -A numeric value is not returned by ASIN unless the switch -ROUNDED is on and its argument has an absolute value less than or -equal to 1. - -\endsection -\item[ASINH] -ASINH (pages 76, 78) - -The ASINH operator returns the hyperbolic arcsine of its argument. -The derivative of ASINH and some simple transformations are known -to the system. - - ASINH(expression) or ASINH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -asinh d; ASINH(D) -asinh(1); ASINH(1) - 2 - 2*SQRT(4*X + 1) -df(asinh(2*x),x); ------------------ - 2 - 4*X + 1 - -You may attach further functionality by defining ASINH to be the -inverse of SINH. This is done by the commands - put('sinh,'inverse,'asinh); - put('asinh,'inverse,'sinh); - -A numeric value is not returned by ASINH unless the switch ROUNDED is -on and its argument evaluates to a number. - -\endsection -\xitem[Assignment] -Assignment, 54, 55, 57, 63, 195, 198 - -\endsection -\item[Asymptotic command] -Asymptotic command (pages 139, 151) - -See WEIGHT and WTLEVEL - -\endsection -\item[ATAN] -ATAN (pages 76, 78, 81) - -The ATAN operator returns the arctangent of its argument. - - ATAN(expression) or ATAN simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -atan(middle); ATAN(MIDDLE) -on rounded; -atan 45; 1.54857776147 -off rounded; - 2 - 2*ATAN(X)*X - LOG(X + 1) -int(atan(x),x); --------------------------- - 2 - 2*Y -df(atan(y**2),y); -------- - 4 - Y + 1 - -A numeric value is not returned by ATAN unless the switch ROUNDED is -on and its argument evaluates to a number. - -\endsection -\item[ATAN2] -ATAN2 (pages 76, 78) - - ATAN2(expression,expression) - -expression is any valid scalar REDUCE expression. In ROUNDED mode, if -a numerical value exists, ATAN2 returns the principal value of the arc -tangent of the second argument divided by the first in the range -[-pi,+pi] radians, using the signs of both arguments to determine the -quadrant of the return value. An expression in terms of ATAN2 is -returned in other cases. - -Examples: -atan2(3,2); ATAN2(3,2); -on rounded; -atan2(3,2); 0.982793723247 -atan2(a,b); ATAN2(a,b); -atan2(1,0); 1.57079632679 - -ATAN2 returns a numeric value only if ROUNDED is on. Then the -arctangent is calculated to the current degree of floating point precision. - -\endsection -\item[ATANH] -ATANH (pages 76, 78) - -The ATANH operator returns the hyperbolic arctangent of its argument. -The derivative of ASINH and some simple transformations are known -to the system. - - ATANH(expression) or ATANH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier -or begin with a prefix operator name. - -Examples: -atanh aa; ATANH(AA) -atanh(1); ATANH(1) - - X -df(atanh(x*y),y); ----------- - 2 2 - X *Y - 1 - -A numeric value is not returned by ASINH unless the switch ROUNDED is -on and its argument evaluates to a number. You may attach additional -functionality by defining ATANH to be the inverse of TANH. This is -done by the commands - put('tanh,'inverse,'atanh); - put('atanh,'inverse,'tanh); - -\endsection -\xitem[AVEC function] -AVEC function, 232 - -\endsection -\item[AVECTOR] -AVECTOR (pages 179, 231) - -Author: David Harper - -A Vector Algebra and Calculus Package. - -This package provides REDUCE with the ability to perform vector -algebra using the same notation as scalar algebra. The basic -algebraic operations are supported, as are differentiation and -integration of vectors with respect to scalar variables, cross -product and dot product, component manipulation and application of -scalar functions (e.g. cosine) to a vector to yield a vector -result. - -\endsection -\item[BALANCED_MOD] -BALANCED_MOD - -MODULAR numbers are normally produced in the range [0,...n), where -n is the current modulus. With BALANCED_MOD on, the range -[-n/2,n/2] is used instead. - -Examples: - setmod 7; 1 - on modular; - 4; 4 - on balanced_mod; - 4; -3 - -\endsection -\item[BEGIN...END] -BEGIN ... END (pages 61, 62, 64) - -BEGIN is used to start a BLOCK statement, which is closed with END. - - BEGIN statement{; statement} END - -statement is any valid REDUCE statement. - -Examples: - begin for i := 1:3 do write i end; 1 - 2 - -begin scalar n;n:=1;b:=for i:=1:4 product(x-i);return n end; - 1 - 4 3 2 -b; X - 10*X + 35*X - 50*X + 24 - -A BEGIN...END block can do actions (such as WRITE), but does not -return a value unless instructed to by a RETURN statement, which must -be the last statement executed in the block. It is unnecessary to -insert a semicolon before the END. - -Local variables, if any, are declared in the first statement -immediately after BEGIN, and may be defined as SCALAR, INTEGER, or -REAL. ARRAY variables declared within a BEGIN...END block are global -in every case, and LET statements have global effects. A LET -statement involving a formal parameter affects the calling parameter -that corresponds to it. LET statements involving local variables make -global assignments, overwriting outside variables by the same name or -creating them if they do not exist. You can use this feature to -affect global variables from procedures, but be careful that you do -not do it inadvertently. - -\endsection -\item[BERNOULLI] -BERNOULLI (pages 185, 393) -[Part of SPECFN package] - -The BERNOULLI operator returns the nth Bernoulli number. - - BERNOULLI(integer) - -Examples: -load_package specfn; (SPECFN) - - - 174611 -bernoulli 20; ----------- - 330 - -bernoulli 17; 0 - -All Bernoulli numbers with odd indices except for 1 are zero. - -The BERNOULLIP operator returns the nth Bernoulli Polynomial evaluated -at x. - - BERNOULLIP(integer,expression) - -Examples: -load_package specfn; (SPECFN) - - 2 - Z*(2*Z - 3*Z + 1) -BernoulliP(3,z); -------------------- - 2 - - 338585 -BernoulliP(10,3); -------- - 66 - -The value of the nth Bernoulli Polynomial at 0 is the nth Bernoulli number. - -\endsection -\item[BESSELI] -BESSELI (pages 185, 396) -[Part of SPECFN package] - -The BESSELI operator returns the modified Bessel function I. - - BESSELI(order,argument) - -Examples: -load_package specfn; (SPECFN) -on rounded; -Besseli (1,1); 0.565159103992 - -The knowledge about the operator BESSELI is currently fairly limited. - -\endsection -\item[BESSELJ] -BESSELJ (pages 185, 396) -[Part of SPECFN package] - -The BESSELJ operator returns the Bessel function of the first kind. - - BESSELJ(order,argument) - -Examples: -load_package specfn; (SPECFN) -BesselJ(1/2,pi); 0 -on rounded; -BesselJ(0,1); 0.765197686558 - -\endsection -\item[BESSELK] -BESSELK (pages 185, 396) -[Part of SPECFN package] - -The BESSELK operator returns the modified Bessel function K. - - BESSELK(order,argument) - -Examples: -load_package specfn; (SPECFN) -df(besselk(0,x),x); - BESSELK(1,X) - -There is currently no numeric support for the operator BesselK. - -\endsection -\item[BESSELY] -BESSELY (pages 185, 396) -[Part of SPECFN package] - -The BESSELY operator returns the Bessel function of the second kind. - BESSELY(order,argument) - -Examples: -load_package specfn; (SPECFN) -Bessely (1/2,pi); - SQRT(2) / PI -on rounded; -Bessely (1,3); 0.324674424792 - -The operator BESSELY is also called Weber's function. - -\endsection -\item[BETA] -BETA (pages 185, 397) -[Part of SPECFN package] - -The BETA operator returns the Beta function defined by - - Beta (z,w) := defint(t**(z-1)* (1 - t)**(w-1),t,0,1) . - - - BETA(expression,expression) - - -Examples: -load_package specfn; (SPECFN) -Beta(2,2); 1 / 6 -Beta(x,y); GAMMA(X)*GAMMA(Y) / GAMMA(X + Y) - -The operator BETA is simplified towards the GAMMA operator. - -\endsection -\item[BFSPACE] -BFSPACE (page 133) - -Floating point numbers are normally printed in a compact notation -(either fixed point or in scientific notation if SCIENTIFIC_NOTATION -has been used). In some (but not all) cases, it helps -comprehensibility if spaces are inserted in the number at regular -intervals. The switch BFSPACE, if on, will cause a blank to be -inserted in the number after every five characters. - -Examples: - on rounded; - 1.2345678; 1.2345678 - on bfspace; - 1.2345678; 1.234 5678 - -BFSPACE is normally off. - -\endsection -\item[BINOMIAL] -BINOMIAL (page 185) - -The BINOMIAL operator returns the Binomial coefficient if both -parameter are integer and expressions involving the Gamma function otherwise. - - BINOMIAL(integer,integer) - -Examples: -Binomial(49,6); 13983816 - - GAMMA(N + 1) -Binomial(n,3); ---------------- - 6*GAMMA(N - 2) - -The operator BINOMIAL evaluates the Binomial coefficients from the -explicit form and therefore it is not the best algorithm if you want -to compute many binomial coefficients with big indices in which case a -recursive algorithm is preferable. - -\endsection -\xitem[Block] -Block, 61, 64 - -\endsection -\xitem[BNDEQ!*] -BNDEQ!*, 257 - -\endsection -\xitem[Boolean] -Boolean, 45 - -\endsection -\item[BOUNDS] -BOUNDS (page 182) - -Upper and lower bounds of a real valued function over an INTERVAL or a -rectangular multivariate domain are computed by the operator -BOUNDS. The algorithmic basis is the computation with inequalities: -starting from the interval(s) of the variables, the bounds are -propagated in the expression using the rules for inequality -computation. Some knowledge about the behavior of special functions -like ABS, SIN, COS, EXP, LOG, fractional exponentials etc. is -integrated and can be evaluated if the operator bounds is called with -rounded mode on (otherwise only algebraic evaluation rules are -available). - -If BOUNDS finds a singularity within an interval, the evaluation is -stopped with an error message indicating the problem part of the -expression. - - BOUNDS(exp,var=(l .. u) [,var=(l .. u) ...]) - BOUNDS(exp,{var=(l .. u) [,var=(l .. u) ...]}) - -where exp is the function to be investigated, var are the variables of -exp, l and u specify the area as set of INTERVAL s. - -BOUNDS computes upper and lower bounds for the expression in the given -area. An INTERVAL is returned. - -Examples: - bounds(sin x,x=(1 .. 2)); - 1 .. 1 - on rounded; - bounds(sin x,x=(1 .. 2)); 0.84147098481 .. 1 - bounds(x**2+x,x=(-0.5 .. 0.5)); - 0.25 .. 0.75 - -\endsection -\xitem[BROEBFULLREDUCTION] -BROEBFULLREDUCTION, 303 - -\endsection -\xitem[Buchberger's Algorithm] -Buchberger's Algorithm, 292, 295 - -\endsection -\item[BYE] -BYE (page 70) - -The BYE command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are -at the top level, the BYE command exits REDUCE. QUIT is a synonym for -BYE. - -\endsection -\xitem[C(I)] -C(I), 379 - -\endsection -\xitem[Call by value] -Call by value, 171, 173 - -\endsection -\xitem[Canonical form] -Canonical form, 97 - -\endsection -\item[CARD_NO] -CARD_NO (page 108) - -CARD_NO sets the total number of cards allowed in a Fortran -output statement when FORT is on. Default is 20. - -Examples: -on fort; -card_no := 4; CARD_NO=4. -z := (x + y)**15; - ANS1=5005.*X**6*Y**9+3003.*X**5*Y**10+1365.*X**4*Y** - . 11+455.*X**3*Y**12+105.*X**2*Y**13+15.*X*Y**14+Y**15 - Z=X**15+15.*X**14*Y+105.*X**13*Y**2+455.*X**12*Y**3+ - . 1365.*X**11*Y**4+3003.*X**10*Y**5+5005.*X**9*Y**6+ - . 6435.*X**8*Y**7+6435.*X**7*Y**8+ANS1 - -Twenty total cards means 19 continuation cards. You may set it for -more if your Fortran system allows more. Expressions are broken apart -in a Fortran-compatible way if they extend for more than CARD_NO -continuation cards. - -\endsection -\xitem[cartesian coordinates] -cartesian coordinates, 354 - -\endsection -\item[CEILING] -CEILING (page 72) - - CEILING(expression) - -This operator returns the ceiling (i.e., the least integer greater -than or equal to its argument) if its argument has a numerical value. -For negative numbers, this is equivalent to FIX. For non-numeric -arguments, the value is an expression in the original operator. - -Examples: -ceiling 3.4; 4 -fix 3.4; 3 -ceiling(-5.2); -5 -fix(-5.2); -5 -ceiling a; CEILING(A) - -\endsection -\item[CENTERED_MOD] -CENTERED_MOD (page 134) - -This is an error in the Reduce manual. It should be BALANCED_MOD. -For more information select that entry. - -\endsection -\xitem[chain rule] -chain rule, 254 - -\endsection -\xitem[Character set] -Character set, 33 - -\endsection -\item[Chebyshev_fit] -Chebyshev fit (page 182) - -The operator family CHEBYSHEV_... implements approximation and -evaluation of functions by the Chebyshev method. Let T(n,a,b,x) be -the Chebyshev polynomial of order n transformed to the interval (a,b). -Then a function f(x) can be approximated in (a,b) by a series - - for i := 0:n sum c(i)*T(i,a,b,x) - -The operator CHEBYSHEV_FIT computes this approximation and returns a -list, which has as first element the sum expressed as a polynomial and -as second element the sequence of Chebyshev coefficients. - -CHEBYSHEV_DF and CHEBYSHEV_INT transform a Chebyshev coefficient list -into the coefficients of the corresponding derivative or integral -respectively. For evaluating a Chebyshev approximation at a given -point in the basic interval the operator CHEBYSHEV_EVAL can be used. - -CHEBYSHEV_EVAL is based on a recurrence relation which is in general -more stable than a direct evaluation of the complete polynomial. - - CHEBYSHEV_FIT(fcn,var=(lo .. hi),n) - - CHEBYSHEV_EVAL(coeffs,var=(lo .. hi), var=pt) - - CHEBYSHEV_DF(coeffs,var=(lo .. hi)) - - CHEBYSHEV_INT(coeffs,var=(lo .. hi)) - - -where fcn is an algebraic expression (the target function), var is the -variable of fcn, lo and hi are numerical real values which describe an -INTERVAL lo < hi, the integer n is the approximation order (set to 20 -if missing), pt is a number in the interval and coeffs is a series of -Chebyshev coefficients. - -Examples: - -on rounded; -w:=chebyshev_fit(sin x/x,x=(1 .. 3),5); - 3 2 - W := {0.0382345446975*X - 0.239802588672*X + 0.0651206939005*X - - + 0.977836217464, - - {0.899091895826,-0.406599215895,-0.00519766024352,0.00946374143 - - 079,-0.0000948947435875}} - -chebyshev_eval(second w, x=(1 .. 3), x=2.1); - 0.411091086819 - -\xitem[Chebyshev Polynomials] -Chebyshev Polynomials, 185 - -\endsection -\item[CHEBYSHEVT] -CHEBYSHEVT (page 185) - -The CHEBYSHEVT operator computes the nth Chebyshev T Polynomial (of the -first kind). - -CHEBYSHEVT(integer,expression) - -Examples: -load_package specfn; (SPECFN) - 2 -ChebyshevT(3,xx); XX*(4*XX - 3) - -ChebyshevT(3,4); 244 - -Chebyshev's T polynomials are computed using the recurrence relation: - -ChebyshevT(n,x) := 2x*ChebyshevT(n-1,x) - ChebyshevT(n-2,x) with -ChebyshevT(0,x) := 0 and ChebyshevT(1,x) := x - -\endsection -\item[CHEBYSHEVU] -CHEBYSHEVU (page 185) - -The CHEBYSHEVU operator returns the nth Chebyshev U Polynomial (of the -second kind). - -CHEBYSHEVU(integer,expression) - -Examples: -load_package specfn; (SPECFN) - 2 -ChebyshevU(3,xx); 4*X*(2*X - 1) - -ChebyshevU(3,4); 496 - -Chebyshev's U polynomials are computed using the recurrence relation: - -ChebyshevU(n,x) := 2x*ChebyshevU(n-1,x) - ChebyshevU(n-2,x) with -ChebyshevU(0,x) := 0 and ChebyshevU(1,x) := 2x - -\endsection -\item[CLEAR] -CLEAR (pages 142, 146) - -The CLEAR command is used to remove assignments or remove substitution -rules from any expression. - -CLEAR identifier{,identifier} or - let-type statement CLEAR identifier - -identifier can be any SCALAR, MATRIX, or ARRAY variable or PROCEDURE -name. let-type statement can be any general or specific LET statement -(see below). - -Examples: -array a(2,3); -a(2,2) := 15; A(2,2) := 15 -clear a; -a(2,2); Declare A operator? (Y or N) -let x = y + z; -sin(x); SIN(Y + Z) -clear x; -sin(x); SIN(X) -let x**5 = 7; -clear x; -x**5; 7 -clear x**5; - 5 -x**5; X - -Although it is not a good idea, operators of the same name but taking -different numbers of arguments can be defined. Using a CLEAR -statement on any of these operators clears every one with the same -name, even if the number of arguments is different. - -The CLEAR command is used to ``forget'' matrices, arrays, operators -and scalar variables, returning their identifiers to the pristine -state to be used for other purposes. When CLEAR is applied to array -elements, the contents of the array element becomes the argument for -CLEAR. Thus, you get an error message if the element contains a -number, or some other expression that is not a legal argument to -CLEAR. If the element contains an identifier, it is cleared. When -clear is applied to matrix elements, an error message is returned if -the element evaluates to a number, otherwise there is no effect. Do -NOT try to use CLEAR to set array or matrix elements to 0. You will -not be pleased with the results. - -If you are trying to clear power or product substitution rules made -with either LET or FORALL...LET, you must reproduce the rule, exactly -as you typed it with the same arguments, up to but not including the -equal sign, using the word CLEAR instead of the word LET. This is -shown in the last example. Any other type of LET or FORALL...LET -substitution can be cleared with just the variable or operator name. -MATCH behaves the same as LET in this situation. There is a more -complicated example under FORALL. - -\endsection -\item[CLEARRULES] -CLEARRULES (page 148) - - CLEARRULES list{,list} - -The operator CLEARRULES is used to remove previously defined -RULE lists from the system. list can be an explicit rule -list, or evaluate to a rule list. - -Examples: -trig1 := {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2, - cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, - sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, - cos(~x)^2 => (1+cos(2*x))/2, - sin(~x)^2 => (1-cos(2*x))/2}$ -let trig1; - COS(A - B) + COS(A + B) -cos(a)*cos(b); ------------------------- - 2 -clearrules trig1; -cos(a)*cos(b); COS(A)*COS(B) - -\endsection -\item[COEFF] -COEFF (page 115) - -The COEFF operator returns the coefficients of the powers of the -specified variable in the given expression, in a list. - - COEFF(expression,variable) - -expression is expected to be a polynomial expression, not a rational -expression. Rational expressions are accepted when the switch RATARG -is on. variable must be a kernel. The results are returned in a -list. - -Examples: - 3 2 -coeff((x+y)**3,x); {Y ,3*Y ,3*Y,1} -coeff((x+2)**4 + sin(x),x); {SIN(X) + 16,32,24,8,1} -high_pow; 4 -low_pow; 0 - 7 9 -ab := x**9 + sin(x)*x**7 + sqrt(y); AB := SQRT(Y) + SIN(X)*X + X -coeff(ab,x); {SQRT(Y),0,0,0,0,0,0,SIN(X),0,1} - -The variables HIGH_POW and LOW_POW are set to the highest and lowest -powers of the variable, respectively, appearing in the expression. - -The coefficients are put into a list, with the coefficient of the -lowest (constant) term first. You can use the usual list access -methods (first, second, third, rest, length, and part) to extract -them. If a power does not appear in the expression, the corresponding -element of the list is zero. Terms involving functions of the -specified variable but not including powers of it (for example in the -expression x**4 + 3*x**2 + tan(x)) are placed in the constant term. - -Since the COEFF command deals with the expanded form of the -expression, you may get unexpected results when EXP is off, or when -FACTOR or IFACTOR are on. - -If you want only a specific coefficient rather than all of them, use the -COEFFN operator. - -\endsection -\item[Coefficient] -Coefficient (pages 132, 134) - -REDUCE allows for a variety of numerical domains for the numerical -coefficients of polynomials used in calculations. The default mode is -integer arithmetic, although the possibility of using real -coefficients has been discussed elsewhere. Rational coefficients have -also been available by using integer coefficients in both the -numerator and denominator of an expression, using the ON DIV option to -print the coefficients as rationals. However, REDUCE includes several -other coefficient options in its basic version. - -See ADJPREC, BFSPACE, COMPLEX, MODULAR, PRECISION, PRINT_PRECISION, -RATIONAL, RATIONALIZE, ROUNDALL, ROUNDBF, ROUNDED and SETMOD. - -\endsection -\item[COEFFN] -COEFFN (page 116) - -The COEFFN operator takes three arguments: an expression, a kernel, -and a non-negative integer. It returns the coefficient of the kernel -to that integer power, appearing in the expression. - - COEFFN(expression,kernel,integer) - -expression must be a polynomial, unless RATARG is on which allows -rational expressions. kernel must be a Kernel, and integer must be a -non-negative integer. - -Examples: - -ff := x**7 + sin(y)*x**5 + y**4 + x + 7$ -coeffn(ff,x,5); SIN(Y) -coeffn(ff,z,3); 0 - 5 7 -coeffn(ff,y,0); SIN(Y)*X + X + X + 7 - -rr := 1/y**2+y**3+sin(y); -on ratarg; - -coeffn(rr,y,-2); ***** -2 invalid as COEFFN index - -coeffn(rr,y,5); 1 - ---- - 2 - y - -If the given power of the kernel does not appear in the expression, -COEFFN returns 0. Negative powers are never detected, even if they -appear in the expression and RATARG are on. COEFFN with an integer -argument of 0 returns any terms in the expression that do not contain -the given kernel. - -\endsection -\item[COFACTOR] -COFACTOR (page 166) - -The operator COFACTOR returns the cofactor of the element in row -row and column column of a MATRIX. Errors occur -if row or column do not evaluate to integer expressions or if -the matrix is not square. - - COFACTOR(matrix_expression,row,column) - -Examples: -cofactor(mat((a,b,c),(d,e,f),(p,q,r)),2,2); A*R - C*P -cofactor(mat((a,b,c),(d,e,f)),1,1); ***** non-square matrix - -\endsection -\xitem[COFRAME] -COFRAME (pages 257, 262, 271) - WITH METRIC (page 263 - WITH SIGNATURE (page 263 - -\endsection -\item[COLLECT] -COLLECT (page 57) - -COLLECT is a key word of the FOR construction. Details are given there. - -\endsection -\item[COMBINEEXPT] -COMBINEEXPT (page 77) - -REDUCE is in general poor at surd simplification. However, when the -switch COMBINEEXPT is on, the system attempts to combine -exponentials whenever possible. - -Example: 1/3 1/6 -3^(1/2)*3^(1/3)*3^(1/6); SQRT(3)*3 *3 -on combineexpt; -ws; 1 - -\endsection -\item[COMBINELOGS] -COMBINELOGS (page 77) - -In many cases it is desirable to expand product arguments of logarithms, -or collect a sum of logarithms into a single logarithm. Since these are -inverse operations, it is not possible to provide rules for doing both at -the same time and preserve the REDUCE concept of idempotent evaluation. -As an alternative, REDUCE provides two switches EXPANDLOGS and -COMBINELOGS to carry out these operations. -Examples: - on expandlogs; - log(x*y); LOG(X) + LOG(Y) - on combinelogs; - ws; LOG(X*Y) - -At the present time, it is possible to have both switches on at once, -which could lead to infinite recursion. However, an expression is -switched from one form to the other in this case. Users should not -rely on this behaviour, since it may change in the next release. - -\endsection -\xitem[COMM] -COMM (page 378 - -\endsection -\xitem[Command] -Command (page 67 - -\endsection -\xitem[Command terminator] -Command terminator (page 153 - -\endsection -\item[COMMENT] -COMMENT (page 38) - -Beginning with the word COMMENT, all text until the next statement -terminator (; or $) is ignored. - -Examples: - 2 -x := a**2 comment--a is the velocity of the particle;; X := A - -Note that the first semicolon ends the comment and the second one -terminates the original REDUCE statement. - -Multiple-line comments are often needed in interactive files. The -COMMENT command allows a normal-looking text to accompany the REDUCE -statements in the file. - -\endsection -\item[COMP] -COMP (page 213) - -(Not available in Personal REDUCE} - -When COMP is on, any succeeding function definitions are compiled -into a faster-running form. Default is OFF. - -Examples: -The following procedure finds Fibonacci numbers recursively. -Create a new file ``refib'' in your current directory with the following -lines in it: - -procedure refib(n); - if fixp n and n >= 0 then - if n <= 1 then 1 - else refib(n-1) + refib(n-2) - else rederr "nonnegative integer only"; - -end; - -{Now load REDUCE and run the following:} - -on time; Time: 100 ms - -in "refib"$ Time: 0 ms - - REFIB - - Time: 260 ms - - Time: 20 ms - -refib(80); 37889062373143906 - - Time: 14840 ms - -on comp; Time: 80 ms - -in "refib"$ Time: 20 ms - - REFIB - - Time: 640 ms - -refib(80); 37889062373143906 - - Time: 10940 ms - -Note that the compiled procedure runs faster. Your time messages will -differ depending upon which system you have. Compiled functions -remain so for the duration of the REDUCE session, and are then lost. -They must be recompiled if wanted in another session. With the switch -TIME on as shown above, the CPU time used in executing the command is -returned in milliseconds. Be careful not to leave COMP on unless you -want it, as it makes the processing of procedures much slower. - -\endsection -\item[COMPACT] -COMPACT (pages 179, 241) - -Author: Anthony C. Hearn - -COMPACT is a package of functions for the reduction of a polynomial in -the presence of side relations. COMPACT applies the side relations to -the polynomial so that an equivalent expression results with as few -terms as possible. For example, the evaluation of - - compact(s*(1-sin x^2)+c*(1-cos x^2)+sin x^2+cos x^2, - {cos x^2+sin x^2=1}); - -yields the result - - 2 2 - SIN(X) *C + COS(X) *S + 1 - -\endsection -\item[Compiler] -Compiler (page 213) - -A compiler is available in the Professional REDUCE to convert -functions into a compiled form for faster execution. See the switch -COMP for more details. - -\endsection -\item[COMPLEX] -COMPLEX (pages 135, 372) - -When the COMPLEX switch is on, full complex arithmetic is used in -simplification, function evaluation, and factorisation. Default is OFF. - -Examples: - 2 2 -factorize(a**2 + b**2); {A + B } -on complex; -factorize(a**2 + b**2); {A - I*B,A + I*B} -(x**2 + y**2)/(x + i*y); X - I*Y -on rounded; *** Domain mode COMPLEX changed to COMPLEX_FLOAT -sqrt(-17); 4.12310562562*I -log(7*i); 1.94591014906 + 1.57079632679*I - -Complex floating-point can be done by turning on ROUNDED in addition -to COMPLEX. With COMPLEX off however, REDUCE knows that i is the -square root of -1 but will not carry out more complicated complex -operations. If you want complex denominators cleared by -multiplication by their conjugates, turn on the switch RATIONALIZE. - -\endsection -\xitem[Compound statement] -Compound statement (pages 61, 63 - -\endsection -\xitem[Conditional statement] -Conditional statement (page 56) - -\endsection -\item[CONJ] -CONJ (page 72) - - CONJ(expression) or CONJ simple_expression - -This operator returns the complex conjugate of an expression, if that -argument has an numerical value. A non-numerical argument is returned as -an expression in the operators REPART and IMPART. - -Examples: -conj(1+i); 1-I -conj(a+i*b); REPART(A) - REPART(B)*I - IMPART(A)*I - IMPART(B) - -\endsection -\xitem[Constructor] -Constructor (page 198) - -\endsection -\item[CONT] -CONT (page 160) - -The command CONT returns control to an interactive file after a -PAUSE command that has been answered with N. - -Examples: -Suppose you are in the middle of an interactive file. - factorize(x**2 + 17*x + 60); {X + 5,X + 12} - pause; Cont? (Y or N) - n - saveas results; - factor1 := first results; FACTOR1 := X + 5 - factor2 := second results; FACTOR2 := X + 12 - cont; -....the file resumes - -A PAUSE allows you to enter your own REDUCE commands, change switch -values, inquire about results, or other such activities. When you -wish to resume operation of the interactive file, use CONT. - -\endsection -\xitem[COORDINATES operator] -COORDINATES operator (page 234) - -\endsection -\xitem[COORDS vector] -COORDS vector (page 234) - -\endsection -\xitem[CORFACTOR] -CORFACTOR (page 350) - -\endsection -\item[COS] -COS (pages 76, 78) - -The COS operator returns the cosine of its argument. - - COS(expression) or COS simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -cos abc; COS(ABC) -cos(pi); -1 -cos 4; COS(4) -on rounded; -cos(4); - 0.653643620864 -cos log 5; - 0.0386319699339 - -COS returns a numeric value only if ROUNDED is on. Then the cosine is -calculated to the current degree of floating point precision. - -\endsection -\item[COSH] -COSH (pages 76, 78) - -The COSH operator returns the hyperbolic cosine of its argument. The -derivative of COSH and some simple transformations are known to the -system. - - COSH(expression) or COSH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -cosh b; COSH(B) -cosh(0); 1 -df(cosh(x*y),x); SINH(X*Y)*Y -int(cosh(x),x); SINH(X) - -You may attach further functionality by defining its inverse (see -ACOSH). A numeric value is not returned by COSH unless the switch -ROUNDED is on and its argument evaluates to a number. - -\endsection -\item[COT] -COT (pages 76, 78) - -COT represents the cotangent of its argument. It takes an arbitrary -scalar expression as its argument. The derivative of ACOT and some -simple properties are known to the system. - - COT(expression) or COT simple_expression - -expression may be any scalar REDUCE expression. simple_expression -must be a single identifier or begin with a prefix operator name. - -Examples: -cot(a)*tan(a); COT(A)*TAN(A)) -cot(1); COT(1) - 2 -df(cot(2*x),x); - 2*(COT(2*X) + 1) - -Numerical values of expressions involving COT may be found by -turning on the switch ROUNDED. - -\endsection -\item[COTH] -COTH (pages 76, 78) - -The COTH operator returns the hyperbolic cotangent of its argument. -The derivative of COTH and some simple transformations are known to -the system. - - COTH(expression) or COTH simple_expression - -expression may be any scalar REDUCE expression. simple_expression -must be a single identifier or begin with a prefix operator name. - -Examples: - 2 -df(coth(x*y),x); - Y*(COTH(X*Y) - 1) -coth acoth z; Z - -You can write LET statements and procedures to add further -functionality to COTH if you wish. Numerical values of expressions -involving COTH may also be found by turning on the switch ROUNDED. - -\endsection -\item[CRAMER] -CRAMER (pages 85, 163) - -When the CRAMER switch is on, MATRIX inversion and linear equation -solving (operator SOLVE) is done by Cramer's rule, through exterior -multiplication. Default is OFF. - -Examples: -on time; Time: 80 ms -off output; Time: 100 ms -mm := mat((a,b,c,d,f),(a,a,c,f,b), - (b,c,a,c,d), (c,c,a,b,f), - (d,a,d,e,f)); - Time: 300 ms -inverse := 1/mm; Time: 18460 -on cramer; Time: 80 ms -cramersinv := 1/mm; Time: 9260 MS - -Your time readings will vary depending on the REDUCE version you use. -After you invert the matrix, turn on OUTPUT and ask for one of the -elements of the inverse matrix, such as CRAMERSINV(3,2), so that you -can see the size of the expressions produced. - -Inversion of matrices and the solution of linear equations with dense -symbolic entries in many variables is generally considerably faster -with CRAMER on. However, inversion of numeric-valued matrices is -slower. Consider the matrices you're inverting before deciding -whether to turn CRAMER on or off. A substantial portion of the time -in matrix inversion is given to formatting the results for printing. -To save this time, turn OUTPUT off, as shown in this example or -terminate the expression with a dollar sign instead of a semicolon. -The results are still available to you in the workspace associated -with your prompt number, or you can assign them to an identifier for -further use. - -\endsection -\item[CREF] -CREF (pages 215, 216) - -The switch CREF invokes the CREF cross-reference program that -processes a set of procedure definitions to produce a summary of their -entry points, undefined procedures, non-local variables and so on. The -program will also check that procedures are called with a consistent -number of arguments, and print a diagnostic message otherwise. - -The output is alphabetised on the first seven characters of each function -name. - -To invoke the cross-reference program, CREF is first turned on. -This causes the program to load and the cross-referencing process to -begin. After all the required definitions are loaded, turning CREF -off will cause a cross-reference listing to be produced. - - - -Algebraic procedures in REDUCE are treated as if they were symbolic, so -that algebraic constructs will actually appear as calls to symbolic -functions, such as AEVAL. - -\endsection -\xitem[CRESYS] -CRESYS (pages 378, 380) - -\endsection -\xitem[CROSS] -CROSS - vector (page 233) - -\endsection -\xitem[cross product] -cross product (pages 233, 357) - -\endsection -\xitem[Cross reference] -Cross reference (page 215) - -\endsection -\item[CSC] -CSC (pages 76, 78) - -The CSC operator returns the cosecant of its argument. The derivative -of CSC and some simple transformations are known to the system. - - CSC(expression) or CSC simple_expression - -expression may be any scalar REDUCE expression. simple_expression -must be a single identifier or begin with a prefix operator name. - -Examples: -csc(q)*sin(q); CSC(Q)*SIN(Q) -df(csc(x*y),x); -COT(X*Y)*CSC(X*Y)*Y - - -You can write LET statements and procedures to add further -functionality to CSC if you wish. Numerical values of expressions -involving CSC may also be found by turning on the switch ROUNDED. - -\endsection -\item[CSCH] -CSCH (pages 76, 78) - -The COSH operator returns the hyperbolic cosecant of its argument. -The derivative of CSCH and some simple transformations are known to -the system. - - CSCH(expression) or CSCH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -csch b; CSCH(B) -csch(0); 0 -df(csch(x*y),x); - COTH(X*Y)*CSCH(X*Y)*Y -int(csch(x),x); INT(CSCH(X),X) - -A numeric value is not returned by CSCH unless the switch ROUNDED is -on and its argument evaluates to a number. - -\endsection -\xitem[CURL operator] -CURL operator (page 234) - -\endsection -\xitem[curl vector field] -curl vector field (page 234) - -\endsection -\xitem[curl operator] -curl operator (page 358) - -\endsection -\xitem[cylindrical coordinates] -cylindrical coordinates (page 355) - -\endsection -\xitem[d exterior differentiation] -d - exterior differentiation (page 271) - -\endsection -\xitem[Declaration] -Declaration (page 67) - -\endsection -\item[DECOMPOSE] -DECOMPOSE (page 127) - -The DECOMPOSE operator takes a multivariate polynomial as argument, -and returns an expression and a LIST of EQUATIONs from which the -original polynomial can be found by composition. - - DECOMPOSE(expression) or DECOMPOSE simple_expression - -Examples: -decompose(x^8-88*x^7+2924*x^6-43912*x^5+263431*x^4- - 218900*x^3+65690*x^2-7700*x+234) - 2 - {U + 35*U + 234, - - 2 - U=V + 10*V, - - 2 - V=X - 22*X } - - 2 -decompose(u^2+v^2+2u*v+1); {W + 1,W=U + V} - -Unlike factorisation, this decomposition is not unique. Further -details can be found in V.S. Alagar, M.Tanh, Fast Polynomial -Decomposition, Proc. EUROCAL 1985, pp 150-153 (Springer) and J. von -zur Gathen, Functional Decomposition of Polynomials: the Tame Case, J. -Symbolic Computation (1990) 9, 281-299. - -\endsection -\item[DEFINE] -DEFINE (page 70) - -The command DEFINE allows you to supply a new name for an identifier -or replace it by any valid REDUCE expression. - - DEFINE identifier = substitution {,identifier = substitution} - -identifier is any valid REDUCE identifier, substitution can be a -number, an identifier, an operator, a reserved word, or an expression. - -Examples: - -define is= :=, xx=y+z; -a is 10; A := 10 - 2 2 -xx**2; Y + 2*Y*Z + Z - -xx := 10; Y + Z := 10 - -The renaming is done at the input level, and therefore takes precedence -over any other replacement or substitution declared for the same identifier. -It remains in effect until the end of the REDUCE session. Be careful with -it, since you cannot easily undo it without ending the session. - -\endsection -\xitem[definite integration (simple)] -definite integration (simple) (page 236) - -\endsection -\xitem[DEFINT function] -DEFINT function (page 236) - -\endsection -\xitem[DEFLINEINT function] -DEFLINEINT function (page 238) - -\endsection -\item[DEFN] -DEFN (pages 197, 218) - -When the switch DEFN is on, the Standard Lisp equivalent of the -input statement or procedure is printed, but not evaluated. Default is -OFF. - -Examples: - -on defn; -17/3; (AEVAL (LIST 'QUOTIENT 17 3)) - -df(sin(x),x,2); (AEVAL (LIST 'DF (LIST 'SIN 'X) 'X 2)) -procedure coshval(a); - begin scalar g; - g := (exp(a) + exp(-a))/2; - return g - end; (AEVAL - (PROGN - (FLAG '(COSHVAL) 'OPFN) - (DE COSHVAL (A) - (PROG (G) - (SETQ G - (AEVAL - (LIST - 'QUOTIENT - (LIST - 'PLUS - (LIST 'EXP A) - (LIST 'EXP (LIST 'MINUS A))) - 2))) - (RETURN G)))) ) -coshval(1); (AEVAL (LIST 'COSHVAL 1)) -off defn; -coshval(1); Declare COSHVAL operator? (Y or N) -n -procedure coshval(a); - begin scalar g; - g := (exp(a) + exp(-a))/2; - return g - end; COSHVAL -on rounded; -coshval(1); 1.54308063482 - -The above function COSHVAL finds the hyperbolic cosine (cosh) of its -argument. When DEFN is on, you can see the Standard Lisp equivalent -of the function, but it is not entered into the system as shown by the -message DECLARE COSHVAL OPERATOR?. It must be reentered with DEFN off -to be recognised. This procedure is used as an example; a more -efficient procedure would eliminate the unnecessary local variable -with - procedure coshval(a); - (exp(a) + exp(-a))/2; - -\endsection -\item[DEFPOLY] -DEFPOLY statement (page 225) - -DEFPOLY is used to introduce a defining polynoimial for an algebraic -number. For example, to define an atom to stand for teh square root -of 2 one would say - - load arnum; - defpoly sqrt2^2 -2; - -This associates a simplification function for the variable and also -generates a power reduction rule used by the operations * and / for -the reduction of their result modulo the defining polynomial. A basis -for the representation of an algebraic number is also set up by the -statement. If the defining polynomial is not monic, it will be made -so by an appropriate substitution. - -\endsection -\item[DEG] -DEG (page 128) - -The operator DEG returns the highest degree of its variable argument -found in its expression argument. - - DEG(expression,kernel) - -expression is expected to be a polynomial expression, not a rational -expression. Rational expressions are accepted when the switch RATARG -is on. variable must be a Kernel. The results are returned in a -list. - -Examples: -deg((x+y)**5,x); 5 -deg((a+b)*(c+2*d)**2,d); 2 -deg(x**2 + cos(y),sin(x)); -deg((x**2 + sin(x))**5,sin(x)); 5 - -\endsection -\xitem[Degree] -Degree (page 128) - -\endsection -\xitem[DELSQ operator] -DELSQ - operator (page 234) - -\endsection -\xitem[delsq operator] -delsq operator (page 358) - -\endsection -\item[DEMO] -DEMO (page 69) - -The DEMO switch is used for interactive files, causing the system -to pause after each command in the file until you type a Return. -Default is OFF. - -The switch DEMO has no effect on top level interactive statements. -Use it when you want to slow down operations in a file so you can see -what is happening. - -You can either include the ON DEMO command in the file, or enter it -from the top level before bringing in any file. Unlike the PAUSE -command, ON DEMO does not permit you to interrupt the file for -questions of your own. - -\endsection -\item[DEN] -DEN (pages 120, 129) - -The DEN operator returns the denominator of its argument. - - DEN(expression) - -expression is ordinarily a rational expression, but may be any valid -scalar REDUCE expression. - -Examples: - 2 -a := x**3 + 3*x**2 + 12*x; A := X*(X + 3*X + 12) -b := 4*x*y + x*sin(x); B := X*(SIN(X) + 4*Y) -den(a/b); SIN(X) + 4*Y -den(aa/4 + bb/5); 20 -den(100/6); 3 -den(sin(x)); 1 - -DEN returns the denominator of the expression after it has been -simplified by REDUCE. As seen in the examples, this includes putting -sums of rational expressions over a common denominator, and reducing -common factors where possible. If the expression does not have any -other denominator, 1 is returned. - -Switch settings, such as MCD or RATIONAL, have an effect on the -denominator of an expression. - -\endsection -\item[DEPEND] -DEPEND (page 95) - -DEPEND declares that its first argument depends on the rest of its -arguments. - - DEPEND kernel{,kernel} - -kernel must be a legal variable name or a prefix operator (see -Kernel). - -Examples: - -depend y,x; -df(y**2,x); 2*DF(Y,X)*Y -depend z,cos(x),y; -df(sin(z),cos(x)); COS(Z)*DF(Z,COS(X)) -df(z**2,x); 2*DF(Z,X)*Z -nodepend z,y; -df(z**2,x); 2*DF(Z,X)*Z -cc := df(y**2,x); CC := 2*DF(Y,X)*Y -y := tan x; Y := TAN(X); - 2 -cc; 2*TAN(X)*(TAN(X) + 1) - -Dependencies can be removed by using the declaration NODEPEND. The -differentiation operator uses this information, as shown in the -examples above. Linear operators also use knowledge of dependencies -(see LINEAR). Note that dependencies can be nested: Having declared y -to depend on x, and z to depend on y, we see that the chain rule was -applied to the derivative of a function of z with respect to x. If -the explicit function of the dependency is later entered into the -system, terms with DF(Y,X), for example, are expanded when they are -displayed again, as shown in the last example. - -\endsection -\xitem[DEPEND statement] -DEPEND statement (page 359) - -\endsection -\xitem[DEQ(I)] -DEQ(I) (page 379) - -\endsection -\xitem[derivative variational] -derivative - variational (page 257) - -\endsection -\item[DET] -DET (pages 97, 163) - -The operator COFACTOR returns the cofactor of the element in row -row and column column of a MATRIX. Errors occur -if row or column do not evaluate to integer expressions or if -the matrix is not square. - - COFACTOR(matrix_expression,row,column) - -Examples: -cofactor(mat((a,b,c),(d,e,f),(p,q,r)),2,2); A*R - C*P -cofactor(mat((a,b,c),(d,e,f)),1,1); ***** non-square matrix - -\endsection -\xitem[determinant] -determinant - in DETM!* (page 263) - -\endsection -\xitem[DETM!*] -DETM!* (page 263) - -\endsection -\item[DF] -DF (pages 79, 80) - -The DF operator finds partial derivatives with respect to one or -more variables. - - DF(expression,var - [,number] - {,var [ ,number] } ) - -expression can be any valid REDUCE algebraic expression. var must be -a Kernel, and is the differentiation variable. number must be a -non-negative integer. - -Examples: -df(x**2,x); 2*X - 2 -df(x**2*y + sin(y),y); COS(Y) + X - -df((x+y)**10,z); 0 - 6 -df(1/x**2,x,2); ---- - 4 - X -df(x**4*y + sin(y),y,x,3); 24*X - -for all x let df(tan(x),x) = sec(x)**2; - 2 -df(tan(3*x),x); 3*SEC(3*X) - -An error message results if a non-kernel is entered as a -differentiation operator. If the optional number is omitted, it is -assumed to be 1. See the declaration DEPEND to establish dependencies -for implicit differentiation. - -You can define your own differentiation rules, expanding REDUCE's -capabilities, using the LET command as shown in the last example -above. Note that once you add your own rule for differentiating a -function, it supersedes REDUCE's normal handling of that function for -the duration of the REDUCE session. If you clear the rule -(CLEARRULES), you don't get back to the previous rule. - -\endsection -\item[DFPRINT] -DFPRINT - -When DFPRINT is on, expressions in the differentiation operator -DF are printed in a more ``natural'' notation, with the -differentiation variables appearing as subscripts. In addition, if the -switch NOARG is on (the default), the arguments of the -differentiated operator are suppressed. - -Examples: -operator f; -df(f x,x); DF(F(X),X); -on dfprint; -ws; F - X -df(f(x,y),x,y); F - X,Y -off noarg; -ws; F(X,Y) - X - -\endsection -\xitem[differential geometry] -differential geometry (page 248) - -\endsection -\xitem[Differentiation] -Differentiation (pages 79, 80, 95) - -\endsection -\xitem[differentiation] -differentiation - partial (page 251) - vector (page 233) - -\endsection -\item[DIGAMMA] -DIGAMMA (page 185, 395) - -See PSI -\endsection -\item[DILOG] -DILOG (pages 76, 81, 185) - -The DILOG operator is known to the differentiation and integration -operators, but has numeric value attached only at DILOG(0). DILOG is -defined by - log(x) - dilog(x) = -int ------ dx - x-1 - - dilog(x) = -int(log(x),x)/(x-1) - -Examples: 2 2 -df(dilog(x**2),x); - (2*LOG(X )*X)/(X - 1) - -int(dilog(x),x); DILOG(X)*X - DILOG(X) + LOG(X)*X - X - 2 -dilog(0); PI /6 - -\endsection -\xitem[dimension] -dimension (page 251) - -\endsection -\xitem[Dirac gamma matrix] -Dirac gamma matrix (page 206) - -\endsection -\item[DISPLAY] -DISPLAY (page 158)) - -When given a numeric argument n, DISPLAY prints the n most recent -input statements, identified by prompt numbers. If an empty pair of -parentheses is given, or if n is greater than the current number of -statements, all the input statements since the beginning of the -session are printed. - - DISPLAY(n) or DISPLAY() - -n should be a positive integer. However, if it is a real number, the -truncated integer value is used, and if a non-numeric argument is -used, all the input statements are printed. - -The statements are displayed in upper case, with lines split at -semicolons or dollar signs, as they are in editing. If long files -have been input during the session, the DISPLAY command is slow to -format these for printing. - -\endsection -\xitem[Display] -Display (page 97) - -\endsection -\xitem[DISPLAYFRAME command] -DISPLAYFRAME command (pages 266, 271) - -\endsection -\xitem[Displaying structure] -Displaying structure (page 112) - -\endsection -\item[DIV] -DIV (pages 103, 132) - -When DIV is on, the system divides any simple factors found in the -denominator of an expression into the numerator. Default is OFF. - -Examples: - -on div; - 2 -2 -a := x**2/y**2; A := X *Y - 1 2 -2 -1 -b := a/(3*z); B := ---*X *Y *Z - 3 -off div; - 2 - X -a; ---- - 2 - Y - - 2 - X -b; -------- - 2 - 3*Y *Z - -The DIV switch only has effect when the PRI switch is on. When PRI is -off, regardless of the setting of DIV, the printing behaviour is as if -DIV were off. - -\endsection -\xitem[DIV operator] -DIV - operator (page 234) - -\endsection -\xitem[div operator] -div operator (page 358) - -\endsection -\xitem[divergence vector field] -divergence - vector field (page 234) - -\endsection -\xitem[DLINEINT] -DLINEINT (page 360) - -\endsection -\xitem[DO] -DO (pages 57--59) - -\endsection -\xitem[Dollar sign] -Dollar sign (page 53) - -\endsection -\item[DOT] -DOT product of vectors (pages 205, 233, 357) - -The . operator is used to denote the scalar product of two Lorentz -four-vectors. - vector . vector - -vector must be an identifier declared to be of type VECTOR to have -the scalar product definition. When applied to arguments that are not -vectors, the CONS operator is used, -whose symbol is also ``dot.'' - -Examples: -vector aa,bb,cc; -let aa.bb = 0; -aa.bb; 0 -aa.cc; AA.CC -q := aa.cc; Q := AA.CC -q; AA.CC - -Since vectors are special high-energy physics entities that do not -contain values, the . product will not return a true scalar product. -You can assign a scalar identifier to the result of a . operation, or -assign a . operation to have the value of the scalar you supply, as -shown above. Note that the result of a . operation is a scalar, not a -vector. - -The metric tensor g(u,v) can be represented by U.V. If contraction -over the indices is required, U and V should be declared to be of type -INDEX. - -The dot operator has the highest precedence of the infix operators, so -expressions involving . and other operators have the scalar product -evaluated first before other operations are done. - -\endsection -\xitem[Dot product] -Dot product (pages 205, 233, 357) - -\endsection -\xitem[DOTGRAD operator] -DOTGRAD operator (page 358) - -\endsection -\xitem[DVINT] -DVINT (page 360) - -\endsection -\xitem[DVOLINT] -DVOLINT (page 360) - -\endsection -\item[E] -E (page 36) - -The constant E is reserved for use as the base of the natural -logarithm. Its value is approximately 2.71828284590, which REDUCE gives -to the current decimal precision when the switch ROUNDED is on. - - -E may be used as an iterative variable in a FOR statement, -or as a local variable or a PROCEDURE. If E is defined as a local -variable inside the procedure, the normal definition as the base of the -natural logarithm would be suspended inside the procedure. - -\endsection -\item[ECHO] -ECHO (page 153) - -The ECHO switch is normally off for top-level entry, and on when files -are brought in. If ECHO is turned on at the top level, your input -statements are echoed to the screen (thus appearing twice). Default -OFF (but note default ON for files). - - -If you want to display certain portions of a file and not others, use the -commands OFF ECHO and ON ECHO inside the file. If you want -no display of the file, use the input command - - IN filename$ - -rather than using the semicolon delimiter. - -Be careful when you use commands within a file to generate another file. -Since ECHO is on for files, the output file echoes input statements -(unlike its behaviour from the top level). You should explicitly turn off -ECHO when writing output, and turn it back on when you're done. - -\endsection -\item[ED] -ED (pages 157, 158) - -The ED command invokes a simple line editor for REDUCE input -statements. - - ED integer or ED - -ED called with no argument edits the last input statement. If integer -is greater than or equal to the current line number, an error message -is printed. Reenter a proper ED command or return to the top level -with a semicolon. - -The editor formats REDUCE's version of the desired input statement, -dividing it into lines at semicolons and dollar signs. The statement -is printed at the beginning of the edit session. The editor works on -one line at a time, and has a pointer (shown by ^) to the current -character of that line. When the session begins, the pointer is at -the left hand side of the first line. The editing prompt is >. - -The following commands are available. They may be entered in either -upper or lower case. All commands are activated by the carriage -return, which also prints out the current line after changes. Several -commands can be placed on a single line, except that commands -terminated by an Cntrl-G must be the last command before the carriage -return. - -b -Move pointer to beginning of current line. - -ddigit -Delete current character and next (digit-1) characters. An error -message is printed if anything other than a single digit follows d. -If there are fewer than digit characters left on the line, all but the -final dollar sign or semicolon is removed. To delete a line -completely, use the k command. - -e -End the current session, causing the edited expression to be reparsed by -REDUCE. - -fchar -Find the next occurrence of the character char to the right of the -pointer on the current line and move the pointer to it. If the -character is not found, an error message is printed and the pointer -remains in its original position. Other lines are not searched. The -f command is not case-sensitive. - -istring{Cntrl-G} -Insert string in front of pointer. The Cntrl-G key is your delimiter for -the input string. No other command may follow this one on the same -line. - -k -Kill rest of the current line, including the semicolon or dollar sign -terminator. If there are characters remaining on the current line, and it -is the last line of the input statement, a semicolon is added to the line -as a terminator for REDUCE. If the current line is now empty, one of the -following actions is performed: If there is a following line, it becomes -the current line and the pointer is placed at its first character. If the -current line was the final line of the statement, and there is a previous -line, the previous line becomes the current line. If the current line was -the only line of the statement, and it is empty, a single semicolon is -inserted for REDUCE to parse. - -l -Finish editing this line and move to the last previous line. An error message -is printed if there is no previous line. - -n -Finish editing this line and move to the next line. An error message is -printed if there is no next line. - -p -Print out all the lines of the statement. Then a dotted line is printed, and -the current line is reprinted, with the pointer under it. - -q -Quit the editing session without saving the changes. If a semicolon is -entered after q, a new line prompt is given, otherwise REDUCE prompts you -for another command. Whatever you type in to the prompt appearing after -the q is entered is stored as the input for the line number in which you -called the edit. Thus if you enter a semicolon, neither INPUT -ED will find anything under the current number. - -rchar -Replace the character at the pointer by char. - -sstring{Cntrl-G} -Search for the first occurrence of string to the right of the -pointer on the current line and move the pointer to its first character. -The Cntrl-G key is your delimiter for the input string. The s function -does not search other lines of the statement. If the string is not found, -an error message is printed and the pointer remains in its original -position. The s command is not case-sensitive. No other command may -follow this one on the same line. - -x or space -Move the pointer one character to the right. If the pointer is already at -the end of the line, an error message is printed. - -- (minus) -Move the pointer one character to the left. If the pointer is already at the -beginning of the line, an error message is printed. - -? -Display the Help menu, showing the commands and their actions. - -Examples: -(Line numbers are shown in the following examples) - 2 -2: x**2 + y; X + Y -3: ed 2; - X**2 + Y; - ^ -For help, type '?' -?- {(Enter three spaces and Return})} - X**2 + Y; - ^ -?- r5 - X**5 + Y; - ^ -?- fY - X**5 + Y; - ^ -?- iabc{(Terminate with Cntrl-G and Return)} - X**5 + abcY; - ^ -?- ---- - X**5 + abcY; - ^ -?- fbd2 - X**5 + aY; - ^ -?- b - X**5 + aY; - ^ 5 -?- e AY + X -4: procedure dumb(a); - write a; -DUMB -5: dumb(17); 17 -6: ed 4; - PROCEDURE DUMB (A); - ^ -WRITE A; -?- fArBn - WRITE A; - ^ -?- ibegin scalar a; a := b + 10;{space Cntrl-G and Return} - begin scalar a; a := b + 10; WRITE A; -?- f;i end {Cntrl-G Return} - begin scalar b; b := a + 10; WRITE A end; - ^ -?- p - PROCEDURE DUMB (B); - begin scalar b; b := a + 10; WRITE A end; - - - - - - - - - - - - begin scalar b; b := a + 10; WRITE A end; - ^ -?- e DUMB -7: dumb(17); 27 -8: - -Note that REDUCE reparsed the procedure DUMB and updated the -definition. - -Since REDUCE divides the expression to be edited into lines at -semicolons or dollar sign terminators, some lines may occupy more than -one line of screen space. If the pointer is directly beneath the last -line of text, it refers to the top line of text. If there is a blank -line between the last line of text and the pointer, it refers to the -second line of text, and likewise for cases of greater than two lines -of text. In other words, the entire REDUCE statement up to the next -terminator is printed, even if it runs to several lines, then the -pointer line is printed. - -You can insert new statements which contain semicolons of their own -into the current line. They are run into the current line where you -placed them until you edit the statement again. REDUCE will -understand the set of statements if the syntax is correct. - -If you leave out needed closing brackets when you exit the editor, a -message is printed allowing you to redo the edit (you can edit the -previous line number and return to where you were). If you leave out -a closing double-quotation mark, an error message is printed, and the -editing must be redone from the original version; the edited version -has been destroyed. Most syntax errors which you inadvertently leave -in an edited statement are caught as usual by the REDUCE parser, and -you will be able to re-edit the statement. - -When the editor processes a previous statement for your editing, -escape characters are removed. Most special characters that you may -use in identifiers are printed in legal fashion, prefixed by the -exclamation point. Be sure to treat the special character and its -escape as a pair in your editing. The characters ( ) # ; ' ` are -different. Since they have special meaning in Lisp, they are -double-escaped in the editor. It is unwise to use these characters -inside identifiers anyway, due to the probability of confusion. - -If you see a Lisp error message during editing, the edit has been -aborted. Enter a semicolon and you will see a new line prompt. - -Since the editor has no dependence on any window system, it can be -used if you are running REDUCE without windows. - -\endsection -\item[EDITDEF] -EDITDEF (page 159) - -The interactive editor ED may be used to edit a user-defined -procedure that has not been compiled. - - EDITDEF(identifier) - -where identifier is the name of the procedure. When EDITDEF is -invoked, the procedure definition will be displayed in editing mode, -and may then be edited and redefined on exiting from the editor using -standard ED commands. - -\endsection -\item[END] -END (page 69) - -The command END has two main uses: - -(i) as the ending of a BEGIN...END BLOCK; and -(ii) to end input from a file. - -In a BEGIN...END BLOCK, there need not be a delimiter (; or $) before -the END, though there must be one after it, or a right bracket -matching an earlier left bracket. - -Files to be read into REDUCE should end with END;, which must be -preceded by a semicolon (usually the last character of the previous -line). The additional semicolon avoids problems with mistakes in the -files. If you have suspended file operation by answering N to a PAUSE -command, you are still, technically speaking, ``in'' the file. Use END -to exit the file. - -An END at the top level of a program is ignored. - -\endsection -\item[EPS] -EPS (pages 207, 267) - -The EPS operator denotes the completely antisymmetric tensor of -order 4 and its contraction with Lorentz four-vectors, as used in -high-energy physics calculations. - - EPS(vector-expr,vector-expr,vector-expr,vector-expr) - -vector-expr must be a valid vector expression, and may be an index. - -Examples: -vector g0,g1,g2,g3; -eps(g1,g0,g2,g3); - EPS(G0,G1,G2,G3); -eps(g1,g2,g0,g3); EPS(G0,G1,G2,G3); -eps(g1,g2,g3,g1); 0 - - -Vector identifiers are ordered alphabetically by REDUCE. When an odd -number of transpositions is required to restore the canonical order to -the four arguments of EPS, the term is ordered and carries a minus -sign. When an even number of transpositions is required, the term is -returned ordered and positive. When one of the arguments is repeated, -the value 0 is returned. A contraction of the form eps(_i j mu nu -p_mu q_nu) is represented by EPS(I,J,P,Q) when I and J have been -declared to be of type INDEX. - -\endsection -\xitem[EPS Levi-Civita tensor] -EPS - Levi-Civita tensor (page 271) - -\endsection -\item[Equation] -Equation (page 47) - -An Equation is an expression where two algebraic expressions -are connected by the (infix) operator EQUAL or by =. -For access to the components of an EQUATION the operators -LHS, RHS or PART can be used. The -evaluation of the left-hand side of an EQUATION is controlled -by the switch EVALLHSEQP, while the right-hand side is -evaluated unconditionally. When an EQUATION is part of a -logical expression, e.g. in a IF or WHILE statement, -the equation is evaluated by subtracting both sides and comparing -the result with zero. - -\endsection -\item[ERF] -ERF (page 81) - -The ERF operator represents the error function, defined by - erf(x) = (2/sqrt(pi))*int(e^(-x^2),x) - -A limited number of its properties are known to the system, including -the fact that it is an odd function. Its derivative is known, and -from this, some integrals may be computed. However, a complete -integration procedure for this operator is not currently included. - -Examples: -erf(0); 0 -erf(-a); - ERF(A) - 4*SQRT(PI)*X -df(erf(x**2),x); -------------- - 4 - X - - 2 - X - E *ERF(X)*PI*X + SQRT(PI) -int(erf(x),x); ---------------------------- - 2 - X - E *PI - -\endsection -\item[ERRCONT] -ERRCONT (page 157) - -When the ERRCONT switch is on, error conditions do not stop file -execution. Error messages will be printed whether ERRCONT is on or off. -Default is OFF. - -The table below shows REDUCE behaviour under the settings of ERRCONT and -INT : - -Behaviour in Case of Error in Files - -errcont int Behaviour when errors in files are encountered - off off Message is printed and parsing continues, but - no further statements are executed; no commands - from keyboard accepted except bye or end - off on Message is printed, and you are asked if you - wish to continue. (This is the default behaviour) - on off Message is printed, and file continues to execute - without pause - on on Message is printed, and file continues to execute - without pause - - -\endsection -\xitem[ETA(ALFA)] -ETA(ALFA) (page 379) - -\endsection -\xitem[euclidean metric] -euclidean metric (page 263) - -\endsection -\item[EULER] -EULER (pages 185, 393) - -The EULER operator returns the nth Euler number. - -EULER(integer) - -Examples: -load_package specfn; (SPECFN) -Euler 20; 370371188237525 -Euler 0; 1 - -The EULER numbers are evaluated by a recursive algorithm which makes -it hard to compute Euler numbers above say 200. - -Euler numbers appear in the coefficients of the power series -representation of 1/cos(z). - -\endsection -\item[EULERP] -Euler Polynomials (page 185) - -The EULERP operator returns the nth Euler Polynomial. - -EULERP(integer,expression) - -Examples: - load_package specfn; (SPECFN) - EulerP(2,xx); XX*(XX - 1) - EulerP(10,3); 2046 - -The Euler numbers are the values of the Euler Polynomials at 1/2 -multiplied by 2**n. - -\endsection -\item[Euler Numbers] -Euler Numbers (pages 185, 393) - -See EULERP. -The Euler numbers are the values of the Euler Polynomials at 1/2 -multiplied by 2**n. - -\endsection -\item[EVAL_MODE] -EVAL_MODE (page 191) - -The constant E is reserved for use as the base of the natural -logarithm. Its value is approximately 2.71828284590, which REDUCE gives -to the current decimal precision when the switch ROUNDED is on. - - -E may be used as an iterative variable in a FOR statement, -or as a local variable or a PROCEDURE. If E is defined as a local -variable inside the procedure, the normal definition as the base of the -natural logarithm would be suspended inside the procedure. - -\endsection -\item[EVALLHSEQP] -EVALLHSEQP (page 47) - -Under normal circumstances, the right-hand-side of an EQUATION is evaluated -but not the left-hand-side. If both sides are to be evaluated, the switch -EVALLHSEQP should be turned on. - -\endsection -\item[EVEN] -EVEN (page 90) - - EVEN identifier{,identifier} - -This declaration is used to declare an operator even in its first -argument. Expressions involving an operator declared in this manner -are transformed if the first argument contains a minus sign. Any -other arguments are not affected. - -Examples: - even f; - f(-a) F(A) - f(-a,-b) F(A,-B) - -\endsection -\xitem[Even operator] -Even operator (page 90) - -\endsection -\item[EVENP] -EVENP (page 46) - -The EVENP logical operator returns TRUE if its argument is an even -integer, and NIL if its argument is an odd integer. An error message -is returned if its argument is not an integer. - - EVENP(integer) or EVENP integer - -integer must evaluate to an integer. - -Examples: -aa := 1782; AA := 1782 -if evenp aa then yes else no; YES -if evenp(-3) then yes else no; NO - -Although you would not ordinarily enter an expression such as the last -example above, note that the negative term must be enclosed in -parentheses to be correctly parsed. The EVENP operator can only be -used in conditional statements such as IF...THEN...ELSE or WHILE...DO. - -\endsection -\item[EXCALC] -EXCALC (pages 180, 247) - -Author: Eberhard Schruefer - -The EXCALC package is designed for easy use by all who are familiar -with the calculus of Modern Differential Geometry. The program is currently -able to handle scalar-valued exterior forms, vectors and operations between -them, as well as non-scalar valued forms (indexed forms). It is thus an ideal -tool for studying differential equations, doing calculations in general -relativity and field theories, or doing simple things such as calculating the -Laplacian of a tensor field for an arbitrary given frame. - -\endsection -\xitem[Exclamation mark] -Exclamation mark (page 33) - -\endsection -\xitem[EXCLUDE] -EXCLUDE (page 368) - -\endsection -\xitem[EXDEGREE] -EXDEGREE (page 271) - -\endsection -\xitem[EXDEGREE command] -EXDEGREE command (page 249) - -\endsection -\item[EXP] -EXP (operator and switch) (pages 76, 78, 81, 120, 124) - -The EXP operator returns E raised to the power of its argument. - - EXP(expression) or EXP simple_expression - -expression can be any valid REDUCE scalar expression. -simple_expression must be a single identifier or begin with a -prefix operator. - -Examples: - SIN X -exp(sin(x)); E - 11 -exp(11); E -on rounded; -exp sin(pi/3); 2.37744267524 - -Numeric values are returned only when ROUNDED is on. The single -letter E with the exponential operator ^ or ** may be substituted for -EXP without change of function. - -EXP switch - -When the EXP switch is on, powers and products of expressions are -expanded. Default is ON. - -Examples: 3 2 -(x+1)**3; X + 3*X + 3*X + 1 -(a + b*i)*(c + d*i); A*C + A*D*I + B*C*I - B*D -off exp; 3 -(x+1)**3; (X + 1) -(a + b*i)*(c + d*i); (A + B*I)*(C + D*I) -length((x+1)**2/(y+1)); 2 - - -Note that REDUCE knows that i^2 = -1. When EXP is off, equivalent -expressions may not simplify to the same form, although zero -expressions still simplify to zero. Several operators that expect a -polynomial argument behave differently when EXP is off, such as -LENGTH. Be cautious about leaving EXP off. - - -\endsection -\item[EXPAND_CASES] -EXPAND_CASES (page 86) - -When a ROOT_OF form in a result of SOLVE has been converted to a -ONE_OF form, EXPAND_CASES can be used to convert this into form -corresponding to the normal explicit results of SOLVE. See ROOT_OF. - -\endsection -\item[EXPANDLOGS] -EXPANDLOGS (page 77) - -In many cases it is desirable to expand product arguments of -logarithms, or collect a sum of logarithms into a single logarithm. -Since these are inverse operations, it is not possible to provide -rules for doing both at the same time and preserve the REDUCE concept -of idempotent evaluation. As an alternative, REDUCE provides two -switches EXPANDLOGS and COMBINELOGS to carry out these operations. -Both are off by default. - -Examples: - on expandlogs; - log(x*y); LOG(X) + LOG(Y) - on combinelogs; - ws; LOG(X*Y) - -At the present time, it is possible to have both switches on at once, -which could lead to infinite recursion. However, an expression is -switched from one form to the other in this case. Users should not rely -on this behaviour, since it may change in the next release. - -\endsection -\item[EXPINT] -EXPINT (page 76) - -The EXPINT operator represents the exponential integral defined by: - - expint(x) = int(e^x,x)/x - -A limited number of its properties are known to the system, including -its derivative. From this, some integrals may be computed. However, -a complete integration procedure for this operator is not currently -included. - -Examples: -expint(0); EXPINT(0) - 2 - X - 2*E -df(expint(x**2),x); ------- - X - X -int(expint(x),x); EXPINT(X)*X - E - -\endsection -\xitem[EXPR] -EXPR (page 196) - -\endsection -\xitem[Expression] -Expression (page 43) - -\endsection -\item[exterior calc] -exterior calculus (page 248) - -See the EXCALC package - -\endsection -\item[exterior df] -exterior differentiation (page 252) - -See the EXCALC package - -\endsection -\xitem[exterior form] -exterior form - declaration (page 249) - vector (page 249) - with indices (pages 249, 259) - -\endsection -\xitem[exterior product] -exterior product (pages 250, 269) - -\endsection -\item[EZGCD] -EZGCD (page 124) - -When EZGCD and GCD are on, greatest common divisors are -computed using the EZ GCD algorithm that uses modular arithmetic (and is -usually faster). Default is OFF. - - -As a side effect of the gcd calculation, the expressions involved are -factored, though not the heavy-duty factoring of FACTORIZE. The -EZ GCD algorithm was introduced in a paper by J. Moses and D.Y.Y. Yun in -Proceedings of the ACM, 1973, pp. 159-166. - -Note that the GCD switch must also be on for EZGCD to have -effect. - -\endsection -\item[FACTOR] -FACTOR (Declaration and Switch) (pages 101, 121, 122) - -When a kernel is declared by FACTOR, all terms involving fixed powers -of that kernel are printed as a product of the fixed powers and the -rest of the terms. - - FACTOR kernel {,kernel} - -kernel must be a Kernel. - -Examples: 2 2 2 -a := (x + y + z)**2; A := X + 2*X*Y + 2*X*Z + Y + 2*Y*Z + Z -factor y; 2 2 2 -a; Y + 2*Y*(X + Z) + X + 2*X*Z + Z -factor sin(x); 4 3 2 -c := df(sin(x)**4*x**2*z,x); C := 2*SIN(X) *X*Z + 4*SIN(X) *COS(X)*X *Z -remfac sin(x); 3 -c; 2*SIN(X) *X*Z*(2*COS(X)*X + SIN(X)) - -Use the FACTOR declaration to display variables of interest so that -you can see their powers more clearly, as shown in the example. -Remove this special treatment with the declaration REMFAC. The FACTOR -declaration is only effective when the switch PRI is on. - -The FACTOR declaration is not a factoring command; to factor -expressions use the FACTOR switch or the FACTORIZE command. - -FACTOR (switch) - -When the FACTOR switch is on, input expressions and results are -automatically factored. - -Examples: - -on factor; -aa := 3*x**3*a + 6*x**2*y*a + 3*x**3*b + 6*x**2*y*b -+ x*y*a + 2*y**2*a + x*y*b + 2*y**2*b; - 2 - AA := (A + B)*(3*X + Y)*(X + 2*Y) -off factor; -aa; - 3 2 2 3 2 2 - 3*A*X + 6*A*X *Y + A*X*Y + 2*A*Y + 3*B*X + 6*B*X *Y + B*X*Y + 2*B*Y -on factor; - 2 -ab := x**2 - 2; AB := X - 2 - -REDUCE factors univariate and multivariate polynomials with integer -coefficients, finding any factors that also have integer coefficients. -The factoring is done by reducing multivariate problems to univariate -ones with symbolic coefficients, and then solving the univariate ones -modulo small primes. The results of these calculations are merged to -determine the factors of the original polynomial. The factoriser -normally selects evaluation points and primes using a random number -generator. Thus, the detailed factoring behaviour may be different -each time any particular problem is tackled. - -When the FACTOR switch is turned on, the EXP switch is turned off, and -when the FACTOR switch is turned off, the EXP switch is turned on, -whether it was on previously or not. - -When the switch TRFAC is on, informative messages are generated at -each call to the factoriser. The TRALLFAC switch causes the -production of a more verbose trace message. It takes precedence over -TRFAC if they are both on. - -To factor a polynomial explicitly and store the results, use the operator -FACTORIZE. - -\endsection -\item[FACTORIAL] -FACTORIAL (pages 72, 174) - -FACTORIAL(expression) - -If the argument of FACTORIAL is a positive integer or zero, its -factorial is returned. Otherwise the result is expressed in terms of -the original operator. For more general operations, the GAMMA -operator is available in the SPECFN package. - -Examples: -factorial 4; 24 -factorial 30 ; 265252859812191058636308480000000 -factorial(a) ; FACTORIAL(A) - -\endsection -\item[Factorization] -Factorization (page 121) - -Operations for factorising expressions exist in REDUCE. See the -operator FACTORIZE and the switch FACTOR. - -The command FACTOR controls output format. - -\endsection -\item[FACTORIZE] -FACTORIZE (pages 121, 122) - -The FACTORIZE operator factors a given expression. - - FACTORIZE(expression) - -expression should be a polynomial, otherwise an error will result. - -Examples: - 2 2 -fff := factorize(x^3 - y^3); FFF := {X - Y,X + X*Y + Y } -fac1 := first fff; FAC1 := X - Y -factorize(x^15 - 1); {X - 1, - - 2 - X + X + 1, - - 4 3 2 - X + X + X + X + 1, - - 8 7 5 4 3 - X - X + X - X + X - X + 1} - - 8 7 5 4 3 -lastone := part(ws,length ws); lastone := x - x + x - x + x - x + 1 -setmod 2; 1 -on modular; -factorize(x^15 - 1); {X + 1, - - 2 - X + X + 1, - - 4 - X + X + 1, - - 4 3 - X + X + 1, - - 4 3 2 - X + X + X + X + 1} - -The FACTORIZE command returns the factors it finds as a LIST. You can -therefore use the usual list access methods (FIRST, SECOND, THIRD, -REST, LENGTH and PART) to extract the factors. - -If the expression given to FACTORIZE is an integer, it will be -factored into its prime components. To factor any integer factor of a -non-numerical expression, the switch IFACTOR should be turned on. Its -default is off. IFACTOR has effect only when factoring is explicitly -done by FACTORIZE, not when factoring is automatically done with the -FACTOR switch. If full factorisation is not needed the switch -LIMITEDFACTORS allows you to reduce the computing time of calls to -FACTORIZE. - -Factoring can be done in a modular domain by calling FACTORIZE when -MODULAR is on. You can set the modulus with the SETMOD command. The -last example above shows factoring modulo 2. - -For general comments on factoring, see comments under the switch -FACTOR. - -\endsection -\item[FAILHARD] -FAILHARD - -When the FAILHARD switch is on, the integration operator INT terminates -with an error message if the integral cannot be done in closed terms. -Default is off. - -Use the FAILHARD switch when you are dealing with complicated integrals -and want to know immediately if REDUCE was unable to handle them. The -integration operator sometimes returns a formal integration form that is -more complicated than the original expression, when it is unable to -complete the integration. - -\endsection -\xitem[Fast loading of code] -Fast loading of code (page 214) - -\endsection -\xitem[FDOMAIN command] -FDOMAIN command (pages 251, 271) - -\endsection -\xitem[FEXPR] -FEXPR (page 196) - -\endsection -\xitem[File handling] -File handling (page 153) - -\endsection -\item[FIRST] -FIRST (page 50) - -The FIRST operator returns the first element of a LIST. - FIRST(list) or FIRST list - -list must be a non-empty list to avoid an error message. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D} -first alist; A -blist := {x,y,{ww,aa,qq},z}; BLIST := {X,Y,{WW,AA,QQ},Z} -first third blist; WW - -\endsection -\item[FIRSTROOT] -FIRSTROOT (page 370) - - FIRSTROOT(expression) - FIRSTROOT simple_exprerssion - -FIRSTROOT is like ROOTS but only the first root determined by ROOTS is -computed. Note that this is not in general the first root that would -be listed in ROOTS output, since the ROOTS outputs are sorted into a -canonical order. Also, in some difficult root finding cases, the -first root computed might be incorrect. - -\endsection -\item[FIX] -FIX (page 73) - FIX(expression) - -The operator FIX returns the integer part of its argument, if that -argument has a numerical value. For positive numbers, this is equivalent -to FLOOR, and, for negative numbers, CEILING. For -non-numeric arguments, the value is an expression in the original operator. - -Examples: -fix 3.4; 3 -floor 3.4; 3 -ceiling 3.4; 4 -fix(-5.2); -5 -floor(-5.2); -6 -ceiling(-5.2); -5 -fix(a); FIX(A) - -\endsection -\item[FIXP] -FIXP (page 46) - -The FIXP logical operator returns true if its argument is an integer. - - FIXP(expression) or FIXP simple_expression - -expression can be any valid REDUCE expression, simple_expression -must be a single identifier or begin with a prefix operator. - -Examples: -if fixp 1.5 then write "ok" else write "not"; not -if fixp(a) then write "ok" else write "not"; not -a := 15; A := 15 -if fixp(a) then write "ok" else write "not"; ok - -Logical operators can only be used inside conditional expressions such as -IF...THEN or WHILE...DO. - -\endsection -\item[FLOOR] -FLOOR (page 73) - - FLOOR(expression) - -This operator returns the floor (i.e., the greatest integer less than -or equal to its argument) if its argument has a numerical value. For -positive numbers, this is equivalent to FIX. For non-numeric -arguments, the value is an expression in the original operator. - -Examples: -floor 3.4; 3 -fix 3.4; 3 -floor(-5.2); -6 -fix(-5.2); -5 -floor a; FLOOR(A) - -\endsection -\item[FOR] -FOR (page 65) - -The FOR command is used for iterative loops. There are many -possible forms it can take. - - / \ - / |STEP UNTIL| \ - |:=| || -FOR| | : | | - | \ / | - |EACH IN | - \ / - - where ::= DO|PRODUCT|SUM|COLLECT|JOIN. - -var can be any valid REDUCE identifier except T or NIL, inc, start and -stop can be any expression that evaluates to a positive or negative -integer. list must be a valid LIST structure. The action taken must -be one of the actions shown above, each of which is followed by a -single REDUCE expression, statement or a GROUP (<<...>>) or BLOCK -(BEGIN...END) statement. - -Examples: -for i := 1:10 sum i; 55 -for a := -2 step 3 until 6 product a; - -8 -a := 3; A := 3 -for iter := 4:a do write iter; -m := 0; M := 0 -for s := 10 step -1 until 3 do - <>; -m; 520 - 2 2 2 -for each x in {q,r,s} sum x**2; Q + R + S - 1 1 1 -for i := 1:4 collect 1/i; {1,---,---,---} - 2 3 4 - -for i := 1:3 join list solve(x**2 + i*x + 1,x); - SQRT(3)*I - 1 - {{X=---------------, - 2 - - - (SQRT(3)*I + 1) - X=--------------------}, - 2 - - {X=-1}, - - SQRT(5) - 3 - SQRT(5) - 3 - {X=-------------,X=----------------}} - 2 2 - -The behaviour of each of the five action words follows: - - Action Word Behaviour -Keyword Argument Type Action - do statement, command, group Evaluates its argument once - or block for each iteration of the loop, - not saving results -collect expression, statement, Evaluates its argument once for - command, group, block, list each iteration of the loop, - storing the results in a list - which is returned by the for - statement when done - join list or an operator which Evaluates its argument once for - produces a list each iteration of the loop, - appending the elements in each - individual result list onto the - overall result list -product expression, statement, Evaluates its argument once for - command, group or block each iteration of the loop, - multiplying the results together - and returning the overall product - sum expression, statement, Evaluates its argument once for - command, group or block each iteration of the loop, - adding the results together and - returning the overall sum - -For number-driven FOR statements, if the ending limit is smaller than -the beginning limit (larger in the case of negative steps) the action -statement is not executed at all. The iterative variable is local to -the FOR statement, and does not affect the value of an identifier with -the same name. For list-driven FOR statements, if the list is empty, -the action statement is not executed, but no error occurs. - -You can use nested FOR statements, with the inner FOR statement after -the action keyword. You must make sure that your inner statement -returns an expression that the outer statement can handle. - -\endsection -\item[FORALL] -FORALL (pages 141, 142) - -See the LET construction. - -\endsection -\item[FOREACH] -FOREACH (page 57--59, 195) - -FOREACH is a synonym for the FOR EACH variant of the -FOR construct. It is designed to iterate down a list, and an -error will occur if a list is not used. The use of FOR EACH is -preferred to FOREACH. - - FOREACH variable in list action expression - where action ::= DO|PRODUCT|SUM|COLLECT|JOIN - -Example: - 2 2 2 -foreach x in {q,r,s} sum x**2; Q + R + S - -\endsection -\xitem[FORDER command] -FORDER command (pages 268, 271) - -\endsection -\item[FORT] -FORT (page 108) - -When FORT is on, output is given Fortran-compatible syntax. Default -is OFF. - -Examples: -on fort; -df(sin(7*x + y),x); ANS=7.*COS(7*X+Y) -on rounded; -b := log(sin(pi/5 + n*pi)); B=LOG(SIN(3.14159265359*N+0.628318530718)) - -REDUCE results can be written to a file (using OUT) and used as data -by Fortran programs when FORT is in effect. FORT knows about correct -statement length, continuation characters, defining a symbol when it -is first used, and other Fortran details. - -The GENTRAN package offers many more possibilities than the FORT -switch. It produces Fortran (or C or Ratfor) code from REDUCE -procedures or structured specifications, including facilities for -producing double precision output. - -\endsection -\item[FORT_WIDTH] -FORT_WIDTH (page 111) - -The FORT_WIDTH variable sets the number of characters in a line of -Fortran-compatible output produced when the FORT switch is on. -Default is 70. - -Examples: -fort_width := 30; FORT_WIDTH := 30 -on fort; -df(sin(x**3*y),x); ANS=3.*COS(X - . **3*Y)*X**2* - . Y - -FORT_WIDTH includes the usually blank characters at the beginning -of the card. As you may notice above, it is conservative and makes the -lines even shorter than it was told. - -\endsection -\item[FORTRAN] -FORTRAN (pages 108, 110) - -REDUCE can produce FORTRAN syntax printed expressions with the switch - ON FORT - -There are also two major packages concerned with generating FORTRAN, -GENTRAN and SCOPE. The first of these is an embedded language for -defining FORTRAN program fragments or program units, with parts -substituted from algebraic calculation. SCOPE is a system for -optimising the form of FORTRAN expressions, usually used in -conjunction with GENTRAN. - -\endsection -\xitem[FRAME command] -FRAME command (pages 265, 271) - -\endsection -\item[FREEOF] -FREEOF (page 46) -The FREEOF logical operator returns TRUE if its first argument does -not contain its second argument anywhere in its structure. - - FREEOF(expression,kernel) or expression FREEOF kernel - -expression can be any valid scalar REDUCE expression, kernel must -be a kernel expression (see Kernel). - -Examples: 2 -a := x + sin(y)**2 + log sin z; A := LOG(SIN(Z)) + SIN(Y) + X -if freeof(a,sin(y)) - then write "free" else write "not free"; - not free -if freeof(a,sin(x)) - then write "free" else write "not free"; - free -if a freeof sin z - then write "free" else write "not free"; - not free - -Logical operators can only be used in conditional expressions such as -IF...THEN or WHILE...DO. - -\endsection -\item[FULLPREC] -FULLPREC - -Trailing zeroes of rounded numbers to the full system precision are -normally not printed. If this information is needed, for example to get a -more understandable indication of the accuracy of certain data, the switch -FULLPREC can be turned on. - -Examples: - on rounded; - 1/2; 0.5 - on fullprec; - ws; 0.500000000000 - -This is just an output options which neither influences the accuracy -of the computation nor does it give additional information about the -precision of the results. See also SCIENTIFIC_NOTATION. - -\endsection -\item[FULLROOTS] -FULLROOTS (page 87) - -Since roots of cubic and quartic polynomials can often be very -messy, a switch FULLROOTS controls the production -of results in closed form. SOLVE will apply the -formulas for explicit forms for degrees 3 and 4 only if -FULLROOTS is ON. Otherwise the result forms -are built using ROOT_OF. Default is OFF. - -\endsection -\xitem[Function] -Function (page 175) - -\endsection -\item[G] -G (page 206) - -G is an n-ary operator used to denote a product of gamma matrices -contracted with Lorentz four-vectors, in high-energy physics. - G(identifier,vector-expr -{,vector-expr}) - -identifier is a scalar identifier representing a fermion line -identifier, vector-expr can be any valid vector expression, -representing a vector or a gamma matrix. - -Examples: -vector aa,bb,cc; -vector a; -g(line1,aa,bb); AA.BB -g(line2,aa,a); 0 -g(id,aa,bb,cc); 0 -g(li1,aa,bb) + k; AA.BB + K -let aa.bb = m*k; -g(ln1,aa)*g(ln1,bb); K*M -g(ln1,aa)*g(ln2,bb); 0 - -The vector A is reserved in arguments of G to denote the special gamma -matrix gamma_5. It must be declared to be a vector before you use it. - -Gamma matrix expressions are associated with fermion lines in a -Feynman diagram. If more than one line occurs in an expression, the -gamma matrices involved are separate (operating in independent spin -space), as shown in the last two example lines above. A product of -gamma matrices associated with a single line can be entered either as -a single G command with several vector arguments, or as products of -separate G commands each with a single argument. - -While the product of vectors is not defined, the product, sum and -difference of several gamma expressions are defined, as is the product -of a gamma expression with a scalar. If an expression involving gamma -matrices includes a scalar, the scalar is treated as if it were the -product of itself with a unit 4 x 4 matrix. - -Dirac expressions are evaluated by computing the trace of the -expression using the commutation algebra of gamma matrices. The -algorithms used are described in articles by J. S. R. Chisholm in Il -Nuovo Cimento X, Vol. 30, p. 426, 1963, and J. Kahane, Journal of -Mathematical Physics, Vol. 9, p. 1732, 1968. The trace is then -divided by 4 to distinguish between the trace of a scalar and the -trace of an expression that is the product of a scalar with a unit 4 x -4 matrix. - -Trace calculations may be prevented over any line identifier by -declaring it to be NOSPUR. If it is later desired to evaluate these -traces, the declaration can be undone with the SPUR declaration. - -The notation of Bjorken and Drell, Relativistic Quantum Mechanics, -1964, is assumed in all operations involving gamma matrices. For an -example of the use of G in a calculation, see the REDUCE -User's Manual. - -\endsection -\item[GAMMA] -GAMMA (pages 185, 394) - -The GAMMA operator returns the Gamma function. - - GAMMA(expression) - -Examples: - load_package specfn; (SPECFN) - gamma(10); 362880 - gamma(1/2); SQRT(PI) - -\endsection -\item[Gamma Function] -Gamma Function (pages 185, 394) - -See GAMMA. - -\endsection -\item[GC] -GC - -With the GC switch, you can turn the garbage collection messages on -or off. The form of the message depends on the particular Lisp used for -the REDUCE implementation. - -See RECLAIM for an explanation of garbage collection. REDUCE does -garbage collection when needed even if you have turned the notices off. - -\endsection -\item[GCD] -GCD (operator and switch) (pages 123, 124) - -The GCD operator returns the greatest common divisor of two -polynomials. - - GCD(expression,expression) - -expression must be a polynomial (or integer), otherwise an error -occurs. - -Examples: -gcd(2*x**2 - 2*y**2,4*x + 4*y); 2*(X + Y) -gcd(sin(x),x**2 + 1); 1 -gcd(765,68); 17 - -The operator GCD described here provides an explicit means to find the -gcd of two expressions. The switch GCD described below simplifies -expressions by finding and cancelling gcd's at every opportunity. When -the switch EZGCD is also on, gcd's are figured using the EZ GCD -algorithm, which is usually faster. - -GCD switch - -With the GC switch, you can turn the garbage collection messages on -or off. The form of the message depends on the particular Lisp used for -the REDUCE implementation. - - -See RECLAIM for an explanation of garbage collection. REDUCE does -garbage collection when needed even if you have turned the notices off. - -\endsection -\item[GDIMENSION] -GDIMENSION (page 300) - - GDIMENSION(bas[,vars]) - -where bas is a GROEBNER basis in the current term order which must be -LEX term order (see IDEAL parameters). GDIMENSION computes the -dimension of the ideal spanned by the given basis. - -GDIMENSION cannot be called with other TERM orders. - -\endsection -\item[GEGENBAUERP] -GEGENBAUERP (page 185) - -The GEGENBAUERP operator computes Gegenbauer's (ultraspherical) -polynomials. - - GEGENBAUERP(integer,expression,expression) - -Examples: - load_package specfn; (SPECFN) - 2 - GegenbauerP(3,2,xx); 4*XX*(8*XX - 3) - - GegenbauerP(3,2,4); 2000 - -\endsection -\xitem[GEN(I)] -GEN(I) (page 379) - -\endsection -\xitem[Generalized Hypergeometric functions] -Generalized Hypergeometric functions (page 187) - -\endsection -\item[GENTRAN] -GENTRAN (page 180) - -Author: Barbara L. Gates - -This package is an automatic code GENerator and TRANslator. It constructs -complete numerical programs based on sets of algorithmic specifications and -symbolic expressions. Formatted FORTRAN, RATFOR or C code can be generated -through a series of interactive commands or under the control of a template -processing routine. Large expressions can be automatically segmented into -subexpressions of manageable size, and a special file-handling mechanism -maintains stacks of open I/O channels to allow output to be sent to any -number of files simultaneously and to facilitate recursive invocation of the -whole code generation process. - -\endsection -\xitem[GETCSYSTEM command] -GETCSYSTEM command (page 235) - -\endsection -\xitem[GETROOT] -GETROOT (page 370) - -\endsection -\xitem[GFNEWT] -GFNEWT (page 371) - -\endsection -\xitem[GFROOT] -GFROOT (page 371) - -\endsection -\item[GINDEPENDENT_SETS] -GINDEPENDENT_SETS (page 300) - - GINDEPENDENT_SETS(bas[,vars]) - -where bas is a GROEBNER basis in LEX term order (which must be the -current TERM order) with the specified variables (see IDEAL -parameters). - -GINDEPENDENT_SETS computes the maximal left independent variable sets -of the ideal, that are the variable sets which play the role of free -parameters in the current ideal basis. Each set is a list which is a -subset of the variable list. The result is a list of these sets. For -an ideal with dimension zero the list is empty. The -Kredel-Weispfenning algorithm is used. - -The operator cannot be called under another TERM order. - -\endsection -\xitem[GL(I)] -GL(I) (page 379) - -\endsection -\item[GLEXCONVERT] -GLEXCONVERT (page 300) - - GLEXCONVERT(bas[,vars][,MAXDEG=mx][,NEWVARS=nv]) - -where bas is a GROEBNER basis in the current term order, mx (optional) -is a positive integer and nvl (optional) is a list of variables (see -IDEAL parameters). - -The operator GLEXCONVERT converts the basis of a zero-dimensional -ideal (finite number of isolated solutions) from arbitrary ordering -into a basis under LEX term order. - -The parameter newvars defines the new variable sequence. If omitted, -the original variable sequence is used. If only a subset of variables -is specified here, the partial ideal basis is evaluated. - -If newvars is a list with one element, the minimal UNIVARIATE -polynomial is computed. - -maxdeg is an upper limit for the degrees. The algorithm stops with an -error message, if this limit is reached. - -A warning occurs, if the ideal is not zero dimensional. - -During the call the TERM order of the input basis must be active. - -\endsection -\item[GLTBASIS] -GLTBASIS (pages 299, 303) - -If GLTBASIS set on, the leading terms of the result basis of a -GROEBNER or GROEBNERF calculation are extracted. They are collected as -a basis of monomials, which is available as value of the global -variable GLTB. - -\endsection -\xitem[GNUPLOT] -GNUPLOT (page 181) - -\endsection -\xitem[GO TO] -GO TO (page 63) - -\endsection -\item[GosperAlg] -Gosper's Algorithm (page 403) - -See SUM and PROD. - -\endsection -\xitem[GRAD operator] -GRAD - operator (page 234) - -\endsection -\xitem[grad operator] -grad operator (page 358) - -\endsection -\xitem[gradient vector field] -gradient - vector field (page 234) - -\endsection -\item[GRADLEX] -GRADLEX (page 293) - -The terms are ordered first with their total degree, and if the total -degree is identical the comparison is LEX term order. With Groebner -basis calculations this term order produces polynomials of lowest -degree. - -\endsection -\item[GRADLEXGRADLEX] -GRADLEXGRADLEX - -The terms are separated into two groups where the second parameter of -the TORDER call determines the length of the first group. For a -comparison first the total degrees of both variable groups are -compared. If both are equal GRADLEX term order comparison is applied -to the first group, and if that does not decide GRADLEX term order is -applied for the second group. This order has the elimination property -for the variable groups. It can be used e.g. for separating variables -from parameters. The terms are ordered first with their total degree, -and if the total degree is identical the comparison is LEX term order. -With Groebner basis calculations this term order produces polynomials -of lowest degree. - -\endsection -\item[GREDUCE] -GREDUCE (page 307) - - GREDUCE(exp, exp1, exp2, ... , expm}[,vars]) - -where exp is an expression, and {exp1, exp2, ... , expm} is a list of -expressions or equations and vars is an optional list of variables -(see IDEAL parameters). - -GREDUCE is functionally equivalent with a call to GROEBNER and then a -call to PREDUCE. - -\endsection -\item[GROEBFULLREDUCTION] -GROEBFULLREDUCTION (page 298) - -If GROEBFULLREDUCTION set off, the polynomial reduction steps during -GROEBNER and GROEBNERF are limited to the pure head term reduction; -subsequent terms are reduced otherwise. - -By default GROEBFULLREDUCTION is on. - -\endsection -\item[GROEBMONFAC] -GROEBMONFAC (page 304) - -The variable GROEBMONFAC is connected to the handling of monomial -factors. A monomial factor is a product of variable powers as a -factor, e.g. x**2*y in x**3*y - 2*x**2*y**2. A monomial factor -represents a solution of the type x = 0 or y = 0 with a certain -multiplicity. With GROEBNERF the multiplicity of monomial factors is -lowered to the value of the shared variable GROEBMONFAC which by -default is 1 (= monomial factors remain present, but their -multiplicity is brought down). With GROEBMONFAC:= 0 the monomial -factors are suppressed completely. - -\endsection -\item[GROEBNER] -GROEBNER (pages 181, 296) - - GROEBNER({exp, ...}[,{var, ...}]) - -where {exp, ... } is a list of expressions or equations, {var, ... } -is an optional list of variables (see IDEAL PARAMETERS). - -The operator GROEBNER implements the Buchberger algorithm for -computing Groebner bases for a given set of expressions with respect -to the given set of variables in the order given. As a side effect, -the sequence of variables is stored as a REDUCE list in the shared -variable GVARSLAST - this is important in cases where the algorithm -rearranges the variable sequence because GROEBOPT is ON. - -Example: - - groebner({x**2+y**2-1,x-y}) {X - Y,2*Y**2 -1} - -See also GROEBNERF, GVARSLAST, GROEBOPT, GROEBPREREDUCE, -GROEBFULLREDUCTION, GLTBASIS, GLTB, GLTERMS, GROEBSTAT, TRGROEB, -TRGROEBS, GROEBPROT, GROEBPROTFILE, GROEBNERT. - -\endsection -\item[Groebner_Bases] -Groebner Bases (page 291) - -The GROEBNER package calculates Groebner bases using the Buchberger -algorithm and provides related algorithms for arithmetic with ideal -bases, such as ideal quotients, Hilbert polynomials, basis conversion, -independent variable set. - -Some routines of the Groebner package are used by SOLVE -- in -that context the package is loaded automatically. However, if you -want to use the package by explict calls you must load it by - - load_package groebner; - -For the common parameter setting of most operators in this package -see IDEAL PARAMETERS. - -\endsection -\item[GROEBNERF] -GROEBNERF (pages 302, 304, 318) - - GROEBNERF({exp, ...}[,{var, ...}] [,{nz, ... }]); - -where {exp, ... } is a list of expressions or equations, {var, ...} is -an optional list of variables (see IDEAL parameters) and {nz,... } is -an optional list of polynomials to be considered as non zero for this -calculation. - -GROEBNERF tries to separate polynomials into individual factors and to -branch the computation in a recursive manner (factorization tree). -The result is a list of partial Groebner bases. Multiplicities (one -factor with a higher power, the same partial basis twice) are deleted -as early as possible in order to speed up the calculation. - -The third parameter of GROEBNERF declares some polynomials -nonzero. If any of these is found in a branch of the calculation -the branch is canceled. - -Example: - -groebnerf({ 3*x**2*y+2*x*y+y+9*x**2+5*x = 3, - 2*x**3*y-x*y-y+6*x**3-2*x**2-3*x = -3, - x**3*y+x**2*y+3*x**3+2*x**2 }, {y,x}); - - {{Y - 3,X}, - - 2 - {2*Y + 2*X - 1,2*X - 5*X - 5}} - -See also GROEBRESMAX, GROEBMONFAC, GROEBRESTRICTION, GROEBNER, -GVARSLAST, GROEBOPT, GROEBPREREDUCE, GROEBFULLREDUCTION, GLTBASIS, -GLTB, GLTERMS, GROEBSTAT, TRGROEB, TRGROEBS, GROEBNERT. - -\endsection -\item[GROEBNERT] -GROEBNERT (page 311) - - GROEBNERT(v}=exp,...}[,vars]) - -where v are KERNELS (simple or indexed variables), exp are polynomials -and optional vars are variables (see IDEAL parameters). - -GROEBNERT is functionally equivalent to a GROEBNER call for {exp,...}, -but the result is a set of equations where the left-hand sides are the -basis elements while the right-hand sides are the same values -expressed as combinations of the input formulas, expressed in terms of -the names v. - -Example: - - groebnert({p1=2*x**2+4*y**2-100,p2=2*x-y+1}); - - GB1 := {2*X - Y + 1=P2, - - 2 - 9*Y - 2*Y - 199= - 2*X*P2 - Y*P2 + 2*P1 + P2} - - -\endsection -\item[GROEBOPT] -GROEBOPT (pages 298, 303) - -If GROEBOPT is set ON, the sequence of variables is optimized with -respect to execution speed of GROEBNER calculations; note that the -final list of variables is available in GVARSLAST. By default -GROEBOPT is off, conserving the original variable sequence. - -An explicitly declared dependency using the DEPEND declaration -superseeds the variable optimization. - -Example: - - depend a, x, y; - -guarantees that a will be placed in front of x and y. - -\endsection -\item[GROEBPREREDUCE] -GROEBPREREDUCE (pages 298, 303) - -If GROEBPREREDUCE set ON, GROEBNER and GROEBNERF try to simplify the -input expressions: if the head term of an input expression is a -multiple of the head term of another expression, it can be reduced; -these reductions are done cyclicly as long as possible in order to -shorten the main part of the algorithm. - -By default GROEBPREREDUCE is off. - -\endsection -\item[GROEBPROT] -GROEBPROT (page 309) - -If GROEBPROT is ON the computation steps during PREDUCE, GREDUCE and -GROEBNER are collected in a list which is assigned to the variable -GROEBPROTFILE. - -\endsection -\item[GROEBPROTFILE] -GROEBPROTFILE (page 309) - -If GROEBPROT is ON the computation steps during PREDUCE, GREDUCE and -GROEBNER are collected in a list which is assigned to the variable -GROEBPROTFILE. - -\endsection -\xitem[GROEBRES] -GROEBRES (page 304) - -\endsection -\item[GROEBRESMAX] -GROEBRESMAX (page 305) - -The variable GROEBRESMAX controls during GROEBNERF calculations the -number of partial results. Its default value is 300. If more partial -results are calculated, the calculation is terminated. - -\endsection -\item[GROEBRESTRICTION] -GROEBRESTRICTION (page 306) - -During GROEBNERF calculations irrelevant branches can be excluded by -setting the variable GROEBRESTRICTION. The following restrictions are -implemented: - - GROEBRESTRICTION := NONNEGATIVE - GROEBRESTRICTION := POSITIVE - -With NONNEGATIVE branches are excluded where one polynomial has no -nonnegative real zeros; with POSITIVE the restriction is sharpened to -positive zeros only. - -\endsection -\item[GROEBSTAT] -GROEBSTAT (pages 299, 303) - -If GROEBSTAT is on, a summary of the GROEBNER or GROEBNERF computation -is printed at the end including the computing time, the number of -intermediate H polynomials and the counters for the criteria hits. - -\endsection -\xitem[GROEPOSTPROC] -GROEPOSTPROC (page 319) - -\endsection -\xitem[GROESOLVE] -GROESOLVE (page 318) - -\endsection -\xitem[Group statement] -Group statement (pages 55, 56, 61) - -\endsection -\xitem[grouped ordering] -grouped ordering (page 315) - -\endsection -\item[GSORT] -GSORT (page 322) - - GSORT(p[,vars]) - -where p is a polynomial or a list of polynomials, vars in an optional -list of variables (see IDEAL parameters). - -The polynomials are reordered and sorted corresponding to the current -TERM order. - -Example: - - torder lex; - 2 2 - gsort(x**2+2x*y+y**2,{y,x}); {Y + 2 * Y * X + X } - -\endsection -\item[GSPLIT] -GSPLIT (page 323) - - GSPLIT(p[,vars]); - -where p is a polynomial or a list of polynomials, vars in an optional -list of variables (see IDEAL parameters). - -The polynomial is reordered corresponding to the the current TERM -order and then separated into leading term and reductum. Result is a -list with the leading term as first and the reductum as second -element. - -Example: - - torder lex; - 2 2 - gsplit(x**2+2x*y+y**2,{y,x}); {Y , 2*Y*X + X } - -\endsection -\item[GSPOLY] -GSPOLY (page 324) - - GSPOLY(p1,p2[,vars]); - -where p1 and p2 are polynomials, vars in an optional list of variables -(see IDEAL parameters). - -The SUBTRACTION polynomial of p1 and p2 is computed corresponding to -the method of the Buchberger algorithm for computing GROEBNER bases: -p1 and p2 are multiplied with terms such that when subtracting them -the leading terms cancel each other. - -\endsection -\item[GVARS] -GVARS (page 296) - - GVARS({exp,exp,... }) - - where exp are expressions or equations. - -GVARS extracts from the expressions the KERNELs which can -play the role of variables for a GROEBNER or GROEBNERF calculation. - -\endsection -\item[GVARSLAST] -GVARSLAST (page 298) - -After a GROEBNER or GROEBNERF calculation the actual variable sequence -is stored in the variable GVARSLAST. If GROEBOPT is ON, GVARSLAST -shows the variable sequence after reordering. - -\endsection -\item[GZERODIM?] -GZERODIM? (page 299) - - GZERODIM!?(basis[,vars]) - -where basis is a Groebner basis in the current -TERM order with the specified variables (see IDEAL parameters). - -GZERODIM!? tests whether the ideal spanned by the given basis -has dimension zero. If yes, the number of zeros is returned, -NIL otherwise. - -\endsection -\item[Hankel Functions] -Hankel Functions (pages 185, 396) - -Part of the SPECFN package. See HANKEL1 and HANKEL2. - -\endsection -\item[HANKEL1] -HANKEL1 (pages 185, 396) - -The HANKEL1 operator returns the Hankel function of the first kind. - -HANKEL1(order,argument) - -Examples: - load_package specfn; (SPECFN) - Hankel1 (1/2,pi); - SQRT(2) / PI - on rounded; - Hankel1 (1,3); 0.324674424792 - -The operator HANKEL1 is also called Bessel function of the third kind. -There is currently no numeric evaluation of Hankel functions. - -\endsection -\item[HANKEL2] -HANKEL2 (pages 185, 396) - -The HANKEL2 operator returns the Hankel function of the second kind. - - HANKEL2(order,argument) - -Examples: - load_package specfn; (SPECFN) - Hankel2 (1/2,pi); - SQRT(2) / PI - on rounded; - Hankel2 (1,3); 0.324674424792 - -The operator HANKEL2 is also called Bessel function of the third kind. -There is currently no numeric evaluation of Hankel functions. - -\endsection -\item[HERMITEP] -HERMITEP (page 185) - -The HERMITEP operator returns the nth Hermite Polynomial. - - HERMITEP(integer,expression) - -Examples: - load_package specfn; (SPECFN) - 2 - HermiteP(3,xx); 4*XX*(2*XX - 3) - HermiteP(3,4); 464 - -Hermite polynomials are computed using the recurrence relation: - -HermiteP(n,x) := 2x*HermiteP(n-1,x) - 2*(n-1)*HermiteP(n-2,x) with -HermiteP(0,x) := 1 and HermiteP(1,x) := 2x - -\endsection -\xitem[HFACTORS scale factors] -HFACTORS scale factors (page 234) - -\endsection -\xitem[High energy trace] -High energy trace (page 209) - -\endsection -\item[HIGH_POW] -HIGH_POW (page 115) - -The variable HIGH_POW is set by COEFF to the highest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. - -Examples: -coeff((x+1)^5*(x*(y+3)^2)^2,x); {0, - - 0, - - 4 3 2 - Y + 12*Y + 54*Y + 108*Y + 81, - - 4 3 2 - 5*(Y + 12*Y + 54*Y + 108*Y + 81), - - 4 3 2 - 10*(Y + 12*Y + 54*Y + 108*Y + 81), - - 4 3 2 - 10*(Y + 12*Y + 54*Y + 108*Y + 81), - - 4 3 2 - 5*(Y + 12*Y + 54*Y + 108*Y + 81), - - 4 3 2 - Y + 12*Y + 54*Y + 108*Y + 81} - -high_pow; 7 - -\endsection -\xitem[HIGHESTDERIV] -HIGHESTDERIV (page 350) - -\endsection -\item[HILBERTPOLYNOMIAL] -HILBERTPOLYNOMIAL (page 321) - - HILBERTPOLYNOMIAL(bas,[vars]) - -where bas is a GROEBNER basis in the current TERM order and vars is an -optional variable list(see IDEAL parameters). - -The degree of the HILBERT polynomial is the dimension of the ideal -spanned by the basis. For an ideal of dimension zero the Hilbert -polynomial is a constant which is the number of common zeros of the -ideal (including eventual multiplicities). The HOLLMANN algorithm is -used. - -\endsection -\xitem[History] -History (page 158) - -\endsection -\xitem[Hodge-* duality operator] -Hodge-* duality operator (pages 256, 266) - -\endsection -\item[HORNER] -HORNER - -When the HORNER switch is on, polynomial expressions are printed -in Horner's form for faster and safer numerical evaluation. Default -is OFF. The leading variable of the expression is selected as -Horner variable. To select the Horner variable explicitly use the -KORDER declaration. - -Examples: -on horner; 3 2 -(13p-4q)^3; ( - 64)*Q + P*(624*Q + P*(( - 2028)*Q + P*2197)) -korder q; - 3 2 -ws; 2197*P + Q*(( - 2028)*P + Q*(624*P + Q*(-64))) - -\endsection -\xitem[HYPERGEOMETRIC] -Hypergeometric Functions (page 397) - -The HYPERGEOMETRIC operator provides simplifications for the -generalised hypergeometric functions. -The HYPERGEOMETRIC operator is included in the package specfn2. - - HYPERGEOMETRIC(list_of_parameters,list_of_parameters,argument) - -Examples: -load_package specfn; (SPECFN) -hypergeometric ({1/2,1},{3/2},-x^2); \rfrac{atan(x)}{x} -hypergeometric ({},{},z); e^z - -The special case with length of the first list equals 2 and -length of the second list equals 1 is often called "hypergeometric function". - -\endsection -\item[HYPOT] -HYPOT (pages 76, 78) - - HYPOT(expression,expression) - -If ROUNDED is on, and the two arguments evaluate to numbers, this -operator returns the square root of the sums of the squares of the -arguments in a manner that avoids intermediate overflow. In other cases, -an expression in the original operator is returned. - -Examples: -hypot(3,4); HYPOT(3,4) -on rounded; -ws; 5.0 -hypot(a,b); HYPOT(A,B) - -\endsection -\item[I] -I (page 36) - -REDUCE knows I is the square root of -1, and that i^2 = -1. - -Examples: -(a + b*i)*(c + d*i); A*C + A*D*I + B*C*I - B*D -i**2; -1 - -I cannot be used as an identifier. It is all right to use I as an -index variable in a FOR loop, or as a local (SCALAR) variable inside a -BEGIN...END block, but it loses its definition as the square root of --1 inside the block in that case. - -Only the simplest properties of i are known by REDUCE unless the -switch COMPLEX is turned on, which implements full complex arithmetic -in factoring, simplification, and functional values. COMPLEX is -ordinarily off. - -\endsection -\xitem[i] -i (page 223) - -\endsection -\xitem[ideal dimension] -ideal dimension (page 300) - -\endsection -\item[IDEAL PARAMETERS] -IDEAL PARAMETERS - -Most operators of the Groebner package compute expressions in a -polynomial ring which given as R[var,var,...] where R is the current -REDUCE coefficient domain. All algebraically exact domains of REDUCE -are supported. The package can operate over rings and fields. The -operation mode is distinguished automatically. In general the ring -mode is a bit faster than the field mode. The factoring variant can -be applied only over domains which allow you factoring of multivariate -polynomials. - -The variable sequence var is either given explicitly as argument in -form of a list, or it is extracted automatically from the -expressions. In the second case the current REDUCE system order is -used (see KORDER) for arranging the variables. If some kernels should -play the role of formal parameters (the ground domain R then is the -polynomial ring over these), the variable sequences must be given -explicitly. - -All REDUCE kernels can be used as variables. But please note, that -all variables are considered as independent; e.g. when using SIN(A) -and COS(A) as variables, the basic relation SIN(A)^2+COS(A)^2-1=0 must -be explicitly added to an equation set because the Groebner operators -do not include such knowledge automatically. - -The terms (monomials) in polynomials are arranged according to the -current TERM ORDER. Note that the algebraic properties of the -computed results only are valid as long as neither the ordering nor -the variable sequence changes. - -The input expressions exp can be polynomials P, rational functions N/D -or equations LH=RH built from polynomials or rational functions. -Apart from the tracing algorithms GROEBNERT and PREDUCET, where the -equations have a specific meaning, equations are converted to simple -expressions by taking the difference of the left-hand and right-hand -sides LH-RH=>P. Rational functions are converted to polynomials by -converting the expression to a common denominator form first, and then -using the numerator only N=>P. So eventual zeros of the denominators -are ignored. - -A basis on input or output of an algorithm is coded as a list -of expressions {exp,exp,...}. - -\endsection -\item[IDEALQUOTIENT] -IDEALQUOTIENT (page 320) - - IDEALQUOTIENT({exp, ...}, d [,{var, ...}]) - -where {exp,...} is a list of expressions or equations, d is a single -expression or equation and {var,...} is an optional list of variables -(see IDEAL parameters). - -IDEALQUOTIENT computes the ideal quotient: ideal spanned by the -expressions {exp,...} divided by the single polynomial/expression -f. The result is the GROEBNER basis of the quotient ideal. - -\endsection -\item[Identifier] -Identifier (page 35) - -Identifiers in REDUCE consist of one or more alphanumeric characters, -of which the first must be alphabetical. The maximum number of -characters allowed is system dependent, but is usually over 100. -However, printing is simplified if they are kept under 25 characters. - -You can also use special characters in your identifiers, but each must be -preceded by an exclamation point ! as an escape character. Useful -special characters are # $ % ^ & * - + = ? < > ~ | / ! and -the space. Note that the use of the exclamation point as a special -character requires a second exclamation point as an escape character. -The underscore _ is special in this regard. It must be preceded -by an escape character in the first position in an identifier, but is -treated like a normal letter within an identifier. - -Other characters, such as ( ) # ; ` ' " can also be used if preceded -by a !, but as they have special meanings to the Lisp reader it is -best to avoid them to avoid confusion. - -Many system identifiers have * before or after their names, or - -between words. If you accidentally pick one of these names for your -own identifier, it could have disastrous effects. For this reason it -is wise not to include * or - anywhere in your identifiers. - -You will notice that REDUCE does not use the escape characters when it -prints identifiers containing special characters; however, you still -must use them when you refer to these identifiers. Be careful when -editing statements containing escaped special characters to treat the -character and its escape as an inseparable pair. - -Identifiers are used for variable names, labels for GO TO statements, -and names of arrays, matrices, operators, and procedures. Once an -identifier is used as a matrix, array, scalar or operator identifier, -it may not be used again as a matrix, array or operator. An operator -or array identifier may later be used as a scalar without problems, -but a matrix identifier cannot be used as a scalar. All procedures -are entered into the system as operators, so the name of a procedure -may not be used as a matrix, array, or operator identifier either. - -\endsection -\item[IF] -IF (pages 55, 56) - -The IF command is a conditional statement that executes a statement -if a condition is true, and optionally another statement if it is not. - - IF condition THEN statement {ELSE statement} - -condition must be a logical or comparison operator that evaluates to -true or false. statement must be a single REDUCE statement or a GROUP -(<<...>>) or BLOCK (BEGIN...END) statement. - -Examples: -if x = 5 then a := b+c else a := d+f; D + F -x := 9; X := 9 -if numberp x and x<20 then y := sqrt(x) else write "illegal"; 3 -clear x; -if numberp x and x<20 then y := sqrt(x) else write "illegal"; illegal -x := 12; X := 12 -a := if x < 5 then 100 else 150; A := 150 -b := u**(if x < 10 then 2); B := 1 - 2 -bb := u**(if x > 10 then 2); BB := U - -An IF statement may be used inside an assignment statement and sets -its value depending on the conditions, or used anywhere else an -expression would be valid, as shown in the last example. If there is -no ELSE clause, the value is 0 if a number is expected, and nothing -otherwise. - -The ELSE clause may be left out if no action is to be taken if the -condition is false. - -The condition may be a compound conditional statement using AND or -OR. If a non-conditional statement, such as a constant, is used by -accident, it is assumed to have value true. - -Be sure to use GROUP or BLOCK statements after THEN or ELSE. - -The IF operator is right associative. The following constructions are -examples: - -(1) - IF condition THEN IF condition THEN action ELSE action - -which is equivalent to - IF condition THEN (IF condition THEN action ELSE action); - -(2) IF condition THEN action ELSE IF condition THEN action ELSE action -which is equivalent to - IF condition THEN action ELSE - (IF condition THEN action ELSE action). - -\endsection -\item[IFACTOR] -IFACTOR (page 121) - -When the IFACTOR switch is on, any integer terms appearing as a result -of the FACTORIZE command are factored themselves into primes. Default -is OFF. If the argument of FACTORIZE is an integer, -IFACTOR has no effect, since the integer is always factored. - -Examples: -factorize(4*x**2 + 28*x + 48); {4,X + 3,X + 4} -factorize(22587); {3,7529} -on ifactor; -factorize(4*x**2 + 28*x + 48); {2,2,X + 4,X + 3} -factorize(22587); {3,7529} - -Constant terms that appear within nonconstant polynomial factors are -not factored. - -The IFACTOR switch affects only factoring done specifically with -FACTORIZE, not on factoring done automatically when the FACTOR switch -is on. - -\endsection -\xitem[imaginary unit] -imaginary unit (page 223) - -\endsection -\item[IMPART] -IMPART (pages 72, 73, 75) - - IMPART(expression) or IMPART simple_expression - -This operator returns the imaginary part of an expression, if that -argument has an numerical value. A non-numerical argument is returned -as an expression in the operators REPART and IMPART. - -Examples: -impart(1+i); 1 -impart(a+i*b); REPART(B) + IMPART(A) - -\endsection -\item[IN] -IN (page 153) - -The IN command takes a list of file names and inputs each file into -the system. - IN filename{,filename} - -filename must be in the current directory, or be a valid pathname. -If the file name is not an identifier, double quote marks (") are -needed around the file name. - - -A message is given if the file cannot be found, or has a mistake -in it. - -Ending the command with a semicolon causes the file to be echoed to the -screen; ending it with a dollar sign does not echo the file. If you want -some but not all of a file echoed, turn the switch ECHO on or off -in the file. - -An efficient way to develop procedures in REDUCE is to write them into a file -using a system editor of your choice, and then input the -files into an active REDUCE session. REDUCE reparses the procedure as -it takes information from the file, overwriting the previous procedure -definition. When it accepts the procedure, it echoes its name to the screen. -Data can also be input to the system from files. - -Files to be read in should always end in END; to avoid -end-of-file problems. Note that this is an additional END; to any -ending procedures in the file. - -\endsection -\item[Indefinite integration] -Indefinite integration (page 80) - -See the INT operator. - -\endsection -\xitem[independent sets] -independent sets (page 300) - -\endsection -\item[INDEX] -INDEX (page 206) - -The declaration INDEX flags a four-vector as an index for subsequent -high-energy physics calculations. - INDEX vector-id{,vector-id} - -vector-id must have been declared of type VECTOR. - -Examples: -vector aa,bb,cc; -index uu; -let aa.bb = 0; -(aa.uu)*(bb.uu); 0 -(aa.uu)*(cc.uu); AA.CC - -Index variables are used to represent contraction over components of -vectors when scalar products are taken by the . operator, as well as -indicating contraction for the EPS operator or metric tensor. - -The special status of a vector as an index can be revoked with the -declaration REMIND. The object remains a vector, however. - -\endsection -\xitem[INDEX_SYMMETRIES command] -INDEX_SYMMETRIES command (page 271) - -\endsection -\xitem[INDEXRANGE command] -INDEXRANGE command (page 271) - -\endsection -\xitem[INDEXSYMMETRIES command] -INDEXSYMMETRIES command (page 262) - -\endsection -\item[INFINITY] -INFINITY (pages 37, 368) - -The name INFINITY is used to represent the infinite positive number. -However, at the present time, arithmetic in terms of this operator reflects -finite arithmetic, rather than true operations on infinity. - -\endsection -\item[INFIX] -INFIX (page 94) - -INFIX declares identifiers to be infix operators. - - INFIX identifier {,identifier} - -identifier can be any valid REDUCE identifier, which has not already -been declared an operator, array or matrix, and is not reserved by the -system. - -Examples: -infix aa; -for all x,y let aa(x,y) = cos(x)*cos(y) - sin(x)*sin(y); -x aa y; COS(X)*COS(Y) - SIN(X)*SIN(Y) - - SQRT(3) -pi/3 aa pi/2; ------------ - 2 -aa(pi,pi); 1 - -A LET statement must be used to attach functionality to the operator. -Note that the operator is defined in prefix form in the LET statement. -After its definition, the operator may be used in either prefix or infix -mode. The above operator aa finds the cosine of the sum of two angles by -the formula - cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y). -Precedence may be attached to infix operators with the PRECEDENCE declaration. - -User-defined infix operators may be used in prefix form. If they are used -in infix form, a space must be left on each side of the operator to avoid -ambiguity. Infix operators are always binary. - -\endsection -\xitem[Infix operator] -Infix operator (pages 38--41) - -\endsection -\xitem[inner product] -inner product (page 357) - -\endsection -\xitem[inner product exterior form] -inner product - exterior form (page 254) - -\endsection -\item[INPUT] -INPUT (page 158) - -The INPUT command returns the input expression to the REDUCE numbered -prompt that is its argument. - INPUT(number) or INPUT number - - -number must be between 1 and the current REDUCE prompt number. - -An expression brought back by INPUT can be re-executed with new -values or switch settings, or used as an argument in another expression. -The command WS brings back the results of a numbered REDUCE -statement. Two lists contain every input and every output statement since -the beginning of the session. If your session is very long, storage space -begins to fill up with these expressions, so it is a good idea to end the -session once in a while, saving needed expressions to files with the -SAVEAS and OUT commands. - -Switch settings and LET statements can also be re-executed by using -INPUT. - -An error message is given if a number is called for that has not yet been used. - -\endsection -\xitem[Input] -Input (page 153) - -\endsection -\xitem[Instant evaluation] -Instant evaluation (pages 68, 117, 140, 162, 164) - -\endsection -\item[INT] -INT (operator and switch) (pages 80, 160) - -The INT operator performs analytic integration on a variety of -functions. - - INT(expression,kernel) - -expression can be any scalar expression. involving polynomials, log -functions, exponential functions, or tangent or arctangent -expressions. INT attempts expressions involving error functions, -dilogarithms and other trigonometric expressions. Integrals involving -algebraic extensions (such as square roots) may not succeed. kernel -must be a REDUCE KERNEL. - -Examples: - 3 - X*(X + 12) -int(x**3 + 3,x); ------------- - 4 - - 2*X - E *( - COS(X) + 2*SIN(x)) -int(sin(x)*exp(2*x),x); ----------------------------- - 5 - - SQRT(2)*(LOG( - SQRT(2) + X) - LOG(SQRT(2) + X)) -int(1/(x^2-2),x); -------------------------------------------------- - 4 - - COS(X) - - ATAN(--------) - 2 -int(sin(x)/(4 + cos(x)**2),x); ------------------- - 2 - - SQRT(x - 1) -int(1/sqrt(x^2-x),x); INT(---------------------,X) - SQRT(X)*X - SQRT(X) - -Note that REDUCE could not handle the last integral with its default -integrator, since the integrand involves a square root. However, the -integral can be found using the ALGINT package. Alternatively, you -could add a rule using the LET statement to evaluate this integral. - -The arbitrary constant of integration is not shown. Definite -integrals can be found by evaluating the result at the limits of -integration (use ROUNDED) and subtracting the lower from the higher. -Evaluation can be easily done by the SUB operator. - -When INT cannot find an integral it returns an expression involving -formal INT expressions unless the switch FAILHARD has been set. If not -all of the expression can be integrated, the switch NOLNR controls -whether a partially integrated result should be returned or not. - -INT switch - -The INT switch specifies an interactive mode of operation. Default -ON. - - -There is no reason to turn INT off during interactive calculations, -since there are no benefits to be gained. If you do have INT off -while inputting a file, and REDUCE finds an error, it prints the message -``Continuing with parsing only''. In this state, REDUCE accepts only -END; or BYE; from the keyboard; -everything else is ignored, even the command ON INT. - -\endsection -\item[INTEGER] -INTEGER (page 61) - -The INTEGER declaration must be made immediately after a BEGIN (or -other variable declaration such as REAL and SCALAR) and declares local -integer variables. They are initialised to 0. - - INTEGER identifier{,identifier} - -identifier may be any valid REDUCE identifier, except T or NIL. - -Integer variables remain local, and do not share values with variables -of the same name outside the BEGIN...END block. When the block is -finished, the variables are removed. You may use the words REAL or -SCALAR in the place of INTEGER. INTEGER does not indicate -type-checking by the current REDUCE; it is only for your own -information. Declaration statements must immediately follow the -BEGIN, without a semicolon between BEGIN and the first variable -declaration. - -Any variables used inside BEGIN...END blocks that were not declared -SCALAR, REAL or INTEGER are global, and any change made to them inside -the block affects their global value. Any ARRAY or MATRIX declared -inside a block is always global. - -\endsection -\xitem[Integer] -Integer (page 44) - -\endsection -\item[Integration] -Integration (pages 80, 92) - -See INT, ALGINT or NUM_INT. -\endsection -\xitem[integration definite (simple)] -integration definite (simple) (page 236) - line (page 238) - volume (page 237) - -\endsection -\xitem[Interactive use] -Interactive use (pages 157, 160) - -\endsection -\item[INTERPOL] -INTERPOL (page 127) - -INTERPOL generates an interpolation polynomial. - - INTERPOL(values,variable,points) - -values and points are LISTs of equal length and variable is an -algebraic expression (preferably a KERNEL). The interpolation -polynomial is generated in the given variable of degree -length(values)-1. The unique polynomial F is defined by the property -that for corresponding elements V of values and P of points the -relation F(P)=V holds. - -Examples: -f := for i:=1:4 collect(i**3-1); F := {0,7,26,63} -p := {1,2,3,4}; P := {1,2,3,4} - 3 -interpol(f,x,p); X - 1 - -The Aitken-Neville interpolation algorithm is used which guarantees a -stable result even with rounded numbers and an ill-conditioned problem. - -\endsection -\item[INTSTR] -INTSTR (page 98) - -If INTSTR (for ``internal structure'') is on, arguments of an -operator are printed in a more structured form. - -Examples: - operator f; - f(2x+2y); F(2*X + 2*Y) - on intstr; - ws; F(2*(X + Y)) - -\endsection -\item[ISOLATER] -ISOLATER (page 369) - - ISOLATER(expression) - ISOLATER simple_expresion - ISOLATER(expression, POSITIVE) - ISOLATER(expression, NEGATIVE) - ISOLATER(expression, lo, hi) - -The ISOLATER function produces a list of rational intervals, each -containing a single real root of the univariate polynomial p, within -the specified region, but does not find the roots. If arg2 and arg3 -are not present, all real roots are found. If the additional -arguments are present, they restrict the region of consideration. - -If arg2=NEGATIVE then only negative roots of p are included; if -arg2=POSITIVE then only positive roots of p are included. Zero roots -are excluded. - -If arguments are (p,arg2,arg3) then Arg2 and Arg3 must be r (a real -number) or EXCLUDE r, or a member of the list POSITIVE, NEGATIVE, -INFINITY, -INFINITY. EXCLUDE r causes the value r to be excluded from -the region. The order of the sequence arg2, arg3 is unimportant. -Assuming that arg2 <= arg3 when both are numeric, then - - {-INFINITY,INFINITY} is equivalent to {} represents all roots; - {arg2,NEGATIVE} represents -1 (cos(x+y)+cos(x-y))/2, - cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, - sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, - cos(~x)^2 => (1+cos(2*x))/2, - sin(~x)^2 => (1-cos(2*x))/2}$ -let trig1; - COS(A - B) + COS(A + B) -cos(a)*cos(b); ------------------------- - 2 - -A LET command returns no value, though the substitution rule is -entered. Assignment rules made by ASSIGN and LET rules are at the -same level, and cancel each other. There is a difference in their -operation, however, as shown in the first example: a LET assignment -tracks the changes in what it is assigned to, while a := assignment is -fixed at the value it originally had. - -The use of expressions as left-hand sides of LET statements is a -little complicated. The rules of operation are: - -(i) Expressions of the form A*B = C do not change A, B or C, but set -A*B to C. - -(ii) Expressions of the form A+B = C substitute C - B for A, but do -not change B or C. - -(iii) Expressions of the form A-B = C substitute B + C for A, but do -not change B or C. - -(iv) Expressions of the form A/B = C substitute B*C for A, but do not -change B or C. - -(v) Expressions of the form A**N = C substitute C for A**N in every -expression of a power of A to N or greater. An asymptotic command -such as A**N = 0 sets all terms involving A to powers greater than or -equal to N to 0. Finite fields may be generated by requiring modular -arithmetic (the MODULAR switch) and defining the primitive polynomial -via a LET statement. - -LET substitutions involving expressions are cleared by using the CLEAR -command with exactly the same expression. - -Note when a simple LET statement is used to assign functionality to an -operator, it is valid only for the exact identifiers used. For the -use of the LET command to attach more general functionality to an -operator, see FORALL. - -Arrays as a whole cannot be arguments to LET statements, but matrices -as a whole can be legal arguments, provided both arguments are -matrices. However, it is important to note that the two matrices are -then linked. Any change to an element of one matrix changes the -corresponding value in the other. Unless you want this behaviour, you -should not use LET for matrices. The assignment operator ASSIGN can -be used for non-tracking assignments, avoiding the side effects. -Matrices are redimensioned as needed in LET statements. - -When array or matrix elements are used as the left-hand side of LET -statements, the contents of that element is used as the argument. -When the contents is a number or some other expression that is not a -valid left-hand side for LET, you get an error message. If the -contents is an identifier or simple expression, the LET rule is -globally attached to that identifier, and is in effect not only inside -the array or matrix, but everywhere. Because of such unwanted side -effects, you should not use LET with array or matrix elements. The -assignment operator := can be used to put values into array or matrix -elements without the side effects. - -Local variables declared inside BEGIN...END blocks cannot be used as -the left-hand side of LET statements. However, BEGIN...END blocks -themselves can be used as the right-hand side of LET statements. The -construction: - FOR ALL vars - LET operator(vars) = block -is an alternative to the - PROCEDURE name(vars); block -construction. One important difference between the two constructions -is that the vars as formal parameters to a procedure have their global -values protected against change by the procedure, while the vars of a -LET statement are changed globally by its actions. - -Be careful in using a construction such as LET x = x + 1 except inside -a controlled loop statement. The process of resubstitution continues -until a stack overflow message is given. - -The LET statement may be used to make global changes to variables from -inside procedures. If X is a formal parameter to a procedure, the -command LET x = ... makes the change to the calling variable. For -example, if a procedure was defined by - procedure f(x,y); - let x = 15; -and the procedure was called as - f(a,b); -A would have its value changed to 15. Be careful when using LET -statements inside procedures to avoid unwanted side effects. - -It is also important to be careful when replacing LET statements with -other LET statements. The overlapping of these substitutions can be -unpredictable. Ordinarily the latest-entered rule is the first to be -applied. Sometimes the previous rule is superseded completely; other -times it stays around as a special case. The order of entering a set -of related LET expressions is very important to their eventual -behaviour. The best approach is to assume that the rules will be -applied in an arbitrary order. - -\endsection -\xitem[Levi-Cevita tensor] -Levi-Cevita tensor (page 267) - -\endsection -\item[LEX] - -The terms are ordered lexicographically: two terms t1 t2 are compared -for their degrees along the fixed variable sequence: t1 is higher than -t2 if the first different degree is higher in t1. This order has the -elimination property for GROEBNER BASIS calculations. If the ideal -has a univariate polynomial in the last variable the groebner basis -will contain such polynomial. LEX is best suited for solving of -polynomial equation systems. - -\endsection -\item[LHS] -LHS (page 47) - -The LHS operator returns the left-hand side of an EQUATION, such as -those returned in a list by SOLVE. - - LHS(equation) or LHS equation - -equation must be an equation of the form - LEFT-HAND SIDE = RIGHT-HAND SIDE. - -Examples: -polly := (x+3)*(x^4+2x+1); - 5 4 2 - POLLY := X + 3*X + 2*X + 7*X + 3 - -pollyroots := solve(polly,x); - 3 2 - POLLYROOTS := {X=ROOT_OF(X_ - X_ + X_ + 1,X_),X=-1,X=-3} - -variable := lhs first pollyroots; - VARIABLE := X - -\endsection -\xitem[LIE Derivative] -Lie Derivative (page 255) - -\endsection -\item[LIMIT] -LIMIT (pages 329, 360) - -LIMITS is a fast limit package for REDUCE for functions which are -continuous except for computable poles and singularities, based on -some earlier work by Ian Cohen and John P. Fitch. The Truncated -Power Series package is used for non-critical points, at which -the value of the function is the constant term in the expansion -around that point. l'Hopital's rule is used in critical cases, -with preprocessing of 1-1 forms and reformatting of product forms -in order to apply l'Hopital's rule. A limited amount of bounded -arithmetic is also employed where applicable. - - LIMIT(expr,var,limpoint) or - LIMIT!+(expr,var,limpoint) or - LIMIT!-(expr,var,limpoint) - -where expr is an expression depending of the variable var (a KERNEL) -and limpoint is the limit point. If the limit depends upon the -direction of approach to the limpoint, the operators LIMIT!+ and -LIMIT!- may be used. - -Examples: - limit(x*cot(x),x,0); 0 - 2 - limit((2x+5)/(3x-2),x,infinity); --- - 3 - -\endsection -\xitem[LIMIT0] -LIMIT0 (page 330) - -\endsection -\xitem[LIMIT1] -LIMIT1 (page 330) - -\endsection -\xitem[LIMIT2] - -\endsection -\xitem[LIMIT2] -LIMIT2 (page 330) - -\endsection -\xitem[LIMITS] -LIMITS (page 181) - -\endsection -\xitem[LIMITS package] -LIMITS package (page 329) - -\endsection -\item[LIMITEDFACTORS] -LIMITEDFACTORS - -To get limited factorisation in cases where it is too expensive to use -full multivariate polynomial factorisation, the switch -LIMITEDFACTORS can be turned on. In that case, only ``inexpensive'' -factoring operations, such as square-free factorisation, will be used -when FACTORIZE is called. - -Examples: -a := (y-x)^2*(y^3+2x*y+5)*(y^2-3x*y+7)$ -factorize a; { - X + Y, - X - Y, - 3 - 2*X*Y + Y + 5, - 2 - 3*X*Y - Y - 7} -on limitedfactors; -factorize a; { - X + Y, - X - Y, - 2 2 4 3 5 3 2 - 6*X *Y + 3*X*Y - 2*X*Y + X*Y - Y - 7*Y - 5*Y - 35} - -\endsection -\xitem[line integrals] -line integrals (page 238) - -\endsection -\item[LINEAR] -LINEAR (page 91) - -An operator can be declared linear in its first argument over powers of -its second argument by the declaration LINEAR. - - LINEAR operator{,operator} - -operator must have been declared to be an operator. Be careful not to -use a system operator name, because this command may change its -definition. The operator being declared must have at least two -arguments, and the second one must be a kernel. - -Examples: -operator f; -linear f; -f(0,x); 0 -f(-y,x); - F(1,X)*Y -f(y+z,x); F(1,X)*(Y + Z) -f(y*z,x); F(1,X)*Y*Z -depend z,x; -f(y*z,x); F(Z,X)*Y - 1 -f(y/z,x); F(---,X)*Y - Z - -depend y,x; - Y -f(y/z,x); F(---,X) - Z -nodepend z,x; - F(Y,X) -f(y/z,x); -------- - Z - - SIN(x) -f(2*e**sin(x),x); 2*F(E ,X) - -Even when the operator has not had its functionality attached, it -exhibits linear properties as shown in the examples. Notice the -difference when dependencies are added. Dependencies are also in -effect when the operator's first argument contains its second, as in -the last line above. - -For a fully-developed example of the use of linear operators, refer to -the article in the Journal of Computational Physics, Vol. 14 -(1974), pp. 301-317, ``Analytic Computation of Some Integrals in -Fourth Order Quantum Electrodynamics'', by J.A. Fox and A.C. Hearn. -The article includes the complete listing of REDUCE procedures used -for this work. - -\endsection -\xitem[Linear operator] -Linear operator (pages 91, 92, 95) - -\endsection -\xitem[LINEINT] -LINEINT (page 360) - -\endsection -\xitem[LINEINT function] -LINEINT function (page 238) - -\endsection -\item[LINELENGTH] -LINELENGTH (page 100) - -The LINELENGTH declaration sets the length of the output line. Default -is 80. - - LINELENGTH integer - -integer must be positive, less than 128 (although this varies from -system to system), and should not be less than 20 or so for proper -operation. - -LINELENGTH returns the previous linelength. If you want the current -linelength value, but not change it, say LINELENGTH NIL. - -\endsection -\item[LISP] -LISP (page 191) - -The LISP command changes REDUCE's mode of operation to symbolic. When -LISP is followed by an expression, that expression is evaluated in -symbolic mode, but REDUCE's mode is not changed. This command is -equivalent to SYMBOLIC. - -Examples: -lisp; NIL -car '(a b c d e); A -algebraic; 2 -c := (lisp car '(first second))**2; C := FIRST - -\endsection -\item[LIST] -LIST (page 103) - -The LIST switch causes REDUCE to print each term in any sum on -separate lines. - -Examples: 2 2 - X*(2*A*X*Y + 4*A*X*Y + Y + Z) -x**2*(y**2 + 2*y) + x*(y**2 + z)/(2*a); --------------------------------- - 2*A -on list; - 2 -ws; (X*(2*A*X*Y - + 4*A*X*Y - 2 - + Y - + Z))/(2*A) - -\endsection -\xitem[List] -List (page 49) - -\endsection -\item[List(operation)] -List operation (pages 49, 51) - -The LIST operator constructs a list from its arguments. - LIST(item {,item}) or - LIST() to construct an empty list. - -item can be any REDUCE scalar expression, including another list. -Left and right curly brackets can also be used instead of the operator -LIST to construct a list. - -Examples: -liss := list(c,b,c,{xx,yy},3x**2+7x+3,df(sin(2*x),x)); - 2 - LISS := {C,B,C,{XX,YY},3*X + 7*X + 3,2*COS(2*X)} -length liss; 6 -liss := {c,b,c,{xx,yy},3x**2+7x+3,df(sin(2*x),x)}; - 2 - LISS := {C,B,C,{XX,YY},3*X + 7*X + 3,2*COS(2*X)} -emptylis := list(); EMPTYLIS := {} -a . emptylis; {A} - -Lists are ordered, hierarchical structures. The elements stay where -you put them, and only change position in the list if you specifically -change them. Lists can have nested sublists to any (reasonable) -level. The PART operator can be used to access elements anywhere -within a list hierarchy. The LENGTH operator counts the number of -top-level elements of its list argument; elements that are themselves -lists still only count as one element. - -\endsection -\item[LISTARGP] -LISTARGP (page 51) - - LISTARGP operator{,operator} - -If an operator other than those specifically defined for lists is -given a single argument that is a LIST, then the result of this -operation will be a list in which that operator is applied to each -element of the list. This process can be inhibited for a specific -operator, or list of operators, by using the declaration LISTARGP. - -Examples: -log {a,b,c}; {LOG(A),LOG(B),LOG(C)} -listargp log; -log {a,b,c}; LOG({A,B,C}) - -It is possible to inhibit such distribution globally by turning on the -switch LISTARGS. In addition, if an operator has more than one -argument, no such distribution occurs, so LISTARGP has no effect. - -\endsection -\item[LISTARGS] -LISTARGS (page 51) - -If an operator other than those specifically defined for lists is given a -single argument that is a list, then the result of this operation will be -a list in which that operator is applied to each element of the list. -This process can be inhibited globally by turning on the switch -LISTARGS. - -Examples: - log {a,b,c}; {LOG(A),LOG(B),LOG(C)} - on listargs; - log {a,b,c}; LOG({A,B,C}) - -It is possible to inhibit such distribution for a specific operator by -using the declaration LISTARGP. In addition, if an operator has -more than one argument, no such distribution occurs, so LISTARGS -has no effect. - -\endsection -\item[LN] -LN (pages 76, 78) - - LN(expression) - -expression can be any valid scalar REDUCE expression. - -The LN operator returns the natural logarithm of its argument. -However, unlike LOG, there are no algebraic rules associated -with it; it will only evaluate when ROUNDED is on, and the -argument is a real number. - -Examples: -ln(x); LN(X) -ln 4; LN(4) -ln(e); LN(E) -df(ln(x),x); DF(LN(X),X) -on rounded; -ln 4; 1.38629436112 -ln e; 1 - -Because of the restricted algebraic properties of LN, users are -advised to use LOG whenever possible. - -\endsection -\xitem[LOAD] -LOAD (page 214) - -\endsection -\item[LOAD_PACKAGE] -LOAD_PACKAGE (pages 177, 188, 215) - -The LOAD_PACKAGE command is used to load REDUCE packages, such as -GENTRAN that are not automatically loaded by the system. - - LOAD_PACKAGE "package_name" - -A package is only loaded once; subsequent calls of LOAD_PACKAGE -for the same package name are ignored. - -\endsection -\item[LOG] -LOG (pages 76, 78, 81) - -The LOG operator returns the natural logarithm of its argument. - - LOG(expression) or LOG expression - -expression can be any valid scalar REDUCE expression. - -Examples: -log(x); LOG(X) -log 4; LOG(4) -log(e); 1 -on rounded; -log 4; 1.38629436112 - -LOG returns a numeric value only when ROUNDED is on. In that case, -use of a negative argument for LOG results in an error message. No -error is given on a negative argument when REDUCE is not in that mode. - -\endsection -\xitem[LOG10] -LOG10 (pages 76, 78) - -\endsection -\item[LOGB] -LOGB (pages 76, 78) - - LOGB(expression,integer) - -expression can be any valid scalar REDUCE expression. - -The LOGB operator returns the logarithm of its first argument using -the second argument as base. However, unlike LOG, there are no -algebraic rules associated with it; it will only evaluate when ROUNDED -is on, and the first argument is a real number. - -Examples: -logb(x,2); LOGB(X,2) -logb(4,3); LOGB(4,3) -logb(2,2); LOGB(2,2) -df(logb(x,3),x); DF(LOGB(X,3),X) -on rounded; -logb(4,3); 1.26185950714 -logb(2,2); 1 - -\endsection -\item[Lommel Functions] -Lommel Functions (pages 185, 397) - -Part of the SPECFN package. See LOMMEL1 and LOMMEL2. - -\endsection -\item[LOMMEL1] -LOMMEL1 (pages 185, 397) - - LOMMEL1(integer, integer, expression) - -The LOMMEL1 function is defined in terms of the BESSELJ and GAMMA -functions for some of its arguments, and the STRUVEH function for -others. There are no rules for differentiation or for numerical -evaluation. - -Examples: - load_package specfn; (SPECFN) - 3 - Lommel1(3,2,xx); - 48*BESSELJ(3,XX) + XX - 15*STRUVEH(3,XX)*PI - Lommel1(3,3,xx); --------------------- - 2 - -\endsection -\item[LOMMEL2] -LOMMEL2 (pages 185, 397) - - LOMMEL2(integer, integer, expression) - -The LOMMEL2 function is defined in terms of the BESSELY, GAMMA and -STRUVEH function for some of its arguments. There are no rules for -differentiation or for numerical evaluation. - -Examples: -load_package specfn; (SPECFN) - 2 -Lommel2(3,2,xx); XX - 15*PI*( - BESSELY(3,XX) + STRUVEH(3,XX)) -Lommel2(3,3,xx); ------------------------------------------ - 2 - -\endsection -\xitem[Loop] -Loop (pages 57, 58) - -\endsection -\item[LOW_POW] -LOW_POW (page 115) - -The variable LOW_POW is set by COEFF to the lowest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. - -Examples: 6 -coeff((x+2*y)**6,y); {X , - 5 - 12*X , - 4 - 60*X , - 3 - 160*X , - 2 - 240*X , - - 192*X, - - 64} - -low_pow; 0 -coeff(x**2*(x*sin(y) + 1),x); {0,0,1,SIN(Y)} -low_pow; 2 - -\endsection -\item[LTERM] -LTERM (pages 130, 203) - -The LTERM operator returns the leading term of an expression with -respect to a kernel. - - LTERM(expression,kernel) - -expression is ordinarily a polynomial. If RATARG is on, a rational -expression may also be used, otherwise an error results. kernel must -be a kernel. - -Examples: 6 -lterm((x+2*y)**6,y); 64*Y - 8 -lterm((x + cos(x))**8 + df(x**2,x),cos(x)); COS(X) -lterm(x**3 + 3*x,y); 0 - -\endsection -\xitem[MACRO] -MACRO (page 196) - -\endsection -\item[MAINVAR] -MAINVAR (page 130) - -The MAINVAR operator returns the main variable (in the system's -internal representation) of its argument. - - MAINVAR(expression) - -expression is usually a polynomial, but may be any valid REDUCE scalar -expression. In the case of a rational function, the main variable of -the numerator is returned. The main variable returned is a KERNEL. - -Examples: 2 2 2 -test := (a + b + c)**2; TEST := A + 2*A*B + 2*A*C + B + 2*B*C + C -mainvar(test); A -korder c,b,a; -mainvar(test); C -mainvar(2*cos(x)**2); COS(X) -mainvar(17); 0 - -The main variable is the first variable in the canonical ordering of -kernels. Generally, alphabetically ordered functions come first, then -alphabetically ordered identifiers (variables). Numbers come last, -and as far as MAINVAR is concerned belong in the family 0. The -canonical ordering can be changed by the declaration KORDER, as shown -above. - -\endsection -\item[MASS] -MASS (pages 208, 210) - -The MASS command associates a scalar variable as a mass with -the corresponding vector variable, in high-energy physics calculations. - MASS vector-var=scalar-var {,vector-var=scalar-var} - -vector-var can be a declared vector variable; MASS will declare -it to be of type VECTOR if it is not. This may override an existing -matrix variable by that name. scalar-var must be a scalar variable. - -Examples: -vector bb,cc; -mass cc=m; -mshell cc; - 2 -cc.cc; M - -Once a mass has been attached to a vector with a MASS declaration, the -MSHELL declaration puts the associated particle ``on the mass shell.'' -Subsequent scalar (.) products of the vector with itself will be -replaced by the square of the mass expression. - -\endsection -\item[MAT] -MAT (pages 161--162) - -The MAT operator is used to represent a two-dimensional -MATRIX. - MAT((expr{,expr}) {(expr{,expr})}) - -expr may be any valid REDUCE scalar expression. - -Examples: -mat((1,2),(3,4)); MAT(1,1) := 1 - MAT(2,3) := 2 - MAT(2,1) := 3 - MAT(2,2) := 4 -mat(2,1); ***** Matrix mismatch - Cont? (Y or N) -matrix qt; -qt := ws; QT(1,1) := 1 - QT(1,2) := 2 - QT(2,1) := 3 - QT(2,2) := 4 -matrix a,b; -a := mat((x),(y),(z)); A(1,1) := X - A(2,1) := Y - A(3,1) := Z -b := mat((sin x,cos x,1)); B(1,1) := SIN(X) - B(1,2) := COS(X) - B(1,3) := 1 - -Matrices need not have a size declared (unlike arrays). MAT -redimensions a matrix variable as needed. It is necessary, of course, -that all rows be the same length. An anonymous matrix, as shown in -the first example, must be named before it can be referenced (note -error message). When using MAT to fill a 1 x n matrix, the row of -values must be inside a second set of parentheses, to eliminate -ambiguity. - -\endsection -\item[MATCH] -MATCH (page 146) - -The MATCH command is similar to the LET command, except -that it matches only explicit powers in substitution. - - MATCH expr = expression{,expr = expression} - -expr is generally a term involving powers, and is limited by the rules -for the LET command. expression may be any valid REDUCE scalar -expression. - -Examples: -match c**2*a**2 = d; - 4 3 3 4 -(a+c)**4; A + 4*A *C + 4*A*C + C + 6*D -match a+b = c; -a + 2*b; B + C - 2 2 2 -(a + b + c)**2; A - B + 2*B*C + 3*C -clear a+b; - 2 2 2 -(a + b + c)**2; A + 2*A*B + 2*A*C + B + 2*B*C + C -let p*r = s; -match p*q = ss; - 2 2 -(a + p*r)**2; A + 2*A*S + S - 2 2 2 -(a + p*q)**2; A + 2*A*SS + P *Q - -Note in the last example that A + B has been explicitly matched after -the squaring was done, replacing each single power of A by C - B. -This kind of substitution, although following the rules, is confusing -and could lead to unrecognisable results. It is better to use MATCH -with explicit powers or products only. MATCH should not be used -inside procedures for the same reasons that LET should not be. - -Unlike LET substitutions, MATCH substitutions are executed after all -other operations are complete. The last example shows the -difference. MATCH commands can be cleared by using CLEAR, with exactly -the expression that the original MATCH took. MATCH commands can also -be done more generally with FOR ALL or FORALL...SUCH THAT commands. - -\endsection -\item[MATEIGEN] -MATEIGEN (page 164) - -The MATEIGEN operator calculates the eigenvalue equation and the -corresponding eigenvectors of a MATRIX. - - MATEIGEN(matrix-id,tag-id) - -matrix-id must be a declared matrix of values, and tag-id must be a -legal REDUCE identifier. - -Examples: -aa := mat((2,5),(1,0))$ - 2 -mateigen(aa,alpha); {{ALPHA - 2*ALPHA - 5, - 1, - 5*ARBCOMPLEX(1) - MAT(1,1) := --------------- - ALPHA - 2 - - MAT(2,1) := ARBCOMPLEX(1) - }} - 2 -charpoly := first first ws; CHARPOLY := ALPHA - 2*ALPHA - 5 - -bb := mat((1,0,1),(1,1,0),(0,0,1))$ - -mateigen(bb,lamb); {{LAMB - 1,3, - - [ 0 ] - [ ] - [ARBCOMPLEX(2)] - [ ] - [ 0 ] - - }} - -The MATEIGEN operator returns a list of lists of three elements. The -first element is a square free factor of the characteristic -polynomial; the second element is its multiplicity; and the third -element is the corresponding eigenvector. If the characteristic -polynomial can be completely factored, the product of the first -elements of all the sublists will produce the minimal polynomial. You -can access the various parts of the answer with the usual list access -operators. - -If the matrix is degenerate, more than one eigenvector can be produced -for the same eigenvalue, as shown by more than one arbitrary variable -in the eigenvector. The identification numbers of the arbitrary -complex variables shown in the examples above may not be the same as -yours. Note that since LAMBDA is a reserved word in REDUCE, you -cannot use it as a tag-id for this operator. - -\endsection -\xitem[Mathematical function] -Mathematical function (page 76) - -\endsection -\item[MATRIX] -MATRIX (page 162) - -Identifiers are declared to be of type MATRIX. - MATRIX identifier (index,index) {,identifier (index,index)} - -identifier must not be an already-defined operator or array or the -name of a scalar variable. Dimensions are optional, and if used -appear inside parentheses. index must be a positive integer. - -Examples: -matrix a,b(1,4),c(4,4); -b(1,1); 0 -a(1,1); ***** Matrix A not set -a := mat((x0,y0),(x1,y1)); A(1,1) := X0 - A(1,2) := Y0 - A(2,1) := X0 - A(2,2) := X1 -length a; {2,2} - 2 -b := a**2; B(1,1) := X0 + X1*Y0 - B(1,2) := Y0*(X0 + Y1) - B(2,1) := X1*(X0 + Y1) - 2 - B(2,2) := X1*Y0 + Y1 - -When a matrix variable has not been dimensioned, matrix elements -cannot be referenced until the matrix is set by the MAT operator. -When a matrix is dimensioned in its declaration, matrix elements are -set to 0. Matrix elements cannot stand for themselves. When you use -LET on a matrix element, there is no effect unless the element -contains a constant, in which case an error message is returned. The -same behaviour occurs with CLEAR. Do not use CLEAR to try to set a -matrix element to 0. LET statements can be applied to matrices as a -whole, if the right-hand side of the expression is a matrix -expression, and the left-hand side identifier has been declared to be -a matrix. - -Arithmetical operators apply to matrices of the correct dimensions. -The operators + and - can be used with matrices of the same -dimensions. The operator * can be used to multiply m x n matrices by -n x p matrices. Matrix multiplication is non-commutative. Scalars -can also be multiplied with matrices, with the result that each -element of the matrix is multiplied by the scalar. The operator / -applied to two matrices computes the first matrix multiplied by the -inverse of the second, if the inverse exists, and produces an error -message otherwise. Matrices can be divided by scalars, which results -in dividing each element of the matrix. Scalars can also be divided -by matrices when the matrices are invertible, and the result is the -multiplication of the scalar by the inverse of the matrix. Matrix -inverses can by found by 1/A or /A, where A is a matrix. Square -matrices can be raised to positive integer powers, and also to -negative integer powers if they are nonsingular. - -When a matrix variable is assigned to the results of a calculation, the -matrix is redimensioned if necessary. - -\endsection -\xitem[Matrix assignment] -Matrix assignment (page 168) - -\endsection -\xitem[Matrix calculations] -Matrix calculations (page 161) - -\endsection -\item[MAX] -MAX (page 73) - -The operator MAX is an n-ary prefix operator, which returns the largest -value in its arguments. - - MAX(expression{,expression}) - -expression must evaluate to a number. MAX of an empty list returns 0. - -Examples: -max(4,6,10,-1); 10 -<>; 46 -max(-5,-10,-a); -5 - -\endsection -\item[MCD] -MCD (pages 123, 125, 126) - -When MCD is on, sums and differences of rational expressions are put -on a common denominator. Default is ON. - -Examples: 5*A + B*X + B -a/(x+1) + b/5; --------------- - 5*(X + 1) -off mcd; - -1 -a/(x+1) + b/5; (X + 1) *A + 1/5*B - -1/6 + 1/7; 13/42 - -Even with MCD off, rational expressions involving only numbers are -still put over a common denominator. - -Turning MCD off is useful when explicit negative powers are needed, or -if no greatest common divisor calculations are desired, or when -differentiating complicated rational expressions. Results when MCD is -off are no longer in canonical form, and expressions equivalent to -zero may not simplify to 0. Some operations, such as factoring cannot -be done while MCD is off. This option should therefore be used with -some caution. Turning MCD off is most valuable in intermediate parts -of a complicated calculation, and should be turned back on for the -last stage. - -\endsection -\xitem[MEIJERG] -Meijer's G function (page 187) - -The MEIJERG operator provides simplifications for Meijer's G -function. The simplifications are performed towards polynomials, -elementary or -special functions or (generalised) HYPERGEOMETRIC functions. - -The MEIJERG operator is included in the package specfn2. - -MEIJERG(list of parameters,list of parameters,argument) -The first element of the lists has to be the list containing the -first group (mostly called "m" and "n") of parameters. This passes -the four parameters of a Meijer's G function implicitly via the -length of the lists. - -Examples: -load specfn2; -MeijerG({{},1},{{0}},x); & heaviside(-x+1) -MeijerG({{}},{{1+1/4},1-1/4},(x^2)/4) * sqrt pi; - & \rfrac{sqrt(2)*sin(x)*x^2}{4*sqrt(x)} - -Many well-known functions can be written as G functions, -e.g. exponentials, logarithms, trigonometric functions, Bessel functions -and hypergeometric functions. -The formulae can be found e.g. in -A.P.Prudnikov, Yu.A.Brychkov, O.I.Marichev: -Integrals and Series, Volume 3: More special functions, -Gordon and Breach Science Publishers (1990). - -\endsection -\xitem[METRIC command] -METRIC command (page 271) - -\endsection -\xitem[metric structure] -metric structure (page 262) - -\endsection -\item[MIN] -MIN (page 73) - -The operator MIN is an n-ary prefix operator, which returns the -smallest value in its arguments. - - MIN(expression{,expression}) - -expression must evaluate to a number. MIN of an empty list -returns 0. - -Examples: -min(-3,0,17,2); -3 -<>; 16 -min(5,10,a); 5 - -\endsection -\xitem[Minimum] -Minimum (page 182) - -\endsection -\item[MKID] -MKID (page 83) - -The MKID command constructs an identifier, given a stem and an identifier -or an integer. - - MKID(stem,leaf) - -stem can be any valid REDUCE identifier that does not include escaped -special characters. leaf may be an integer, including one given by a -local variable in a FOR loop, or any other legal group of characters. - -Examples: -mkid(x,3); X3 -factorize(x^15 - 1); {X - 1, - - 2 - X + X + 1, - - 4 3 2 - X + X + X + X + 1, - - 8 7 5 4 3 - X - X + X - X + X - X + 1} - -for i := 1:length ws do write set(mkid(f,i),part(ws,i)); - X - 1 - - 2 - X + X + 1 - - 4 3 2 - X + X + X + X + 1 - - 8 7 5 4 3 - X - X + X - X + X - X + 1 - -You can use MKID to construct identifiers from inside procedures. This -allows you to handle an unknown number of factors, or deal with variable -amounts of data. It is particularly helpful to attach identifiers to the -answers returned by FACTORIZE and SOLVE. - -\endsection -\item[MKPOLY] -MKPOLY (page 370) - -Given a roots list as returned by ROOTS, the operator MKPOLY -constructs a polynomial which has these numbers as roots. - - MKPOLY rl - -where rl is a LIST with equations, which all have the same KERNEL on -their left-hand sides and numbers as right-hand sides. - -Examples: - 4 3 2 - mkpoly{x=1,x=-2,x=i,x=-i}; X + X - X + X - 2 - - -Note that this polynomial is unique only up to a numeric factor. - -\endsection -\xitem[MM] -MM (page 379) - -\endsection -\xitem[Mode] -Mode (page 68) - -\endsection -\xitem[Mode communication] -Mode communication (page 197) - -\endsection -\item[MODULAR] -MODULAR (page 134) - -When MODULAR is on, polynomial coefficients are reduced by the -modulus set by SETMOD. If no modulus has been set, MODULAR -has no effect. - -Examples: -setmod 2; 1 -on modular; - 2 2 -(x+y)**2; X + Y - 2 -145*x**2 + 20*x**3 + 17 + 15*x*y; X + X*Y + 1 - -Modular operations are only conducted on the coefficients, not the -exponents. The modulus is not restricted to being prime. When the -modulus is prime, division by a number not relatively prime to the -modulus results in a Zero divisor error message. When the modulus is -a composite number, division by a power of the modulus results in an -error message, but division by an integer which is a factor of the -modulus does not. The representation of modular number can be -influenced by BALANCED_MOD. - -\endsection -\xitem[Modular coefficient] -Modular coefficient (page 134) - -\endsection -\item[MSG] -MSG (page 218) - -When MSG is off, the printing of warning messages is suppressed. Error -messages are still printed. - -Warning messages include those about redimensioning an ARRAY or declaring -an OPERATOR where one is expected. - -\endsection -\item[MSHELL] -MSHELL (page 210) - -The MSHELL command puts particles on the mass shell in high-energy -physics calculations. - MSHELL vector-var{,vector-var} - -vector-var must have had a mass attached to it by a MASS -declaration. - -Examples: -vector v1,v2; -mass v1=m,v2=q; -mshell v1; - 2 -v1.v1; M -v2.v2; V2.V2 -mshell v2; - 2 2 -v1.v1*v2.v2; M *Q - -Even though a mass is attached to a vector variable representing a -particle, the replacement does not take place until the MSHELL -declaration is given for that vector variable. - -\endsection -\xitem[Multiple assignment statement] -Multiple assignment statement (page 54) - -\endsection -\item[MULTIPLICITIES] -MULTIPLICITIES (page 86) - -When MSG is off, the printing of warning messages is suppressed. Error -messages are still printed. - -Warning messages include those about redimensioning an ARRAY or declaring -an OPERATOR where one is expected. - -\endsection -\xitem[MULTIROOT] -MULTIROOT (page 373) - -\endsection -\item[NAT] -NAT (page 111, 259) - -When NAT is on, output is printed to the screen in natural form, with -raised exponents. NAT should be turned off when outputting expressions -to a file for future input. Default is ON. - -Examples: 3 2 2 3 -(x + y)**3; X + 3*X *Y + 3*X*Y + Y -off nat; -(x + y)**3; X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3$ -on fort; -(x + y)**3; ANS=X**3+3.*X**2*Y+3.*X*Y**2+Y**3 - -With NAT off, a dollar sign is printed at the end of each expression. -An output file written with NAT off is ready to be read into REDUCE -using the command IN. - -\endsection -\item[NEARESTROOT] -NEARESTROOT (pages 370, 372) - -The operator NEARESTROOT finds one root of a polynomial with an -iteration using a given starting point. - - NEARESTROOT(p,pt) - -where p is a univariate polynomial and pt is a number. - -Example: - - nearestroot(x^2+2,2); {X=1.41421*I} - -The minimal accuracy of the result values is controlled by ROOTACC. - -\endsection -\xitem[NEARESTROOTS] -NEARESTROOTS (page 370) - -\endsection -\xitem[NEGATIVE] -NEGATIVE (page 368) - -\endsection -\item[NERO] -NERO (page 108) - -When NERO is on, zero assignments (such as matrix elements) are not -printed. - -Examples: -matrix a; -a := mat((1,0),(0,1)); A(1,1) := 1 - A(1,2) := 0 - A(2,1) := 0 - A(2,2) := 1 -on nero; -a; MAT(1,1) := 1 - MAT(2,2) := 1 -a(1,2); {nothing is printed.} -b := 0; {nothing is printed.} -off nero; -b := 0; B := 0 - -NERO is often used when dealing with large sparse matrices, to avoid -being overloaded with zero assignments. - -\endsection -\xitem[Newton's method] -Newton's method (page 182) - -\endsection -\item[NEXTPRIME] -NEXTPRIME (page 74) - - NEXTPRIME(expression) - -If the argument of NEXTPRIME is an integer, the least prime greater -than that argument is returned. Otherwise, a type error results. - -Examples: -nextprime 5001; 5003 -nextprime(10^30); 1000000000000000000000000000057 -nextprime a; ***** A invalid as integer - -\endsection -\xitem[NN] -NN (page 379) - -\endsection -\item[NOARG] -NOARG - -When DFPRINT is on, expressions in the differentiation operator -DF are printed in a more ``natural'' notation, with the -differentiation variables appearing as subscripts. When NOARG -is on (the default), the arguments of the differentiated operator are also -suppressed. - -Examples: -operator f; -df(f x,x); DF(F(X),X); -on dfprint; -ws; F - X -df(f(x,y),x,y); F - X,Y -off noarg; -ws; F(X) - X - -\endsection -\item[NODEPEND] -NODEPEND (page 95) - -The NODEPEND declaration removes the dependency declared with DEPEND. - - NODEPEND dep-kernel{,kernel} - -dep-kernel -must be a kernel that has had a dependency declared upon -the one or more other kernels that are its other arguments. - -Examples: -depend y,x,z; -df(sin y,x); COS(Y)*DF(Y,X) -df(sin y,x,z); COS(Y)*DF(Y,X,Z) - DF(Y,X)*DF(Y,Z)*SIN(Y) -nodepend y,z; -df(sin y,x); COS(Y)*DF(Y,X) -df(sin y,x,z); 0 - -A warning message is printed if the dependency had not been declared by -DEPEND. - -\endsection -\xitem[NOETHER function] -NOETHER function (pages 258, 271) - -\endsection -\xitem[Non-commuting operator] -Non-commuting operator (page 92) - -\endsection -\item[NOLNR] -NOLNR - -When NOLNR is on, the linear properties of the integration operator -INT are suppressed if the integral cannot be found in closed terms. - - -REDUCE uses the linear properties of integration to attempt to break down -an integral into manageable pieces. If an integral cannot be found in -closed terms, these pieces are returned. When the NOLNR switch is off, -as many of the pieces as possible are integrated. When it is on, if any piece -fails, the rest of them remain unevaluated. - -\endsection -\item[NONCOM] -NONCOM (page 92) - -NONCOM declares that already-declared operators are noncommutative -under multiplication. - - NONCOM operator{,operator} - -operator must have been declared an OPERATOR, or a warning message is -given. - -Examples: -operator f,h; -noncom f; -f(a)*f(b) - f(b)*f(a); F(A)*F(B) - F(B)*F(A) -h(a)*h(b) - h(b)*h(a); 0 -operator comm; -for all x,y such that x neq y and ordp(x,y) - let f(x)*f(y) = f(y)*f(x) + comm(x,y); -f(1)*f(2); F(1)*F(2) -f(2)*f(1); COMM(2,1) + F(1)*F(2) - -The last example introduces the commutator of f(x) and f(y) for all x -and y. The equality check is to prevent an infinite loop. The -operator f can have other functionality attached to it if desired, or -it can remain an indeterminate operator. - -\endsection -\item[NONZERO] -NONZERO (page 90) - - NONZERO identifier{,identifier} - -If an operator F is declared ODD, then F(0) is replaced by zero unless -F is also declared non zero by the declaration NONZERO. - -Examples: - odd f; - f(0) 0 - nonzero f; - f(0) F(0) - -\endsection -\item[NOSPLIT] -NOSPLIT (page 103) - -Under normal circumstances, the printing routines try to break an expression -across lines at a natural point. This is a fairly expensive process. If -you are not overly concerned about where the end-of-line breaks come, you -can speed up the printing of expressions by turning off the switch -NOSPLIT. This switch is normally on. - -\endsection -\item[NOSPUR] -NOSPUR (page 210) - -The NOSPUR declaration prevents the trace calculation over the given -line identifiers in high-energy physics calculations. - NOSPUR line-id{,line-id} - - -line-id is a scalar identifier that will be used as a line identifier. - -Examples: -vector a1,b1,c1; -g(line1,a1,b1)*g(line2,b1,c1); A1.B1*B1.C1 -nospur line2; -g(line1,a1,b1)*g(line2,b1,c1); A1.B1*G(LINE2,B1,C1) - -Nospur declarations can be removed by making the declaration SPUR. - -\endsection -\xitem[NOSUM command] -NOSUM command (pages 262, 271) - -\endsection -\xitem[NOSUM switch] -NOSUM switch (page 262) - -\endsection -\item[NOXPND @] -NOXPND @ (pages 254, 271) -NOXPND D (pages 253, 271) - -(Part of the EXCALC package) - -There are two forms of the NOXPND command, which controls the use of -the product rule for the d operator and the expansion into partial -derivatives. The default for both these is OFF. - - noxpnd d; - noxpnd @; - -Example: - load_package excalc; *** ^ redefined - (excalc) - pform x=0,y=k,z=m; - K - d(y^z); ( - 1) *Y^d Z + d Y^Z - - noxpnd d; - d(y^z); d(Y^Z) - - -See also XPND -\endsection -\xitem[NS dummy variable] -NS dummy variable (page 260) - -\endsection -\item[NULLSPACE] -NULLSPACE (page 166) - -NULLSPACE(matrix_expression) - -nullspace calculates for its MATRIX argument, A, a list of linear -independent vectors (a basis) whose linear combinations satisfy the -equation A x = 0. The basis is provided in a form such that as many -upper components as possible are isolated. - -Examples: -nullspace mat((1,2,3,4),(5,6,7,8)); { - [ 1 ] - [ ] - [ 0 ] - [ ] - [ - 3] - [ ] - [ 2 ] - , - [ 0 ] - [ ] - [ 1 ] - [ ] - [ - 2] - [ ] - [ 1 ] - } - -Note that with B := NULLSPACE A, the expression LENGTH B is the -nullity of A, and that SECOND LENGTH A - LENGTH B calculates the rank -of A. The rank of a matrix expression can also be found more directly -by the RANK operator. - -In addition to the REDUCE matrix form, NULLSPACE accepts as input a -matrix given as a LIST of lists, that is interpreted as a row matrix. If -that form of input is chosen, the vectors in the result will be -represented by lists as well. This additional input syntax facilitates -the use of NULLSPACE in applications different from classical linear -algebra. - -\endsection -\item[NUM] -NUM (page 131) -The NUM operator returns the numerator of its argument. - - NUM(expression) or NUM simple_expression - -expression can be any valid REDUCE scalar expression. - -Examples: -num(100/6); 50 -num(a/5 + b/6); 6*A + 5*B -num(sin(x)); SIN(X) - -NUM returns the numerator of the expression after it has been simplified -by REDUCE. As seen in the examples, this includes putting sums of rational -expressions over a common denominator, and reducing common factors where -possible. If the expression is not a rational expression, it is returned -unchanged. - -\endsection -\item[NUMVAL] -NUMVAL - -With ROUNDED on, elementary functions with numerical arguments -will return a numerical answer where appropriate. If you wish to inhibit -this evaluation, NUMVAL should be turned off. It is normally on. - -Examples: - on rounded; - cos 3.4; - 0.966798192579 - off numval; - cos 3.4; COS(3.4) - -\endsection -\item[NUM_INT] -NUM_INT (page 182) - -For the numerical evaluation of univariate integrals over a finite -interval the following strategy is used: If INT finds a formal -antiderivative which is bounded in the integration interval, this is -evaluated and the end points and the difference is returned. -Otherwise a Chebyshev fit is computed, starting with order 20, -eventually up to order 80. If that is recognized as sufficiently -convergent it is used for computing the integral by directly -integrating the coefficient sequence. If none of these methods is -successful, an adaptive multilevel quadrature algorithm is used. - -For multivariate integrals only the adaptive quadrature is used. This -algorithm tolerates isolated singularities. The value ITERATIONS here -limits the number of local interval intersection levels. a is a -measure for the relative total discretization error (comparison of -order 1 and order 2 approximations). - -NUM_INT(exp,var=(l .. u) [,var=(l .. u),...] [,accuracy=a][,iterations=i]) - -where exp is the function to be integrated, var are the integration -variables, l are the lower bounds, u are the upper bounds. - -Result is the value of the integral. - -Example: - on rounded; - num_int(sin x,x=(0 .. pi)); 2.0 - -\endsection -\item[NUM_MIN] -NUM_MIN (page 182) - -The Fletcher Reeves version of the STEEPEST_DESCENT algorithms is used -to find the minimum of a function of one or more variables. The -function must have continuous partial derivatives with respect to all -variables. The starting point of the search can be specified; if not, -random values are taken instead. The steepest descent algorithms in -general find only local minima. - -Syntax: - - NUM_MIN(exp, var[=val] [,var[=val] ... [,accuracy=a] [,iterations=i]) -NUM_MIN(exp, {var[=val] [,var[=val} ...] } [,accuracy=a] [,iterations=i]) - -where exp is a function expression, var are the variables in exp and -val are the (optional) start values. For a and i see NUMERIC_ACCURACY. - -NUM_MIN tries to find the next local minimum along the descending path -starting at the given point. The result is a LIST with the minimum -function value as first element followed by a list of equations, where -the variables are equated to the coordinates of the result point. - -Examples: - load numeric; - num_min(sin(x)+x/5, x); { - 0.0775892231689,{x=4.51200216375}} - num_min(sin(x)+x/5, x=0); { - 1.33416631212,{x= - 1.78326532423}} - -\endsection -\item[NUM_ODESOLVE] -NUM_ODESOLVE (page 182) - -The Runge-Kutta method of order 3 finds an approximate graph for the -solution of real ODE initial value problem. - -NUM_ODESOLVE(exp,depvar=start, indep=(from .. to) [,accuracy=a][,iterations=i]) -NUM_ODESOLVE({exp,exp,...},{depvar=start,depvar=start,...} indep=(from .. to) - [,accuracy=a][,iterations=i]) - -where depvar and start specify the dependent variable(s) and the -starting point value (vector), indep, from and to specify the -independent variable and the integration interval (starting point and -end point), exp are equations or expressions which contain the first -derivative of the independent variable with respect to the dependent -variable. - -The ODEs are converted to an explicit form, which then is used for a -Runge Kutta iteration over the given range. The number of steps is -controlled by the value of i (default: 20). If the steps are too -coarse to reach the desired accuracy in the neighborhood of the -starting point, the number is increased automatically. - -Result is a list of pairs, each representing a point of the -approximate solution of the ODE problem. - -Example: -num_odesolve(df(y,x)=y,y=1,x=(0 .. 1), iterations=5); - - {{0.0,1.0},{0.2,1.2214},{0.4,1.49181796},{0.6,1.8221064563}, - {0.8,2.2255208258},{1.0,2.7182511366}} - -If in exp the differential is not isolated on the left-hand side, -please ensure that the dependent variable is explicitly declared using -a DEPEND otherwise the formal derivative will be computed to zero by -REDUCE. - -The operator SOLVE is used to convert the form into an explicit -ODE. If that process fails or has no unique result, the evaluation is -stopped with an error message. - -\endsection -\item[NUM_SOLVE] -NUM_SOLVE (page 182) - -An adaptively damped Newton iteration is used to find an approximative -root of a function (function vector) or the solution of an EQUATION -(equation system). The expressions must have continuous derivatives -for all variables. A starting point for the iteration can be -given. If not given random values are taken instead. When the number -of forms is not equal to the number of variables, the Newton method -cannot be applied. Then the minimum of the sum of absolute squares is -located instead. - -With COMPLEX on, solutions with imaginary parts can be found, if -either the expression(s) or the starting point contain a nonzero -imaginary part. - - NUM_SOLVE(exp, var[=val][,accuracy=a][,iterations=i]) - NUM_SOLVE({exp,...,exp}, var[=val],...,var[=val] [,accuracy=a][,iterations=i]) - NUM_SOLVE({exp,...,exp}, {var[=val],...,var[=val]} - [,accuracy=a][,iterations=i]) - -where exp are function expressions, - var are the variables, - val are optional start values. -For a and i see NUMERIC_ACCURACY. - -NUM_SOLVE tries to find a zero/solution of the expression(s). Result -is a list of equations, where the variables are equated to the -coordinates of the result point. - -The Jacobian matrix is stored as side effect the shared jacobian. - -Examples: -num_solve({sin x=cos y, x + y = 1},{x=1,y=2}); - {X= - 52.1216769476,Y=53.1216769476} - [COS(X) SIN(Y)] -jacobian; [ ] - [ 1 1 ] -\endsection -\xitem[Number] -Number (pages 34, 35) - -\endsection -\item[NUMBERP] -NUMBERP (page 46) -The NUMBERP operator returns TRUE if its argument is a number, -and NIL otherwise. - - NUMBERP(expression) or NUMBERP expression - -expression can be any REDUCE scalar expression. - -Examples: -cc := 15.3; CC := 15.3 -if numberp(cc) then write "number" else write "nonnumber"; number -if numberp(cb) then write "number" else write "nonnumber"; nonnumber - -Logical operators can only be used in conditional expressions, such as -IF...THEN...ELSE and WHILE...DO. - -\endsection -\item[NUMERIC package] -NUMERIC package (page 337) - -The numeric package supplies algorithms based on approximation -techniques of numerical mathematics. The algorithms use the ROUNDED -mode arithmetic of REDUCE, including the variable precision feature -which is exploited in some algorithms in an adaptive manner in order -to reach the desired accuracy. - -\endsection -\xitem[Numerical operator] -Numerical operator (page 71) - -\endsection -\xitem[Numerical precision] -Numerical precision (page 36) - -\endsection -\item[ODD] -ODD (page 90) - - ODD identifier{,identifier} - -This declaration is used to declare an operator odd in its first -argument. Expressions involving an operator declared in this manner -are transformed if the first argument contains a minus sign. Any -other arguments are not affected. - -Examples: - odd f; - f(-a) -F(A) - f(-a,-b) -F(A,-B) - f(a,-b) F(A,-B) - -If say F is declared odd, then F(0) is replaced by zero unless F is -also declared non zero by the declaration NONZERO. - -\endsection -\xitem[ODEDEGREE] -ODEDEGREE (page 350) - -\endsection -\xitem[ODELINEARITY] -ODELINEARITY (page 350) - -\endsection -\xitem[ODEORDER] -ODEORDER (page 350) - -\endsection -\item[ODESOLVE] -ODESOLVE (pages 183, 349) - -Main Author: Malcolm A.H. MacCallum -Other contributors: Francis Wright, Alan Barnes - -Ordinary Differential Equations Solver. - -The ODESOLVE package is a solver for ordinary differential -equations. At the present time it has very limited capabilities. -It can handle only a single scalar equation presented as an -algebraic expression or equation, and it can solve only first- -order equations of simple types, linear equations with constant -coefficients and Euler equations. These solvable types are exactly -those for which Lie symmetry techniques give no useful information. - -For example, the evaluation of - depend(y,x); - odesolve(df(y,x)=x**2+e**x,y,x); -yields the result - X 3 - 3*E + 3*ARBCONST(1) + X - {Y=---------------------------} - 3 - -\endsection -\item[OFF] -OFF (pages 68, 69) - -The OFF command is used to turn switches off. - - OFF switch{,switch} - -switch can be any SWITCH name. There is no problem if the switch is -already off. If the switch name is mistyped, an error message is -given. - -\endsection -\item[ON] -ON (pages 68, 69) - -The ON command is used to turn switches on. - - ON switch{,switch} - -switch can be any SWITCH name. There is no problem if the switch is -already on. If the switch name is mistyped, an error message is -given. - -\endsection -\item[ONE_OF] -ONE_OF (page 86) -The operator ONE_OF is used to represent an indefinite choice -of one element from a finite set of objects. - -Example: - x=one_of{1,2,5} - -This equation encodes that x can take one of the values 1,2 or 5 - -REDUCE generates a ONE_OF form in cases when an implicit ROOT_OF -expression could be converted to an explicit solution set. A ONE_OF -form can be converted to a SOLVE solution using EXPAND_CASES. See -ROOT_OF. - -\endsection -\item[OPERATOR] -OPERATOR (page 202) - -Use the OPERATOR declaration to declare your own operators. - - OPERATOR identifier{,identifier} - -identifier can be any valid REDUCE identifier, which is not the name -of a MATRIX, ARRAY, scalar variable or previously-defined operator. - -Examples: -operator dis,fac; -let dis(~x,~y) = sqrt(x^2 + y^2); -dis(1,2); SQRT(5) - 2 -dis(a,10); SQRT(A + 100) -on rounded; -dis(1.5,7.2); 7.35459040329 -let fac(~n) = - if n=0 then 1 - else if not(fixp n and n>0) - then rederr "choose non-negative integer" - else for i := 1:n product i; - -fac(5); 120 -fac(-2); ***** choose non-negative integer - -The first operator is the Euclidean distance metric, the distance of -point (x,y) from the origin. The second operator is the factorial. - -Operators can have various properties assigned to them; they can be -declared INFIX, LINEAR, SYMMETRIC, ANTISYMMETRIC, or NONCOMmutative. -The default operator is prefix, nonlinear, and commutative. -Precedence can also be assigned to operators using the declaration -PRECEDENCE. - -Functionality is assigned to an operator by a LET statement or a -FORALL...LET statement, (or possibly by a procedure with the name of -the operator). Be careful not to redefine a system operator by -accident. REDUCE permits you to redefine system operators, giving you -a warning message that the operator was already defined. This -flexibility allows you to add mathematical rules that do what you want -them to do, but can produce odd or erroneous behaviour if you are not -careful. - -You can declare operators from inside PROCEDUREs, as long as they are -not local variables. Operators defined inside procedures are global. -A formal parameter may be declared as an operator, and has the effect -of declaring the calling variable as the operator. - -\endsection -\xitem[Operator precedence] -Operator precedence (page 39, 41) - -\endsection -\item[ORDER] -ORDER (pages 101, 114) - -The ORDER declaration changes the order of precedence of kernels for -display purposes only. - - ORDER identifier{,identifier} - -kernel must be a valid KERNEL or OPERATOR name complete with argument. - -Examples: -x + y + z + cos(a); COS(A) + X + Y + Z -order z,y,x,cos(a); -x + y + z + cos(a); Z + Y + X + COS(A) - 2 2 -(x + y)**2; Y + 2*Y*X + X -order nil; - 2 2 -(z + cos(z))**2; COS(Z) + 2*COS(Z)*Z + Z - -ORDER affects the printing order of the identifiers only; internal -order is unchanged. Change internal order of evaluation with the -declaration KORDER. You can use ORDER to feature variables or -functions you are particularly interested in. - -Declarations made with ORDER are cumulative: kernels in new order -declarations are ordered behind those in previous declarations, and -previous declarations retain their relative order. Of course, -specific kernels named in new declarations are removed from previous -ones and given the new priority. Return to the standard canonical -printing order with the statement ORDER NIL. - -The print order specified by ORDER commands is not in effect if the -switch PRI is off. - -\endsection -\xitem[ordering exterior form] -ordering - exterior form (page 268) - -\endsection -\xitem[ordinary differential equations] -ordinary differential equations (page 349) - -\endsection -\item[ORDP] -ORDP (pages 46, 92) - -The ORDP logical operator returns TRUE if its first argument is -ordered ahead of its second argument in canonical internal ordering, -or is identical to it. - - ORDP(expression1,expression2) - -expression1 and expression2 can be any valid REDUCE scalar expression. - -Examples: -if ordp(x**2 + 1,x**3 + 3) then write "yes" else write "no"; no -if ordp(101,100) then write "yes" else write "no"; yes -if ordp(x,x) then write "yes" else write "no"; yes - -Logical operators can only be used in conditional expressions, such as -IF...THEN...ELSE and WHILE...DO. - -\endsection -\item[ORTHOVEC] -ORTHOVEC (pages 184, 353) - -Author: James W. Eastwood - -A Package for the Manipulation of Scalars and Vectors. - -ORTHOVEC is a collection of REDUCE procedures and operations which -provide a simple-to-use environment for the manipulation of scalars -and vectors. Operations include addition, subtraction, dot and -cross products, division, modulus, div, grad, curl, laplacian, -differentiation, integration, and Taylor expansion. - -\endsection -\item[OUT] -OUT (pages 153, 154) - -The OUT command directs output to the filename that is its argument, -until another OUT changes the output file, or SHUT closes it. - OUT filename or OUT "pathname " or OUT T - -filename must be in the current directory, or be a valid complete -file description for your system. If the file name is not -in the current directory, quote marks are needed around the file name. -If the file already exists, a message is printed allowing you to decide -whether to supersede the contents of the file with new material. - - -To restore output to the terminal, type OUT T, or SHUT the -file. When you use OUT T, the file remains available, and if you -open it again (with another OUT), new material is appended rather -than overwriting. - -To write a file using OUT that can be input at a later time, the -switch NAT must be turned off, so that the standard linear form -is saved that can be read in by IN. If NAT is on, exponents -are printed on the line above the expression, which causes trouble -when REDUCE tries to read the file. - -There is a slight complication if you are using the OUT command from -inside a file to create another file. The ECHO switch is normally -off at the top-level and on while reading files (so you can see what is -being read in). If you create a file using OUT at the top-level, -the result lines are printed into the file as you want them. But if you -create such a file from inside a file, the ECHO switch is on, and -every line is echoed, first as you typed it, then as REDUCE parsed it, and -then once more for the file. Therefore, when you create a file from -a file, you need to turn ECHO off explicitly before the OUT -command, and turn it back on when you SHUT the created file, so your -executing file echoes as it should. This behaviour also means that as you -watch the file execute, you cannot see the lines that are being put into -the OUT file. As soon as you turn ECHO on, you can see -output again. - -\endsection -\item[OUTPUT] -OUTPUT (page 100) - -When OUTPUT is off, no output is printed from any REDUCE calculation. -The calculations have their usual effects other than printing. Default is -ON. - - -Turn output OFF if you do not wish to see output when executing -large files, or to save the time REDUCE spends formatting large expressions -for display. Results are still available with WS, or in their -assigned variables. - -\endsection -\xitem[Output] -Output (pages 105, 110) - -\endsection -\xitem[Output declaration] -Output declaration (pages 100, 101) - -\endsection -\item[OVERVIEW] -OVERVIEW - -When OVERVIEW is on, the amount of detail reported by the factoriser -switches TRFAC and TRALLFAC is reduced. - - -\endsection -\item[PART] -PART (pages 49, 113, 116) -The operator PART permits the extraction of various parts or -operators of expressions and LISTS. - - PART(expression,integer{,integer}) - -expression can be any valid REDUCE expression or a list, integer may -be an expression that evaluates to a positive or negative integer or -0. A positive integer n picks up the nth term, counting from the -first term toward the end. A negative integer n picks up the nth -term, counting from the back toward the front. The integer 0 picks up -the operator (which is LIST when the expression is a list). - -Examples: - 2 3 -part((x + y)**5,4); 10*X *Y - - 2 -part((x + y)**5,4,2); X - -part((x + y)**5,4,2,1); X -part((x + y)**5,0); PLUS - 4 -part((x + y)**5,-5); 5*x *y - - 5 4 3 2 4 5 -part((x + y)**5,4) := sin(x); x + 5*x *y + 10*x *y + sin(x) + 5*x*y + y - -alist := {x,y,{aa,bb,cc},x**2*sqrt(y)}; - ALIST := {X, - Y, - {AA,BB,CC}, - 2 - SQRT(Y)*X } -part(alist,3,2); BB -part(alist,4,0); TIMES - -Additional integer arguments after the first one examine the terms -recursively, as shown above. In the third line, the fourth term is -picked from the original polynomial, 10x^2y^3, then the second term -from that, x^2, and finally the first component, x. If an integer's -absolute value is too large for the appropriate expression, a message -is given. - -PART works on the form of the expression as printed, or as it would -have been printed at that point of the calculation, bearing in mind -the current switch settings. It is important to realise that the -switch settings change the operation of PART. PRI must be on when -PART is used. - -When PART is used on a polynomial expression that has minus signs, the -+ is always returned as the top-level operator. The minus is found as -a unary operator attached to the negative term. - -PART can also be used to change the relevant part of the expression or -list as shown in the sixth example line. The PART operator returns the -changed expression, though original expression is not changed. You can -also use PART to change the operator. - -\endsection -\xitem[partial differentiation] -partial differentiation (page 251) - -\endsection -\item[PAUSE] -PAUSE (page 160)) -The PAUSE command, given in an interactive file, stops operation and -asks if you want to continue or not. - -Examples: -An interactive file is running, and at some point you see the -question - Cont? (Y or N) -If you type y {Return} -the file continues to run until the next pause or the end. -If you type n {Return} - -you will get a numbered REDUCE prompt, and be allowed to enter and -execute any REDUCE statements. If you later wish to continue with the -file, type - cont; -and the file resumes. - -To use PAUSE in your own interactive files, type - -PAUSE; - -in the file wherever you want it. - -PAUSE does not allow you to continue without typing either Y or N. -Its use is to slow down scrolling of interactive files, or to let you -change parameters or switch settings for the calculations. - -If you have stopped an interactive file at a PAUSE, and do not wish to -resume the file, type END;. This does not end the REDUCE session, but -stops input from the file. A second END; ends the REDUCE session. -However, if you have pauses from more than one file stacked up, an -END; brings you back to the top level, not the file directly above. - -A PAUSE typed from the terminal has no effect. - -\endsection -\xitem[PCLASS] -PCLASS (pages 379, 380, 383) - -\endsection -\xitem[Percent sign] -Percent sign (page 38) - -\endsection -\item[PERIOD] -PERIOD (page 111) - -When PERIOD is on, periods are added after integers in -Fortran-compatible output (when FORT is on). There is no effect -when FORT is off. Default is ON. - -\endsection -\item[PF] -PF (page 83) - - PF(expression,variable) - -PF transforms expression into a LIST of partial fractions with respect -to the main variable, variable. PF does a complete partial fraction -decomposition, and as the algorithms used are fairly unsophisticated -(factorisation and the extended Euclidean algorithm), the code may be -unacceptably slow in complicated cases. - -Examples: - 2 - 2 2 -pf(2/((x+1)^2*(x+2)),x); {-------,-------,--------------} - x + 2 x + 1 2 - x + 2*x + 1 -off exp; -pf(2/((x+1)^2*(x+2)),x); - 2 - 2 2 -pf(2/((x+1)^2*(x+2)),x); {-------,-------,----------} - x + 2 x + 1 2 - (x + 1) - - 2 -for each j in ws sum j; ------------------ - 2 - (x + 2)*(x + 1) - -If you want the denominators in factored form, turn EXP off, as shown -in the second example above. As shown in the final example, the FOR -EACH construct can be used to recombine the terms. Alternatively, one -can use the operations on lists to extract any desired term. - -\endsection -\xitem[PFORM command] -PFORM command (page 271) - -\endsection -\xitem[PFORM statement] -PFORM statement (page 249) - -\endsection -\item[PI] -PI (page 37) - -The identifier PI is reserved for use as the circular constant. -Its value is given by 3.14159265358..., which REDUCE gives to the current -decimal precision when REDUCE is in a floating-point mode. - - -PI may be used as a looping variable in a FOR statement, -or as a local variable in a PROCEDURE. Its value in such cases will be -taken from the local environment. - -\endsection -\xitem[PLOT] -PLOT (page 181) - -\endsection -\item[POCHHAMMER] -POCHHAMMER (pages 185, 394) - -The POCHHAMMER operator implements the Pochhammer notation -(shifted factorial). - - POCHHAMMER(expression,expression) - -Examples: -load_package specfn; (SPECFN) -pochhammer(17,4); 116280 - - FACTORIAL(2*Z) -pochhammer(1/2,z); ------------------- - 2*Z - 2 *FACTORIAL(Z) - -A number of complex rules for POCHHAMMER are inactive, because they -cause a huge system load in algebraic mode. If one wants to use more -rules for the simplification of Pochhammer's notation, one can do: - let special!*pochhammer!*rules; - -\endsection -\item[POLYGAMMA] -POLYGAMMA (pages 185, 395) - -The POLYGAMMA operator returns the Polygamma function. - - Polygamma(n,x) := df(Psi(z),z,n); - - POLYGAMMA(integer,expression) - -Examples: - load_package specfn; (SPECFN) - PI - 6 - Polygamma(1,2); --------- - 6 - on rounded; - Polygamma(1,2.35); 0.52849689109 - -The POLYGAMMA function is used for simplification of the ZETA function -for some arguments. - -\endsection -\xitem[Polynomial] -Polynomial (page 119) - -\endsection -\xitem[Polynomial equations] -Polynomial equations (page 181) - -\endsection -\xitem[POSITIVE] -POSITIVE (page 368) - -\endsection -\xitem[power series] -power series (page 413) - -\endsection -\xitem[power series arithmetic] -power series - arithmetic (page 422) - composition (page 420) - differentiation (page 422) - of integral (page 415) - of user defined function (page 415) - -\endsection -\item[PRECEDENCE] -PRECEDENCE (page 94) - -The PRECEDENCE declaration attaches a precedence to an infix operator. - - PRECEDENCE operator, known_operator - -operator should have been declared an operator but may be a REDUCE -identifier that is not already an operator, array, or matrix. -known_operator must be a system infix operator or have had its -precedence already declared. - -Examples: -operator f,h; -precedence f,+; -precedence h,*; -a + f(1,2)*c; (1 F 2)*C + A -a + h(1,2)*c; 1 H 2*C + A -a*1 f 2*c; A F 2*C -a*1 h 2*c; 1 H 2*A*C - -The operator whose precedence is being declared is inserted into the -infix operator precedence list at the next higher place than -known-operator. - -Attaching a precedence to an operator has the side effect of declaring -the operator to be infix. If the identifier argument for PRECEDENCE -has not been declared to be an operator, an attempt to use it causes -an error message. After declaring it to be an operator, it becomes an -infix operator with the precedence previously given. Infix operators -may be used in prefix form; if they are used in infix form, a space -must be left on each side of the operator to avoid ambiguity. -Declared infix operators are always binary. - -To see the infix operator precedence list, enter symbolic mode and -type PRECLIS!*;. The lowest precedence operator is listed first. - -All prefix operators have precedence higher than infix operators. - -\endsection -\item[PRECISE] -PRECISE (page 78) - -When the PRECISE switch is on, simplification of roots of even -powers returns absolute values, a more precise answer mathematically. - -Examples: -sqrt(x**2); X -(x**2)**(1/4); SQRT(X) -on precise; -sqrt(x**2); ABS(X) -(x**2)**(1/4); SQRT(ABS(X)) - -In many types of mathematical work, simplification of powers and surds -can proceed by the fastest means of simplifying the exponents -arithmetically. When it is important to you that the positive root be -returned, turn PRECISE on. One situation where this is important is -when graphing square-root expressions such as sqrt(x^2+y^2) to avoid a -spike caused by REDUCE simplifying sqrt(y^2) to y when x is zero. - -\endsection -\item[PRECISION] -PRECISION (pages 132, 374) - -The PRECISION declaration sets the number of decimal places used when -ROUNDED is on. Default is system dependent, and normally about 12. - - PRECISION(integer) or PRECISION integer - -integer must be a positive integer. When integer is 0, the current -precision is displayed, but not changed. There is no upper limit, but -precision of greater than several hundred causes unpleasantly slow -operation on numeric calculations. - -Examples: -on rounded; -7/9; 0.777777777778 -precision 20; 20 -7/9; 0.77777777777777777778 -sin(pi/4); 0.7071067811865475244 - -Trailing zeroes are dropped, so sometimes fewer than 20 decimal places -are printed as in the last example. Turn on the switch FULLPREC if -you want to print all significant digits. The ROUNDED mode carries -calculations to two more places than given by PRECISION, and rounds -off. - -\endsection -\item[PREDUCE] -PREDUCE (page 308) - - PREDUCE(p, {exp, ... }[,vars]) - -where p is an expression, and {exp, ... } is a list of expressions or -equations and vars is an optional list of variables (see IDEAL -parameters). - -PREDUCE computes the remainder of EXP modulo the given set of -polynomials resp. equations. This result is unique (canonical) only -if the given set is a GROEBNER basis under the current TERM order. - -see also: PREDUCET operator. - -\endsection -\item[PREDUCET] -PREDUCET (page 311) - - PREDUCE(p,{v=exp...}[,vars]) - -where p is an expression, v are kernels (simple or indexed variables), -EXP are polynomials and optional vars is a variable list (see IDEAL -parameters). - -PREDUCET computes the remainder of p modulo {exp,...} similar to -PREDUCE, but the result is an equation which expresses the remainder -as combination of the polynomials. - -Example: - - gb2 := {g1=2*x - y + 1,g2=9*y**2 - 2*y - 199}$ - preducet(q=x**2,gb2); - - - 16*Y + 208= - 18*X*G1 - 9*Y*G1 + 36*Q + 9*G1 - G2 - - -\endsection -\xitem[Prefix] -Prefix (pages 71, 93, 95) - -\endsection -\xitem[Prefix operator] -Prefix operator (page 38, 39) - -\endsection -\item[PRET] -PRET (pages 217, 218) - -When PRET is on, input is printed in standard REDUCE format and then -evaluated. - -Examples: -on pret; -(x+1)^3; (x + 1)**3; - 3 2 - X + 3*X + 3*X + 1 - -procedure fac(n); - if not (fixp(n) and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n-1 product i+1; - procedure fac n; - if not (fixp n and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n - 1 product i + 1; - - FAC - -fac 5; fac 5; - 120 - -Note that all input is converted to lower case except strings (which -keep the same case) all operators with a single argument have had the -parentheses removed, and all infix operators have had a space added on -each side. In addition, syntactical constructs like IF...THEN...ELSE -are printed in a standard format. - -\endsection -\xitem[PRETTYPRINT] -PRETTYPRINT (page 218) - -\endsection -\xitem[Prettyprinting] -Prettyprinting (pages 217, 218) - -\endsection -\xitem[PRGEN] -PRGEN (page 378) - -\endsection -\item[PRI] -PRI (page 101) - -When PRI is on, the declarations ORDER and FACTOR can -be used, and the switches ALLFAC, DIV, RAT, -and REVPRI take effect when they are on. Default is ON. - - -Printing of expressions is faster with PRI off. The expressions are -then returned in one standard form, without any of the display options that -can be used to feature or display various parts of the expression. You can -also gain insight into REDUCE's representation of expressions with -PRI off. - -\endsection -\item[PRIMEP] -PRIMEP (page 46) - - PRIMEP(expression) or PRIMEP simple_expression - -If expression evaluates to a integer, PRIMEP returns TRUE if -expression is a prime number and NIL otherwise. If expression does -not have an integer value, a type error occurs. - -Examples: -if primep 3 then write "yes" else write "no"; YES -if primep a then 1; ***** A invalid as integer - -\endsection -\item[PRINT_PRECISION] -PRINT_PRECISION (page 133) - - PRINT_PRECISION(integer) or PRINT_PRECISION integer - -In ROUNDED mode, numbers are normally printed to the specified -precision. If the user wishes to print such numbers with less -precision, the printing precision can be set by the declaration -PRINT_PRECISION. - -Examples: -on rounded; -1/3; 0.333333333333 -print_precision 5; -1/3 0.33333 - -\endsection -\item[PROCEDURE] -PROCEDURE (page 169) - -The PROCEDURE command allows you to define a mathematical operation as a -function with arguments. - PROCEDURE identifier (arg{,arg});body - -The option may be ALGEBRAIC or SYMBOLIC, indicating the mode under -which the procedure is executed, or REAL or INTEGER, indicating the -type of answer expected. The default is algebraic. Real or integer -procedures are subtypes of algebraic procedures; type-checking is done -on the results of integer procedures, but not on real procedures (in -the current REDUCE release). identifier may be any valid REDUCE -identifier that is not already a procedure name, operator, ARRAY or -MATRIX. arg is a formal parameter that may be any valid REDUCE -identifier. body is a single statement (a GROUP or BLOCK statement -may be used) with the desired activities in it. - -Examples: - -procedure fac(n); - if not (fixp(n) and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n-1 product i+1; - FAC -fac(0); 1 -fac(5); 120 -fac(-5); ***** choose nonneg. integer only - -Procedures are automatically declared as operators upon definition. -When REDUCE has parsed the procedure definition and successfully -converted it to a form for its own use, it prints the name of the -procedure. Procedure definitions cannot be nested. Procedures can -call other procedures, or can recursively call themselves. Procedure -identifiers can be cleared as you would clear an operator. Unlike LET -statements, new definitions under the same procedure name replace the -previous definitions completely. - -Be careful not to use the name of a system operator for your own -procedure. REDUCE may or may not give you a warning message. If you -redefine a system operator in your own procedure, the original -function of the system operator is lost for the remainder of the -REDUCE session. - -Procedures may have none, one, or more than one parameter. A REDUCE -parameter is a formal parameter only; the use of x as a parameter in a -PROCEDURE definition has no connection with a value of x in the REDUCE -session, and the results of calling a procedure have no effect on the -value of x. If a procedure is called with x as a parameter, the -current value of x is used as specified in the computation, but is not -changed outside the procedure. Making an assignment statement by := -with a formal parameter on the left-hand side only changes the value -of the calling parameter within the procedure. - -Using a LET statement inside a procedure always changes the value -globally: a LET with a formal parameter makes the change to the -calling parameter. LET statements cannot be made on local variables -inside BEGIN...END BLOCKS. When CLEAR statements are used on formal -parameters, the calling variables associated with them are cleared -globally too. The use of LET or CLEAR statements inside procedures -should be done with extreme caution. - -Arrays and operators may be used as parameters to procedures. The -body of the procedure can contain statements that appropriately -manipulate these arguments. Changes are made to values of the calling -arrays or operators. Simple expressions can also be used as -arguments, in the place of scalar variables. Matrices may not be used -as arguments to procedures. - -A procedure that has no parameters is called by the procedure name, -immediately followed by empty parentheses. The empty parentheses may -be left out when writing a procedure with no parameters, but must -appear in a call of the procedure. If this is a nuisance to you, use -a LET statement on the name of the procedure (i.e., LET NOARGS = -NOARGS()) after which you can call the procedure by just its name. - -Procedures that have a single argument can leave out the parentheses -around it both in the definition and procedure call. (You can use the -parentheses if you wish.) Procedures with more than one argument must -use parentheses, with the arguments separated by commas. - -Procedures often have a BEGIN...END block in them. Inside the block, -local variables are declared using SCALAR, REAL or INTEGER -declarations. The declarations must be made immediately after the -word BEGIN, and if more than one type of declaration is made, they are -separated by semicolons. REDUCE currently does no type checking on -local variables; REAL and INTEGER are treated just like SCALAR. -Actions take place as specified in the statements inside the block -statement. Any identifiers that are not formal parameters or local -variables are treated as global variables, and activities involving -these identifiers are global in effect. - -If a return value is desired from a procedure call, a specific RETURN -command must be the last statement executed before exiting from the -procedure. If no RETURN is used, a procedure returns a zero or no -value. - -Procedures are often written in a file using an editor, then the file -is input using the command IN. This method allows easy changes in -development, and also allows you to load the named procedures whenever -you like, by loading the files that contain them. - -\endsection -\xitem[Procedure body] -Procedure body (pages 171--173) - -\endsection -\xitem[Procedure heading] -Procedure heading (page 170) - -\endsection -\item[PROD] -PROD operator (page 403) - -The operator PROD returns -the indefinite or definite product of a given expression. - - -PROD(expr,k[,lolim [,uplim ]]) - - -where expr is the expression to be multiplied, k is the -control variable (a KERNEL), and lolim and uplim -uplim are the optional lower and upper limits. If uplim is -not supplied the upper limit is taken as k. The GOSPER -algorithm is used. If there is no closed form solution, the operator -returns the input unchanged. - -Examples: -prod(k/(k-2),k); k*( - k + 1) - -\endsection -\item[PRODUCT] -PRODUCT (page 57, 58) - -See the FOR loop construction. - -\endsection -\xitem[Program] -Program (page 38) - -\endsection -\xitem[Program structure] -Program structure (page 33) - -\endsection -\xitem[Proper statement] -Proper statement (pages 48, 53, 54) - -\endsection -\xitem[PRSYS] -PRSYS (pages 378, 382) - -\endsection -\xitem[PS] -PS (page 188) - -\endsection -\xitem[PS operator] -PS operator (page 414) - -\endsection -\xitem[PSCHANGEVAR operator] -PSCHANGEVAR operator (page 418) - -\endsection -\xitem[PSCOMPOSE operator] -PSCOMPOSE operator (page 420) - -\endsection -\xitem[PSDEPVAR operator] -PSDEPVAR operator (page 418) - -\endsection -\xitem[PSEXPANSIONPT operator] -PSEXPANSIONPT operator (page 418) - -\endsection -\xitem[PSEXPLIM operator] -PSEXPLIM operator (pages 414, 416) - -\endsection -\xitem[PSFUNCTION operator] -PSFUNCTION operator (page 418) - -\endsection -\item[PSI] -PSI (pages 185, 395) - -The PSI operator returns the Psi (or DiGamma) function. - - Psi(x) := df(Gamma(z),z)/ Gamma (z) - - GAMMA(expression) - -Examples: - load_package specfn; - 1 - 2*LOG(2) + PSI(---) + PSI(1) + 3 - 2 - Psi(3); ---------------------------------- - 2 - - on rounded; - - Psi(1); 0.577215664902 - -Euler's constant can be found as - Psi(1). - -\endsection -\xitem[PSINTCONST (shared)] -PSINTCONST (shared) (page 415) - -\endsection -\xitem[PSORDER operator] -PSORDER operator (page 417) - -\endsection -\xitem[PSORDLIM operator] -PSORDLIM operator (page 416) - -\endsection -\xitem[PSREVERSE operator] -PSREVERSE operator (page 419) - -\endsection -\xitem[PSSETORDER operator] -PSSETORDER operator (page 417) - -\endsection -\xitem[PSSUM operator] -PSSUM operator (page 421) - -\endsection -\xitem[PSTERM operator] -PSTERM operator (page 417) - -\endsection -\xitem[Puiseux expansion] -Puiseux expansion (page 419) - -\endsection -\xitem[PUTCSYSTEM command] -PUTCSYSTEM command (page 235) - -\endsection -\xitem[Quadrature] -Quadrature (page 182) - -\endsection -\item[QUIT] -QUIT (page 70) - -The QUIT command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are -at the top level, the QUIT command exits REDUCE. BYE is a synonym for -QUIT. - -\endsection -\xitem[QUOTE] -QUOTE (page 193) - -\endsection -\xitem[RANDOM] -RANDOM (page 74) - -\endsection -\xitem[RANDOM_NEW_SEED] -RANDOM_NEW_SEED (page 75) - -\endsection -\item[RANK] -RANK (page 167) - - RANK(matrix_expression) -RANK calculates the rank of its matrix argument. - -Examples: - rank mat((a,b,c),(d,e,f)); 2 - -The argument to RANK can also be a LIST of lists, interpreted either -as a row matrix or a set of equations. If that form of input is -chosen, the vectors in the result will be represented by lists as -well. This additional input syntax facilitates the use of RANK in -applications different from classical linear algebra. - -\endsection -\item[RAT] -RAT (page 104) - -When the RAT switch is on, and kernels have been selected to display -with the FACTOR declaration, the denominator is printed with each -term rather than one common denominator at the end of an expression. - -Examples: 3 - SIN(Y)*X + SIN(Y) + X -(x+1)/x + x**2/sin y; ------------------------ - SIN(Y)*X -factor x; - 3 - X + X*SIN(Y) + SIN(Y) -(x+1)/x + x**2/sin y; ------------------------ - X*SIN(Y) -on rat; - 2 - X -1 -(x+1)/x + x**2/sin y; -------- + 1 + X - SIN(Y) - -The RAT switch only has effect when the PRI switch is on. -When PRI is off, regardless of the setting of RAT, the -printing behaviour is as if RAT were off. RAT only has -effect upon the display of expressions, not their internal form. - -\endsection -\item[RATARG] -RATARG (pages 115, 128) - -When RATARG is on, rational expressions can be given to operators -such as COEFF and LTERM that normally require -polynomials in one of their arguments. When RATARG is off, rational -expressions cause an error message. - -Examples: 3 2 3 - X + X*Y + Y -aa := x/y**2 + 1/x + y/x**2; AA := ---------------- - 2 2 - X *Y - 3 2 3 - X + X*Y + Y -coeff(aa,x); ***** ---------------- invalid as POLYNOMIAL - 2 2 - X *Y -on ratarg; - Y 1 1 -coeff(aa,x); {----,----,0,-------} - 2 2 2 2 - X X X *Y - -\endsection -\item[RATIONAL] -RATIONAL (page 132) - -When RATIONAL is on, polynomial expressions with rational coefficients -are produced. - -Examples: - 2*X + 3*Y -x/2 + 3*y/4; ----------- - 4 - 2 - X + 5*X + 17 -(x**2 + 5*x + 17)/2; --------------- - 2 -on rational; - 1 3 -x/2 + 3y/4; ---*(X + ---*Y) - 2 2 - - 1 2 -(x**2 + 5*x + 17)/2; ---*(X + 5*X + 17) - 2 - -By using RATIONAL, polynomial expressions with rational coefficients -can be used in some commands that expect polynomials. With RATIONAL -off, such a polynomial becomes a rational expression, with denominator -the least common multiple of the denominators of the rational number -coefficients. - -\endsection -\xitem[Rational coefficient] -Rational coefficient (page 132) - -\endsection -\xitem[Rational function] -Rational function (page 119) - -\endsection -\item[RATIONALIZE] -RATIONALIZE (page 135) - -When the RATIONALIZE switch is on, denominators of rational expressions -that contain complex numbers or root expressions are simplified by -multiplication by their conjugates. - -Examples: - SQRT(3) + 1 -qq := (1+sqrt(3))/(sqrt(3)-7); QQ := ------------- - SQRT(3) - 7 -on rationalize; - - 4*SQRT(3) - 5 -qq; ------------------ - 23 - 2/3 1/3 - 6 - 4*6 + 16 -2/(4 + 6**(1/3)); -------------------- - 35 -on complex; - 1 - 2*i -(i-1)/(i+3); --------- - 5 - - -\endsection -\item[RATPRI] -RATPRI (page 104) - -When the RATPRI switch is on, rational expressions and fractions are -printed as two lines separated by a fraction bar, rather than in a linear -style. Default is ON. - -Examples: - 3 -3/17; ---- - 17 - 3*B + 2*Y -2/b + 3/y; ----------- - B*Y -off ratpri; -3/17; 3/17 -2/b + 3/y; (3*B + 2*Y)/(B*Y) - -\endsection -\xitem[RATROOT] -RATROOT (page 373) - -\endsection -\item[REAL] -REAL (page 61) - -The REAL declaration must be made immediately after a BEGIN (or other -variable declaration such as INTEGER and SCALAR) and declares local -integer variables. They are initialised to zero. - - REAL identifier{,identifier} - -identifier may be any valid REDUCE identifier, except T or NIL. - -Real variables remain local, and do not share values with variables of -the same name outside the BEGIN...END block. When the block is -finished, the variables are removed. You may use the words INTEGER or -SCALAR in the place of REAL. REAL does not indicate type-checking by -the current REDUCE; it is only for your own information. Declaration -statements must immediately follow the BEGIN, without a semicolon -between BEGIN and the first variable declaration. - -Any variables used inside a BEGIN...END BLOCK that were not declared -SCALAR, REAL or INTEGER are global, and any change made to them inside -the block affects their global value. Any array or matrix declared -inside a block is always global. - -\endsection -\xitem[Real] -Real (pages 34, 35) - -\endsection -\xitem[Real coefficient] -Real coefficient (page 132) - -\endsection -\item[REALROOTS] -REALROOTS (pages 369, 370) - -The operator REALROOTS finds that real roots of a polynomial to an -accuracy that is sufficient to separate them and which is a minimum of -6 decimal places. - - REALROOTS(p) - REALROOTS(p,from,to) - -where p is a univariate polynomial. The optional parameters from and -to classify an interval: if given, exactly the real roots in this -interval will be returned. from and to can also take the values -INFINITY or -INFINITY. If omitted all real roots will be returned. -Result is a LIST of equations which represent the roots of the -polynomial at the given accuracy. - -Examples: - realroots(x^5-2); {X=1.1487} - realroots(x^3-104*x^2+403*x-300,2,infinity); {X=3.0,X=100.0} - realroots(x^3-104*x^2+403*x-300,-infinity,2); {X=1} - -The minimal accuracy of the result values is controlled by ROOTACC. - -\endsection -\xitem[REDERR] -REDERR (page 173) - -\endsection -\item[REDUCT] -REDUCT (page 131) -The REDUCT operator returns the remainder of its expression after the -leading term is removed. - - REDUCT(expression,kernel) - -expression is ordinarily a polynomial. If RATARG is on, a rational -expression may also be used, otherwise an error results. kernel must -be a KERNEL. - -Examples: - 3 -reduct((x+y)**3,x); (x + y) - -reduct(x + sin(x)**3,sin(x)); x - 3 -reduct(x + sin(x)**3,y); sin(x) + x - -If the expression does not contain the kernel, REDUCT returns the -expression. - -\endsection -\xitem[side relations] -relations - side (page 241) - -\endsection -\item[REMAINDER] -REMAINDER (page 126) -The REMAINDER operator returns the remainder after its first -argument is divided by its second argument. - - REMAINDER(expression,expression) - -expression can be any valid REDUCE polynomial, and is not limited -to numeric values. - -Examples: -remainder(13,6); 1 -remainder(x**2 + 3*x + 2,x+1); 0 -remainder(x**3 + 12*x + 4,x**2 + 1); 11*X + 4 -remainder(sin(2*x),x*y); SIN(2*X) - -If the first argument to REMAINDER contains a denominator not equal to -1, an error occurs. - -\endsection -\item[REMFAC] -REMFAC (page 102) - -The REMFAC declaration removes the special factoring treatment of its -arguments that was declared with FACTOR. - -REMFAC kernel{,kernel} - -kernel must be a KERNEL or OPERATOR name that was declared as special -with the FACTOR declaration. - -\endsection -\xitem[REMFORDER command] -REMFORDER command (pages 268, 271) - -\endsection -\item[REMIND] -REMIND (page 206) - -The REMIND declaration removes the special status of its arguments -as indices, which was set in the INDEX declaration, in -high-energy physics calculations. - REMIND identifier{,identifier} - -identifier must have been declared to be of type INDEX. - -\endsection -\xitem[RENOSUM command] -RENOSUM command (pages 262, 271) - -\endsection -\item[REPART] -REPART (pages 72, 73, 75) - - REPART(expression) or REPART simple_expression - -This operator returns the real part of an expression, if that argument -has an numerical value. A non-numerical argument is returned as an -expression in the operators REPART and IMPART. - -Examples: -repart(1+i); 1 -repart(a+i*b); REPART(A) - IMPART(B) - -\endsection -\item[REPEAT] -REPEAT (pages 60, 61, 63, 65) - -The REPEAT command causes repeated execution of a statement UNTIL -the given condition is found to be true. The statement is always executed -at least once. - REPEAT statement UNTIL condition - -statement can be a single statement, GROUP statement, or -a BEGIN...END BLOCK. condition must be a logical -operator that evaluates to rue or nil. - -Examples: -<> until m = 0>>; - 400*X - 300*X - 200*X - 100*X - -<> until m <= 0>>; - -1 - -REPEAT must always be followed by an UNTIL with a condition. Be -careful not to generate an infinite loop with a condition that is -never true. In the second example, if the condition had been M = 0, -it would never have been true since M already had value -2 when the -condition was first evaluated. - -\endsection -\xitem[Reserved variable] -Reserved variable (pages 36, 37) - -\endsection -\item[REST] -REST (page 50) - -The REST operator returns a LIST containing all but the first element of -the list it is given. - REST(list) or REST list - - -list must be a non-empty list, but need not have more than one element. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D}; -rest alist; {B,C,D} -blist := {x,y,{aa,bb,cc},z}; BLIST := {X,Y,{AA,BB,CC},Z} -second rest blist; {AA,BB,CC} -clist := {c}; CLIST := C -rest clist; {} - -\endsection -\xitem[RESULT] -RESULT (page 378) - -\endsection -\item[RESULTANT] -RESULTANT (page 126) -The RESULTANT operator computes the resultant of two polynomials with -respect to a given variable. If the resultant is 0, the polynomials -have a root in common. - RESULTANT(expression,expression,kernel) - -expression must be a polynomial containing kernel ; -kernel must be a KERNEL. - -Examples: -resultant(x**2 + 2*x + 1,x+1,x); 0 -resultant(x**2 + 2*x + 1,x-3,x); 16 -resultant(z**3 + z**2 + 5*z + 5, - z**4 - 6*z**3 + 16*z**2 - 30*z + 55, - z); 0 -resultant(x**3*y + 4*x*y + 10,y**2 + 6*y + 4,y); - 6 4 3 2 - 4*(x + 8*x - 15*x + 16*x - 60*x + 25) - -The resultant is the determinant of the Sylvester matrix, formed from the -coefficients of the two polynomials in the following way: - -Given two polynomials: - - n n-1 - a x + a1 x + ... + an - -and - m m-1 - b x + b1 x + ... + bm - -form the (m+n)x(m+n-1) Sylvester matrix by the following means: - - 0.......0 a a1 .......... an - 0....0 a a1 .......... an 0 - . . . . - a0 a1 .......... an 0.......0 - 0.......0 b b1 .......... bm - 0....0 b b1 .......... bm 0 - . . . . - b b1 .......... bm 0.......0 - -If the determinant of this matrix is 0, the two polynomials have a -common root. Finding the resultant of large expressions is -time-consuming, due to the time needed to find a large determinant. - -The sign conventions RESULTANT uses are those given in the article, -``Computing in Algebraic Extensions,'' by R. Loos, appearing in -Computer Algebra--Symbolic and Algebraic Computation, 2nd ed., edited -by B. Buchberger, G.E. Collins and R. Loos, and published by - -Springer-Verlag, 1983. - -These are: - resultant(p(x),q(x),x) = (-1)^{deg p(x)*deg q(x)} * resultant(q(x),p(x),x), - resultant(a,p(x),x) = a^{deg p(x)}, - resultant(a,b,x) = 1 - -where p(x) and q(x) are polynomials which have x as a variable, and -a and b are free of x. - -Error messages are given if RESULTANT is given a non-polynomial -expression, or a non-kernel variable. - -\endsection -\item[RETRY] -RETRY (page 157) -The RETRY command allows you to retry the latest statement that resulted -in an error message. - -Examples: -matrix a; -det a; ***** Matrix A not set -a := mat((1,2),(3,4)); A(1,1) := 1 - A(1,2) := 2 - A(2,1) := 3 - A(2,2) := 4 -retry; -2 - -RETRY remembers only the most recent statement that resulted in an -error message. It allows you to stop and fix something obvious, then -continue on your way without retyping the original command. - -\endsection -\item[RETURN] -RETURN (pages 62--64) - -The RETURN command causes a value to be returned from inside a -BEGIN...END BLOCK. - BEGIN statements RETURN (expression) - END - -statements can be any valid REDUCE statements. The value of -expression is returned. - -Examples: -begin write "yes"; return a end; yes - A -procedure dumb(a); - begin if numberp(a) then return a - else return 10 end; - DUMB -dumb(x); 10 -dumb(-5); -5 -procedure dumb2(a); - begin c := a**2 + 2*a + 1; - d := 17; c*d; return end; - DUMB2 -dumb2(4); -c; 25 -d; 17 - -Note in DUMB2 above that the assignments were made as requested, but -the product C*D cannot be accessed. Changing the procedure to read -RETURN C*D would remedy this problem. - -The RETURN statement is always the last statement executed before -leaving the block. If RETURN has no argument, the block is exited but -no value is returned. A block statement does not need a RETURN ; the -statements inside terminate in their normal fashion without one. In -that case no value is returned, although the specified actions inside -the block take place. - -The RETURN command can be used inside <<...>> GROUP statements and -IF...THEN...ELSE commands that are inside BEGIN...END BLOCKs. It is -not valid in these constructions that are not inside a BEGIN...END -block. It is not valid inside FOR, REPEAT...UNTIL or WHILE...DO loops -in any construction. To force early termination from loops, the GO -TO(GOTO) command must be used. When you use nested block statements, -a RETURN from an inner block exits returning a value to the -next-outermost block, rather than all the way to the outside. - -\endsection -\item[REVERSE] -REVERSE (page 51) - -The REVERSE operator returns a LIST that is the reverse of the list it -is given. - REVERSE(list) or REVERSE list - -list must be a LIST. - -Examples: - 2 3 -aa := {c,b,a,{x**2,z**3},y}; AA := {C,B,A,{X ,Z },Y} - 2 3 -reverse aa; {Y,{X ,Z},A,B,C} - 2 3 -reverse(q . reverse aa); {C,B,A,{X ,Z },Y,Q} - -REVERSE and CONS can be used together to add a new element to the end -of a list (. adds its new element to the beginning). The REVERSE -operator uses a noticeable amount of system resources, especially if -the list is long. If you are doing much heavy-duty list manipulation, -you should probably design your algorithms to avoid much reversing of -lists. A moderate amount of list reversing is no problem. - -\endsection -\item[REVGRADLEX] -REVGRADLEX (page 293) - -The terms are ordered first with their total degree (degree sum), and -if the total degree is identical the comparison is the inverse of LEX -term order. With GROEBNER and GROEBNERF calculations this term order -is similar to GRADLEX term order; it is known as most efficient -ordering with respect to computing time. - -\endsection -\item[REVPRI] -REVPRI (page 105) - -When the REVPRI switch is on, terms are printed in reverse order from -the normal printing order. - -Examples: - 5 2 -x**5 + x**2 + 18 + sqrt(y); SQRT(Y) + X + X + 18 - -a + b + c + w; A + B + C + W - -on revpri; - 2 5 -x**5 + x**2 + 18 + sqrt(y); 17 + X + X + SQRT(Y) - -a + b + c + w; W + C + B + A - -Turn REVPRI on when you want to display a polynomial in ascending -rather than descending order. - -\endsection -\item[RHS] -RHS (page 47) -The RHS operator returns the right-hand side of an EQUATION, such as -those returned in a LIST by SOLVE. - - RHS(equation) or RHS equation - -equation must be an equation of the form left-hand side = right-hand side. - -Examples: - roots := solve(x**2 + 6*x*y + 5x + 3y**2,x); - - 2 - - (SQRT(24*Y + 60*Y + 25) + 6*Y + 5) - ROOTS := {X=----------------------------------------, - 2 - - 2 - SQRT(24*Y + 60*Y + 25) - 6*Y - 5 - X=-----------------------------------} - 2 - -root1 := rhs first roots; - - (SQRT(24*Y + 60*Y + 25) + 6*Y + 5) - ROOT1 := ---------------------------------------- - 2 -root2 := rhs second roots; - 2 - SQRT(24*Y + 60*Y + 25) - 6*Y - 5 - ROOT2 := ----------------------------------- - 2 - -An error message is given if RHS is applied to something other than an -equation. - -\endsection -\xitem[Riemann Zeta Function] -Riemann Zeta Function (pages 185, 395) - -\endsection -\xitem[RIEMANNCONX command] -RIEMANNCONX command (pages 267, 271) - -\endsection -\xitem[Riemannian Connections] -Riemannian Connections (page 267) - -\endsection -\xitem[Rlisp] -Rlisp (page 213) - -\endsection -\item[RLISP88] -RLISP88 (page 204) - -Rlisp '88 is a superset of the Rlisp that has been traditionally used -for the support of REDUCE. It is fully documented in the book Marti, -J.B., ``RLISP '88: An Evolutionary Approach to Program Design and -Reuse'', World Scientific, Singapore (1993). It supports different -looping constructs from the traditional Rlisp, and treats ``-'' as a -letter unless separated by spaces. Turning on the switch RLISP88 -converts to Rlisp '88 parsing conventions in symbolic mode, and -enables the use of Rlisp '88 extensions. Turning off the switch -reverts to the traditional Rlisp and the previous mode (SYMBOLIC or -ALGEBRAIC) in force before RLISP88 was turned on. - -\endsection -\item[RLROOTNO] -RLROOTNO (page 369) - -The function RLROOTNO computes the number of real roots of p in the -specified region, but does not find the roots. - - RLROOTNO(expression) - RLROOTNO(expression, POSITIVE) - RLROOTNO(expression, NEGATIVE) - RLROOTNO(expression, lo, hi) - -For more details on the specification of an interval, see ISOLATER. - -Examples: - load_package roots; - rlrootno (x^3-3x^2+2x+10); 1 - rlrootno(x^3-3x^2+2x+10,positive); 0 -\endsection -\xitem[root finding] -root finding (page 367) - -\endsection -\item[ROOT_OF] -ROOT_OF (pages 85, 86) - -When the operator SOLVE is unable to find an explicit solution or if -that solution would be too complicated, the result is presented as -formal root expression using the internal operator ROOT_OF and a new -local variable. An expression with a top level ROOT_OF is implicitly a -list with an unknown number of elements since we can't always know how -many solutions an equation has. If a substitution is made into such an -expression, closed form solutions can emerge. If this occurs, the -ROOT_OF construct is replaced by an operator ONE_OF. At this point it -is of course possible to transform the result if the original SOLVE -operator expression into a standard SOLVE solution. To effect this, -the operator EXPAND_CASES can be used. - -Examples: 7 2 -solve(a*x^7-x^2+1,x); {x=root_of(a*x_ - x_ + 1,x_)} -sub(a=0,ws); {x=one_of(1,-1)} -expand_cases ws; {x=1,x=-1} - -The components of ROOT_OF and ONE_OF expressions can be processed as -usual with operators ARGLENGTH and PART. - -\endsection -\item[ROOT_MULTIPLICITES] -ROOT_MULTIPLICITES - -The ROOT_MULTIPLICITIES variable is set to the list of the -multiplicities of the roots of an equation by the SOLVE operator. - - -SOLVE returns its solutions in a list. The multiplicities of -each solution are put in the corresponding locations of the list -ROOT_MULTIPLICITIES. - -\endsection -\xitem[ROOT_VAL] -ROOT_VAL (page 370) - -\endsection -\item[ROOTACC] -ROOTACC (page 373) - -The operator ROOTACC allows you to set the accuracy up to which the -roots package computes its results. - - ROOTACC(n) - -Here n is an integer value. The internal accuracy of the ROOTS package -is adjusted to a value of MAX(6,N). The default value is 6. - -\endsection -\xitem[ROOTMSG] -ROOTMSG (page 373) - -\endsection -\xitem[ROOTPREC] -ROOTPREC (page 374) - -\endsection -\item[ROOTS] -ROOTS (pages 184, 369, 370) - -The operator ROOTS is the main top level function of the roots -package. It will find all roots, real and complex, of the polynomial -p to an accuracy that is sufficient to separate them and which is a -minimum of 6 decimal places. - - ROOTS(p) - -where p is a univariate polynomial. Result is a LIST of equations -which represent the roots of the polynomial at the given accuracy. In -addition, ROOTS stores separate lists of real roots and complex roots -in the global variables ROOTSREAL and ROOTSCOMPLEX. - -Examples: - - roots(x^5-2); {X=-0.929316 + 0.675188*I, - X=-0.929316 - 0.675188*I, - X=0.354967 + 1.09248*I, - X=0.354967 - 1.09248*I, - X=1.1487} - -The minimal accuracy of the result values is controlled by -ROOTACC. - -\endsection -\xitem[ROOTS package] -ROOTS package (page 367) - -\endsection -\xitem[ROOTS_AT_PREC] -ROOTS_AT_PREC (page 370) - -\endsection -\item[ROOTSCOMPLEX] -ROOTSCOMPLEX (page 369) - -When the operator ROOTS is called the complex roots are collected in -the global variable ROOTSCOMPLEX as LIST. - -\endsection -\item[ROOTSREAL] -ROOTSREAL (page 369) - -When the operator ROOTS is called the real roots are collected in the -global variable ROOTREAL as LIST. - -\endsection -\item[ROUND] -ROUND (page 75) - - ROUND(expression) - -If its argument has a numerical value, ROUND rounds it to the nearest -integer. For non-numeric arguments, the value is an expression in the -original operator. - -Examples: -round 3.4; 3 -round 3.5; 4 -round a; ROUND(A) - -\endsection -\item[ROUNDALL] -ROUNDALL (page 133) - -In ROUNDED mode, rational numbers are normally converted to a -floating point representation. If ROUNDALL is off, this conversion -does not occur. ROUNDALL is normally ON. - -Examples: -on rounded; -1/2; 0.5 -off roundall; - 1 -1/2; --- - 2 - -\endsection -\item[ROUNDBF] -ROUNDBF (page 133) - -When ROUNDED is on, the normal defaults cause underflows to be -converted to zero. If you really want the small number that results -in such cases, ROUNDBF can be turned on. - -Examples: -on rounded; -exp(-100000.1^2); 0 -on roundbf; -exp(-100000.1^2); 1.18441281937E-4342953505 - -If a polynomial is input in ROUNDED mode at the default precision into -any ROOTS function, and it is not possible to represent any of the -coefficients of the polynomial precisely in the system floating point -representation, the switch ROUNDBF will be automatically turned on. -All rounded computation will use the internal bigfloat representation -until the user subsequently turns ROUNDBF off. (A message is output to -indicate that this condition is in effect.) - -\endsection -\item[ROUNDED] -ROUNDED (pages 36, 44, 78, 108, 132, 372) - -When ROUNDED is on, floating-point arithmetic is enabled, with -precision initially at a system default value, which is usually 12 -digits. The precise number can be found by the command PRECISION(0). - -Examples: -pi; PI - - 5 -35/217; ---- - 31 -on rounded; -pi; 3.14159265359 - -35/217; 0.161 - -sqrt(3); 1.73205080756 - -If more than the default number of decimal places are required, use the -PRECISION command to set the required number. - -\endsection -\item[Rule lists] -Rule lists (page 147) - -A RULE is an instruction to replace an algebraic expression -or a part of an expression by another one. - lhs => rhs or - lhs => rhs WHEN cond -lhs is an algebraic expression used as search pattern and -rhs is an algebraic expression which replaces matches of -rhs. => is the operator REPLACE. - -lsh can contain free variables which are preceded by a tilde ~ in -their leftmost position in lhs. If a rule has a WHEN cond part it -will fire only if the evaluation of cond has a result TRUE. cond may -contain references to free variables of lhs. - -Rules can be collected in a LIST which then forms a RULE LIST. RULE -LISTS can be used to collect algebraic knowledge for a specific -evaluation context. - -RULES and RULE LISTS are globally activated and deactivated by LET, -FORALL, CLEARRULES. For a single evaluation they can be locally -activate by WHERE. The active rules for an operator can be visualised -by SHOWRULES. - -Examples: -operator f,g,h; -let f(x) => x^2; - 2 -f(x); X -g_rules:={g(~n,~x)=>h(n/2,x) when evenp n, -g(~n,~x)=>h((1-n)/2,x) when not evenp n}$ -let g_rules; -g(3,x); H(-1,X) - -\endsection -\item[SAVEAS] -SAVEAS (page 99)) -The SAVEAS command saves the current workspace under the name of its -argument. - - SAVEAS identifier - -identifier can be any valid REDUCE identifier. - -Examples: - -(The numbered prompts are shown below, unlike in most examples) -1: solve(x^2-3); {x=sqrt(3),x= - sqrt(3)} -2: saveas rts(0)$ -3: rts(0); {x=sqrt(3),x= - sqrt(3)} - -SAVEAS works only for the current workspace, the last algebraic -expression produced by REDUCE. This allows you to save a result that -you did not assign to an identifier when you originally typed the -input. For access to previous output use WS. - -\endsection -\xitem[savesfs] -savesfs (page 393) - -\endsection -\item[SAVESTRUCTR] -SAVESTRUCTR (page 113) - -When SAVESTRUCTR is on, results of the STRUCTR command are returned as -a list whose first element is the representation for the expression -and the remaining elements are equations showing the relationships of -the generated variables. - -Examples: -off exp; - -structr((x+y)^3 + sin(x)^2); ANS3 - where - 3 2 - ANS3 := ANS1 + ANS2 - - ANS2 := SIN(X) - - ANS1 := X + Y - -ans3; ANS3 -on savestructr; - 3 2 -structr((x+y)^3 + sin(x)^2); {ANS3,ANS3=ANS1 + ANS2 ,ANS2=SIN(X),ANS1=X + Y} - 3 2 -ans3 where rest ws; (X + Y) + SIN(X) - -In normal operation, STRUCTR is only a display command. With -SAVESTRUCTR on, you can access the various parts of the expression -produced by STRUCTR. - -The generic system names use the stem ANS. You can change this to your -own stem by the command VARNAME. REDUCE adds integers to this stem -to make unique identifiers. - -\endsection -\xitem[Saving an expression] -Saving an expression (page 111) - -\endsection -\item[SCALAR] -SCALAR (pages 61, 62) - -The SCALAR declaration must be made immediately after a BEGIN (or -other variable declaration such as INTEGER and REAL) and declares -local scalar variables. They are initialised to 0. - - SCALAR identifier{,identifier} - -identifier may be any valid REDUCE identifier, except T or NIL. - -Scalar variables remain local, and do not share values with variables -of the same name outside the BEGIN...END BLOCK. When the block is -finished, the variables are removed. You may use the words REAL or -INTEGER in the place of SCALAR. REAL and INTEGER do not indicate -type-checking by the current REDUCE; they are only for your own -information. Declaration statements must immediately follow the -BEGIN, without a semicolon between BEGIN and the first variable -declaration. - -Any variables used inside BEGIN...END blocks that were not declared -SCALAR, REAL or INTEGER are global, and any change made to them inside -the block affects their global value. Arrays declared inside a block -are always global. - -\endsection -\xitem[Scalar] -Scalar (page 43) - -\endsection -\xitem[SCALEFACTORS operator] -SCALEFACTORS operator (page 234) - -\endsection -\item[SCIENTIFIC_NOTATION] -SCIENTIFIC_NOTATION (page 34) - - SCIENTIFIC_NOTATION(m) or SCIENTIFIC_NOTATION(m,n) - -m and n are positive integers. SCIENTIFIC_NOTATION controls the -output format of floating point numbers. At the default settings, any -number with five or less digits before the decimal point is printed in -a fixed-point notation, e.g., 12345.6. Numbers with more than five -digits are printed in scientific notation, e.g., 1.234567E+5. -Similarly, by default, any number with eleven or more zeros after the -decimal point is printed in scientific notation. - -When SCIENTIFIC_NOTATION is called with the numerical argument m a -number with more than m digits before the decimal point, or m or more -zeros after the decimal point, is printed in scientific notation. -When SCIENTIFIC_NOTATION is called with a list {m, n}, a number with -more than m digits before the decimal point, or n or more zeros after -the decimal point is printed in scientific notation. - -Examples: - -on rounded; -12345.6; 12345.6 - -123456.5; 1.234565e+5 - -0.00000000000000012; 1.2e-16 - -scientific_notation 20; {5,11} - -5: 123456.7; 123456.7 - -0.00000000000000012; 0.00000000000000012 - -\endsection -\item[SCOPE] -SCOPE (page 185) - -Author: J.A. van Hulzen - -REDUCE Source Code Optimization Package. - -SCOPE is a package for the production of an optimised form of a -set of expressions. It applies an heuristic search for common -(sub)expressions to almost any set of proper REDUCE assignment -statements. The output is obtained as a sequence of assignment -statements. GENTRAN is used to facilitate expression output. - -\endsection -\xitem[SDER(I)] -SDER(I) (page 379) - -\endsection -\item[SEC] -SEC (pages 76, 78) - -The SEC operator returns the secant of its argument. - - SEC(expression) or SEC simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -sec abc; SEC(ABC) -sec(pi); -1 -sec 4; SEC(4) -on rounded; -sec(4); - 1.52988565647 -sec log 5; - 25.8852966005 - -SEC returns a numeric value only if ROUNDED is on. Then the secant is -calculated to the current degree of floating point precision. - -\endsection -\item[SECH] -SECH (pages 76, 78) - -The SECH operator returns the hyperbolic secant of its argument. - - SECH(expression) or SECH simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -sech abc; SECH(ABC) -sech(0); 1 -sech 4; SECH(4) -on rounded; -sech(4); 0.0366189934737 -sech log 5; 0.384615384615 - -SECH returns a numeric value only if ROUNDED is on. Then the -expression is calculated to the current degree of floating point -precision. - -\endsection -\item[SECOND] -SECOND (page 50) - -The SECOND operator returns the second element of a list. - SECOND(list) or SECOND list - -list must be a list with at least two elements, to avoid an error -message. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D} -second alist; B -blist := {x,{aa,bb,cc},z}; BLIST := {X,{AA,BB,CC},Z} -second second blist; BB - -\endsection -\xitem[Selector] -Selector (page 198) - -\endsection -\xitem[Semicolon] -Semicolon (page 53) - -\endsection -\item[SET] -SET (pages 55, 83) - -The SET operator is used for assignments when you want both sides of -the assignment statement to be evaluated. - - SET(restricted_expression,expression) - -expression can be any REDUCE expression; restricted_expression -must be an identifier or an expression that evaluates to an identifier. - -Examples: -a := y; A := Y - 2 -set(a,sin(x^2)); SIN(X ) - 2 -a; SIN(X ) - 2 -y; SIN(X ) - -a := b + c; A := B + C - -set(a-c,z); Z - -b; Z - -Using an ARRAY or MATRIX reference as the first argument to SET has -the result of setting the contents of the designated element to SET's -second argument. You should be careful to avoid unwanted side effects -when you use this facility. - -\endsection -\item[SETMOD] -SETMOD (page 134) - -The SETMOD command sets the modulus value for subsequent MODULAR -arithmetic. - - SETMOD integer - -integer must be positive, and greater than 1. It need not be a prime -number. - -Examples: -setmod 6; 1 -on modular; -16; 4 - 2 -x^2 + 5x + 7; X + 5*X + 1 - X -x/3; --- - 3 -setmod 2; 6 - 4 -(x+1)^4; X + 1 -x/3; X - -SETMOD returns the previous modulus, or 1 if none has been set before. -SETMOD only has effect when MODULAR is on. - -Modular operations are done only on numbers such as coefficients of -polynomials, not on the exponents. The modulus need not be prime. -Attempts to divide by a power of the modulus produces an error -message, since the operation is equivalent to dividing by 0. However, -dividing by a factor of a non-prime modulus does not produce an error -message. - -\endsection -\xitem[SGN indeterminate sign] -SGN - indeterminate sign (page 257) - -\endsection -\item[SHARE] -SHARE (page 197) - -The SHARE declaration allows access to its arguments by both -algebraic and symbolic modes. - - SHARE identifier{,identifier} - -identifier can be any valid REDUCE identifier. - -Programming in SYMBOLIC as well as algebraic mode allows you a wider -range of techniques than just algebraic mode alone. Expressions do -not cross the boundary since they have different representations, -unless the SHARE declaration is used. For more information on using -symbolic mode, see the REDUCE User's Manual, and the Standard Lisp -Report. - -You should be aware that a previously-declared array is destroyed by -the SHARE declaration. Scalar variables retain their values. You can -share a declared MATRIX that has not yet been dimensioned so that it -can be used by both modes. Values that are later put into the matrix -are accessible from symbolic mode too, but not by the usual matrix -reference mechanism. In symbolic mode, a matrix is stored as a list -whose first element is MAT, and whose next elements are the rows of -the matrix stored as lists of the individual elements. Access in -symbolic mode is by the operators FIRST, SECOND, THIRD and REST. - -\endsection -\item[SHOWRULES] -SHOWRULES (page 150) - - SHOWRULES(expression) or SHOWRULES simple_expression - -SHOWRULES returns in RULE-LIST form any OPERATOR rules associated with -its argument. - -Examples: -showrules log; {log(e) => 1, - - log(1) => 0, - - ~x - log(e ) => ~x, - - 1 - df(log(~x),~x) => ----} - ~x - -Such rules can then be manipulated further as with any LIST. For example -RHS FIRST WS; has the value 1. - -An operator may have properties that cannot be displayed in such a form, -such as the fact it is an odd function, or has a definition defined as a -procedure. - -\endsection -\item[SHOWTIME] -SHOWTIME (page 70) - -The SHOWTIME command prints the elapsed system time since the last -call of this command or since the beginning of the session, if it has -not been called before. - -Examples: -showtime; Time: 1020 ms - 2 -factorize(x^4 - 8x^4 + 8x^2 - 136x - 153); {X - 9,X + 17,X + 1} -showtime; Time: 920 ms - -The time printed is either the elapsed cpu time or the elapsed wall -clock time, depending on your system. SHOWTIME allows you to see the -system time resources REDUCE uses in its calculations. Your time -readings will of course vary from this example according to the system -you use. - -\endsection -\item[SHUT] -SHUT (pages 153--155) - -The SHUT command closes output files. - SHUT filename{,filename} - -filename must have been a file opened by OUT. - - -A file that has been opened by OUT must be SHUT before it is -brought in by IN. Files that have been opened by OUT should -always be SHUT before the end of the REDUCE session, to avoid either -loss of information or the printing of extraneous information into the file. -In most systems, terminating a session by BYE closes all open -output files. - -\endsection -\xitem[Side effect] -Side effect (page 48) - -\endsection -\xitem[side relations] -side relations (page 241) - -\endsection -\item[SIGN] -SIGN (page 75) - - SIGN expression - -SIGN tries to evaluate the sign of its argument. If this is possible -SIGN returns one of 1, 0 or -1. Otherwise, the result is the original -form or a simplified variant. - -Examples: - sign(-5) -1 - sign(-a^2*b) -SIGN(B) - -Even powers of formal expressions are assumed to be positive only as long -as the switch COMPLEX is off. - -\endsection -\xitem[SIGNATURE command] -SIGNATURE command (page 271) - -\endsection -\xitem[Simplification] -Simplification (pages 44, 97) - -\endsection -\xitem[SIMPSYS] -SIMPSYS (pages 378, 380, 383) - -\endsection -\item[SIN] -SIN (pages 76, 78) - -The SIN operator returns the sine of its argument. - - SIN(expression) or SIN simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -sin aa; SIN(AA) -sin(pi/2); 1 -on rounded; -sin 3; 0.14112000806 -sin(pi/2); 1.0 - -SIN returns a numeric value only if ROUNDED is on. Then the sine is -calculated to the current degree of floating point precision. The -argument in this case is assumed to be in radians. - -\endsection -\item[SINH] -SINH (pages 76, 78) - -The SINH operator returns the hyperbolic sine of its argument. The -derivative of SINH and some simple transformations are known to the -system. - - SINH(expression) or SINH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -sinh b; SINH(B) -sinh(0); 0 - 2 -df(sinh(x**2),x); 2*COSH(X )*X - COSH(4*X) -int(sinh(4*x),x); ----------- - 4 -on rounded; -sinh 4; 27.2899171971 - - -You may attach further functionality by defining its inverse (see -ASINH). A numeric value is not returned by SINH unless the switch -ROUNDED is on and its argument evaluates to a number. - -\endsection -\xitem[SMACRO] -SMACRO (page 196) - -\endsection -\item[SOLVE] -SOLVE (pages 84, 85, 90, 181) - -The SOLVE operator solves a single algebraic EQUATION or a system of -simultaneous equations. - - SOLVE(expression [ , kernel]) or - - SOLVE({expression,...} [ ,{ kernel ,...}] ) - -If the number of equations equals the number of distinct kernels, the -optional kernel argument(s) may be omitted. expression is either a -scalar expression or an EQUATION. When more than one expression is -given, the LIST of expressions is surrounded by curly braces. The -optional list of KERNELs follows, also in curly braces. - -Examples: -sss := solve(x^2 + 7); Unknown: X - SSS := {X= - SQRT(7)*I, - X=SQRT(7)*I} -rhs first sss; - SQRT(7)*I -solve(sin(x^2*y),y); - PI*(2*ARBINT(1) + 1) - {Y=----------------------, - 2 - X - - 2*ARBINT(1)*PI - Y=----------------} - 2 - X - -off allbranch; -solve(sin(x**2*y),y); {Y=0} -solve({3x + 5y = -4,2*x + y = -10},{x,y}); - 46 22 - {{x=-------,y=----}} - 7 7 -solve({x + a*y + z,2x + 5},{x,y}); - 5 - 2*z + 5 - {{x=------,y=------------}} - 2 2*a -ab := (x+2)^2*(x^6 + 17x + 1); - 8 7 6 3 2 - ab := x + 4*x + 4*x + 17*x + 69*x + 72*x + 4 - - 6 -www := solve(ab,x); {X=ROOT_OF(X_ + 17*X_ + 1),X=-2} -root_multiplicities; {1,2} - -Results of the SOLVE operator are returned as EQUATIONS in a LIST. -You can use the usual list access methods (FIRST, SECOND, THIRD, REST -and PART) to extract the desired equation, and then use the operators -RHS and LHS to access the right-hand or left-hand expression of the -equation. When SOLVE is unable to solve an equation, it returns the -unsolved part as the argument of ROOT_OF, with the variable renamed to -avoid confusion, as shown in the last example above. - -For one equation, SOLVE uses square-free factorisation, roots of -unity, and the known inverses of the LOG, SIN, COS, ACOS, ASIN, and -exponentiation operators. The quadratic, cubic and quartic formulas -are used if necessary, but these are applied only when the switch -FULLROOTS is set on; otherwise or when no closed form is available the -result is returned as ROOT_OF expression. The switch TRIGFORM -determines which type of cubic and quartic formula is used. The -multiplicity of each solution is given in a list as the system -variable ROOT_MULTIPLICITIES. For systems of simultaneous linear -equations, matrix inversion is used. For nonlinear systems, the -Groebner basis method is used. - -Linear equation system solving is influenced by the switch CRAMER. - -Singular systems can be solved when the switch SOLVESINGULAR is on, -which is the default setting. A message is given if the system of -equations is inconsistent. - -Related: ALLBRANCH switch, FULLROOTS switch, ROOTS operator, ROOT_OF -operator, TRIGFORM switch. - -\endsection -\item[SOLVESINGULAR] -SOLVESINGULAR (page 89) - -When SOLVESINGULAR is on, singular or under determined systems of -linear equations are solved, using arbitrary real, complex or integer -variables in the answer. Default is ON. - -Examples: - - ARBCOMPLEX(1) -solve({2x + y,4x + 2y},{x,y}); {{X=------------------,Y=ARBCOMPLEX(1)}} - 2 - - 8*arbcomplex(2) -solve({7x + 15y - z,x - y - z},{x,y,z});{{x=-----------------, - 11 - - - 3*ARBCOMPLEX(2) - Y=--------------------, - 11 - - Z=ARBCOMPLEX(2)}} - -off solvesingular; -solve({2x + y,4x + 2y},{x,y}); ***** SOLVE given singular equations -solve({7x + 15y - z,x - y - z},{x,y,z});***** SOLVE given singular equations - -The integer following the identifier ARBCOMPLEX above is assigned by -the system, and serves to identify the variable uniquely. It has no other -significance. - -\endsection -\xitem[SORTOUTODE] -SORTOUTODE (page 350) - -\endsection -\xitem[SPACEDIM command] -SPACEDIM command (pages 251, 271) - -\endsection -\item[SPDE] -SPDE (page 185) - -Author: Fritz Schwartz - -The package SPDE provides a set of functions which may be used to -determine the symmetry group of Lie- or point-symmetries of a given system of -partial differential equations. In many cases the determining system is -solved completely automatically. In other cases the user has to provide -additional input information for the solution algorithm to terminate. - - -\endsection -\xitem[SPECFN] -SPECFN (page 185) - -\endsection -\xitem[SPECFN package] -SPECFN package (page 391) - -\endsection -\xitem[SPECFN2] -SPECFN2 (page 187) - -\endsection -\xitem[spherical coordinates] -spherical coordinates (pages 265, 355) - -\endsection -\item[SPLIT_FIELD] -SPLIT_FIELD function (page 227) - -SPLIT_FIELD is part of the ARNUM package for algebraic numbers. It -calculates a primitive element of minimal degree for which a given -polynomial splits into linear factors. The algorithm as described by -Trager. - -Example: - load arnum; - split!_field(x**3-3*x+7); - - *** Splitting field is generated by: - - 6 4 2 - A5 - 18*A5 + 81*A5 + 1215 - - - - 4 2 - {1/126*A5 - 5/42*A5 - 1/2*A5 + 2/7, - - - 4 2 - - (1/63*A5 - 5/21*A5 + 4/7), - - - 4 2 - 1/126*A5 - 5/42*A5 + 1/2*A5 + 2/7} - - - for each j in ws product (x-j); - - 3 - X - 3*X + 7 - - -\endsection -\item[SPUR] -SPUR (page 210) - -The SPUR declaration removes the special exemption from trace -calculations that was declared by NOSPUR, in high-energy physics -calculations. - SPUR line-id{,line-id} - -line-id must be a line-identifier that has previously been declared -NOSPUR. - -\endsection -\xitem[SQFRF] -SQFRF (page 373) - -\endsection -\item[SQRT] -SQRT (pages 76, 78) - -The SQRT operator returns the square root of its argument. - - SQRT(expression) - -expression can be any REDUCE scalar expression. - -Examples: -sqrt(16*a^3); 4*SQRT(A)*A -sqrt(17); SQRT(17) -on rounded; -sqrt(17); 4.12310562562 -off rounded; 2 -sqrt(a*b*c^5*d^3*27); 3*SQRT(D)*SQRT(C)*SQRT(B)*SQRT(A)*SQRT(3)*C *D - -SQRT checks its argument for squared factors and removes them. - -Numeric values for square roots that are not exact integers are given -only when ROUNDED is on. - -Please note that SQRT(A**2) is given as A, which may be incorrect if A -eventually has a negative value. If you are programming a calculation -in which this is a concern, you can turn on the PRECISE switch, which -causes the absolute value of the square root to be returned. - -\endsection -\xitem[Standard form] -Standard form (page 198) - -\endsection -\xitem[Standard quotient] -Standard quotient (page 198) - -\endsection -\xitem[Statement] -Statement (page 53) - -\endsection -\xitem[Stirling Numbers] -Stirling Numbers (page 185, 394) - -\endsection -\item[STIRLING1] -STIRLING1 (pages 185, 394) - -The STIRLING1 operator returns the Stirling Numbers S(n,m) of the -first kind, i.e. the number of permutations of n symbols which have -exactly m cycles (divided by (-1)**(n-m)). - - STIRLING1(integer,integer) - -Examples: - load_package specfn; (SPECFN) - Stirling1 (17,4); -87077748875904 - - GAMMA(N + 1) - Stirling1 (n,n-1); ----------------- - 2*GAMMA(N - 1) - -The operator STIRLING1 evaluates the Stirling numbers of the first -kind by rulesets for special cases or by a computing the closed form, -which is a series involving the operators BINOMIAL and STIRLING2. - -\endsection -\item[STIRLING2] -STIRLING2 (pages 185, 394) - -The STIRLING1 operator returns the Stirling Numbers S(n,m) of the -second kind, i.e. the number of ways of partitioning a set of n elements -into m non-empty subsets. - - STIRLING2(integer,integer) - -Examples: - load_package specfn; (SPECFN) - Stirling2 (17,4); 694337290 - GAMMA(N + 1) - Stirling2 (n,n-1); ---------------- - 2*GAMMA(N - 1) - -The operator STIRLING2 evaluates the Stirling numbers of the second -kind by rulesets for special cases or by a computing the closed form. - -\endsection -\item[String] -String (page 37)) -A STRING is any collection of characters enclosed in double quotation -marks ("). It may be used as an argument for a variety of commands -and operators, such as IN, REDERR and WRITE. -Examples: -write "this is a string"; this is a string -write a, " ", b, " ",c,"!"; A B C! - -\endsection -\item[STRUCTR] -STRUCTR (pages 112, 113) - -The STRUCTR operator breaks its argument expression into named -subexpressions. - - STRUCTR(expression [,identifier[,identifier ...]]) - -expression may be any valid REDUCE scalar expression. identifier may -be any valid REDUCE IDENTIFIER. The first identifier is the stem for -subexpression names, the second is the name to be assigned to the -structured expression. - -Examples: -structr(sqrt(x**2 + 2*x) + sin(x**2*z)); ANS1*ANS3 + ANS2 - - WHERE - - 1/2 - ANS3 := X - - 2 - ANS2 := SIN(X *Z) - - 1/2 - ANS1 := (X + 2) - -ans3; ANS3 -on fort; -structr((x+1)**5 + tan(x*y*z),var,aa); - VAR1=TAN(X*Y*Z) - AA=VAR1+X**5+5.*X**4+10.*X**3+10.*X**2+5.*X+1. - -The second argument to STRUCTR is optional. If it is not given, the -default stem ANS is used by REDUCE to construct names for the -subexpression. The names are only for display purposes: REDUCE does -not store the names and their values unless the switch SAVESTRUCTR is -on. - -If a third argument is given, the structured expression as a whole is -named by this argument, when FORT is on. The expression is not stored -under this name. You can send these structured Fortran expressions to -a file with the OUT command. - -\endsection -\xitem[Structuring] -Structuring (page 97) - -\endsection -\xitem[Struve Functions] -Struve Functions (pages 185, 397) - -\endsection -\item[STRUVEH] -STRUVEH (pages 185, 397) - -The STRUVEH operator returns Struve's H function. - - STRUVEH(order,argument) - -Examples: -load_package specfn; (SPECFN) - - 3 - - BESSELJ(---,X) - 2 -struveh(-3/2,x); ------------------- - I - - -There is currently no numeric support for the operator STRUVEH. - -\endsection -\item[STRUVEL] -STRUVEL (pages 185, 397) - -The STRUVEL operator returns the modified Struve L function . - - STRUVEL(order,argument) - -Examples: - load_package specfn; (SPECFN); - 3 - struvel(-3/2,x); BESSELI(---,X) - 2 - -There is currently no numeric support for the operator STRUVEL. - -\endsection -\xitem[Sturm Sequences] -Sturm Sequences (page 369) - -\endsection -\item[SUB] -SUB (page 137) - -The SUB operator substitutes a new expression for a kernel in an -expression. - - SUB(kernel=expression {,kernel=expression} expression) - or - SUB({kernel=expression, kernel=EXPRESSION},expression}) - -kernel must be a KERNEL, expression can be any REDUCE scalar -expression. - -Examples: -sub(x=3,y=4,(x+y)**3); 343 -x; X -sub({cos=sin,sin=cos},cos a+sin b} COS(B) + SIN(A) - -Note in the second example that operators can be replaced using the -SUB operator. - -\endsection -\xitem[SUCH THAT] -SUCH THAT (page 142) - -\endsection -\item[SUM] -SUM (pages 57, 58, 187) - -The operator SUM returns -the indefinite or definite summation of a given expression. - - -SUM(expr,k[,lolim [,uplim ]]) - - -where expr is the expression to be added, k is the -control variable (a KERNEL), and lolim and uplim -uplim are the optional lower and upper limits. If uplim is -not supplied the upper limit is taken as k. The GOSPER -algorithm is used. If there is no closed form solution, the operator -returns the input unchanged. - -Examples: - 2 2 -sum(4n**3,n); N *(N + 2*N + 1) - -sum(2a+2k*r,k,0,n-1); N*(2*A + N*R - R) - -\endsection -\xitem[SUM-SQ] -SUM-SQ (page 404) - -\endsection -\xitem[SVEC] -SVEC (page 355) - -\endsection -\xitem[Switch] -Switch (pages 68, 69) - -\endsection -\item[SYMBOLIC] -SYMBOLIC (page 191) - -The SYMBOLIC command changes REDUCE's mode of operation to symbolic. -When SYMBOLIC is followed by an expression, that expression is -evaluated in symbolic mode, but REDUCE's mode is not changed. It is -equivalent to the LISP command. - -Examples: -symbolic; NIL -cdr '(a b c); (B C) -algebraic; -x + symbolic car '(y z); X + Y - -\endsection -\xitem[Symbolic mode] -Symbolic mode (pages 191, 193, 197, 198) - -\endsection -\xitem[Symbolic procedure] -Symbolic procedure (page 196) - -\endsection -\item[SYMMETRIC] -SYMMETRIC (page 93) - -When an operator is declared SYMMETRIC, its arguments are reordered -to conform to the internal ordering of the system. - - SYMMETRIC identifier{,identifier} - -identifier is an identifier that has been declared an operator. - -Examples: -operator m,n; -symmetric m,n; -m(y,a,sin(x)); M(SIN(X),A,Y) -n(z,m(b,a,q)); N(M(A,B,Q),Z) - -If identifier has not been declared to be an operator, the flag -SYMMETRIC is still attached to it. When identifier is subsequently -used as an operator, the message - DECLARE identifier OPERATOR ? (Y OR N) -is printed. If the user replies Y, the symmetric property of the -operator is used. - -\endsection -\xitem[system precision] -system precision (page 374) - -\endsection -\item[T] -T (page 37) - -The constant T stands for the truth value true. It cannot be used as -a scalar variable in a BLOCK, as a looping variable in a FOR statement -or as an OPERATOR name. - -\endsection -\item[TAN] -TAN (pages 76, 78, 81) - -The TAN operator returns the tangent of its argument. - - TAN(expression) or TAN simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -tan a; TAN(A) -tan(pi/3); SQRT(3) -on rounded; -tan(pi/3); 1.73205080757 - -TAN returns a numeric value only if ROUNDED is on. Then the tangent -is calculated to the current degree of floating point accuracy. - -When ON ROUNDED is in force, no check is made to see if the argument -to TAN is a multiple of pi/2, for which the tangent goes to positive -or negative infinity. (Of course, since REDUCE uses a fixed-point -representation of pi/2, it produces a large but not infinite number). -You need to make a check for multiples of pi/ in any program you use -that might possibly ask for the tangent of such a quantity. - -\endsection -\xitem[tangent vector] -tangent vector (page 252) - -\endsection -\item[TANH] -TANH (pages 76, 78) - -The TANH operator returns the hyperbolic tangent of its argument. The -derivative of TANH and some simple transformations are known to the -system. - - TANH(expression) or TANH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -tanh b; TANH(B) -tanh(0); 0 - 2 -df(tanh(x*y),x); Y*( - TANH(X*Y) + 1) - 2*X -int(tanh(x),x); LOG(E + 1) - X -on rounded; -tanh 2; 0.964027580076 - -You may attach further functionality by defining its inverse (see -ATANH). A numeric value is not returned by TANH unless the switch -ROUNDED is on and its argument evaluates to a number. - -\endsection -\item[TAYLOR] -TAYLOR (page 188, 406) - -The TAYLOR operator is used for expanding an expression into a Taylor -series. - -TAYLOR(expression, var, expression, number) -TAYLOR(expression, var, expression, number {,var, expression, number}) - -expression can be any valid REDUCE algebraic expression. var must be -a KERNEL, and is the expansion variable. The expression following it -denotes the point about which the expansion is to take place. number -must be a non-negative integer and denotes the maximum expansion -order. If more than one triple is specified TAYLOR will expand its -first argument independently with respect to all the variables. Note -that once the expansion has been done it is not possible to calculate -higher orders. - -Instead of a KERNEL, var may also be a list of kernels. In this case -expansion will take place in a way so that the sum of the degrees of -the kernels does not exceed the maximum expansion order. If the -expansion point evaluates to the special identifier INFINITY, TAYLOR -tries to expand in a series in 1/var. - -The expansion is performed variable per variable, i.e. in the example -below by first expanding exp(x^2+y^2) with respect to x and then -expanding every coefficient with respect to y. - -Examples: - 2 2 2 2 3 3 -taylor(e^(x^2+y^2),x,0,2,y,0,2); 1 + Y + X + Y *X + O(X ,Y ) - - 2 2 3 -taylor(e^(x^2+y^2),{x,y},0,2); 1 + Y + X + O({X,Y} ) - -taylor(x*y/(x+y),x,0,2,y,0,2); ***** Not a unit in argument to quottaylor - -Note that it is not generally possible to apply the standard REDUCE -operators to a Taylor kernel. For example, PART, COEFF, or COEFFN -cannot be used. Instead, the expression at hand has to be converted -to standard form first using the TAYLORTOSTANDARD operator. - -Differentiation of a Taylor expression is possible. If you -differentiate with respect to one of the Taylor variables the order -will decrease by one. - -Substitution is a bit restricted: Taylor variables can only be -replaced by other kernels. There is one exception to this rule: you -can always substitute a Taylor variable by an expression that -evaluates to a constant. Note that REDUCE will not always be able to -determine that an expression is constant: an example is sin(acos(4)). - -Only simple taylor kernels can be integrated. More complicated -expressions that contain Taylor kernels as parts of themselves are -automatically converted into a standard representation by means of the -TAYLORTOSTANDARD operator. In this case a suitable warning is -printed. - -\endsection -\xitem[TAYLOR package] -TAYLOR package (page 405) - -\endsection -\xitem[Taylor series arithmetic] -Taylor series - arithmetic (page 407) - differentiation (page 408) - integration (page 408) - reversion (page 408) - substitution (page 408) - -\endsection -\item[TAYLORAUTOCOMBINE] -TAYLORAUTOCOMBINE switch (page 408) - -If you set TAYLORAUTOCOMBINE to ON, REDUCE automatically combines -Taylor expressions during the simplification process. This is -equivalent to applying TAYLORCOMBINE to every expression that contains -Taylor kernels. Default is ON. - -\endsection -\item[TAYLORAUTOEXPAND] -TAYLORAUTOEXPAND switch (pages 408, 409) - -TAYLORAUTOEXPAND makes Taylor expressions ``contagious'' in the sense -that TAYLORCOMBINE tries to Taylor expand all non-Taylor -subexpressions and to combine the result with the rest. Default is -OFF. - -\endsection -\item[TAYLORCOMBINE] -TAYLORCOMBINE (page 407) - -This operator tries to combine all Taylor kernels found in its -argument into one. Operations currently possible are: - -Addition, subtraction, multiplication, and division. -Roots, exponentials, and logarithms. -Trigonometric and hyperbolic functions and their inverses. - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 - 3 -taylorcombine log hugo; X + O(X ) - - 1 2 3 -taylorcombine(hugo + x); (1 + X + ---*X + O(X )) + X - 2 -on taylorautoexpand; - 1 2 3 -taylorcombine(hugo + x); 1 + 2*X + ---*X + O(X ) - 2 - -Application of unary operators like LOG and ATAN will nearly always -succeed. For binary operations their arguments have to be Taylor -kernels with the same template. This means that the expansion -variable and the expansion point must match. Expansion order is not -so important, different order usually means that one of them is -truncated before doing the operation. - -If TAYLORKEEPORIGINAL is set to ON and if all Taylor kernels in its -argument have their original expressions kept TAYLORCOMBINE will also -combine these and store the result as the original expression of the -resulting Taylor kernel. There is also the switch TAYLORAUTOEXPAND. - -There are a few restrictions to avoid mathematically undefined -expressions: it is not possible to take the logarithm of a Taylor -kernel which has no terms (i.e. is zero), or to divide by such a -beast. There are some provisions made to detect singularities during -expansion: poles that arise because the denominator has zeros at the -expansion point are detected and properly treated, i.e. the Taylor -kernel will start with a negative power. (This is accomplished by -expanding numerator and denominator separately and combining the -results.) Essential singularities of the known functions (see above) -are handled correctly. - -\endsection -\item[TAYLORKEEPORIGINAL] -TAYLORKEEPORIGINAL (pages 406, 407, 409, 411) - -TAYLORKEEPORIGINAL, if set to ON, forces the TAYLOR and all Taylor -kernel manipulation operators to keep the original expression, -i.e. the expression that was Taylor expanded. All operations -performed on the Taylor kernels are also applied to this expression -which can be recovered using the operator TAYLORORIGINAL. Default is -OFF. - -\endsection -\item[TAYLORORIGINAL] -TAYLORORIGINAL (pages 411, 412) - -TAYLORORINAL can recover the original expression (the one that was -expanded) from the Taylor kernel that is given as its argument. - - TAYLORORIGINAL(expression) - TAYLORORIGINAL simple_expression - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 -taylororiginal hugo; - ***** Taylor kernel doesn't have an original part in taylororiginal - -on taylorkeeporiginal; - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 - X -taylororiginal hugo; E - -An error is signalled if the argument is not a Taylor kernel or if the -original expression was not kept, i.e. if TAYLORKEEPORIGINAL was set -OFF during expansion. - -\endsection -\item[TAYLORPRINTORDER] -TAYLORPRINTORDER switch (page 409) - -TAYLORPRINTORDER, if set to ON, causes the remainder to be printed in -big-O notation. Otherwise, three dots are printed. Default is -ON. - -\endsection -\item[TAYLORPRINTTERMS] -TAYLORPRINTTERMS (pages 406, 412) - -Only a certain number of (non-zero) coefficients are printed. If there -are more, an expression of the form N TERMS is printed to indicate how -many non-zero terms have been suppressed. The number of terms printed -is given by the value of the shared algebraic variable -TAYLORPRINTTERMS. Allowed values are integers and the special -identifier ALL. The latter setting specifies that all terms are to be -printed. The default setting is 5. - -Examples: -taylor(e^(x^2+y^2),x,0,4,y,0,4); - 2 1 4 2 2 2 5 5 - 1 + Y + ---*Y + X + Y *X + (4 TERMS) + O(X ,Y ) - 2 -taylorprintterms := all; - ALL -taylor(e^(x^2+y^2),x,0,4,y,0,4); - 2 1 4 2 2 2 1 4 2 1 4 1 2 4 - 1 + y + ---*y + x + y *x + ---*y *x + ---*x + ---*y *x - 2 2 2 2 - - 1 4 4 5 5 - + ---*y *x + O(x ,y ) - 4 - -\endsection -\item[TAYLORREVERT] -TAYLORREVERT (page 411) - -TAYLORREVERT allows reversion of a Taylor series of a function f, -i.e., to compute the first terms of the expansion of the inverse of f -from the expansion of f. - - TAYLORREVERT(expression, var, var) - -The first argument must evaluate to a Taylor kernel with the second -argument being one of its expansion variables. - -Examples: - 2 6 -taylor(u - u**2,u,0,5); U - U + O(U ) - 2 3 4 5 6 -taylorrevert(ws,u,x); X + X + 2*X + 5*X + 14*X + O(X ) - -\endsection -\item[TAYLORSERIESP] -TAYLORSERIESP (page 407) - -The TAYLORSERIESP operator may be used to determine if its argument is -a Taylor kernel. - - TAYLORSERIESP(expression) - TAYLORSERIESP simple_expression - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 -if taylorseriesp hugo then OK; OK -if taylorseriesp(hugo + y) then OK else NO; NO - -Note that this operator is subject to the same restrictions as, e.g., -ORDP or NUMBERP, i.e. it may only be used in boolean expressions in IF -or LET statements. -\endsection -\item[TAYLORTEMPLATE] -TAYLORTEMPLATE (pages 407, 412) - -The template of a Taylor kernel, i.e. the list of all variables with -respect to which expansion took place together with expansion point -and order can be extracted using - - TAYLORTEMPLATE(expression) - TAYLORTEMPLATE simple_expression - -The operator returns a list of lists with the three elements -(VAR,VAR0,ORDER). An error is signalled if the argument is not a -Taylor kernel. - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 -taylortemplate hugo; {{X,0,2}} - -\endsection -\item[TAYLORTOSTANDARD] -TAYLORTOSTANDARD (page 407) - -The TAYLORTOSTANDARD operator converts all Taylor kernels in its -argument into standard form and resimplifies the result. - - TAYLORTOSTANDARD(expression) - TAYLORTOSTANDARD simple_expression - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 - 2 - X + 2*X + 2 -taylortostandard hugo; -------------- - 2 -\endsection -\xitem[Terminator] -Terminator (page 53) - -\endsection -\item[THIRD] -THIRD (page 50) - -The THIRD operator returns the third item of a LIST. - THIRD(list) or THIRD list - - - -list must be a list containing at least three items to avoid an error -message. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D} -third alist; C -blist := {x,{aa,bb,cc},y,z}; BLIST := {X,{AA,BB,CC},Y,Z}; -third second blist; CC -third blist; Y - -\endsection -\item[TIME] -TIME (page 68) - -When TIME is on, the system time used in executing each REDUCE -statement is printed after the answer is printed. - -Examples: -on time; Time: 4940 ms - 2 -df(sin(x**2 + y),y); COS(X + Y ) - Time: 180 ms -solve(x**2 - 6*y,x); {X= - SQRT(Y)*SQRT(6), - X=SQRT(Y)*SQRT(6)} - Time: 320 ms - -When TIME is first turned on, the time since the beginning of the -REDUCE session is printed. After that, the time used in computation, -(usually in milliseconds, though this is system dependent) is printed -after the results of each command. Idle time or time spent typing in -commands is not counted. If TIME is turned off, the first reading -after it is turned on again gives the time elapsed since it was turned -off. The time printed is CPU or wall clock time, depending on the -system. - -\endsection -\item[TORDER] -TORDER (pages 296, 315, 316) - -The operator TORDER sets the actual term order. - -1. simple term order: - TORDER m - -where m is the name of a term order mode LEX term order, GRADLEX term -order, REVGRADLEX term order or another implemented parameterless -mode. - -2. stepped term order: - TORDER m,n - TORDER {m,n} - -where m is the name of a two step term order, one of GRADLEXGRADLEX -term order, GRADLEXREVGRADLEX term order, LEXGRADLEX term order or -LEXREVGRADLEX term order, and n is a positive integer. - -3. weighted term order - TORDER WEIGHTED, n,n,... - TORDER WEIGHTED, {n,n,...} - -where the n are positive integers, see weighted term order. - -TORDER sets the term order mode. The default mode is LEX. The -previous order mode is returned. - -\endsection -\item[TP] -TP (page 165) - -The TP operator returns the transpose of its MATRIX - argument. - TP identifier or TP(identifier) - -identifier must be a matrix, which either has had its dimensions set -in its declaration, or has had values put into it by MAT. - -Examples: -matrix m,n; -m := mat((1,2,3),(4,5,6))$ -n := tp m; N(1,1) := 1 - N(1,2) := 4 - N(2,1) := 2 - N(2,2) := 5 - N(3,1) := 3 - N(3,2) := 6 - -In an assignment statement involving TP, the matrix identifier on the -left-hand side is redimensioned to the correct size for the transpose. - -\endsection -\item[TPS] -TPS (pages 188, 330) - -Authors: Alan Barnes and Julian Padget - -A Truncated Power Series Package. - -This package implements formal Laurent series expansions in one -variable using the domain mechanism of REDUCE. This means that power -series objects can be added, multiplied, differentiated etc., like -other first class objects in the system. A lazy evaluation scheme -is used and thus terms of the series are not evaluated until they -are required for printing or for use in calculating terms in other -power series. The series are extendible giving the user the -impression that the full infinite series is being manipulated. The -errors that can sometimes occur using series that are truncated at -some fixed depth (for example when a term in the required series -depends on terms of an intermediate series beyond the truncation -depth) are thus avoided. - -\endsection -\xitem[TRA] -TRA (page 178) - -\endsection -\item[TRACE] -TRACE (page 166) - -The TRACE operator finds the trace of its MATRIX argument. - TRACE(expression) or TRACE simple_expression - -expression or simple_expression must evaluate to a square -matrix. - -Examples: -matrix a; -a := mat((x1,y1),(x2,y2))$ -trace a; X1 + Y2 - -The trace is the sum of the entries along the diagonal of a square matrix. -Given a non-matrix expression, or a non-square matrix, TRACE returns -an error message. - -\endsection -\xitem[tracing EXCALC] -tracing - EXCALC (page 266) - ODESOLVE (page 351) - ROOTS package (page 373) - SPDE package (page 380) - SUM package (page 404) - -\endsection -\item[TRALLFAC] -TRALLFAC - -When TRALLFAC is on, a more detailed trace of factoriser calls is -generated. - - -The TRALLFAC switch takes precedence over TRFAC if they are -both on. TRFAC gives a factorisation trace with less detail in it. -When the FACTOR switch is on also, all input polynomials are sent to -the factoriser automatically and trace information is generated. The -OUT command saves the results of the factoring, but not the trace. - - -\endsection -\item[TRFAC] -TRFAC (page 122) - -When TRFAC is on, a narrative trace of any calls to the factoriser is -generated. Default is OFF. - - -When the switch FACTOR is on, and TRFAC is on, every input -polynomial is sent to the factoriser, and a trace generated. With -FACTOR off, only polynomials that are explicitly factored with the -command FACTORIZE generate trace information. - -The OUT command saves the results of the factoring, but not -the trace. The TRALLFAC switch gives trace information to a -greater level of detail. - -\endsection -\item[TRGROEB] -TRGROEB (pages 299, 303) - -If TRGROEB is on, intermediate H polynomials are printed during a -GROEBNER or GROEBNERF calculation. - -\endsection -\xitem[TRGROEB1] -TRGROEB1 (pages 299, 303) - -\endsection -\xitem[TRGROEBR] -TRGROEBR (page 304) - -\endsection -\item[TRGROEBS] -TRGROEBS (pages 299, 303) - -If TRGROEBS is on, intermediate H and S polynomials are printed during -a GROEBNER or GROEBNERF calculation. - -\endsection -\item[TRIGFORM] -TRIGFORM (page 87) - -When FULLROOTS is on, SOLVE will compute the -roots of a cubic or quartic polynomial is closed form. When -TRIGFORM is on, the roots will be expressed by trigonometric -forms. Otherwise nested surds are used. Default is ON. - -\endsection -\item[TRINT] -TRINT (page 178) - -When TRINT is on, a narrative tracing various steps in the -integration process is produced. - -The OUT command saves the results of the integration, but not the -trace. - -\endsection -\item[TRNONLNR] -TRNONLNR - -When TRNONLNR is on, a narrative tracing various steps in -the process for solving non-linear equations is produced. - - -TRNONLNR can only be used after the solve package has been loaded -(e.g., by an explicit call of LOAD_PACKAGE). The OUT -command saves the results of the equation solving, but not the trace. - -\endsection -\xitem[TRODE] -TRODE (page 351) - -\endsection -\xitem[TRROOT] -TRROOT (page 373) - -\endsection -\xitem[TRSUM] -TRSUM (page 404) - -\endsection -\xitem[truncated power series] -truncated power series (page 413) - -\endsection -\xitem[TVECTOR command] -TVECTOR command (pages 249, 271) - -\endsection -\xitem[U(ALFA)] -U(ALFA) (page 379) - -\endsection -\xitem[U(ALFA] -U(ALFA,I) (page 379) - -\endsection -\item[UNTIL] -UNTIL (page 57) - -See the FOR loop construction. -\endsection -\xitem[User packages] -User packages (page 177) - -\endsection -\xitem[VARDF] -VARDF (pages 257, 271) - -\endsection -\xitem[Variable] -Variable (page 36) - -\endsection -\xitem[Variable elimination] -Variable elimination (page 181) - -\endsection -\xitem[variational derivative] -variational derivative (page 257) - -\endsection -\item[VARNAME] -VARNAME (pages 111, 112) - -The declaration VARNAME instructs REDUCE to use its argument as the -default Fortran (when FORT is on) or STRUCTR identifier and identifier -stem, rather than using ANS. - - VARNAME identifier - -identifier can be any combination of one or more alphanumeric -characters. Try to avoid REDUCE reserved words. - -Examples: -varname ident; IDENT -on fort; -x**2 + 1; IDENT=X**2+1. - -off fort,exp; 3 -structr(((x+y)**2 + z)**3); IDENT2 - where - 2 - IDENT2 := IDENT1 + Z - IDENT1 := X + Y - -EXP was turned off so that STRUCTR could show the structure. If EXP -had been on, the expression would have been expanded into a -polynomial. - -\endsection -\xitem[VDF] -VDF (page 359) - -\endsection -\xitem[VEC command] -VEC command (page 232) - -\endsection -\item[VECDIM] -VECDIM (page 212) - -The SPUR declaration removes the special exemption from trace -calculations that was declared by NOSPUR, in high-energy physics -calculations. - SPUR line-id{,line-id} - -line-id must be a line-identifier that has previously been declared -NOSPUR. - -\endsection -\item[VECTOR] -VECTOR (High Energy Physics) (page 208) - -The VECTOR declaration declares that its arguments are of type VECTOR. - VECTOR identifier{,identifier} - -identifier must be a valid REDUCE identifier. It may have already -been used for a matrix, array, operator or scalar variable. After an -identifier has been declared to be a vector, it may not be used as a -scalar variable. - -Vectors are special entities for high-energy physics calculations. -You cannot put values into their coordinates; they do not have -coordinates. They are legal arguments for the high-energy physics -operators EPS, G and . (dot). Vector variables are used to represent -gamma matrices and gamma matrices contracted with Lorentz 4-vectors, -since there are no Dirac variables per se in the system. Vectors do -follow the usual vector rules for arithmetic operations: + and - -operate upon two or more vectors, producing a vector; * and / cannot -be used between vectors; the scalar product is represented by the -. operator; and the product of a scalar and vector expression is well -defined, and is a vector. - -You can represent components of vectors by including representations -of unit vectors in your system. For instance, letting E0 represent -the unit vector (1,0,0,0), the command - -V1.E0 := 0; - -would set up the substitution of zero for the first component of the -vector V1. - -Identifiers that are declared by the INDEX and MASS declarations are -automatically declared to be vectors. - -The following errors can occur in calculations using the high energy -physics package: - -A REPRESENTS ONLY GAMMA5 IN VECTOR EXPRESSIONS -You have tried to use A in some way other than gamma5 in a high-energy -physics expression. - -GAMMA5 NOT ALLOWED UNLESS VECDIM IS 4 -You have used gamma_5 in a high-energy physics computation involving a -vector dimension other than 4. - -ID HAS NO MASS -One of the arguments to MSHELL has had no mass assigned to it, in -high-energy physics calculations. - -MISSING ARGUMENTS FOR G OPERATOR -A line symbol is missing in a gamma matrix expression in high-energy physics -calculations. - -UNMATCHED INDEX list -The parser has found unmatched indices during the evaluation of a -gamma matrix expression in high-energy physics calculations. - -\endsection -\xitem[vector] -vector , integration 233 - addition (page 356) - cross product (page 357) - differentiation (page 233) - division (page 357) - dot product (page 357) - exponentiation (page 357) - inner product (page 357) - modulus (page 357) - multiplication (page 357) - subtraction (page 356) - -\endsection -\xitem[vector algebra] -vector algebra (page 231) - -\endsection -\xitem[VECTORADD] -VECTORADD (page 356) - -\endsection -\xitem[VECTORCROSS] -VECTORCROSS (page 357) - -\endsection -\xitem[VECTORDIFFERENCE] -VECTORDIFFERENCE (page 356) - -\endsection -\xitem[VECTOREXPT] -VECTOREXPT (page 357) - -\endsection -\xitem[VECTORMINUS] -VECTORMINUS (page 356) - -\endsection -\xitem[VECTORPLUS] -VECTORPLUS (page 356) - -\endsection -\xitem[VECTORQUOTIENT] -VECTORQUOTIENT (page 357) - -\endsection -\xitem[VECTORRECIP] -VECTORRECIP (page 357) - -\endsection -\xitem[VECTORTIMES] -VECTORTIMES (page 357) - -\endsection -\xitem[VERBOSELOAD switch] -VERBOSELOAD switch (page 409) - -\endsection -\xitem[VINT] -VINT (page 360) - -\endsection -\xitem[VMOD] -VMOD (page 357) - -\endsection -\xitem[VMOD operator] -VMOD operator (page 233) - -\endsection -\xitem[VOLINT] -VOLINT (page 360) - -\endsection -\xitem[VOLINTEGRAL function] -VOLINTEGRAL function (page 237) - -\endsection -\xitem[VOLINTORDER vector] -VOLINTORDER vector (page 237) - -\endsection -\xitem[VORDER] -VORDER (page 359) - -\endsection -\xitem[VOUT] -VOUT (page 355) - -\endsection -\xitem[VSTART] -VSTART (page 354) - -\endsection -\xitem[VTAYLOR] -VTAYLOR (page 359) - -\endsection -\xitem[wedge] -wedge (page 271) - -\endsection -\item[WEIGHT] -WEIGHT (page 152) - -The WEIGHT command is used to attach weights to kernels for asymptotic -constraints. - - WEIGHT kernel = number - -kernel must be a REDUCE KERNEL, number must be a positive integer, not -0. - -Examples: 4 3 2 2 3 4 -a := (x+y)**4; A := X + 4*X *Y + 6*X *Y + 4*X*Y + Y -weight x=2,y=3; -wtlevel 8; - 4 -a; X -wtlevel 10; - 2 2 2 -a; X *(6*Y + 4*X*Y + X ) -int(x**2,x); ***** X invalid as KERNEL - -Weights and WTLEVEL are used for asymptotic constraints, where -higher-order terms are considered insignificant. - -Weights are originally equivalent to 0 until set by a WEIGHT command. -To remove a weight from a kernel, use the CLEAR command. Weights once -assigned cannot be changed without clearing the identifier. Once a -weight is assigned to a kernel, it is no longer a kernel and cannot be -used in any REDUCE commands or operators that require kernels, until -the weight is cleared. Note that terms are ordered by greatest -weight. - -The weight level of the system is set by WTLEVEL, initially at 2. -Since no kernels have weights, no effect from WTLEVEL can be seen. -Once you assign weights to kernels, you must set WTLEVEL correctly for -the desired operation. When weighted variables appear in a term, -their weights are summed for the total weight of the term (powers of -variables multiply their weights). When a term exceeds the weight -level of the system, it is discarded from the result expression. - -\endsection -\xitem[weighted ordering] -weighted ordering (page 316) - -\endsection -\item[WHEN] -WHEN (page 147) - -The WHEN operator is used inside a RULE to make the -execution of the rule depend on a boolean condition which is -evaluated at execution time. For the use see RULE. - -\endsection -\item[WHERE] -WHERE (page 148) - -The WHERE operator provides an infix notation for one-time -substitutions for kernels in expressions. - - expression WHERE kernel = expression{,kernel = expression} - -expression can be any REDUCE scalar expression, kernel must be a -KERNEL. Alternatively a RULE or a RULE LIST can be a member of the -right-hand part of a WHERE expression. - -Examples: -x**2 + 17*x*y + 4*y**2 where x=1,y=2; - 51 -for i := 1:5 collect x**i*q where q= for j := 1:i product j; - 2 3 4 5 - {X,2*X ,6*X ,24*X ,120*X } - 2 3 -x**2 + y + z where z=y**3,y=3; X + Y + 3 - -Substitution inside a WHERE expression has no effect upon the values -of the kernels outside the expression. The WHERE operator has the -lowest precedence of all the infix operators, which are lower than -prefix operators, so that the substitutions apply to the entire -expression preceding the WHERE operator. However, WHERE is applied -before command keywords such as THEN, REPEAT, or DO. - -A RULE or a RULE SET in the right-hand part of the WHERE expression -act as if the rules were activated by LET immediately before the -evaluation of the expression and deactivated by CLEARRULES immediately -afterwards. - -WHERE gives you a natural notation for auxiliary variables in -expressions. As the second example shows, the substitute expression -can be a command to be evaluated. The substitute assignments are made -in parallel, rather than sequentially, as the last example shows. The -expression resulting from the first round of substitutions is not -reexamined to see if any further such substitutions can be made. -WHERE can also be used to define auxiliary variables in PROCEDURE -definitions. - -\endsection -\item[WHILE] -WHILE (pages 59, 61, 63, 65) - -The WHILE command causes a statement to be repeatedly executed until a -given condition is true. If the condition is initially false, the -statement is not executed at all. - - WHILE condition DO statement - -condition is given by a logical operator, statement must be a single -REDUCE statement, or a GROUP (<<...>>) or BEGIN...END block. - -Examples: -a := 10; A := 10 -while a <= 12 do <>; 10 - 11 - 12 -while a < 5 do <>; .... nothing is printed - -\endsection -\xitem[WHITTAKERM] -WHITTAKERM (pages 185, 397) - -\endsection -\item[WHITTAKERW] -WHITTAKERW (pages 185, 397) - -The WHITTAKERW operator returns Whittaker's W function. - - WHITTAKERW(parameter,parameter,argument) - -Examples: -load_package specfn; (SPECFN) - 1 - 4*SQRT(2)*KUMMERU(---,5,2) - 2 -WhittakerW(2,2,2); ---------------------------- - E - -Whittaker's W function is one of the Confluent Hypergeometric functions. -For reference see the HYPERGEOMETRIC operator. - -\endsection -\xitem[Workspace] -Workspace (page 99) - -\endsection -\item[WRITE] -WRITE (page 105)) - -The WRITE command explicitly writes its arguments to the output device -(terminal or file). - - WRITE item{,item} - -item can be an expression, an assignment or a STRING enclosed in -double quotation marks ("). - -Examples: -write a, sin x, "this is a string"; ASIN(X)this is a string -write a," ",sin x," this is a string"; A SIN(X) this is a string -if not numberp(a) then write "the symbol ",a; the symbol A -array m(10); -for i := 1:5 do write m(i) := 2*i; - M(1) := 2 - M(2) := 4 - M(3) := 6 - M(4) := 8 - M(5) := 10 -m(4); 8 - -The items specified by a single WRITE statement print on a single line -unless they are too long. A printed line is always ended with a carriage -return, so the next item printed starts a new line. - -When an assignment statement is printed, the assignment is also made. -This allows you to get feedback on filling slots in an array with a -FOR statement, as shown in the last example above. - -\endsection -\item[WS] -WS (pages 29, 158) - -The WS operator alone returns the last result; WS with a number -argument returns the results of the REDUCE statement executed after -that numbered prompt. - - WS or WS(number) - -number must be an integer between 1 and the current REDUCE prompt number. - -Examples: -(In the following examples, unlike most others, the numbered -prompt is shown.) -1: df(sin y,y); COS(Y) - 2 -2: ws^2; COS(Y) - -3: df(ws 1,y); -SIN(Y) - -WS and WS(number) can be used anywhere the expression they stand for -can be used. Calling a number for which no result was produced, such -as a switch setting, will give an error message. - -The current workspace always contains the results of the last REDUCE -command that produced an expression, even if several input statements -that do not produce expressions have intervened. For example, if you -do a differentiation, producing a result expression, then change -several switches, the operator WS; returns the results of the -differentiation. The current workspace (WS) can also be used inside -files, though the numbered workspace contains only the IN command that -input the file. - -There are three history lists kept in your REDUCE session. The first -stores raw input, suitable for the statement editor. The second -stores parsed input, ready to execute and accessible by INPUT. The -third stores results, when they are produced by statements, which are -accessible by the WS n operator. If your session is very long, -storage space begins to fill up with these expressions, so it is a -good idea to end the session once in a while, saving needed -expressions to files with the SAVEAS and OUT commands. - -An error message is given if a reference number has not yet been used. - -\endsection -\item[WTLEVEL] -WTLEVEL (page 152) - -In conjunction with WEIGHT, WTLEVEL is used to implement asymptotic -constraints. Default value is 2. - - WTLEVEL integer - -integer is a positive integer that is the greatest weight term to be -retained in expressions involving kernels with weight assignments. - -Examples: 4 3 2 2 3 4 -(x+y)**4; X + 4*X *Y + 6*X *Y + 4*X*Y + Y -weight x=2,y=3; -wtlevel 8; - 4 -(x+y)**4; X -wtlevel 10; - 2 2 2 -(x+y)**4; X *(6*Y + 4*X*Y + X ) -int(x**2,x); ***** X invalid as KERNEL - -WTLEVEL is used in conjunction with the command WEIGHT to enable -asymptotic constraints. Weight of a term is computed by multiplying -the weights of each variable in it by the power to which it has been -raised, and adding the resulting weights for each variable. If the -weight of the term is greater than WTLEVEL, the term is dropped from -the expression, and not used in any further computation involving the -expression. - -Once a weight has been attached to a KERNEL, it is no longer -recognised by the system as a kernel, though still a variable. It -cannot be used in REDUCE commands and operators that need kernels. -The weight attachment can be undone with a CLEAR command. WTLEVEL can -be changed as desired. - -\endsection -\xitem[X(I)] -X(I) (page 379) - -\endsection -\xitem[XI(I)] -XI(I) (page 379) - -\endsection -\item[XPND command] -XPND command (pages 253, 254, 271) - -(Part of the EXCALC package) - -There are two forms of the XPND command, which controls the use of the -product rule for the d operator and the expansion into partial -derivatives. The default for both these is ON. - - xpnd d; - xpnd @; - -Example: - load_package excalc; *** ^ redefined - (excalc) - pform x=0,y=k,z=m; - K - d(y^z); ( - 1) *Y^d Z + d Y^Z - - noxpnd d; - d(y^z); d(Y^Z) - - -See also NOXPND - -\endsection -\item[ZETA] -ZETA (pages 185, 395) - -The ZETA operator returns Riemann's Zeta function, - - Zeta (z) := sum(1/(k**z),k,1,infinity) - - ZETA(expression) - -Examples: - load_package specfn; (SPECFN) - 2 - PI - Zeta(2); ----- - 6 - on rounded; - Zeta 1.01; 100.577943338 - -Numerical computation for the Zeta function for arguments close to 1 -are tedious, because the series is converging very slowly. 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The maximum number -of characters allowed is system dependent, but is usually over 100. -However, printing is simplified if they are kept under 25 characters. - - You can also use special characters in your identifiers, but each -must be preceded by an exclamation point ! as an escape character. -Useful special characters are # $ % ^ & * - + = ? < > ^ / ! and the -space. Note that the use of the exclamation point as a special -character requires a second exclamation point as an escape character. -The underscore _ is special in this regard. It must be preceded by an -escape character in the first position in an identifier, but is treated -like a normal letter within an identifier. - - Other characters, such as ( ) # ; ' " can also be used if preceded -by a ! , but as they have special meanings to the Lisp reader it is -best to avoid them to avoid confusion. - - Many system identifiers have * before or after their names, or - -between words. If you accidentally pick one of these names for your own -identifier, it could have disastrous effects. For this reason it is -wise not to include * or - anywhere in your identifiers. - - You will notice that REDUCE does not use the escape characters when -it prints identifiers containing special characters; however, you still -must use them when you refer to these identifiers. Be careful when -editing statements containing escaped special characters to treat the -character and its escape as an inseparable pair. - - Identifiers are used for variable names, labels for GO TO statements, -and names of arrays, matrices, operators, and procedures. Once an -identifier is used as a matrix, array, scalar or operator identifier, -it may not be used again as a matrix, array or operator. An operator or -array identifier may later be used as a scalar without problems, but a -matrix identifier cannot be used as a scalar. All procedures are -entered into the system as operators, so the name of a procedure may -not be used as a matrix, array, or operator identifier either. - - -File: redhelp, Node: KERNEL, Next: STRING, Prev: IDENTIFIER, Up: Concepts section - - KERNEL type - - A KERNEL is a form that cannot be modified further by the REDUCE -canonical simplifier. Scalar variables are always kernels. The other -important class of kernels are operators with their arguments. Some -examples should help clarify this concept: - - ____________________________________________________________ - - Expression Kernel? - - x Yes - varname Yes - cos(a) Yes - log(sin(x**2)) Yes - a*b No - (x+y)**4 No - matrix-identifier No - ____________________________________________________________ - Many REDUCE operators expect kernels among their arguments. Error -messages result from attempts to use non-kernel expressions for these -arguments. - - -File: redhelp, Node: STRING, Prev: KERNEL, Up: Concepts section - - STRING type - - A STRING is any collection of characters enclosed in double quotation -marks (" ). It may be used as an argument for a variety of commands and -operators, such as IN , REDERR and WRITE . - -examples: - - ____________________________________________________________ - - write "this is a string"; - - this is a string - - - write a, " ", b, " ",c,"!"; - - A B C! - - ____________________________________________________________ - - -File: redhelp, Node: Concepts section, Next: Variables section, Up: Top - - Concepts section - -* Menu: - -* IDENTIFIER:: type -* KERNEL:: type -* STRING:: type - - -File: redhelp, Node: assumptions, Next: CARD_NO, Up: Variables section - - ASSUMPTIONS variable - - After solving a linear or polynomial equation system with -parameters, the variable ASSUMPTIONS contains a list of side relations -for the parameters. The solution is valid only as long as none of these -expression is zero. - -examples: - - ____________________________________________________________ - - solve({a*x-b*y+x,y-c},{x,y}); - - b*c - {{x=-----,y=c}} - a + 1 - - - assumptions; - - {a + 1} - - ____________________________________________________________ - - -File: redhelp, Node: CARD_NO, Next: E, Prev: assumptions, Up: Variables section - - CARD_NO variable - - CARD_NO sets the total number of cards allowed in a Fortran output -statement when FORT is on. Default is 20. - -examples: - - ____________________________________________________________ - - on fort; - - card_no := 4; - - CARD_NO=4. - - - z := (x + y)**15; - - ANS1=5005.*X**6*Y**9+3003.*X**5*Y**10+1365.*X**4*Y** - . 11+455.*X**3*Y**12+105.*X**2*Y**13+15.*X*Y**14+Y**15 - Z=X**15+15.*X**14*Y+105.*X**13*Y**2+455.*X**12*Y**3+ - . 1365.*X**11*Y**4+3003.*X**10*Y**5+5005.*X**9*Y**6+ - . 6435.*X**8*Y**7+6435.*X**7*Y**8+ANS1 - - ____________________________________________________________ - Twenty total cards means 19 continuation cards. You may set it for -more if your Fortran system allows more. Expressions are broken apart -in a Fortran-compatible way if they extend for more than CARD_NO -continuation cards. - - -File: redhelp, Node: E, Next: EVAL_MODE, Prev: CARD_NO, Up: Variables section - - E constant - - The constant E is reserved for use as the base of the natural -logarithm. Its value is approximately 2.71828284590, which REDUCE gives -to the current decimal precision when the switch [*note ROUNDED::.] is -on. - - E may be used as an iterative variable in a [*note FOR::.] statement, -or as a local variable or a [*note PROCEDURE::.] . If E is defined as a -local variable inside the procedure, the normal definition as the base -of the natural logarithm would be suspended inside the procedure. - - -File: redhelp, Node: EVAL_MODE, Next: FORT_WIDTH, Prev: E, Up: Variables section - - EVAL_MODE variable - - The system variable EVAL_MODE contains the current mode, either -[*note ALGEBRAIC::.] or [*note SYMBOLIC::.] . - -examples: - - ____________________________________________________________ - - EVAL_MODE; - - ALGEBRAIC - - ____________________________________________________________ - Some commands do not behave the same way in algebraic and symbolic -modes. - - -File: redhelp, Node: FORT_WIDTH, Next: HIGH_POW, Prev: EVAL_MODE, Up: Variables section - - FORT_WIDTH variable - - The FORT_WIDTH variable sets the number of characters in a line of -Fortran-compatible output produced when the [*note FORT::.] switch is -on. Default is 70. - -examples: - - ____________________________________________________________ - - fort_width := 30; - - FORT_WIDTH := 30 - - - on fort; - - df(sin(x**3*y),x); - - ANS=3.*COS(X - . **3*Y)*X**2* - . Y - - ____________________________________________________________ - FORT_WIDTH includes the usually blank characters at the beginning of -the card. As you may notice above, it is conservative and makes the -lines even shorter than it was told. - - -File: redhelp, Node: HIGH_POW, Next: I, Prev: FORT_WIDTH, Up: Variables section - - HIGH_POW variable - - The variable HIGH_POW is set by [*note COEFF::.] to the highest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. - -examples: - - ____________________________________________________________ - - coeff((x+1)^5*(x*(y+3)^2)^2,x); - - {0, - 0, - 4 3 2 - Y + 12*Y + 54*Y + 108*Y + 81, - 4 3 2 - 5*(Y + 12*Y + 54*Y + 108*Y + 81), - 4 3 2 - 10*(Y + 12*Y + 54*Y + 108*Y + 81), - 4 3 2 - 10*(Y + 12*Y + 54*Y + 108*Y + 81), - 4 3 2 - 5*(Y + 12*Y + 54*Y + 108*Y + 81), - 4 3 2 - Y + 12*Y + 54*Y + 108*Y + 81} - - - high_pow; - - 7 - - ____________________________________________________________ - - -File: redhelp, Node: I, Next: INFINITY, Prev: HIGH_POW, Up: Variables section - - I constant - - REDUCE knows I is the square root of -1, and that i^2 = -1. - -examples: - - ____________________________________________________________ - - (a + b*i)*(c + d*i); - - A*C + A*D*I + B*C*I - B*D - - - i**2; - - -1 - - ____________________________________________________________ - I cannot be used as an identifier. It is all right to use I as an -index variable in a FOR loop, or as a local (SCALAR ) variable inside a -BEGIN...END block, but it loses its definition as the square root of -1 -inside the block in that case. - - Only the simplest properties of i are known by REDUCE unless the -switch [*note COMPLEX::.] is turned on, which implements full complex -arithmetic in factoring, simplification, and functional values. -COMPLEX is ordinarily off. - - -File: redhelp, Node: INFINITY, Next: LOW_POW, Prev: I, Up: Variables section - - INFINITY constant - - The name INFINITY is used to represent the infinite positive number. -However, at the present time, arithmetic in terms of this operator -reflects finite arithmetic, rather than true operations on infinity. - - -File: redhelp, Node: LOW_POW, Next: NIL, Prev: INFINITY, Up: Variables section - - LOW_POW variable - - The variable LOW_POW is set by [*note COEFF::.] to the lowest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. - -examples: - - ____________________________________________________________ - - coeff((x+2*y)**6,y); - - 6 - {X , - 5 - 12*X , - 4 - 60*X , - 3 - 160*X , - 2 - 240*X , - 192*X, - 64} - - - low_pow; - - 0 - - - coeff(x**2*(x*sin(y) + 1),x); - - - - {0,0,1,SIN(Y)} - - - low_pow; - - 2 - - ____________________________________________________________ - - -File: redhelp, Node: NIL, Next: PI, Prev: LOW_POW, Up: Variables section - - NIL constant - - NIL represents the truth value false in symbolic mode, and is a -synonym for 0 in algebraic mode. It cannot be used for any other -purpose, even inside procedures or [*note FOR::.] loops. - - -File: redhelp, Node: PI, Next: requirements, Prev: NIL, Up: Variables section - - PI constant - - The identifier PI is reserved for use as the circular constant. Its -value is given by 3.14159265358..., which REDUCE gives to the current -decimal precision when REDUCE is in a floating-point mode. - - PI may be used as a looping variable in a [*note FOR::.] statement, -or as a local variable in a [*note PROCEDURE::.] . Its value in such -cases will be taken from the local environment. - - -File: redhelp, Node: requirements, Next: ROOT_MULTIPLICITIES, Prev: PI, Up: Variables section - - REQUIREMENTS variable - - After an attempt to solve an inconsistent equation system with -parameters, the variable REQUIREMENTS contains a list of expressions. -These expressions define a set of conditions implicitly equated with -zero. Any solution to this system defines a setting for the parameters -sufficient to make the original system consistent. - -examples: - - ____________________________________________________________ - - solve({x-a,x-y,y-1},{x,y}); - - {} - - - requirements; - - {a - 1} - - ____________________________________________________________ - - -File: redhelp, Node: ROOT_MULTIPLICITIES, Next: T, Prev: requirements, Up: Variables section - - ROOT_MULTIPLICITIES variable - - The ROOT_MULTIPLICITIES variable is set to the list of the -multiplicities of the roots of an equation by the [*note SOLVE::.] -operator. - - [*note SOLVE::.] returns its solutions in a list. The multiplicities -of each solution are put in the corresponding locations of the list -ROOT_MULTIPLICITIES . - - -File: redhelp, Node: T, Prev: ROOT_MULTIPLICITIES, Up: Variables section - - T constant - - The constant T stands for the truth value true. It cannot be used as -a scalar variable in a [*note block::.] , as a looping variable in a -[*note FOR::.] statement or as an [*note OPERATOR::.] name. - - -File: redhelp, Node: Variables section, Next: Syntax section, Prev: Concepts section, Up: Top - - Variables section - -* Menu: - -* assumptions:: variable -* CARD_NO:: variable -* E:: -* EVAL_MODE:: variable -* FORT_WIDTH:: variable -* HIGH_POW:: variable -* I:: -* INFINITY:: -* LOW_POW:: variable -* NIL:: -* PI:: -* requirements:: variable -* ROOT_MULTIPLICITIES:: variable -* T:: - - -File: redhelp, Node: semicolon, Next: dollar, Up: Syntax section - - ; SEMICOLON command - - The semicolon is a statement delimiter, indicating results are to be -printed when used in interactive mode. - -examples: - - ____________________________________________________________ - - (x+1)**2; - - 2 - X + 2*X + 1 - - - df(x**2 + 1,x); - - 2*X - - ____________________________________________________________ - Entering a RETURN without a semicolon or dollar sign results in a -prompt on the following line. A semicolon or dollar sign can be added -at this point to execute the statement. In interactive mode, a -statement that is ended with a semicolon and RETURN has its results -printed on the screen. - - Inside a group statement << ...>> or a BEGIN ...END block, a -semicolon or dollar sign separates individual REDUCE statements. Since -results are not printed from a block without a specific RETURN -statement, there is no difference between using the semicolon or dollar -sign. In a group statement, the last value produced is the value -returned by the group statement. Thus, if a semicolon or dollar sign is -placed between the last statement and the ending brackets, the group -statement returns the value 0 or nil, rather than the value of the last -statement. - - -File: redhelp, Node: dollar, Next: percent, Prev: semicolon, Up: Syntax section - - $ DOLLAR command - - The dollar sign is a statement delimiter, indicating results are not -to be printed when used in interactive mode. - -examples: - - ____________________________________________________________ - - - (x+1)**2$ - ____________________________________________________________ - The workspace is set to x^2 + 2x + 1 but nothing shows on the screen - ____________________________________________________________ - - - - ws; - - 2 - X + 2*X + 1 - - ____________________________________________________________ - - Entering a RETURN without a semicolon or dollar sign results in a -prompt on the following line. A semicolon or dollar sign can be added -at this point to execute the statement. In interactive mode, a -statement that ends with a dollar sign $ and a RETURN is executed, but -the results not printed. - - Inside a [*note group::.] statement << ...>> or a BEGIN ...END -[*note block::.] , a semicolon or dollar sign separates individual -REDUCE statements. Since results are not printed from a [*note -block::.] without a specific [*note RETURN::.] statement, there is no -difference between using the semicolon or dollar sign. - - In a group statement, the last value produced is the value returned -by the group statement. Thus, if a semicolon or dollar sign is placed -between the last statement and the ending brackets, the group statement -returns the value 0 or nil, rather than the value of the last statement. - - -File: redhelp, Node: percent, Next: dot, Prev: dollar, Up: Syntax section - - % PERCENT command - - The percent sign is used to precede comments; everything from a -percent to the end of the line is ignored. - -examples: - - ____________________________________________________________ - - - df(x**3 + y,x);% This is a comment (Key){Return} - - - 2 - 3*X - - - int(3*x**2,x) %This is a comment; (Key){Return} - ____________________________________________________________ - A prompt is given, waiting for the semicolon that was not detected -in the comment - ____________________________________________________________ - ____________________________________________________________ - - Statement delimiters ; and $ are not detected between a percent sign -and the end of the line. - - -File: redhelp, Node: dot, Next: assign, Prev: percent, Up: Syntax section - - . DOT operator - - The . (dot) infix binary operator adds a new item to the beginning -of an existing [*note LIST::.] . In high energy physics expressions, it -can also be used to represent the scalar product of two Lorentz -four-vectors. - -syntax: - - . - - can be any REDUCE scalar expression, including a list; -must be a [*note LIST::.] to avoid producing an error message. The dot -operator is right associative. - -examples: - - ____________________________________________________________ - - - liss := a . {}; - - LISS := {A} - - - liss := b . liss; - - LISS := {B,A} - - - newliss := liss . liss; - - NEWLISS := {{B,A},B,A} - - - firstlis := a . b . {c}; - - FIRSTLIS := {A,B,C} - - - secondlis := x . y . {z}; - - SECONDLIS := {X,Y,Z} - - - for i := 1:3 sum part(firstlis,i)*part(secondlis,i); - - - - A*X + B*Y + C*Z - - ____________________________________________________________ - - -File: redhelp, Node: assign, Next: equalsign, Prev: dot, Up: Syntax section - - := ASSIGN operator - - The := is the assignment operator, assigning the value on the -right-hand side to the identifier or other valid expression on the -left-hand side. - -syntax: - - := - - is ordinarily a single identifier, though -simple expressions may be used (see Comments below). is any -valid REDUCE expression. If is a [*note MATRIX::.] -identifier, then can be a matrix identifier -(redimensioned if necessary) which has each element set to the -corresponding elements of the identifier on the right-hand side. - -examples: - - ____________________________________________________________ - - a := x**2 + 1; - - 2 - A := X + 1 - - - a; - - 2 - X + 1 - - - first := second := third; - - FIRST := SECOND := THIRD - - - first; - - THIRD - - - second; - - THIRD - - - b := for i := 1:5 product i; - - B := 120 - - - b; - - 120 - - - w + (c := x + 3) + z; - - W + X + Z + 3 - - - c; - - X + 3 - - - y + b := c; - - Y + B := C - - - y; - - - (B - C) - - ____________________________________________________________ - The assignment operator is right associative, as shown in the second -and third examples. A string of such assignments has all but the last -item set to the value of the last item. Embedding an assignment -statement in another expression has the side effect of making the -assignment, as well as causing the given replacement in the expression. - - Assignments of values to expressions rather than simple identifiers -(such as in the last example above) can also be done, subject to the -following remarks: - - (i) If the left-hand side is an identifier, an operator, or a power, -the substitution rule is added to the rule table. - - (ii) If the operators - + / appear on the left-hand side, all but -the first term of the expression is moved to the right-hand side. - - (iii) If the operator * appears on the left-hand side, any constant -terms are moved to the right-hand side, but the symbolic factors remain. - - Assignment is valid for [*note ARRAY::.] elements, but not for -entire arrays. The assignment operator can also be used to attach -functionality to operators. - - A recursive construction such as A := A + B is allowed, but when A -is referenced again, the process of resubstitution continues until the -expression stack overflows (you get an error message). Recursive -assignments can be done safely inside controlled loop expressions, such -as [*note FOR::.] ... or [*note REPEAT::.] ...UNTIL . - - -File: redhelp, Node: equalsign, Next: replace, Prev: assign, Up: Syntax section - - = EQUALSIGN operator - - The = operator is a prefix or infix equality comparison operator. - -syntax: - - = (, ) or = - - can be any REDUCE scalar expression. - -examples: - - ____________________________________________________________ - - a := 4; - - A := 4 - - - if =(a,10) then write "yes" else write "no"; - - - - no - - - b := c; - - B := C - - - if b = c then write "yes" else write "no"; - - - - yes - - - on rounded; - - if 4.0 = 4 then write "yes" else write "no"; - - - - yes - - ____________________________________________________________ - This logical equality operator can only be used inside a conditional -statement, such as [*note IF::.] ...THEN ...ELSE or [*note REPEAT::.] -...UNTIL . In other places the equal sign establishes an algebraic -object of type [*note EQUATION::.] . - - -File: redhelp, Node: replace, Next: plussign, Prev: equalsign, Up: Syntax section - - => REPLACE operator - - The => operator is a binary operator used in [*note RULE::.] lists to -denote replacements. - -examples: - - ____________________________________________________________ - - operator f; - - let f(x) => x^2; - - f(x); - - 2 - x - - ____________________________________________________________ - - -File: redhelp, Node: plussign, Next: minussign, Prev: replace, Up: Syntax section - - + PLUSSIGN operator - - The + operator is a prefix or infix n-ary addition operator. - -syntax: - - + + - - or + ( ,+) - - may be any valid REDUCE expression. - -examples: - - ____________________________________________________________ - - x**4 + 4*x**2 + 17*x + 1; - - 4 2 - X + 4*X + 17*X + 1 - - - 14 + 15 + x; - - X + 29 - - - +(1,2,3,4,5); - - 15 - - ____________________________________________________________ - + is also valid as an addition operator for [*note MATRIX::.] -variables that are of the same dimensions and for [*note EQUATION::.] s. - - -File: redhelp, Node: minussign, Next: asterisk, Prev: plussign, Up: Syntax section - - - MINUSSIGN operator - - The - operator is a prefix or infix binary subtraction operator, as -well as the unary minus operator. - -syntax: - - - or - (,) - - may be any valid REDUCE expression. - -examples: - - ____________________________________________________________ - - 15 - 4; - - 11 - - - x*(-5); - - - 5*X - - - a - b - 15; - - A - B - 15 - - - -(a,4); - - A - 4 - - ____________________________________________________________ - The subtraction operator is left associative, so that a - b - c is -equivalent to (a - b) - c, as shown in the third example. The -subtraction operator is also valid with [*note MATRIX::.] expressions -of the correct dimensions and with [*note EQUATION::.] s. - - -File: redhelp, Node: asterisk, Next: slash, Prev: minussign, Up: Syntax section - - * ASTERISK operator - - The * operator is a prefix or infix n-ary multiplication operator. - -syntax: - - * + - - or * ( ,+) - - may be any valid REDUCE expression. - -examples: - - ____________________________________________________________ - - 15*3; - - 45 - - - 24*x*yvalue*2; - - 48*X*YVALUE - - - *(6,x); - - 6*X - - - on rounded; - - 3*1.5*x*x*x; - - 3 - 4.5*X - - - off rounded; - - 2x**2; - - 2 - 2*X - - ____________________________________________________________ - REDUCE assumes you are using an implicit multiplication operator -when an identifier is preceded by a number, as shown in the last line -above. Since no valid identifiers can begin with numbers, there is no -ambiguity in making this assumption. - - The multiplication operator is also valid with [*note MATRIX::.] -expressions of the proper dimensions: matrices A and B can be -multiplied if A is n x m and B is m x p. Matrices and [*note -EQUATION::.] s can also be multiplied by scalars: the result is as if -each element was multiplied by the scalar. - - -File: redhelp, Node: slash, Next: power, Prev: asterisk, Up: Syntax section - - / SLASH operator - - The / operator is a prefix or infix binary division operator or -prefix unary [*note RECIP::.] rocal operator. - -syntax: - - / or / - - or / (,) - - may be any valid REDUCE expression. - -examples: - - ____________________________________________________________ - - 20/5; - - 4 - - - 100/6; - - 50 - -- - 3 - - - 16/2/x; - - 8 - - - X - - - /b; - - 1 - - - B - - - /(y,5); - - Y - - - 5 - - - on rounded; - - 35/4; - - 8.75 - - - /20; - - 0.05 - - ____________________________________________________________ - The division operator is left associative, so that A/B/C is -equivalent to (A/B)/C . The division operator is also valid with square -[*note MATRIX::.] expressions of the same dimensions: With A and B -both n x n matrices and B invertible, A/B is given by A*B^-1. Division -of a matrix by a scalar is defined, with the results being the division -of each element of the matrix by the scalar. Division of a scalar by a -matrix is defined if the matrix is invertible, and has the effect of -multiplying the scalar by the inverse of the matrix. When / is used as -a reciprocal operator for a matrix, the inverse of the matrix is -returned if it exists. - - -File: redhelp, Node: power, Next: caret, Prev: slash, Up: Syntax section - - ** POWER operator - - The ** operator is a prefix or infix binary exponentiation operator. - -syntax: - - ** or ** (,) - - may be any valid REDUCE expression. - -examples: - - ____________________________________________________________ - - x**15; - - 15 - X - - - x**y**z; - - Y*Z - X - - - x**(y**z); - - Z - Y - X - - - **(y,4); - - 4 - Y - - - on rounded; - - 2**pi; - - 8.82497782708 - - ____________________________________________________________ - The exponentiation operator is left associative, so that A**B**C is -equivalent to (A**B)**C , as shown in the second example. Note that -this is not A**(B**C) , which would be right associative. - - When [*note NAT::.] is on (the default), REDUCE output produces -raised exponents, as shown. The symbol ^ , which is the upper-case 6 on -most keyboards, may be used in the place of ** . - - A square [*note MATRIX::.] may also be raised to positive and -negative powers with the exponentiation operator (negative powers -require the matrix to be invertible). Scalar expressions and [*note -EQUATION::.] s may be raised to fractional and floating-point powers. - - -File: redhelp, Node: caret, Next: geqsign, Prev: power, Up: Syntax section - - ^ CARET operator - - The ^ operator is a prefix or infix binary exponentiation operator. -It is equivalent to [*note power::.] or **. - -syntax: - - ^ or ^ (,) - - may be any valid REDUCE expression. - -examples: - - ____________________________________________________________ - - x^15; - - 15 - X - - - x^y^z; - - Y*Z - X - - - x^(y^z); - - Z - Y - X - - - ^(y,4); - - 4 - Y - - - on rounded; - - 2^pi; - - 8.82497782708 - - ____________________________________________________________ - The exponentiation operator is left associative, so that A^B^C is -equivalent to (A^B)^C , as shown in the second example. Note that this -is A^(B^C) , which would be right associative. - - When [*note NAT::.] is on (the default), REDUCE output produces -raised exponents, as shown. - - A square [*note MATRIX::.] may also be raised to positive and -negative powers with the exponentiation operator (negative powers -require the matrix to be invertible). Scalar expressions and [*note -EQUATION::.] s may be raised to fractional and floating-point powers. - - -File: redhelp, Node: geqsign, Next: greater, Prev: caret, Up: Syntax section - - >= GEQSIGN operator - - >= is an infix binary comparison operator, which returns true if -its first argument is greater than or equal to its second argument. - -syntax: - - >= - - must evaluate to an integer or floating-point number. - -examples: - - ____________________________________________________________ - - if (3 >= 2) then yes; - - yes - - - a := 15; - - A := 15 - - - if a >= 20 then big else small; - - - small - - ____________________________________________________________ - The binary comparison operators can only be used for comparisons -between numbers or variables that evaluate to numbers. The truth values -returned by such a comparison can only be used inside programming -constructs, such as [*note IF::.] ...THEN ...ELSE or [*note REPEAT::.] -...UNTIL or [*note WHILE::.] ...DO . - - -File: redhelp, Node: greater, Next: leqsign, Prev: geqsign, Up: Syntax section - - > GREATER operator - - The > is an infix binary comparison operator that returns true if -its first argument is strictly greater than its second. - -syntax: - - > - - must evaluate to a number, e.g., integer, rational or -floating point number. - -examples: - - ____________________________________________________________ - - on rounded; - - if 3.0 > 3 then write "different" else write "same"; - - - same - - - off rounded; - - a := 20; - - A := 20 - - - if a > 20 then write "bigger" else write "not bigger"; - - - not bigger - - ____________________________________________________________ - The binary comparison operators can only be used for comparisons -between numbers or variables that evaluate to numbers. The truth values -returned by such a comparison can only be used inside programming -constructs, such as [*note IF::.] ...THEN ...ELSE or [*note REPEAT::.] -...UNTIL or [*note WHILE::.] ...DO . - - -File: redhelp, Node: leqsign, Next: less, Prev: greater, Up: Syntax section - - <= LEQSIGN operator - - <= is an infix binary comparison operator that returns true if its -first argument is less than or equal to its second argument. - -syntax: - - <= - - must evaluate to a number, e.g., integer, rational or -floating point number. - -examples: - - ____________________________________________________________ - - a := 10; - - A := 10 - - - if a <= 10 then true; - - true - - ____________________________________________________________ - The binary comparison operators can only be used for comparisons -between numbers or variables that evaluate to numbers. The truth values -returned by such a comparison can only be used inside programming -constructs, such as [*note IF::.] ...THEN ...ELSE or [*note REPEAT::.] -...UNTIL or [*note WHILE::.] ...DO . - - -File: redhelp, Node: less, Next: tilde, Prev: leqsign, Up: Syntax section - - < LESS operator - - < is an infix binary logical comparison operator that returns true -if its first argument is strictly less than its second argument. - -syntax: - - < - - must evaluate to a number, e.g., integer, rational or -floating point number. - -examples: - - ____________________________________________________________ - - f := -3; - - F := -3 - - - if f < -3 then write "yes" else write "no"; - - - no - - ____________________________________________________________ - The binary comparison operators can only be used for comparisons -between numbers or variables that evaluate to numbers. The truth values -returned by such a comparison can only be used inside programming -constructs, such as [*note IF::.] ...THEN ...ELSE or [*note REPEAT::.] -...UNTIL or [*note WHILE::.] ...DO . - - -File: redhelp, Node: tilde, Next: group, Prev: less, Up: Syntax section - - ~ TILDE operator - - The ^ is used as a unary prefix operator in the left-hand sides of -[*note RULE::.] s to mark [*note Free Variable::.] s. A double tilde -marks an optional [*note Free Variable::.] . - - -File: redhelp, Node: group, Next: AND, Prev: tilde, Up: Syntax section - - << GROUP command - - The << ...>> command is a group statement, used to group statements -together where REDUCE expects a single statement. - -syntax: - - << ; OR * >> - - may be any valid REDUCE statement or expression. - -examples: - - ____________________________________________________________ - - a := 2; - - A := 2 - - - if a < 5 then <>; - - - 12 - - - <>; - - - 2 - C + 90*C + 202 - ---------------- - 225 - - ____________________________________________________________ - The value returned from a group statement is the value of the last -individual statement executed inside it. Note that when a semicolon is -placed between the last statement and the closing brackets, 0 or nil -is returned. Group statements are often used in the consequence -portions of [*note IF::.] ...THEN , [*note REPEAT::.] ...UNTIL , and -[*note WHILE::.] ...DO clauses. They may also be used in interactive -operation to execute several statements at one time. Statements inside -the group statement are separated by semicolons or dollar signs. - - -File: redhelp, Node: AND, Next: BEGIN, Prev: group, Up: Syntax section - - AND operator - - The AND binary logical operator returns true if both of its -arguments are true. - -syntax: - - AND - - must evaluate to true or nil. - -examples: - - ____________________________________________________________ - - a := 12; - - A := 12 - - - if numberp a and a < 15 then write a**2 else write "no"; - - - - 144 - - - clear a; - - if numberp a and a < 15 then write a**2 else write "no"; - - - - no - - ____________________________________________________________ - Logical operators can only be used inside conditional statements, -such as [*note WHILE::.] ...DO or [*note IF::.] ...THEN ...ELSE . AND -examines each of its arguments in order, and quits, returning nil, on -finding an argument that is not true. An error results if it is used in -other contexts. - - AND is left associative: X AND Y AND Z is equivalent to (X AND Y) -AND Z . - - -File: redhelp, Node: BEGIN, Next: block, Prev: AND, Up: Syntax section - - BEGIN command - - BEGIN is used to start a [*note block::.] statement, which is -closed with END . - -syntax: - - BEGIN ; * END - - is any valid REDUCE statement. - -examples: - - ____________________________________________________________ - - begin for i := 1:3 do write i end; - - - 1 - 2 - 3 - - - begin scalar n;n:=1;b:=for i:=1:4 product(x-i);return n end; - - - - 1 - - - b; - - 4 3 2 - X - 10*X + 35*X - 50*X + 24 - - ____________________________________________________________ - A BEGIN ...END block can do actions (such as WRITE ), but does not -return a value unless instructed to by a [*note RETURN::.] statement, -which must be the last statement executed in the block. It is -unnecessary to insert a semicolon before the END . - - Local variables, if any, are declared in the first statement -immediately after BEGIN , and may be defined as SCALAR, INTEGER, or -REAL . [*note ARRAY::.] variables declared within a BEGIN ...END block -are global in every case, and [*note LET::.] statements have global -effects. A [*note LET::.] statement involving a formal parameter affects -the calling parameter that corresponds to it. [*note LET::.] statements -involving local variables make global assignments, overwriting outside -variables by the same name or creating them if they do not exist. You -can use this feature to affect global variables from procedures, but be -careful that you do not do it inadvertently. - - -File: redhelp, Node: block, Next: COMMENT, Prev: BEGIN, Up: Syntax section - - BLOCK command - - A BLOCK is a sequence of statements enclosed by commands [*note -BEGIN::.] and [*note END::.] . - -syntax: - - BEGIN ; * END - - For more details see [*note BEGIN::.] . - - -File: redhelp, Node: COMMENT, Next: CONS, Prev: block, Up: Syntax section - - COMMENT command - - Beginning with the word COMMENT , all text until the next statement -terminator (; or $ ) is ignored. - -examples: - - ____________________________________________________________ - - - x := a**2 comment--a is the velocity of the particle;; - - - - 2 - X := A - - ____________________________________________________________ - Note that the first semicolon ends the comment and the second one -terminates the original REDUCE statement. - - Multiple-line comments are often needed in interactive files. The -COMMENT command allows a normal-looking text to accompany the REDUCE -statements in the file. - - -File: redhelp, Node: CONS, Next: END, Prev: COMMENT, Up: Syntax section - - CONS operator - - The CONS operator adds a new element to the beginning of a [*note -LIST::.] . Its operation is identical to the symbol [*note dot::.] -(dot). It can be used infix or prefix. - -syntax: - - CONS (,) or CONS - - can be any REDUCE scalar expression, including a list; -must be a list. - -examples: - - ____________________________________________________________ - - - liss := cons(a,{b}); - - {A,B} - - - - liss := c cons liss; - - {C,A,B} - - - - newliss := for each y in liss collect cons(y,list x); - - - - NEWLISS := {{C,X},{A,X},{B,X}} - - - - for each y in newliss sum (first y)*(second y); - - - - X*(A + B + C) - - ____________________________________________________________ - If you want to use CONS to put together two elements into a new list, -you must make the second one into a list with curly brackets or the LIST -command. You can also start with an empty list created by [] . - - The CONS operator is right associative: A CONS B CONS C is valid if -C is a list; B need not be a list. The list produced is [A,B,C] . - - -File: redhelp, Node: END, Next: EQUATION, Prev: CONS, Up: Syntax section - - END command - - The command END has two main uses: - - (i) as the ending of a [*note BEGIN::.] ...END [*note block::.] ; -and - - (ii) to end input from a file. - - In a BEGIN ...END [*note block::.] , there need not be a delimiter -(; or $ ) before the END , though there must be one after it, or a -right bracket matching an earlier left bracket. - - Files to be read into REDUCE should end with END; , which must be -preceded by a semicolon (usually the last character of the previous -line). The additional semicolon avoids problems with mistakes in the -files. If you have suspended file operation by answering N to a PAUSE -command, you are still, technically speaking, "in" the file. Use END -to exit the file. - - An END at the top level of a program is ignored. - - -File: redhelp, Node: EQUATION, Next: FIRST, Prev: END, Up: Syntax section - - EQUATION type - - An EQUATION is an expression where two algebraic expressions are -connected by the (infix) operator [*note EQUAL::.] or by = . For -access to the components of an EQUATION the operators [*note LHS::.] , -[*note RHS::.] or [*note PART::.] can be used. The evaluation of the -left-hand side of an EQUATION is controlled by the switch [*note -EVALLHSEQP::.] , while the right-hand side is evaluated -unconditionally. When an EQUATION is part of a logical expression, e.g. -in a [*note IF::.] or [*note WHILE::.] statement, the equation is -evaluated by subtracting both sides can comparing the result with zero. - - Equations occur in many contexts, e.g. as arguments of the [*note -SUB::.] operator and in the arguments and the results of the operator -[*note SOLVE::.] . An equation can be member of a [*note LIST::.] and -you may assign an equation to a variable. Elementary arithmetic is -supported for equations: if [*note EVALLHSEQP::.] is on, you may add -and subtract equations, and you can combine an equation with a scalar -expression by addition, subtraction, multiplication, division and raise -an equation to a power. - -examples: - - ____________________________________________________________ - - on evallhseqp; - - u:=x+y=1$ - - v:=2x-y=0$ - - 2*u-v; - - - 3*y=-2 - - - ws/3; - - 2 - y=-- - 3 - - ____________________________________________________________ - - Important: the equation must occur in the leftmost term of such an -expression. For other operations, e.g. taking function values of both -sides, use the [*note MAP::.] operator. - - -File: redhelp, Node: FIRST, Next: FOR, Prev: EQUATION, Up: Syntax section - - FIRST operator - - The FIRST operator returns the first element of a [*note LIST::.] . - -syntax: - - FIRST () or FIRST - - must be a non-empty list to avoid an error message. - -examples: - - ____________________________________________________________ - - alist := {a,b,c,d}; - - ALIST := {A,B,C,D} - - - first alist; - - A - - - blist := {x,y,{ww,aa,qq},z}; - - BLIST := {X,Y,{WW,AA,QQ},Z} - - - first third blist; - - WW - - ____________________________________________________________ - - -File: redhelp, Node: FOR, Next: FOREACH, Prev: FIRST, Up: Syntax section - - FOR command - - The FOR command is used for iterative loops. There are many possible -forms it can take. - - ____________________________________________________________ - - / - / |STEP UNTIL| - |:=| || - FOR| | : | | - | / | - |EACH IN | - / - - where ::= DO|PRODUCT|SUM|COLLECT|JOIN. - ____________________________________________________________ - can be any valid REDUCE identifier except T or NIL , , - and can be any expression that evaluates to a positive -or negative integer. must be a valid [*note LIST::.] structure. -The action taken must be one of the actions shown above, each of which -is followed by a single REDUCE expression, statement or a [*note -group::.] (<< ...>> ) or [*note block::.] ([*note BEGIN::.] ...[*note -END::.] ) statement. - -examples: - - ____________________________________________________________ - - for i := 1:10 sum i; - - - - 55 - - - for a := -2 step 3 until 6 product a; - - - - -8 - - - a := 3; - - A := 3 - - - for iter := 4:a do write iter; - - m := 0; - - M := 0 - - - for s := 10 step -1 until 3 do <>; - - m; - - 520 - - - for each x in {q,r,s} sum x**2; - - 2 2 2 - Q + R + S - - - for i := 1:4 collect 1/i; - - - - 1 1 1 - {1,-,-,-} - 2 3 4 - - - for i := 1:3 join list solve(x**2 + i*x + 1,x); - - - - SQRT(3)*I + 1 - {{X= --------------, - 2 - SQRT(3)*I - 1 - X= --------------} - 2 - {X=-1}, - SQRT(5) + 3 SQRT(5) - 3 - {X= - -----------,X=-----------}} - 2 2 - - ____________________________________________________________ - The behavior of each of the five action words follows: - - ____________________________________________________________ - - Action Word Behavior - Keyword Argument Type Action - do statement, command, group Evaluates its argument once - or block for each iteration of the loop, - not saving results - collect expression, statement, Evaluates its argument once for - command, group, block, list each iteration of the loop, - storing the results in a list - which is returned by the for - statement when done - join list or an operator which Evaluates its argument once for - produces a list each iteration of the loop, - appending the elements in each - individual result list onto the - overall result list - product expression, statement, Evaluates its argument once for - command, group or block each iteration of the loop, - multiplying the results together - and returning the overall product - sum expression, statement, Evaluates its argument once for - command, group or block each iteration of the loop, - adding the results together and - returning the overall sum - ____________________________________________________________ - For number-driven FOR statements, if the ending limit is smaller -than the beginning limit (larger in the case of negative steps) the -action statement is not executed at all. The iterative variable is -local to the FOR statement, and does not affect the value of an -identifier with the same name. For list-driven FOR statements, if the -list is empty, the action statement is not executed, but no error -occurs. - - You can use nested FOR statements, with the inner FOR statement -after the action keyword. You must make sure that your inner statement -returns an expression that the outer statement can handle. - - -File: redhelp, Node: FOREACH, Next: GEQ, Prev: FOR, Up: Syntax section - - FOREACH command - - FOREACH is a synonym for the FOR EACH variant of the [*note FOR::.] -construct. It is designed to iterate down a list, and an error will -occur if a list is not used. The use of FOR EACH is preferred to -FOREACH . - -syntax: - - FOREACH in - - where ::= DO PRODUCT SUM COLLECT JOIN - -examples: - - ____________________________________________________________ - - foreach x in {q,r,s} sum x**2; - - 2 2 2 - Q + R + S - - ____________________________________________________________ - - -File: redhelp, Node: GEQ, Next: GOTO, Prev: FOREACH, Up: Syntax section - - GEQ operator - - The GEQ operator is a binary infix or prefix logical operator. It -returns true if its first argument is greater than or equal to its -second argument. As an infix operator it is identical with >= . - -syntax: - - GEQ (,) or GEQ - - can be any valid REDUCE expression that evaluates to a -number. - -examples: - - ____________________________________________________________ - - a := 20; - - A := 20 - - - if geq(a,25) then write "big" else write "small"; - - - - small - - - if a geq 20 then write "big" else write "small"; - - - - big - - - if (a geq 18) then write "big" else write "small"; - - - - big - - ____________________________________________________________ - Logical operators can only be used in conditional statements such as - - [*note IF::.] ...THEN ...ELSE or [*note REPEAT::.] ...UNTIL . - - -File: redhelp, Node: GOTO, Next: GREATERP, Prev: GEQ, Up: Syntax section - - GOTO command - - Inside a BEGIN ...END [*note block::.] , GOTO , or preferably, GO -TO , transfers flow of control to a labeled statement. - -syntax: - - GO TO or GOTO - - is of the form >; - - - - 46 - - - max(-5,-10,-a); - - -5 - - ____________________________________________________________ - - -File: redhelp, Node: MIN, Next: MINUS, Prev: MAX, Up: Arithmetic Operations section - - MIN operator - - The operator MIN is an n-ary prefix operator, which returns the -smallest value in its arguments. - -syntax: - - MIN (,*) - - must evaluate to a number. MIN of an empty list returns -0. - -examples: - - ____________________________________________________________ - - min(-3,0,17,2); - - -3 - - - <>; - - - - 16 - - - min(5,10,a); - - 5 - - ____________________________________________________________ - - -File: redhelp, Node: MINUS, Next: NEXTPRIME, Prev: MIN, Up: Arithmetic Operations section - - MINUS operator - - The MINUS operator is a unary minus, returning the negative of its -argument. It is equivalent to the unary - . - -syntax: - - MINUS () - - may be any scalar REDUCE expression. - -examples: - - ____________________________________________________________ - - minus(a); - - - A - - - minus(-1); - - 1 - - - minus((x+1)**4); - - 4 3 2 - - (X + 4*X + 6*X + 4*X + 1) - - ____________________________________________________________ - - -File: redhelp, Node: NEXTPRIME, Next: NOCONVERT, Prev: MINUS, Up: Arithmetic Operations section - - NEXTPRIME operator - -syntax: - - NEXTPRIME () - - If the argument of NEXTPRIME is an integer, the least prime greater -than that argument is returned. Otherwise, a type error results. - -examples: - - ____________________________________________________________ - - nextprime 5001; - - 5003 - - - nextprime(10^30); - - 1000000000000000000000000000057 - - - nextprime a; - - ***** A invalid as integer - - ____________________________________________________________ - - -File: redhelp, Node: NOCONVERT, Next: NORM, Prev: NEXTPRIME, Up: Arithmetic Operations section - - NOCONVERT switch - - Under normal circumstances when ROUNDED is on, REDUCE converts the -number 1.0 to the integer 1. If this is not desired, the switch -NOCONVERT can be turned on. - -examples: - - ____________________________________________________________ - - on rounded; - - 1.0000000000001; - - 1 - - - on noconvert; - - 1.0000000000001; - - 1.0 - - ____________________________________________________________ - - -File: redhelp, Node: NORM, Next: PERM, Prev: NOCONVERT, Up: Arithmetic Operations section - - NORM operator - -syntax: - - NORM () - - If ROUNDED is on, and the argument is a real number, returns -its absolute value. If COMPLEX is also on, returns the square -root of the sum of squares of the real and imaginary parts of the -argument. In all other cases, a result is returned in terms of the -original operator. - -examples: - - ____________________________________________________________ - - norm (-2); - - NORM(-2) - - - on rounded; - - ws; - - 2.0 - - - norm(3+4i); - - NORM(4*I+3) - - - on complex; - - ws; - - 5.0 - - ____________________________________________________________ - - -File: redhelp, Node: PERM, Next: PLUS, Prev: NORM, Up: Arithmetic Operations section - - PERM operator - -syntax: - - perm(,) - - If and evaluate to positive integers, -PERM returns the number of permutations possible in selecting - objects from objects. In other cases, an -expression in the original operator is returned. - -examples: - - ____________________________________________________________ - - perm(1,1); - - 1 - - - perm(3,5); - - 60 - - - perm(-3,5); - - PERM(-3,5) - - - perm(a,b); - - PERM(A,B) - - ____________________________________________________________ - - -File: redhelp, Node: PLUS, Next: QUOTIENT, Prev: PERM, Up: Arithmetic Operations section - - PLUS operator - - The PLUS operator is both an infix and prefix n-ary addition -operator. It exists because of the way in which REDUCE handles such -operators internally, and is not recommended for use in algebraic mode -programming. [*note plussign::.] , which has the identical effect, -should be used instead. - -syntax: - - PLUS (,, *) or - - PLUS PLUS * - - can be any valid REDUCE expression, including matrix -expressions of the same dimensions. - -examples: - - ____________________________________________________________ - - a plus b plus c plus d; - - A + B + C + D - - - 4.5 plus 10; - - 29 - -- - 2 - - - - plus(x**2,y**2); - - 2 2 - X + Y - - ____________________________________________________________ - - -File: redhelp, Node: QUOTIENT, Next: RAD2DEG, Prev: PLUS, Up: Arithmetic Operations section - - QUOTIENT operator - - The QUOTIENT operator is both an infix and prefix binary operator -that returns the quotient of its first argument divided by its second. -It is also a unary [*note RECIP::.] rocal operator. It is identical to -/ and [*note slash::.] . - -syntax: - - QUOTIENT (,) or QUOTIENT - or QUOTIENT () or QUOTIENT - - can be any valid REDUCE scalar expression. Matrix -expressions can also be used if the second expression is invertible and -the matrices are of the correct dimensions. - -examples: - - ____________________________________________________________ - - quotient(a,x+1); - - A - ----- - X + 1 - - - 7 quotient 17; - - 7 - -- - 17 - - - on rounded; - - 4.5 quotient 2; - - 2.25 - - - quotient(x**2 + 3*x + 2,x+1); - - X + 2 - - - matrix m,inverse; - - m := mat((a,b),(c,d)); - - M(1,1) := A; - M(1,2) := B; - M(2,1) := C - M(2,2) := D - - - - inverse := quotient m; - - D - INVERSE(1,1) := ---------- - A*D - B*C - B - INVERSE(1,2) := - ---------- - A*D - B*C - C - INVERSE(2,1) := - ---------- - A*D - B*C - A - INVERSE(2,2) := ---------- - A*D - B*C - - ____________________________________________________________ - - The QUOTIENT operator is left associative: A QUOTIENT B QUOTIENT C -is equivalent to (A QUOTIENT B) QUOTIENT C . - - If a matrix argument to the unary QUOTIENT is not invertible, or if -the second matrix argument to the binary quotient is not invertible, an -error message is given. - - -File: redhelp, Node: RAD2DEG, Next: RAD2DMS, Prev: QUOTIENT, Up: Arithmetic Operations section - - RAD2DEG operator - -syntax: - - RAD2DEG () - - In [*note ROUNDED::.] mode, if is a real number, the -operator RAD2DEG will interpret it as radians, and convert it to the -equivalent degrees. In all other cases, an expression in terms of the -original operator is returned. - -examples: - - ____________________________________________________________ - - rad2deg 1; - - RAD2DEG(1) - - - on rounded; - - ws; - - 57.2957795131 - - - rad2deg a; - - RAD2DEG(A) - - ____________________________________________________________ - - -File: redhelp, Node: RAD2DMS, Next: RECIP, Prev: RAD2DEG, Up: Arithmetic Operations section - - RAD2DMS operator - -syntax: - - RAD2DMS () - - In [*note ROUNDED::.] mode, if is a real number, the -operator RAD2DMS will interpret it as radians, and convert it to a list -containing the equivalent degrees, minutes and seconds. In all other -cases, an expression in terms of the original operator is returned. - -examples: - - ____________________________________________________________ - - rad2dms 1; - - RAD2DMS(1) - - - on rounded; - - ws; - - {57,17,44.8062470964} - - - rad2dms a; - - RAD2DMS(A) - - ____________________________________________________________ - - -File: redhelp, Node: RECIP, Next: REMAINDER, Prev: RAD2DMS, Up: Arithmetic Operations section - - RECIP operator - - RECIP is the alphabetical name for the division operator / or -[*note slash::.] used as a unary operator. The use of / is preferred. - -examples: - - ____________________________________________________________ - - recip a; - - 1 - - - A - - - recip 2; - - 1 - -- - 2 - - ____________________________________________________________ - - -File: redhelp, Node: REMAINDER, Next: ROUND, Prev: RECIP, Up: Arithmetic Operations section - - REMAINDER operator - - The REMAINDER operator returns the remainder after its first -argument is divided by its second argument. - -syntax: - - REMAINDER (,) - - can be any valid REDUCE polynomial, and is not limited -to numeric values. - -examples: - - ____________________________________________________________ - - remainder(13,6); - - 1 - - - remainder(x**2 + 3*x + 2,x+1); - - 0 - - - remainder(x**3 + 12*x + 4,x**2 + 1); - - - 11*X + 4 - - - remainder(sin(2*x),x*y); - - SIN(2*X) - - ____________________________________________________________ - In the default case, remainders are calculated over the integers. If -you need the remainder with respect to another domain, it must be -declared explicitly. - - If the first argument to REMAINDER contains a denominator not equal -to 1, an error occurs. - - -File: redhelp, Node: ROUND, Next: SETMOD, Prev: REMAINDER, Up: Arithmetic Operations section - - ROUND operator - -syntax: - - ROUND () - - If its argument has a numerical value, ROUND rounds it to the -nearest integer. For non-numeric arguments, the value is an expression -in the original operator. - -examples: - - ____________________________________________________________ - - round 3.4; - - 3 - - - round 3.5; - - 4 - - - round a; - - ROUND(A) - - ____________________________________________________________ - - -File: redhelp, Node: SETMOD, Next: SIGN, Prev: ROUND, Up: Arithmetic Operations section - - SETMOD command - - The SETMOD command sets the modulus value for subsequent [*note -MODULAR::.] arithmetic. - -syntax: - - SETMOD - - must be positive, and greater than 1. It need not be a -prime number. - -examples: - - ____________________________________________________________ - - setmod 6; - - 1 - - - on modular; - - 16; - - 4 - - - x^2 + 5x + 7; - - 2 - X + 5*X + 1 - - - x/3; - - X - - - 3 - - - setmod 2; - - 6 - - - (x+1)^4; - - 4 - X + 1 - - - x/3; - - X - - ____________________________________________________________ - SETMOD returns the previous modulus, or 1 if none has been set -before. SETMOD only has effect when [*note MODULAR::.] is on. - - Modular operations are done only on numbers such as coefficients of -polynomials, not on the exponents. The modulus need not be prime. -Attempts to divide by a power of the modulus produces an error message, -since the operation is equivalent to dividing by 0. However, dividing -by a factor of a non-prime modulus does not produce an error message. - - -File: redhelp, Node: SIGN, Next: SQRT, Prev: SETMOD, Up: Arithmetic Operations section - - SIGN operator - -syntax: - - SIGN - - SIGN tries to evaluate the sign of its argument. If this is possible -SIGN returns one of 1, 0 or -1. Otherwise, the result is the original -form or a simplified variant. - -examples: - - ____________________________________________________________ - - sign(-5) - - -1 - - - sign(-a^2*b) - - -SIGN(B) - - ____________________________________________________________ - Even powers of formal expressions are assumed to be positive only as -long as the switch [*note COMPLEX::.] is off. - - -File: redhelp, Node: SQRT, Next: TIMES, Prev: SIGN, Up: Arithmetic Operations section - - SQRT operator - - The SQRT operator returns the square root of its argument. - -syntax: - - SQRT () - - can be any REDUCE scalar expression. - -examples: - - ____________________________________________________________ - - sqrt(16*a^3); - - 4*SQRT(A)*A - - - sqrt(17); - - SQRT(17) - - - on rounded; - - sqrt(17); - - 4.12310562562 - - - off rounded; - - sqrt(a*b*c^5*d^3*27); - - 2 - 3*SQRT(D)*SQRT(C)*SQRT(B)*SQRT(A)*SQRT(3)*C *D - - ____________________________________________________________ - SQRT checks its argument for squared factors and removes them. - - Numeric values for square roots that are not exact integers are -given only when [*note ROUNDED::.] is on. - - Please note that SQRT(A**2) is given as A , which may be incorrect -if A eventually has a negative value. If you are programming a -calculation in which this is a concern, you can turn on the [*note -PRECISE::.] switch, which causes the absolute value of the square root -to be returned. - - -File: redhelp, Node: TIMES, Prev: SQRT, Up: Arithmetic Operations section - - TIMES operator - - The TIMES operator is an infix or prefix n-ary multiplication -operator. It is identical to * . - -syntax: - - TIMES TIMES * - - or TIMES (, ,*) - - can be any valid REDUCE scalar or matrix expression. -Matrix expressions must be of the correct dimensions. Compatible scalar -and matrix expressions can be mixed. - -examples: - - ____________________________________________________________ - - var1 times var2; - - VAR1*VAR2 - - - times(6,5); - - 30 - - - matrix aa,bb; - - aa := mat((1),(2),(x))$ - - bb := mat((0,3,1))$ - - aa times bb times 5; - - [0 15 5 ] - [ ] - [0 30 10 ] - [ ] - [0 15*X 5*X] - - ____________________________________________________________ - - -File: redhelp, Node: Arithmetic Operations section, Next: Boolean Operators section, Prev: Syntax section, Up: Top - - Arithmetic Operations section - -* Menu: - -* ARITHMETIC_OPERATIONS:: introduction -* ABS:: operator -* ADJPREC:: switch -* ARG:: operator -* CEILING:: operator -* CHOOSE:: operator -* DEG2DMS:: operator -* DEG2RAD:: operator -* DIFFERENCE:: operator -* DILOG:: operator -* DMS2DEG:: operator -* DMS2RAD:: operator -* FACTORIAL:: operator -* FIX:: operator -* FIXP:: operator -* FLOOR:: operator -* EXPT:: operator -* GCD:: operator -* LN:: operator -* LOG:: operator -* LOGB:: operator -* MAX:: operator -* MIN:: operator -* MINUS:: operator -* NEXTPRIME:: operator -* NOCONVERT:: switch -* NORM:: operator -* PERM:: operator -* PLUS:: operator -* QUOTIENT:: operator -* RAD2DEG:: operator -* RAD2DMS:: operator -* RECIP:: operator -* REMAINDER:: operator -* ROUND:: operator -* SETMOD:: command -* SIGN:: operator -* SQRT:: operator -* TIMES:: operator - - -File: redhelp, Node: boolean value, Next: EQUAL, Up: Boolean Operators section - - BOOLEAN VALUE - - There are no extra symbols for the truth values true and false. -Instead, [*note NIL::.] and the number zero are interpreted as truth -value false in algebraic programs (see [*note false::.] ), while any -different value is considered as true (see [*note TRUE::.] ). - - -File: redhelp, Node: EQUAL, Next: EVENP, Prev: boolean value, Up: Boolean Operators section - - EQUAL operator - - The operator EQUAL is an infix binary comparison operator. It is -identical with = . It returns [*note TRUE::.] if its two arguments are -equal. - -syntax: - - EQUAL - - Equality is given between floating point numbers and integers that -have the same value. - -examples: - - ____________________________________________________________ - - on rounded; - - a := 4; - - A := 4 - - - b := 4.0; - - B := 4.0 - - - if a equal b then write "true" else write "false"; - - - - true - - - if a equal 5 then write "true" else write "false"; - - - - false - - - if a equal sqrt(16) then write "true" else write "false"; - - - - true - - ____________________________________________________________ - Comparison operators can only be used as conditions in conditional -commands such as IF ...THEN and REPEAT ...UNTIL . can also be -used as a prefix operator. However, this use is not encouraged. - - -File: redhelp, Node: EVENP, Next: false, Prev: EQUAL, Up: Boolean Operators section - - EVENP operator - - The EVENP logical operator returns [*note TRUE::.] if its argument -is an even integer, and [*note NIL::.] if its argument is an odd -integer. An error message is returned if its argument is not an integer. - -syntax: - - EVENP () or EVENP - - must evaluate to an integer. - -examples: - - ____________________________________________________________ - - aa := 1782; - - AA := 1782 - - - if evenp aa then yes else no; - - YES - - - if evenp(-3) then yes else no; - - NO - - ____________________________________________________________ - Although you would not ordinarily enter an expression such as the -last example above, note that the negative term must be enclosed in -parentheses to be correctly parsed. The EVENP operator can only be used -in conditional statements such as IF ...THEN ...ELSE or WHILE ...DO . - - -File: redhelp, Node: false, Next: FREEOF, Prev: EVENP, Up: Boolean Operators section - - FALSE - - The symbol [*note NIL::.] and the number zero are considered as -[*note boolean value::.] false if used in a place where a boolean value -is required. Most builtin operators return [*note NIL::.] as false -value. Algebraic programs use better zero. Note that NIL is not -printed when returned as result to a top level evaluation. - - -File: redhelp, Node: FREEOF, Next: LEQ, Prev: false, Up: Boolean Operators section - - FREEOF operator - - The FREEOF logical operator returns [*note TRUE::.] if its first -argument does not contain its second argument anywhere in its structure. - -syntax: - - FREEOF (,) or FREEOF - - can be any valid scalar REDUCE expression, must -be a kernel expression (see KERNEL ). - -examples: - - ____________________________________________________________ - - a := x + sin(y)**2 + log sin z; - - - - 2 - A := LOG(SIN(Z)) + SIN(Y) + X - - - if freeof(a,sin(y)) then write "free" else write "not free"; - - - - not free - - - if freeof(a,sin(x)) then write "free" else write "not free"; - - - - free - - - if a freeof sin z then write "free" else write "not free"; - - - - not free - - ____________________________________________________________ - Logical operators can only be used in conditional expressions such as - - IF ...THEN or WHILE ...DO . - - -File: redhelp, Node: LEQ, Next: LESSP, Prev: FREEOF, Up: Boolean Operators section - - LEQ operator - - The LEQ operator is a binary infix or prefix logical operator. It -returns [*note TRUE::.] if its first argument is less than or equal to -its second argument. As an infix operator it is identical with <= . - -syntax: - - LEQ (,) or LEQ - - can be any valid REDUCE expression that evaluates to a -number. - -examples: - - ____________________________________________________________ - - a := 15; - - A := 15 - - - if leq(a,25) then write "yes" else write "no"; - - - - yes - - - if leq(a,15) then write "yes" else write "no"; - - - - yes - - - if leq(a,5) then write "yes" else write "no"; - - - - no - - ____________________________________________________________ - Logical operators can only be used in conditional statements such as - - IF ...THEN ...ELSE or WHILE ...DO . - - -File: redhelp, Node: LESSP, Next: MEMBER, Prev: LEQ, Up: Boolean Operators section - - LESSP operator - - The LESSP operator is a binary infix or prefix logical operator. It -returns [*note TRUE::.] if its first argument is strictly less than its -second argument. As an infix operator it is identical with < . - -syntax: - - LESSP (,) or LESSP - - can be any valid REDUCE expression that evaluates to a -number. - -examples: - - ____________________________________________________________ - - a := 15; - - A := 15 - - - if lessp(a,25) then write "yes" else write "no"; - - - - yes - - - if lessp(a,15) then write "yes" else write "no"; - - - - no - - - if lessp(a,5) then write "yes" else write "no"; - - - - no - - ____________________________________________________________ - Logical operators can only be used in conditional statements such as - - IF ...THEN ...ELSE or WHILE ...DO . - - -File: redhelp, Node: MEMBER, Next: NEQ, Prev: LESSP, Up: Boolean Operators section - - MEMBER operator - -syntax: - - MEMBER - - MEMBER is an infix binary comparison operator that evaluates to -[*note TRUE::.] if is [*note EQUAL::.] to a member of the -[*note LIST::.] . - -examples: - - ____________________________________________________________ - - if a member {a,b} then 1 else 0; - - 1 - - - if 1 member(1,2,3) then a else b; - - a - - - if 1 member(1.0,2) then a else b; - - b - - ____________________________________________________________ - Logical operators can only be used in conditional statements such as - - IF ...THEN ...ELSE or WHILE ...DO . can also be used as a -prefix operator. However, this use is not encouraged. Finally, [*note -EQUAL::.] (= ) is used for the test within the list, so expressions -must be of the same type to match. - - -File: redhelp, Node: NEQ, Next: NOT, Prev: MEMBER, Up: Boolean Operators section - - NEQ operator - - The operator NEQ is an infix binary comparison operator. It returns -[*note TRUE::.] if its two arguments are not [*note EQUAL::.] . - -syntax: - - NEQ - - An inequality is satisfied between floating point numbers and -integers that have the same value. - -examples: - - ____________________________________________________________ - - on rounded; - - a := 4; - - A := 4 - - - b := 4.0; - - B := 4.0 - - - if a neq b then write "true" else write "false"; - - - - false - - - if a neq 5 then write "true" else write "false"; - - - - true - - ____________________________________________________________ - Comparison operators can only be used as conditions in conditional -commands such as IF ...THEN and REPEAT ...UNTIL . can also be -used as a prefix operator. However, this use is not encouraged. - - -File: redhelp, Node: NOT, Next: NUMBERP, Prev: NEQ, Up: Boolean Operators section - - NOT operator - - The NOT operator returns [*note TRUE::.] if its argument evaluates to -[*note NIL::.] , and NIL if its argument is TRUE . - -syntax: - - NOT () - -examples: - - ____________________________________________________________ - - if not numberp(a) then write "indeterminate" else write a; - - - - indeterminate; - - - a := 10; - - A := 10 - - - if not numberp(a) then write "indeterminate" else write a; - - - - 10 - - - if not(numberp(a) and a < 0) then write "positive number"; - - - - positive number - - ____________________________________________________________ - Logical operators can only be used in conditional statements such as - - IF ...THEN ...ELSE or WHILE ...DO . - - -File: redhelp, Node: NUMBERP, Next: ORDP, Prev: NOT, Up: Boolean Operators section - - NUMBERP operator - - The NUMBERP operator returns [*note TRUE::.] if its argument is a -number, and [*note NIL::.] otherwise. - -syntax: - - NUMBERP () or NUMBERP - - can be any REDUCE scalar expression. - -examples: - - ____________________________________________________________ - - cc := 15.3; - - CC := 15.3 - - - if numberp(cc) then write "number" else write "nonnumber"; - - - number - - - if numberp(cb) then write "number" else write "nonnumber"; - - - nonnumber - - ____________________________________________________________ - Logical operators can only be used in conditional expressions, such -as - - IF ...THEN ...ELSE and WHILE ...DO . - - -File: redhelp, Node: ORDP, Next: PRIMEP, Prev: NUMBERP, Up: Boolean Operators section - - ORDP operator - - The ORDP logical operator returns [*note TRUE::.] if its first -argument is ordered ahead of its second argument in canonical internal -ordering, or is identical to it. - -syntax: - - ORDP (,) - - and can be any valid REDUCE scalar -expression. - -examples: - - ____________________________________________________________ - - if ordp(x**2 + 1,x**3 + 3) then write "yes" else write "no"; - - - - no - - - if ordp(101,100) then write "yes" else write "no"; - - - - yes - - - if ordp(x,x) then write "yes" else write "no"; - - - - yes - - ____________________________________________________________ - Logical operators can only be used in conditional expressions, such -as - - IF ...THEN ...ELSE and WHILE ...DO . - - -File: redhelp, Node: PRIMEP, Next: TRUE, Prev: ORDP, Up: Boolean Operators section - - PRIMEP operator - -syntax: - - PRIMEP () or PRIMEP - - If evaluates to a integer, PRIMEP returns [*note -TRUE::.] if is a prime number and [*note NIL::.] otherwise. -If does not have an integer value, a type error occurs. - -examples: - - ____________________________________________________________ - - if primep 3 then write "yes" else write "no"; - - - YES - - - if primep a then 1; - - ***** A invalid as integer - - ____________________________________________________________ - - -File: redhelp, Node: TRUE, Prev: PRIMEP, Up: Boolean Operators section - - TRUE - - Any value of the boolean part of a logical expression which is -neither [*note NIL::.] nor 0 is considered as TRUE . Most builtin test -and compare functions return [*note T::.] for TRUE and [*note NIL::.] -for FALSE . - -examples: - - ____________________________________________________________ - - if member(3,{1,2,3}) then 1 else -1; - - - 1 - - - if floor(1.7) then 1 else -1; - - 1 - - - if floor(0.7) then 1 else -1; - - -1 - - ____________________________________________________________ - - -File: redhelp, Node: Boolean Operators section, Next: General Commands section, Prev: Arithmetic Operations section, Up: Top - - Boolean Operators section - -* Menu: - -* boolean value:: concept -* EQUAL:: operator -* EVENP:: operator -* false:: concept -* FREEOF:: operator -* LEQ:: operator -* LESSP:: operator -* MEMBER:: operator -* NEQ:: operator -* NOT:: operator -* NUMBERP:: operator -* ORDP:: operator -* PRIMEP:: operator -* TRUE:: concept - - -File: redhelp, Node: BYE, Next: CONT, Up: General Commands section - - BYE command - - The BYE command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are at -the top level, the BYE command exits REDUCE. QUIT is a synonym for BYE . - - -File: redhelp, Node: CONT, Next: DISPLAY, Prev: BYE, Up: General Commands section - - CONT command - - The command CONT returns control to an interactive file after a -[*note PAUSE::.] command that has been answered with N . - -examples: - - ____________________________________________________________ - ____________________________________________________________ - Suppose you are in the middle of an interactive file. - ____________________________________________________________ - - - - - factorize(x**2 + 17*x + 60); - - - - - {X + 5,X + 12} - - - pause; - - Cont? (Y or N) - - - n - - saveas results; - - factor1 := first results; - - FACTOR1 := X + 5 - - - factor2 := second results; - - FACTOR2 := X + 12 - - - cont; - ____________________________________________________________ - the file resumes - ____________________________________________________________ - - - ____________________________________________________________ - - A [*note PAUSE::.] allows you to enter your own REDUCE commands, -change switch values, inquire about results, or other such activities. -When you wish to resume operation of the interactive file, use CONT . - - -File: redhelp, Node: DISPLAY, Next: LOAD_PACKAGE, Prev: CONT, Up: General Commands section - - DISPLAY command - - When given a numeric argument , DISPLAY prints the most -recent input statements, identified by prompt numbers. If an empty pair -of parentheses is given, or if is greater than the current number -of statements, all the input statements since the beginning of the -session are printed. - -syntax: - - DISPLAY () or DISPLAY () - - should be a positive integer. However, if it is a real number, -the truncated integer value is used, and if a non-numeric argument is -used, all the input statements are printed. - - The statements are displayed in upper case, with lines split at -semicolons or dollar signs, as they are in editing. If long files have -been input during the session, the DISPLAY command is slow to format -these for printing. - - -File: redhelp, Node: LOAD_PACKAGE, Next: PAUSE, Prev: DISPLAY, Up: General Commands section - - LOAD_PACKAGE command - - The LOAD_PACKAGE command is used to load REDUCE packages, such as -GENTRAN that are not automatically loaded by the system. - -syntax: - - LOAD_PACKAGE " " - - A package is only loaded once; subsequent calls of LOAD_PACKAGE for -the same package name are ignored. - - -File: redhelp, Node: PAUSE, Next: QUIT, Prev: LOAD_PACKAGE, Up: General Commands section - - PAUSE command - - The PAUSE command, given in an interactive file, stops operation and -asks if you want to continue or not. - -examples: - - ____________________________________________________________ - ____________________________________________________________ - An interactive file is running, and at some point you see the -question - ____________________________________________________________ - - - Cont? (Y or N) - ____________________________________________________________ - If you type - ____________________________________________________________ - - - y(Key){Return} - ____________________________________________________________ - the file continues to run until the next pause or the end. - ____________________________________________________________ - - ____________________________________________________________ - If you type - ____________________________________________________________ - - - n(Key){Return} - ____________________________________________________________ - you will get a numbered REDUCE prompt, and be allowed to enter and -execute any REDUCE statements. If you later wish to continue with the -file, type - ____________________________________________________________ - - - cont; - ____________________________________________________________ - and the file resumes. - ____________________________________________________________ - ____________________________________________________________ - - To use PAUSE in your own interactive files, type - - PAUSE; in the file wherever you want it. - - PAUSE does not allow you to continue without typing either Y or N . -Its use is to slow down scrolling of interactive files, or to let you -change parameters or switch settings for the calculations. - - If you have stopped an interactive file at a PAUSE, and do not wish -to resume the file, type END; . This does not end the REDUCE session, -but stops input from the file. A second END; ends the REDUCE session. -However, if you have pauses from more than one file stacked up, an END; -brings you back to the top level, not the file directly above. - - A PAUSE typed from the terminal has no effect. - - -File: redhelp, Node: QUIT, Next: RECLAIM, Prev: PAUSE, Up: General Commands section - - QUIT command - - The QUIT command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are at -the top level, the QUIT command exits REDUCE. [*note BYE::.] is a -synonym for QUIT . - - -File: redhelp, Node: RECLAIM, Next: REDERR, Prev: QUIT, Up: General Commands section - - RECLAIM operator - - REDUCE's memory is in a storage structure called a heap. As REDUCE -statements execute, chunks of memory are used up. When these chunks are -no longer needed, they remain idle. When the memory is almost full, the -system executes a garbage collection, reclaiming space that is no -longer needed, and putting all the free space at one end. Depending on -the size of the image REDUCE is using, garbage collection needs to be -done more or less often. A larger image means fewer but longer garbage -collections. Regardless of memory size, if you ask REDUCE to do -something ridiculous, like FACTORIAL(2000) , it may garbage collect -many times. - - -File: redhelp, Node: REDERR, Next: RETRY, Prev: RECLAIM, Up: General Commands section - - REDERR command - - The REDERR command allows you to print an error message from inside -a [*note PROCEDURE::.] or a [*note block::.] statement. The -calculation is gracefully terminated. - -syntax: - - REDERR - - is an error message, usually inside double quotation marks -(a [*note STRING::.] ). - -examples: - - ____________________________________________________________ - - procedure fac(n); - if not (fixp(n) and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n-1 product i+1; - - - fac - - - fac a; - - ***** Choose nonneg. integer only - - - fac 5; - - 120 - - ____________________________________________________________ - The above procedure finds the factorial of its argument. If n is -not a positive integer or 0, an error message is returned. - - If your procedure is executed in a file, the usual error message is -printed, followed by CONT? (Y OR N) , just as any other error does from -a file. Although the procedure is gracefully terminated, any switch -settings or variable assignments you made before the error occurred are -not undone. If you need to clean up such items before exiting, use a -group statement, with the REDERR command as its last statement. - - -File: redhelp, Node: RETRY, Next: SAVEAS, Prev: REDERR, Up: General Commands section - - RETRY command - - The RETRY command allows you to retry the latest statement that -resulted in an error message. - -examples: - - ____________________________________________________________ - - matrix a; - - det a; - - ***** Matrix A not set - - - a := mat((1,2),(3,4)); - - A(1,1) := 1 - A(1,2) := 2 - A(2,1) := 3 - A(2,2) := 4 - - - retry; - - -2 - - ____________________________________________________________ - RETRY remembers only the most recent statement that resulted in an -error message. It allows you to stop and fix something obvious, then -continue on your way without retyping the original command. - - -File: redhelp, Node: SAVEAS, Next: SHOWTIME, Prev: RETRY, Up: General Commands section - - SAVEAS command - - The SAVEAS command saves the current workspace under the name of its -argument. - -syntax: - - SAVEAS - - can be any valid REDUCE identifier. - -examples: - - ____________________________________________________________ - ____________________________________________________________ - (The numbered prompts are shown below, unlike in most examples) - ____________________________________________________________ - - - 1: solve(x^2-3); - - {x=sqrt(3),x= - sqrt(3)} - - - 2: saveas rts(0)$ - - 3: rts(0); - - {x=sqrt(3),x= - sqrt(3)} - - ____________________________________________________________ - - SAVEAS works only for the current workspace, the last algebraic -expression produced by REDUCE. This allows you to save a result that you -did not assign to an identifier when you originally typed the input. -For access to previous output use [*note WS::.] . - - -File: redhelp, Node: SHOWTIME, Next: WRITE, Prev: SAVEAS, Up: General Commands section - - SHOWTIME command - - The SHOWTIME command prints the elapsed system time since the last -call of this command or since the beginning of the session, if it has -not been called before. - -examples: - - ____________________________________________________________ - - showtime; - - Time: 1020 ms - - - factorize(x^4 - 8x^4 + 8x^2 - 136x - 153); - - - - 2 - {X - 9,X + 17,X + 1} - - - showtime; - - Time: 920 ms - - ____________________________________________________________ - The time printed is either the elapsed cpu time or the elapsed wall -clock time, depending on your system. SHOWTIME allows you to see the -system time resources REDUCE uses in its calculations. Your time -readings will of course vary from this example according to the system -you use. - - -File: redhelp, Node: WRITE, Prev: SHOWTIME, Up: General Commands section - - WRITE command - - The WRITE command explicitly writes its arguments to the output -device (terminal or file). - -syntax: - - WRITE ,* - - can be an expression, an assignment or a [*note STRING::.] -enclosed in double quotation marks (" ). - -examples: - - ____________________________________________________________ - - write a, sin x, "this is a string"; - - - ASIN(X)this is a string - - - write a," ",sin x," this is a string"; - - - A SIN(X) this is a string - - - if not numberp(a) then write "the symbol ",a; - - - - the symbol A - - - array m(10); - - for i := 1:5 do write m(i) := 2*i; - - - M(1) := 2 - M(2) := 4 - M(3) := 6 - M(4) := 8 - M(5) := 10 - - - m(4); - - 8 - - ____________________________________________________________ - The items specified by a single WRITE statement print on a single -line unless they are too long. A printed line is always ended with a -carriage return, so the next item printed starts a new line. - - When an assignment statement is printed, the assignment is also -made. This allows you to get feedback on filling slots in an array with -a [*note FOR::.] statement, as shown in the last example above. - - -File: redhelp, Node: General Commands section, Next: Algebraic Operators section, Prev: Boolean Operators section, Up: Top - - General Commands section - -* Menu: - -* BYE:: command -* CONT:: command -* DISPLAY:: command -* LOAD_PACKAGE:: command -* PAUSE:: command -* QUIT:: command -* RECLAIM:: operator -* REDERR:: command -* RETRY:: command -* SAVEAS:: command -* SHOWTIME:: command -* WRITE:: command - - -File: redhelp, Node: APPEND, Next: ARBINT, Up: Algebraic Operators section - - APPEND operator - - The APPEND operator constructs a new [*note LIST::.] from the -elements of its two arguments (which must be lists). - -syntax: - - APPEND (,) - - must be a list, though it may be the empty list ([] ). Any -arguments beyond the first two are ignored. - -examples: - - ____________________________________________________________ - - alist := {1,2,{a,b}}; - - ALIST := {1,2,{A,B}} - - - blist := {3,4,5,sin(y)}; - - BLIST := {3,4,5,SIN(Y)} - - - append(alist,blist); - - {1,2,{A,B},3,4,5,SIN(Y)} - - - append(alist,{}); - - {1,2,{A,B}} - - - append(list z,blist); - - {Z,3,4,5,SIN(Y)} - - ____________________________________________________________ - The new list consists of the elements of the second list appended to -the elements of the first list. You can APPEND new elements to the -beginning or end of an existing list by putting the new element in a -list (use curly braces or the operator LIST ). This is particularly -helpful in an iterative loop. - - -File: redhelp, Node: ARBINT, Next: ARBCOMPLEX, Prev: APPEND, Up: Algebraic Operators section - - ARBINT operator - - The operator ARBINT is used to express arbitrary integer parts of an -expression, e.g. in the result of [*note SOLVE::.] when [*note -ALLBRANCH::.] is on. - -examples: - - ____________________________________________________________ - - - solve(log(sin(x+3)),x); - - {X=2*ARBINT(1)*PI - ASIN(1) - 3, - X=2*ARBINT(1)*PI + ASIN(1) + PI - 3} - - ____________________________________________________________ - - -File: redhelp, Node: ARBCOMPLEX, Next: ARGLENGTH, Prev: ARBINT, Up: Algebraic Operators section - - ARBCOMPLEX operator - - The operator ARBCOMPLEX is used to express arbitrary scalar parts of -an expression, e.g. in the result of [*note SOLVE::.] when the solution -is parametric in one of the variable. - -examples: - - ____________________________________________________________ - - - solve({x+3=y-2z,y-3x=0},{x,y,z}); - - - 2*ARBCOMPLEX(1) + 3 - {X=-------------------, - 2 - 3*ARBCOMPLEX(1) + 3 - Y=-------------------, - 2 - Z=ARBCOMPLEX(1)} - - ____________________________________________________________ - - -File: redhelp, Node: ARGLENGTH, Next: COEFF, Prev: ARBCOMPLEX, Up: Algebraic Operators section - - ARGLENGTH operator - - The operator ARGLENGTH returns the number of arguments of the -top-level operator in its argument. - -syntax: - - ARGLENGTH () - - can be any valid REDUCE algebraic expression. - -examples: - - ____________________________________________________________ - - arglength(a + b + c + d); - - 4 - - - arglength(a/b/c); - - 2 - - - arglength(log(sin(df(r**3*x,x)))); - - - 1 - - ____________________________________________________________ - In the first example, + is an n-ary operator, so the number of terms -is returned. In the second example, since / is a binary operator, the -argument is actually (a/b)/c, so there are two terms at the top level. -In the last example, no matter how deeply the operators are nested, -there is still only one argument at the top level. - - -File: redhelp, Node: COEFF, Next: COEFFN, Prev: ARGLENGTH, Up: Algebraic Operators section - - COEFF operator - - The COEFF operator returns the coefficients of the powers of the -specified variable in the given expression, in a [*note LIST::.] . - -syntax: - - COEFF (, ) - - is expected to be a polynomial expression, not a -rational expression. Rational expressions are accepted when the switch -[*note RATARG::.] is on. must be a kernel. The results are -returned in a list. - -examples: - - ____________________________________________________________ - - coeff((x+y)**3,x); - - 3 2 - {Y ,3*Y ,3*Y,1} - - - coeff((x+2)**4 + sin(x),x); - - {SIN(X) + 16,32,24,8,1} - - - high_pow; - - 4 - - - low_pow; - - 0 - - - ab := x**9 + sin(x)*x**7 + sqrt(y); - - - - 7 9 - AB := SQRT(Y) + SIN(X)*X + X - - - coeff(ab,x); - - {SQRT(Y),0,0,0,0,0,0,SIN(X),0,1} - - ____________________________________________________________ - The variables [*note HIGH_POW::.] and [*note LOW_POW::.] are set to -the highest and lowest powers of the variable, respectively, appearing -in the expression. - - The coefficients are put into a list, with the coefficient of the -lowest (constant) term first. You can use the usual list access methods -(FIRST , SECOND , THIRD , REST , LENGTH , and PART ) to extract them. -If a power does not appear in the expression, the corresponding element -of the list is zero. Terms involving functions of the specified -variable but not including powers of it (for example in the expression -X**4 + 3*X**2 + TAN(X) ) are placed in the constant term. - - Since the COEFF command deals with the expanded form of the -expression, you may get unexpected results when [*note EXP::.] is off, -or when [*note FACTOR::.] or [*note IFACTOR::.] are on. - - If you want only a specific coefficient rather than all of them, use -the [*note COEFFN::.] operator. - - -File: redhelp, Node: COEFFN, Next: CONJ, Prev: COEFF, Up: Algebraic Operators section - - COEFFN operator - - The COEFFN operator takes three arguments: an expression, a kernel, -and a non-negative integer. It returns the coefficient of the kernel to -that integer power, appearing in the expression. - -syntax: - - COEFFN (,,) - - must be a polynomial, unless [*note RATARG::.] is on -which allows rational expressions. must be a kernel, and - must be a non-negative integer. - -examples: - - ____________________________________________________________ - - - ff := x**7 + sin(y)*x**5 + y**4 + x + 7; - - - 5 7 4 - FF := SIN(Y)*X + X + X + Y + 7 - - - coeffn(ff,x,5); - - SIN(Y) - - - coeffn(ff,z,3); - - 0 - - - coeffn(ff,y,0); - - 5 7 - SIN(Y)*X + X + X + 7 - - - - rr := 1/y**2+y**3+sin(y); - - 2 5 - SIN(Y)*Y + Y + 1 - RR := -------------------- - 2 - Y - - - on ratarg; - - - coeffn(rr,y,-2); - - ***** -2 invalid as COEFFN index - - - - coeffn(rr,y,5); - - 1 - --- - 2 - Y - - ____________________________________________________________ - If the given power of the kernel does not appear in the expression, -COEFFN returns 0. Negative powers are never detected, even if they -appear in the expression and [*note RATARG::.] are on. COEFFN with an -integer argument of 0 returns any terms in the expression that do not -contain the given kernel. - - -File: redhelp, Node: CONJ, Next: CONTINUED_FRACTION, Prev: COEFFN, Up: Algebraic Operators section - - CONJ operator - -syntax: - - CONJ () or CONJ - - This operator returns the complex conjugate of an expression, if that -argument has an numerical value. A non-numerical argument is returned as -an expression in the operators [*note REPART::.] and [*note IMPART::.] . - -examples: - - ____________________________________________________________ - - conj(1+i); - - 1-I - - - conj(a+i*b); - - REPART(A) - REPART(B)*I - IMPART(A)*I - IMPART(B) - - ____________________________________________________________ - - -File: redhelp, Node: CONTINUED_FRACTION, Next: DECOMPOSE, Prev: CONJ, Up: Algebraic Operators section - - CONTINUED_FRACTION operator - -syntax: - - CONTINUED_FRACTION () or CONTINUED_FRACTION ( ,) - - This operator approximates the real number ( [*note -RATIONAL::.] number, [*note ROUNDED::.] number) into a continued -fraction. The result is a list of two elements: the first one is the -rational value of the approximation, the second one is the list of -terms of the continued fraction which represents the same value -according to the definition T0 +1/(T1 + 1/(T2 + ...)) . Precision: the -second optional parameter is an upper bound for the absolute -value of the result denominator. If omitted, the approximation is -performed up to the current system precision. - -examples: - - ____________________________________________________________ - - continued_fraction pi; - - - 1146408 - {-------,{3,7,15,1,292,1,1,1,2,1}} - 364913 - - - continued_fraction(pi,100); - - - 22 - {--,{3,7}} - 7 - - ____________________________________________________________ - - -File: redhelp, Node: DECOMPOSE, Next: DEG, Prev: CONTINUED_FRACTION, Up: Algebraic Operators section - - DECOMPOSE operator - - The DECOMPOSE operator takes a multivariate polynomial as argument, -and returns an expression and a [*note LIST::.] of [*note EQUATION::.] -s from which the original polynomial can be found by composition. - -syntax: - - DECOMPOSE () or DECOMPOSE - -examples: - - ____________________________________________________________ - - decompose(x^8-88*x^7+2924*x^6-43912*x^5+263431*x^4- - 218900*x^3+65690*x^2-7700*x+234) - - - - 2 2 2 - U + 35*U + 234, U=V + 10*V, V=X - 22*X - - - decompose(u^2+v^2+2u*v+1) - - 2 - W + 1, W=U + V - - ____________________________________________________________ - Unlike factorization, this decomposition is not unique. Further -details can be found in V.S. Alagar, M.Tanh, , Proc. EUROCAL 1985, pp 150-153 (Springer) and J. von zur -Gathen, , J. -Symbolic Computation (1990) 9, 281-299. - - -File: redhelp, Node: DEG, Next: DEN, Prev: DECOMPOSE, Up: Algebraic Operators section - - DEG operator - - The operator DEG returns the highest degree of its variable argument -found in its expression argument. - -syntax: - - DEG (,) - - is expected to be a polynomial expression, not a -rational expression. Rational expressions are accepted when the switch -[*note RATARG::.] is on. must be a [*note KERNEL::.] . The -results are returned in a list. - -examples: - - ____________________________________________________________ - - - deg((x+y)**5,x); - - 5 - - - - deg((a+b)*(c+2*d)**2,d); - - 2 - - - - deg(x**2 + cos(y),sin(x)); - - - deg((x**2 + sin(x))**5,sin(x)); - - 5 - - ____________________________________________________________ - - -File: redhelp, Node: DEN, Next: DF, Prev: DEG, Up: Algebraic Operators section - - DEN operator - - The DEN operator returns the denominator of its argument. - -syntax: - - DEN () - - is ordinarily a rational expression, but may be any -valid scalar REDUCE expression. - -examples: - - ____________________________________________________________ - - - a := x**3 + 3*x**2 + 12*x; - - 2 - A := X*(X + 3*X + 12) - - - - b := 4*x*y + x*sin(x); - - B := X*(SIN(X) + 4*Y) - - - - den(a/b); - - SIN(X) + 4*Y - - - - den(aa/4 + bb/5); - - 20 - - - - den(100/6); - - 3 - - - - den(sin(x)); - - 1 - - ____________________________________________________________ - DEN returns the denominator of the expression after it has been -simplified by REDUCE. As seen in the examples, this includes putting -sums of rational expressions over a common denominator, and reducing -common factors where possible. If the expression does not have any -other denominator, 1 is returned. - - Switch settings, such as [*note MCD::.] or [*note RATIONAL::.] , -have an effect on the denominator of an expression. - - -File: redhelp, Node: DF, Next: EXPAND_CASES, Prev: DEN, Up: Algebraic Operators section - - DF operator - - The DF operator finds partial derivatives with respect to one or -more variables. - -syntax: - - DF (, [, ] , [ , ] -) - - can be any valid REDUCE algebraic expression. -must be a [*note KERNEL::.] , and is the differentiation variable. - must be a non-negative integer. - -examples: - - ____________________________________________________________ - - - df(x**2,x); - - 2*X - - - - df(x**2*y + sin(y),y); - - 2 - COS(Y) + X - - - - df((x+y)**10,z); - - 0 - - - - - df(1/x**2,x,2); - - 6 - --- - 4 - X - - - - df(x**4*y + sin(y),y,x,3); - - 24*X - - - - for all x let df(tan(x),x) = sec(x)**2; - - - df(tan(3*x),x); - - 2 - 3*SEC(3*X) - - ____________________________________________________________ - An error message results if a non-kernel is entered as a -differentiation operator. If the optional number is omitted, it is -assumed to be 1. See the declaration [*note DEPEND::.] to establish -dependencies for implicit differentiation. - - You can define your own differentiation rules, expanding REDUCE's -capabilities, using the [*note LET::.] command as shown in the last -example above. Note that once you add your own rule for differentiating -a function, it supersedes REDUCE's normal handling of that function for -the duration of the REDUCE session. If you clear the rule ([*note -CLEARRULES::.] ), you don't get back to the previous rule. - - -File: redhelp, Node: EXPAND_CASES, Next: EXPREAD, Prev: DF, Up: Algebraic Operators section - - EXPAND_CASES operator - - When a [*note ROOT_OF::.] form in a result of [*note SOLVE::.] has -been converted to a [*note ONE_OF::.] form, EXPAND_CASES can be used to -convert this into form corresponding to the normal explicit results of -[*note SOLVE::.] . See [*note ROOT_OF::.] . - - -File: redhelp, Node: EXPREAD, Next: FACTORIZE, Prev: EXPAND_CASES, Up: Algebraic Operators section - - EXPREAD operator - -syntax: - - EXPREAD () - - EXPREAD reads one well-formed expression from the current input -buffer and returns its value. - -examples: - - ____________________________________________________________ - - expread(); a+b; - - A + B - - ____________________________________________________________ - - -File: redhelp, Node: FACTORIZE, Next: HYPOT, Prev: EXPREAD, Up: Algebraic Operators section - - FACTORIZE operator - - The FACTORIZE operator factors a given expression. - -syntax: - - FACTORIZE () - - should be a polynomial, otherwise an error will result. - -examples: - - ____________________________________________________________ - - - fff := factorize(x^3 - y^3); - - 2 2 - {X - Y,X + X*Y + Y } - - - fac1 := first fff; - - FAC1 := X - Y - - - factorize(x^15 - 1); - - {X - 1, - 2 - X + X + 1, - 4 3 2 - X + X + X + X + 1, - 8 7 6 5 4 - X - X + X - X + X - X + 1} - - - lastone := part(ws,length ws); - - 8 7 6 5 4 - LASTONE := X - X + X - X + X - X + 1 - - - setmod 2; - - 1 - - - on modular; - - factorize(x^15 - 1); - - {X + 1, - 2 - X + X + 1, - 4 - X + X + 1, - 4 3 - X + X + 1, - 4 3 2 - X + X + X + X + 1} - - ____________________________________________________________ - The FACTORIZE command returns the factors it finds as a [*note -LIST::.] . You can therefore use the usual list access methods ([*note -FIRST::.] , [*note SECOND::.] , [*note THIRD::.] , [*note REST::.] , -[*note LENGTH::.] and [*note PART::.] ) to extract the factors. - - If the given to FACTORIZE is an integer, it will be -factored into its prime components. To factor any integer factor of a -non-numerical expression, the switch [*note IFACTOR::.] should be -turned on. Its default is off. [*note IFACTOR::.] has effect only when -factoring is explicitly done by FACTORIZE , not when factoring is -automatically done with the [*note FACTOR::.] switch. If full -factorization is not needed the switch [*note LIMITEDFACTORS::.] allows -you to reduce the computing time of calls to FACTORIZE . - - Factoring can be done in a modular domain by calling FACTORIZE when -[*note MODULAR::.] is on. You can set the modulus with the [*note -SETMOD::.] command. The last example above shows factoring modulo 2. - - For general comments on factoring, see comments under the switch -[*note FACTOR::.] . - - -File: redhelp, Node: HYPOT, Next: IMPART, Prev: FACTORIZE, Up: Algebraic Operators section - - HYPOT operator - -syntax: - - hypot(,) - - If ROUNDED is on, and the two arguments evaluate to numbers, this -operator returns the square root of the sums of the squares of the -arguments in a manner that avoids intermediate overflow. In other cases, -an expression in the original operator is returned. - -examples: - - ____________________________________________________________ - - hypot(3,4); - - HYPOT(3,4) - - - on rounded; - - ws; - - 5.0 - - - hypot(a,b); - - HYPOT(A,B) - - ____________________________________________________________ - - -File: redhelp, Node: IMPART, Next: INT, Prev: HYPOT, Up: Algebraic Operators section - - IMPART operator - -syntax: - - IMPART () or IMPART - - This operator returns the imaginary part of an expression, if that -argument has an numerical value. A non-numerical argument is returned as -an expression in the operators [*note REPART::.] and IMPART . - -examples: - - ____________________________________________________________ - - impart(1+i); - - 1 - - - impart(a+i*b); - - REPART(B) + IMPART(A) - - ____________________________________________________________ - - -File: redhelp, Node: INT, Next: INTERPOL, Prev: IMPART, Up: Algebraic Operators section - - INT operator - - The INT operator performs analytic integration on a variety of -functions. - -syntax: - - INT (,) - - can be any scalar expression. involving polynomials, log -functions, exponential functions, or tangent or arctangent expressions. -INT attempts expressions involving error functions, dilogarithms and -other trigonometric expressions. Integrals involving algebraic -extensions (such as square roots) may not succeed. must be a -REDUCE [*note KERNEL::.] . - -examples: - - ____________________________________________________________ - - int(x**3 + 3,x); - - 3 - X*(X + 12) - ----------- - 4 - - - - int(sin(x)*exp(2*x),x); - - - 2*X - E *(COS(X) - 2*SIN(X)) - - ------------------------ - 5 - - - int(1/(x^2-2),x); - - - SQRT(2)*(LOG( - SQRT(2) + X) - LOG(SQRT(2) + X)) - ------------------------------------------------ - 4 - - - int(sin(x)/(4 + cos(x)**2),x); - - - COS(X) - ATAN(------) - 2 - - ------------ - 2 - - - - int(1/sqrt(x^2-x),x); - - SQRT(X)*SQRT(X - 1) - INT(-------------------,X) - 2 - X -X - - ____________________________________________________________ - Note that REDUCE couldn't handle the last integral with its default -integrator, since the integrand involves a square root. However, the -integral can be found using the [*note ALGINT::.] package. -Alternatively, you could add a rule using the [*note LET::.] statement -to evaluate this integral. - - The arbitrary constant of integration is not shown. Definite -integrals can be found by evaluating the result at the limits of -integration (use [*note ROUNDED::.] ) and subtracting the lower from -the higher. Evaluation can be easily done by the [*note SUB::.] -operator. - - When INT cannot find an integral it returns an expression involving -formal INT expressions unless the switch [*note FAILHARD::.] has been -set. If not all of the expression can be integrated, the switch [*note -NOLNR::.] controls whether a partially integrated result should be -returned or not. - - -File: redhelp, Node: INTERPOL, Next: LCOF, Prev: INT, Up: Algebraic Operators section - - INTERPOL operator - - INTERPOL generates an interpolation polynomial. - -syntax: - - interpol(,,) - - and are [*note LIST::.] s of equal length and - is an algebraic expression (preferably a [*note KERNEL::.] ). -The interpolation polynomial is generated in the given variable of -degree length()-1. The unique polynomial F is defined by the -property that for corresponding elements V of and P of - the relation F(P)=V holds. - -examples: - - ____________________________________________________________ - - f := for i:=1:4 collect(i**3-1); - - F := 0,7,26,63 - - - p := {1,2,3,4}; - - P := 1,2,3,4 - - - interpol(f,x,p); - - 3 - X - 1 - - ____________________________________________________________ - The Aitken-Neville interpolation algorithm is used which guarantees a -stable result even with rounded numbers and an ill-conditioned problem. - - -File: redhelp, Node: LCOF, Next: LENGTH, Prev: INTERPOL, Up: Algebraic Operators section - - LCOF operator - - The LCOF operator returns the leading coefficient of a given -expression with respect to a given variable. - -syntax: - - LCOF (,) - - is ordinarily a polynomial. If [*note RATARG::.] is on, -a rational expression may also be used, otherwise an error results. - must be a [*note KERNEL::.] . - -examples: - - ____________________________________________________________ - - lcof((x+2*y)**5,y); - - 32 - - - lcof((x + y*sin(x))**2 + cos(x)*sin(x)**2,sin(x)); - - - - 2 - COS(X) + Y - - - lcof(x**2 + 3*x + 17,y); - - 2 - X + 3*X + 17 - - ____________________________________________________________ - If the kernel does not appear in the expression, LCOF returns the -expression. - - -File: redhelp, Node: LENGTH, Next: LHS, Prev: LCOF, Up: Algebraic Operators section - - LENGTH operator - - The LENGTH operator returns the number of items in a [*note LIST::.] -, the number of terms in an expression, or the dimensions of an array -or matrix. - -syntax: - - LENGTH () or LENGTH - - can be a list structure, an array, a matrix, or a scalar -expression. - -examples: - - ____________________________________________________________ - - alist := {a,b,{ww,xx,yy,zz}}; - - ALIST := {A,B,{WW,XX,YY,ZZ}} - - - length alist; - - 3 - - - length third alist; - - 4 - - - dlist := {d}; - - DLIST := {D} - - - length rest dlist; - - 0 - - - matrix mmm(4,5); - - length mmm; - - {4,5} - - - array aaa(5,3,2); - - length aaa; - - {6,4,3} - - - eex := (x+3)**2/(x-y); - - 2 - X + 6*X + 9 - EEX := ------------ - X - Y - - - length eex; - - 5 - - ____________________________________________________________ - An item in a list that is itself a list only counts as one item. An -error message will be printed if LENGTH is called on a matrix which has -not had its dimensions set. The LENGTH of an array includes the zeroth -element of each dimension, showing the full number of elements -allocated. (Declaring an array A with n elements allocates -A(0),A(1),...,A(n).) The LENGTH of an expression is the total number -of additive terms appearing in the numerator and denominator of the -expression. Note that subtraction of a term is represented internally -as addition of a negative term. - - -File: redhelp, Node: LHS, Next: LIMIT, Prev: LENGTH, Up: Algebraic Operators section - - LHS operator - - The LHS operator returns the left-hand side of an [*note -EQUATION::.] , such as those returned in a list by [*note SOLVE::.] . - -syntax: - - LHS () or LHS - - must be an equation of the form - - LEFT-HAND SIDE = RIGHT-HAND SIDE . - -examples: - - ____________________________________________________________ - - polly := (x+3)*(x^4+2x+1); - - 5 4 2 - POLLY := X + 3*X + 2*X + 7*X + 3 - - - pollyroots := solve(polly,x); - - POLLYROOTS := {X=ROOT F(X3 - X2 + X + 1,X , - O ) - X=-1, - X=-3} - - - variable := lhs first pollyroots; - - VARIABLE := X - - ____________________________________________________________ - - -File: redhelp, Node: LIMIT, Next: LPOWER, Prev: LHS, Up: Algebraic Operators section - - LIMIT operator - - LIMITS is a fast limit package for REDUCE for functions which are -continuous except for computable poles and singularities, based on some -earlier work by Ian Cohen and John P. Fitch. The Truncated Power Series -package is used for non-critical points, at which the value of the -function is the constant term in the expansion around that point. -l'Hopital's rule is used in critical cases, with preprocessing of 1-1 -forms and reformatting of product forms in order to apply l'Hopital's -rule. A limited amount of bounded arithmetic is also employed where -applicable. - -syntax: - - LIMIT (,,) or - - LIMIT!+ (,,) or - - LIMIT!- (,,) - - where is an expression depending of the variable (a -[*note KERNEL::.] ) and is the limit point. If the limit -depends upon the direction of approach to the , the operators -LIMIT!+ and LIMIT!- may be used. - -examples: - - ____________________________________________________________ - - limit(x*cot(x),x,0); - - 0 - - - limit((2x+5)/(3x-2),x,infinity); - - 2 - -- - 3 - - ____________________________________________________________ - - -File: redhelp, Node: LPOWER, Next: LTERM, Prev: LIMIT, Up: Algebraic Operators section - - LPOWER operator - - The LPOWER operator returns the leading power of an expression with -respect to a kernel. 1 is returned if the expression does not depend on -the kernel. - -syntax: - - LPOWER (,) - - is ordinarily a polynomial. If [*note RATARG::.] is on, -a rational expression may also be used, otherwise an error results. - must be a [*note KERNEL::.] . - -examples: - - ____________________________________________________________ - - lpower((x+2*y)**6,y); - - 6 - Y - - - lpower((x + cos(x))**8 + df(x**2,x),cos(x)); - - - - 8 - COS(X) - - - lpower(x**3 + 3*x,y); - - 1 - - ____________________________________________________________ - - -File: redhelp, Node: LTERM, Next: MAINVAR, Prev: LPOWER, Up: Algebraic Operators section - - LTERM operator - - The LTERM operator returns the leading term of an expression with -respect to a kernel. The expression is returned if it does not depend on -the kernel. - -syntax: - - LTERM (,) - - is ordinarily a polynomial. If [*note RATARG::.] is on, -a rational expression may also be used, otherwise an error results. - must be a [*note KERNEL::.] . - -examples: - - ____________________________________________________________ - - lterm((x+2*y)**6,y); - - 6 - 64*Y - - - lterm((x + cos(x))**8 + df(x**2,x),cos(x)); - - - - 8 - COS(X) - - - lterm(x**3 + 3*x,y); - - 3 - X + 3X - - ____________________________________________________________ - - -File: redhelp, Node: MAINVAR, Next: MAP, Prev: LTERM, Up: Algebraic Operators section - - MAINVAR operator - - The MAINVAR operator returns the main variable (in the system's -internal representation) of its argument. - -syntax: - - MAINVAR () - - is usually a polynomial, but may be any valid REDUCE -scalar expression. In the case of a rational function, the main variable -of the numerator is returned. The main variable returned is a [*note -KERNEL::.] . - -examples: - - ____________________________________________________________ - - test := (a + b + c)**2; - - 2 2 2 - TEST := A + 2*A*B + 2*A*C + B + 2*B*C + C - - - mainvar(test); - - A - - - korder c,b,a; - - mainvar(test); - - C - - - mainvar(2*cos(x)**2); - - COS(X) - - - mainvar(17); - - 0 - - ____________________________________________________________ - The main variable is the first variable in the canonical ordering of -kernels. Generally, alphabetically ordered functions come first, then -alphabetically ordered identifiers (variables). Numbers come last, and -as far as MAINVAR is concerned belong in the family 0 . The canonical -ordering can be changed by the declaration [*note KORDER::.] , as shown -above. - - -File: redhelp, Node: MAP, Next: MKID, Prev: MAINVAR, Up: Algebraic Operators section - - MAP operator - - The MAP operator applies a uniform evaluation pattern to all members -of a composite structure: a [*note MATRIX::.] , a [*note LIST::.] or -the arguments of an [*note OPERATOR::.] expression. The evaluation -pattern can be a unary procedure, an operator, or an algebraic -expression with one free variable. - -syntax: - - MAP (,) - - is a list, a matrix or an operator expression. - - is the name of an operator for a single argument: the -operator is evaluated once with each element of as its single -argument, - - or an algebraic expression with exactly one [*note Free Variable::.] -, that is a variable preceded by the tilde symbol: the expression is -evaluated for each element of where the element is -substituted for the free variable, - - or a replacement [*note RULE::.] of the form - -syntax: - - VAR => REP - - where is a variable (a without subscript) and -is an expression which contains . Here REP is evaluated for each -element of where the element is substituted for VAR . VAR may -be optionally preceded by a tilde. - - The rule form for is needed when more than one free -variable occurs. - -examples: - - ____________________________________________________________ - - map(abs,{1,-2,a,-a}); - - 1,2,abs(a),abs(a) - - - map(int(~w,x), mat((x^2,x^5),(x^4,x^5))); - - - [ 3 6 ] - [ x x ] - [---- ----] - [ 3 6 ] - [ ] - [ 5 6 ] - [ x x ] - [---- ----] - [ 5 6 ] - - - map(~w*6, x^2/3 = y^3/2 -1); - - 2 3 - 2*x =3*(y -2) - - ____________________________________________________________ - You can use MAP in nested expressions. It is not allowed to apply -MAP for a non-composed object, e.g. an identifier or a number. - - -File: redhelp, Node: MKID, Next: NPRIMITIVE, Prev: MAP, Up: Algebraic Operators section - - MKID command - - The MKID command constructs an identifier, given a stem and an -identifier or an integer. - -syntax: - - MKID (,) - - can be any valid REDUCE identifier that does not include -escaped special characters. may be an integer, including one -given by a local variable in a [*note FOR::.] loop, or any other legal -group of characters. - -examples: - - ____________________________________________________________ - - mkid(x,3); - - X3 - - - factorize(x^15 - 1); - - {X - 1, - 2 - X + X + 1, - 4 3 2 - X + X + X + X + 1, - 8 7 5 4 3 - X - X + X - X + X - X + 1} - - - - for i := 1:length ws do write set(mkid(f,i),part(ws,i)); - - - - 8 7 5 4 3 - X - X + X - X + X - X + 1 - 4 3 2 - X + X + X + X + 1 - 2 - X + X + 1 - X - 1 - - ____________________________________________________________ - You can use MKID to construct identifiers from inside procedures. -This allows you to handle an unknown number of factors, or deal with -variable amounts of data. It is particularly helpful to attach -identifiers to the answers returned by FACTORIZE and SOLVE . - - -File: redhelp, Node: NPRIMITIVE, Next: NUM, Prev: MKID, Up: Algebraic Operators section - - NPRIMITIVE operator - -syntax: - - NPRIMITIVE () or NPRIMITIVE - - This operator returns the numerically-primitive part of any scalar -expression. In other words, any overall integer factors in the -expression are removed. - -examples: - - ____________________________________________________________ - - nprimitive((2x+2y)^2); - - 2 2 - X + 2*X*Y + Y - - - nprimitive(3*a*b*c); - - 3*A*B*C - - ____________________________________________________________ - - -File: redhelp, Node: NUM, Next: ODESOLVE, Prev: NPRIMITIVE, Up: Algebraic Operators section - - NUM operator - - The NUM operator returns the numerator of its argument. - -syntax: - - NUM () or NUM - - can be any valid REDUCE scalar expression. - -examples: - - ____________________________________________________________ - - num(100/6); - - 50 - - - num(a/5 + b/6); - - 6*A + 5*B - - - num(sin(x)); - - SIN(X) - - ____________________________________________________________ - NUM returns the numerator of the expression after it has been -simplified by REDUCE. As seen in the examples, this includes putting -sums of rational expressions over a common denominator, and reducing -common factors where possible. If the expression is not a rational -expression, it is returned unchanged. - - -File: redhelp, Node: ODESOLVE, Next: ONE_OF, Prev: NUM, Up: Algebraic Operators section - - ODESOLVE operator - - The ODESOLVE package is a solver for ordinary differential -equations. At the present time it has still limited capabilities: - - 1. it can handle only a single scalar equation presented as an -algebraic expression or equation, and - - 2. it can solve only first-order equations of simple types, linear -equations with constant coefficients and Euler equations. - - These solvable types are exactly those for which Lie symmetry -techniques give no useful information. - -syntax: - - ODESOLVE (,,) - - is a single scalar expression such that =0 is the -ordinary differential equation (ODE for short) to be solved, or is an -equivalent [*note EQUATION::.] . - - is the name of the dependent variable, is the name of -the independent variable. - - A differential in is expressed using the [*note DF::.] -operator. Note that in most cases you must declare explicitly to -depend of using a [*note DEPEND::.] declaration - otherwise the -derivative might be evaluated to zero on input to ODESOLVE . - - The returned value is a list containing the equation giving the -general solution of the ODE (for simultaneous equations this will be a -list of equations eventually). It will contain occurrences of the -operator ARBCONST for the arbitrary constants in the general solution. -The arguments of ARBCONST should be new. A counter !!ARBCONST is used -to arrange this. - -examples: - - ____________________________________________________________ - - depend y,x; - - % A first-order linear equation, with an initial condition - - ode:=df(y,x) + y * sin x/cos x - 1/cos x$ - - odesolve(ode,y,x); - - {y=arbconst(1)*cos(x) + sin(x)} - - ____________________________________________________________ - - -File: redhelp, Node: ONE_OF, Next: PART, Prev: ODESOLVE, Up: Algebraic Operators section - - ONE_OF type - - The operator ONE_OF is used to represent an indefinite choice of one -element from a finite set of objects. - -examples: - - ____________________________________________________________ - - x=one_of{1,2,5} - ____________________________________________________________ - this equation encodes that x can take one of the values 1,2 or 5 - ____________________________________________________________ - - ____________________________________________________________ - - REDUCE generates a ONE_OF form in cases when an implicit ROOT_OF -expression could be converted to an explicit solution set. A ONE_OF -form can be converted to a SOLVE solution using [*note EXPAND_CASES::.] -. See [*note ROOT_OF::.] . - - -File: redhelp, Node: PART, Next: PF, Prev: ONE_OF, Up: Algebraic Operators section - - PART operator - - The operator PART permits the extraction of various parts or -operators of expressions and [*note LIST::.] S . - -syntax: - - PART (,*) - - can be any valid REDUCE expression or a list, integer -may be an expression that evaluates to a positive or negative integer -or 0. A positive integer picks up the n th term, counting from the -first term toward the end. A negative integer n picks up the n th term, -counting from the back toward the front. The integer 0 picks up the -operator (which is LIST when the expression is a [*note LIST::.] ). - -examples: - - ____________________________________________________________ - - part((x + y)**5,4); - - 2 3 - 10*X *Y - - - part((x + y)**5,4,2); - - 2 - X - - - part((x + y)**5,4,2,1); - - X - - - part((x + y)**5,0); - - PLUS - - - part((x + y)**5,-5); - - 4 - 5*X *Y - - - part((x + y)**5,4) := sin(x); - - 5 4 3 2 4 5 - X + 5*X *Y + 10*X *Y + SIN(X) + 5*X*Y + Y - - - alist := {x,y,{aa,bb,cc},x**2*sqrt(y)}; - - - 2 - ALIST := {X,Y,{AA,BB,CC},SQRT(Y)*X } - - - part(alist,3,2); - - BB - - - part(alist,4,0); - - TIMES - - ____________________________________________________________ - Additional integer arguments after the first one examine the terms -recursively, as shown above. In the third line, the fourth term is -picked from the original polynomial, 10x^2y^3, then the second term -from that, x^2, and finally the first component, x. If an integer's -absolute value is too large for the appropriate expression, a message -is given. - - PART works on the form of the expression as printed, or as it would -have been printed at that point of the calculation, bearing in mind the -current switch settings. It is important to realize that the switch -settings change the operation of PART . [*note PRI::.] must be on when -PART is used. - - When PART is used on a polynomial expression that has minus signs, -the + is always returned as the top-level operator. The minus is found -as a unary operator attached to the negative term. - - PART can also be used to change the relevant part of the expression -or list as shown in the sixth example line. The PART operator returns -the changed expression, though original expression is not changed. You -can also use PART to change the operator. - - -File: redhelp, Node: PF, Next: PROD, Prev: PART, Up: Algebraic Operators section - - PF operator - -syntax: - - pf(,) - - PF transforms into a [*note LIST::.] of partial fraction -s with respect to the main variable, . PF does a complete -partial fraction decomposition, and as the algorithms used are fairly -unsophisticated (factorization and the extended Euclidean algorithm), -the code may be unacceptably slow in complicated cases. - -examples: - - ____________________________________________________________ - - pf(2/((x+1)^2*(x+2)),x); - - 2 -2 2 - {-----,-----,------------} - X + 2 X + 1 2 - X + 2*X + 1 - - - off exp; - - pf(2/((x+1)^2*(x+2)),x); - - - 2 - 2 2 - {-----,-----,--------} - X + 2 X + 1 2 - (X + 1) - - - for each j in ws sum j; - - 2 - ---------------- - 2 - ( + 2)*(X + 1) - - ____________________________________________________________ - - If you want the denominators in factored form, turn [*note EXP::.] -off, as shown in the second example above. As shown in the final -example, the [*note FOR::.] EACH construct can be used to recombine -the terms. Alternatively, one can use the operations on lists to -extract any desired term. - - -File: redhelp, Node: PROD, Next: REDUCT, Prev: PF, Up: Algebraic Operators section - - PROD operator - - The operator PROD returns the indefinite or definite product of a -given expression. - -syntax: - - PROD (,[, [, ]]) - - where is the expression to be multiplied, is the control -variable (a [*note KERNEL::.] ), and and uplim are the -optional lower and upper limits. If is not supplied the upper -limit is taken as . The Gosper algorithm is used. If there is no -closed form solution, the operator returns the input unchanged. - -examples: - - ____________________________________________________________ - - prod(k/(k-2),k); - - k*( - k + 1) - - ____________________________________________________________ - - -File: redhelp, Node: REDUCT, Next: REPART, Prev: PROD, Up: Algebraic Operators section - - REDUCT operator - - The REDUCT operator returns the remainder of its expression after the -leading term with respect to the kernel in the second argument is -removed. - -syntax: - - REDUCT (,) - - is ordinarily a polynomial. If [*note RATARG::.] is on, -a rational expression may also be used, otherwise an error results. - must be a [*note KERNEL::.] . - -examples: - - ____________________________________________________________ - - reduct((x+y)**3,x); - - 2 2 - Y*(3*X + 3*X*Y + Y ) - - - reduct(x + sin(x)**3,sin(x)); - - X - - - reduct(x + sin(x)**3,y); - - 0 - - ____________________________________________________________ - If the expression does not contain the kernel, REDUCT returns 0. - - -File: redhelp, Node: REPART, Next: RESULTANT, Prev: REDUCT, Up: Algebraic Operators section - - REPART operator - -syntax: - - REPART () or REPART - - This operator returns the real part of an expression, if that -argument has an numerical value. A non-numerical argument is returned -as an expression in the operators REPART and [*note IMPART::.] . - -examples: - - ____________________________________________________________ - - repart(1+i); - - 1 - - - repart(a+i*b); - - REPART(A) - IMPART(B) - - ____________________________________________________________ - - -File: redhelp, Node: RESULTANT, Next: RHS, Prev: REPART, Up: Algebraic Operators section - - RESULTANT operator - - The RESULTANT operator computes the resultant of two polynomials with -respect to a given variable. If the resultant is 0, the polynomials have -a root in common. - -syntax: - - RESULTANT (,,) - - must be a polynomial containing ; -must be a [*note KERNEL::.] . - -examples: - - ____________________________________________________________ - - resultant(x**2 + 2*x + 1,x+1,x); - - 0 - - - resultant(x**2 + 2*x + 1,x-3,x); - - 16 - - - resultant(z**3 + z**2 + 5*z + 5, - z**4 - 6*z**3 + 16*z**2 - 30*z + 55, - z); - - - 0 - - - resultant(x**3*y + 4*x*y + 10,y**2 + 6*y + 4,y); - - - 6 5 4 3 2 - Y + 18*Y + 120*Y + 360*Y + 480*Y + 288*Y + 64 - - ____________________________________________________________ - The resultant is the determinant of the Sylvester matrix, formed -from the coefficients of the two polynomials in the following way: - - Given two polynomials: - - ____________________________________________________________ - - n n-1 - a x + a1 x + ... + an - - ____________________________________________________________ - and - - ____________________________________________________________ - - m m-1 - b x + b1 x + ... + bm - - ____________________________________________________________ - form the (m+n)x(m+n-1) Sylvester matrix by the following means: - - ____________________________________________________________ - - 0.......0 a a1 .......... an - 0....0 a a1 .......... an 0 - . . . . - a0 a1 .......... an 0.......0 - 0.......0 b b1 .......... bm - 0....0 b b1 .......... bm 0 - . . . . - b b1 .......... bm 0.......0 - - ____________________________________________________________ - If the determinant of this matrix is 0, the two polynomials have a -common root. Finding the resultant of large expressions is -time-consuming, due to the time needed to find a large determinant. - - The sign conventions RESULTANT uses are those given in the article, -"Computing in Algebraic Extensions," by R. Loos, appearing in , 2nd ed., edited by B. -Buchberger, G.E. Collins and R. Loos, and published by Springer-Verlag, -1983. These are: - - ____________________________________________________________ - - resultant(p(x),q(x),x) = (-1)^{deg p(x)*deg q(x)} * resultant(q(x),p(x),x), - resultant(a,p(x),x) = a^{deg p(x)}, - resultant(a,b,x) = 1 - ____________________________________________________________ - where p(x) and q(x) are polynomials which have x as a variable, and -a and b are free of x. - - Error messages are given if RESULTANT is given a non-polynomial -expression, or a non-kernel variable. - - -File: redhelp, Node: RHS, Next: ROOT_OF, Prev: RESULTANT, Up: Algebraic Operators section - - RHS operator - - The RHS operator returns the right-hand side of an [*note -EQUATION::.] , such as those returned in a [*note LIST::.] by [*note -SOLVE::.] . - -syntax: - - RHS () or RHS - - must be an equation of the form left-hand side = -right-hand side. - -examples: - - ____________________________________________________________ - - roots := solve(x**2 + 6*x*y + 5x + 3y**2,x); - - - 2 - SQRT(24*Y + 60*Y + 25) + 6*Y + 5 - ROOTS := {X= - ---------------------------------, - 2 - 2 - SQRT(24*Y + 60*Y + 25) - 6*Y - 5 - X= ---------------------------------} - 2 - - - root1 := rhs first roots; - - 2 - SQRT(24*Y + 60*Y + 25) + 6*Y + 5 - ROOT1 := - --------------------------------- - 2 - - - root2 := rhs second roots; - - 2 - SQRT(24*Y + 60*Y + 25) - 6*Y - 5 - ROOT2 := ---------------------------------- - 2 - - ____________________________________________________________ - An error message is given if RHS is applied to something other than -an equation. - - -File: redhelp, Node: ROOT_OF, Next: SELECT, Prev: RHS, Up: Algebraic Operators section - - ROOT_OF operator - - When the operator [*note SOLVE::.] is unable to find an explicit -solution or if that solution would be too complicated, the result is -presented as formal root expression using the internal operator ROOT_OF -and a new local variable. An expression with a top level ROOT_OF is -implicitly a list with an unknown number of elements since we can't -always know how many solutions an equation has. If a substitution is -made into such an expression, closed form solutions can emerge. If this -occurs, the ROOT_OF construct is replaced by an operator [*note -ONE_OF::.] . At this point it is of course possible to transform the -result if the original SOLVE operator expression into a standard SOLVE -solution. To effect this, the operator [*note EXPAND_CASES::.] can be -used. - -examples: - - ____________________________________________________________ - - solve(a*x^7-x^2+1,x); - - 7 2 - {x=root_of(a*x_ - x_ + 1,x_)} - - - sub(a=0,ws); - - {x=one_of(1,-1)} - - - expand_cases ws; - - x=1,x=-1 - - ____________________________________________________________ - The components of ROOT_OF and ONE_OF expressions can be processed as -usual with operators [*note ARGLENGTH::.] and [*note PART::.] . A -higher power of a ROOT_OF expression with a polynomial as first -argument is simplified by using the polynomial as a side relation. - - -File: redhelp, Node: SELECT, Next: SHOWRULES, Prev: ROOT_OF, Up: Algebraic Operators section - - SELECT operator - - The SELECT operator extracts from a list or from the arguments of an -n-ary operator elements corresponding to a boolean predicate. The -predicate pattern can be a unary procedure, an operator or an algebraic -expression with one [*note Free Variable::.] . - -syntax: - - SELECT (,) - - is a [*note LIST::.] . - - is the name of an operator for a single argument: the -operator is evaluated once with each element of as its single -argument, - - or an algebraic expression with exactly one [*note Free Variable::.] -, that is a variable preceded by the tilde symbol: the expression is -evaluated for each element of where the element is -substituted for the free variable, - - or a replacement [*note RULE::.] of the form - -syntax: - - VAR => REP - - where is a variable (a without subscript) and -is an expression which contains . Here REP is evaluated for each -element of where the element is substituted for VAR . VAR may -be optionally preceded by a tilde. - - The rule form for is needed when more than one free -variable occurs. The evaluation result of is interpreted as -[*note boolean value::.] corresponding to the conventions of REDUCE. -The result value is built with the leading operator of the input -expression. - -examples: - - ____________________________________________________________ - - select( ~w>0 , {1,-1,2,-3,3}) - - {1,2,3} - - - q:=(part((x+y)^5,0):=list) - - select(evenp deg(~w,y),q); - - 5 3 2 4 - {x ,10*x *y ,5*x*y } - - - select(evenp deg(~w,x),2x^2+3x^3+4x^4); - - - 2 4 - 2x +4x - - ____________________________________________________________ - - -File: redhelp, Node: SHOWRULES, Next: SOLVE, Prev: SELECT, Up: Algebraic Operators section - - SHOWRULES operator - -syntax: - - SHOWRULES () or SHOWRULES - - SHOWRULES returns in [*note RULE::.] -LIST form any [*note -OPERATOR::.] rules associated with its argument. - -examples: - - ____________________________________________________________ - - showrules log; - - {LOG(E) => 1, - LOG(1) => 0, - ~X - LOG(E ) => ~X, - 1 - DF(LOG(~X),~X) => --} - ~X - - ____________________________________________________________ - Such rules can then be manipulated further as with any [*note -LIST::.] . For example RHS FIRST WS; has the value 1. - - An operator may have properties that cannot be displayed in such a -form, such as the fact it is an [*note ODD::.] function, or has a -definition defined as a procedure. - - -File: redhelp, Node: SOLVE, Next: SORT, Prev: SHOWRULES, Up: Algebraic Operators section - - SOLVE operator - - The SOLVE operator solves a single algebraic [*note EQUATION::.] or a -system of simultaneous equations. - -syntax: - - SOLVE ( [ , ]) or - - SOLVE (,... [ , ,...] ) - - If the number of equations equals the number of distinct kernels, the -optional kernel argument(s) may be omitted. is either a -scalar expression or an [*note EQUATION::.] . When more than one -expression is given, the [*note LIST::.] of expressions is surrounded -by curly braces. The optional list of [*note KERNEL::.] s follows, -also in curly braces. - -examples: - - ____________________________________________________________ - - sss := solve(x^2 + 7); - - Unknown: X - SSS := {X= - SQRT(7)*I, - X=SQRT(7)*I} - - - rhs first sss; - - - SQRT(7)*I - - - solve(sin(x^2*y),y); - - 2*ARBINT(1)*PI - {Y=--------------- - 2 - X - PI*(2*ARBINT(1) + 1) - Y=--------------------} - 2 - X - - - off allbranch; - - solve(sin(x**2*y),y); - - {Y=0} - - - solve({3x + 5y = -4,2*x + y = -10},{x,y}); - - - - 22 46 - {{X= - --,Y=--}} - 7 7 - - - solve({x + a*y + z,2x + 5},{x,y}); - - - - 5 2*Z - 5 - {{X= - -,Y= - -------}} - 2 2*A - - - ab := (x+2)^2*(x^6 + 17x + 1); - - - 8 7 6 3 2 - AB := X + 4*X + 4*X + 17*X + 69*X + 72*X + 4 - - - www := solve(ab,x); - - {X=ROOT F(X6 + 17*X + 1),X=-2} - O - - - root_multiplicities; - - {1,2} - - ____________________________________________________________ - Results of the SOLVE operator are returned as [*note EQUATION::.] S -in a [*note LIST::.] . You can use the usual list access methods -([*note FIRST::.] , [*note SECOND::.] , [*note THIRD::.] , [*note -REST::.] and [*note PART::.] ) to extract the desired equation, and -then use the operators [*note RHS::.] and [*note LHS::.] to access the -right-hand or left-hand expression of the equation. When SOLVE is -unable to solve an equation, it returns the unsolved part as the -argument of ROOT_OF , with the variable renamed to avoid confusion, as -shown in the last example above. - - For one equation, SOLVE uses square-free factorization, roots of -unity, and the known inverses of the [*note LOG::.] , [*note SIN::.] , -[*note COS::.] , [*note ACOS::.] , [*note ASIN::.] , and exponentiation -operators. The quadratic, cubic and quartic formulas are used if -necessary, but these are applied only when the switch [*note -FULLROOTS::.] is set on; otherwise or when no closed form is available -the result is returned as [*note ROOT_OF::.] expression. The switch -[*note TRIGFORM::.] determines which type of cubic and quartic formula -is used. The multiplicity of each solution is given in a list as the -system variable [*note ROOT_MULTIPLICITIES::.] . For systems of -simultaneous linear equations, matrix inversion is used. For nonlinear -systems, the Groebner basis method is used. - - Linear equation system solving is influenced by the switch [*note -CRAMER::.] . - - Singular systems can be solved when the switch [*note -SOLVESINGULAR::.] is on, which is the default setting. An empty list is -returned the system of equations is inconsistent. For a linear -inconsistent system with parameters the variable [*note -requirements::.] constraints conditions for the system to become -consistent. - - For a solvable linear and polynomial system with parameters the -variable [*note assumptions::.] contains a list side relations for the -parameters: the solution is valid only as long as none of these -expressions is zero. - - If the switch [*note VAROPT::.] is on (default), the system -rearranges the variable sequence for minimal computation time. Without -VAROPT the user supplied variable sequence is maintained. - - If the solution has free variables (dimension of the solution is -greater than zero), these are represented by [*note ARBCOMPLEX::.] -expressions as long as the switch [*note ARBVARS::.] is on (default). -Without ARBVARS no explicit equations are generated for free variables. - -related: - - [*note ALLBRANCH::.] switch - - [*note ARBVARS::.] switch - - [*note assumptions::.] variable - - [*note FULLROOTS::.] switch - - [*note requirements::.] variable - - [*note ROOTS::.] operator - - [*note ROOT_OF::.] operator - - [*note TRIGFORM::.] switch - - [*note VAROPT::.] switch - - -File: redhelp, Node: SORT, Next: STRUCTR, Prev: SOLVE, Up: Algebraic Operators section - - SORT operator - - The SORT operator sorts the elements of a list according to an -arbitrary comparison operator. - -syntax: - - SORT (,) - - is a [*note LIST::.] of algebraic expressions. is a -comparison operator which defines a partial ordering among the members -of . may be one of the builtin comparison operators like < -([*note LESSP::.] ), <= ([*note LEQ::.] ) etc., or may be the -name of a comparison procedure. Such a procedure has two arguments, -and it returns [*note TRUE::.] if the first argument ranges before the -second one, and 0 or [*note NIL::.] otherwise. The result of SORT is a -new list which contains the elements of in a sequence -corresponding to . - -examples: - - ____________________________________________________________ - - procedure ce(a,b); - - if evenp a and not evenp b then 1 else 0; - - for i:=1:10 collect random(50)$ - - sort(ws,>=); - - {41,38,33,30,28,25,20,17,8,5} - - - sort(ws,<); - - {5,8,17,20,25,28,30,33,38,41} - - - sort(ws,ce); - - {8,20,28,30,38,5,17,25,33,41} - - - procedure cd(a,b); - - if deg(a,x)>deg(b,x) then 1 else - - if deg(a,x)deg(b,y) then 1 else 0; - - sort({x^2,y^2,x*y},cd); - - 2 2 - {x ,x*y,y } - - ____________________________________________________________ - - -File: redhelp, Node: STRUCTR, Next: SUB, Prev: SORT, Up: Algebraic Operators section - - STRUCTR operator - - The STRUCTR operator breaks its argument expression into named -subexpressions. - -syntax: - - STRUCTR ( [,[, ...]]) - - may be any valid REDUCE scalar expression. - may be any valid REDUCE IDENTIFIER . The first identifier -is the stem for subexpression names, the second is the name to be -assigned to the structured expression. - -examples: - - ____________________________________________________________ - - structr(sqrt(x**2 + 2*x) + sin(x**2*z)); - - - ANS1 + ANS2 - where - 2 - ANS2 := SIN(X *Z) - 1/2 - ANS1 := ((X + 2)*X) - - - ans3; - - ANS3 - - - on fort; - - structr((x+1)**5 + tan(x*y*z),var,aa); - - - VAR1=TAN(X*Y*Z) - AA=VAR1+X**5+5.*X**4+10.*X**3+10.X**2+5.*X+1 - - ____________________________________________________________ - The second argument to STRUCTR is optional. If it is not given, the -default stem ANS is used by REDUCE to construct names for the -subexpression. The names are only for display purposes: REDUCE does not -store the names and their values unless the switch [*note -SAVESTRUCTR::.] is on. - - If a third argument is given, the structured expression as a whole -is named by this argument, when [*note FORT::.] is on. The expression -is not stored under this name. You can send these structured Fortran -expressions to a file with the OUT command. - - -File: redhelp, Node: SUB, Next: SUM, Prev: STRUCTR, Up: Algebraic Operators section - - SUB operator - - The SUB operator substitutes a new expression for a kernel in an -expression. - -syntax: - - SUB (= ,= *, - ) or - - SUB (= *, = EXPRESSION -,) - - must be a [*note KERNEL::.] , can be any REDUCE -scalar expression. - -examples: - - ____________________________________________________________ - - sub(x=3,y=4,(x+y)**3); - - 343 - - - x; - - X - - - sub({cos=sin,sin=cos},cos a+sin b} - - - COS(B) + SIN(A) - - ____________________________________________________________ - Note in the second example that operators can be replaced using the -SUB operator. - - -File: redhelp, Node: SUM, Next: WS, Prev: SUB, Up: Algebraic Operators section - - SUM operator - - The operator SUM returns the indefinite or definite summation of a -given expression. - -syntax: - - SUM (,[, [, ]]) - - where is the expression to be added, is the control -variable (a [*note KERNEL::.] ), and and are the -optional lower and upper limits. If is not supplied the upper -limit is taken as . The Gosper algorithm is used. If there is no -closed form solution, the operator returns the input unchanged. - -examples: - - ____________________________________________________________ - - sum(4n**3,n); - - 2 2 - n *(n + 2*n + 1) - - - sum(2a+2k*r,k,0,n-1); - - n*(2*a + n*r - r) - - ____________________________________________________________ - - -File: redhelp, Node: WS, Prev: SUM, Up: Algebraic Operators section - - WS operator - - The WS operator alone returns the last result; WS with a number -argument returns the results of the REDUCE statement executed after -that numbered prompt. - -syntax: - - WS or WS () - - must be an integer between 1 and the current REDUCE prompt -number. - -examples: - - ____________________________________________________________ - ____________________________________________________________ - (In the following examples, unlike most others, the numbered prompt -is shown.) - ____________________________________________________________ - - - 1: df(sin y,y); - - COS(Y) - - - 2: ws^2; - - 2 - COS(Y) - - - 3: df(ws 1,y); - - -SIN(Y) - - ____________________________________________________________ - - WS and WS ( ) can be used anywhere the expression they -stand for can be used. Calling a number for which no result was -produced, such as a switch setting, will give an error message. - - The current workspace always contains the results of the last REDUCE -command that produced an expression, even if several input statements -that do not produce expressions have intervened. For example, if you do -a differentiation, producing a result expression, then change several -switches, the operator WS; returns the results of the differentiation. -The current workspace (WS ) can also be used inside files, though the -numbered workspace contains only the IN command that input the file. - - There are three history lists kept in your REDUCE session. The first -stores raw input, suitable for the statement editor. The second stores -parsed input, ready to execute and accessible by [*note INPUT::.] . The -third stores results, when they are produced by statements, which are -accessible by the WS < n> operator. If your session is very long, -storage space begins to fill up with these expressions, so it is a good -idea to end the session once in a while, saving needed expressions to -files with the [*note SAVEAS::.] and [*note OUT::.] commands. - - An error message is given if a reference number has not yet been -used. - - -File: redhelp, Node: Algebraic Operators section, Next: Declarations section, Prev: General Commands section, Up: Top - - Algebraic Operators section - -* Menu: - -* APPEND:: operator -* ARBINT:: operator -* ARBCOMPLEX:: operator -* ARGLENGTH:: operator -* COEFF:: operator -* COEFFN:: operator -* CONJ:: operator -* CONTINUED_FRACTION:: operator -* DECOMPOSE:: operator -* DEG:: operator -* DEN:: operator -* DF:: operator -* EXPAND_CASES:: operator -* EXPREAD:: operator -* FACTORIZE:: operator -* HYPOT:: operator -* IMPART:: operator -* INT:: operator -* INTERPOL:: operator -* LCOF:: operator -* LENGTH:: operator -* LHS:: operator -* LIMIT:: operator -* LPOWER:: operator -* LTERM:: operator -* MAINVAR:: operator -* MAP:: operator -* MKID:: command -* NPRIMITIVE:: operator -* NUM:: operator -* ODESOLVE:: operator -* ONE_OF:: type -* PART:: operator -* PF:: operator -* PROD:: operator -* REDUCT:: operator -* REPART:: operator -* RESULTANT:: operator -* RHS:: operator -* ROOT_OF:: operator -* SELECT:: operator -* SHOWRULES:: operator -* SOLVE:: operator -* SORT:: operator -* STRUCTR:: operator -* SUB:: operator -* SUM:: operator -* WS:: operator - - -File: redhelp, Node: ALGEBRAIC, Next: ANTISYMMETRIC, Up: Declarations section - - ALGEBRAIC command - - The ALGEBRAIC command changes REDUCE's mode of operation to -algebraic. When ALGEBRAIC is used as an operator (with an argument -inside parentheses) that argument is evaluated in algebraic mode, but -REDUCE's mode is not changed. - -examples: - - ____________________________________________________________ - - algebraic; - - symbolic; - - NIL - - - algebraic(x**2); - - 2 - X - - - x**2; - - ***** The symbol X has no value. - - ____________________________________________________________ - REDUCE's symbolic mode does not know about most algebraic commands. -Error messages in this mode may also depend on the particular Lisp used -for the REDUCE implementation. - - -File: redhelp, Node: ANTISYMMETRIC, Next: ARRAY, Prev: ALGEBRAIC, Up: Declarations section - - ANTISYMMETRIC declaration - - When an operator is declared ANTISYMMETRIC , its arguments are -reordered to conform to the internal ordering of the system. If an odd -number of argument interchanges are required to do this ordering, the -sign of the expression is changed. - -syntax: - - ANTISYMMETRIC , * - - is an identifier that has been declared as an operator. - -examples: - - ____________________________________________________________ - - operator m,n; - - antisymmetric m,n; - - m(x,n(1,2)); - - - M( - N(2,1),X) - - - operator p; - - antisymmetric p; - - p(a,b,c); - - P(A,B,C) - - - p(b,a,c); - - - P(A,B,C) - - ____________________________________________________________ - If has not been declared an operator, the flag -ANTISYMMETRIC is still attached to it. When is -subsequently used as an operator, the message DECLARE -OPERATOR? (Y OR N) is printed. If the user replies Y , the -antisymmetric property of the operator is used. - - Note in the first example, identifiers are customarily ordered -alphabetically, while numbers are ordered from largest to smallest. -The operators may have any desired number of arguments (less than 128). - - -File: redhelp, Node: ARRAY, Next: CLEAR, Prev: ANTISYMMETRIC, Up: Declarations section - - ARRAY declaration - - The ARRAY declaration declares a list of identifiers to be of type -ARRAY , and sets all their entries to 0. - -syntax: - - ARRAY () , ()* - - may be any valid REDUCE identifier. If the identifier -was already an array, a warning message is given that the array has been -redefined. are of form ,*. - -examples: - - ____________________________________________________________ - - array a(2,5),b(3,3,3),c(200); - - array a(3,5); - - *** ARRAY A REDEFINED - - - a(3,4); - - 0 - - - length a; - - 4,6 - - ____________________________________________________________ - Arrays are always global, even if defined inside a procedure or block -statement. Their status as an array remains until the variable is reset -by [*note CLEAR::.] . Arrays may not have the same names as operators, -procedures or scalar variables. - - Array elements are referred to by the usual notation: A(I,J) returns -the jth element of the ith row. The [*note assign::.] ment operator := -is used to put values into the array. Arrays as a whole cannot be -subject to assignment by [*note LET::.] or := ; the assignment operator -:= is only valid for individual elements. - - When you use [*note LET::.] on an array element, the contents of that -element become the argument to LET . Thus, if the element contains a -number or some other expression that is not a valid argument for this -command, you get an error message. If the element contains an -identifier, the identifier has the substitution rule attached to it -globally. The same behavior occurs with [*note CLEAR::.] . If the array -element contains an identifier or simple_expression, it is cleared. Do - use CLEAR to try to set an array element to 0. Because of the -side effects of either LET or CLEAR , it is unwise to apply either of -these to array elements. - - Array indices always start with 0, so that the declaration ARRAY A(5) -sets aside 6 units of space, indexed from 0 through 5, and initializes -them to 0. The [*note LENGTH::.] command returns a list of the true -number of elements in each dimension. - - -File: redhelp, Node: CLEAR, Next: CLEARRULES, Prev: ARRAY, Up: Declarations section - - CLEAR command - - The CLEAR command is used to remove assignments or remove -substitution rules from any expression. - -syntax: - - CLEAR ,+ or - - CLEAR - - can be any SCALAR , [*note MATRIX::.] , or [*note -ARRAY::.] variable or [*note PROCEDURE::.] name. -can be any general or specific [*note LET::.] statement (see below in -Comments). - -examples: - - ____________________________________________________________ - - array a(2,3); - - a(2,2) := 15; - - A(2,2) := 15 - - - clear a; - - a(2,2); - - Declare A operator? (Y or N) - - - let x = y + z; - - sin(x); - - SIN(Y + Z) - - - clear x; - - sin(x); - - SIN(X) - - - let x**5 = 7; - - clear x; - - x**5; - - 7 - - - clear x**5; - - x**5; - - 5 - X - - ____________________________________________________________ - Although it is not a good idea, operators of the same name but taking -different numbers of arguments can be defined. Using a CLEAR statement -on any of these operators clears every one with the same name, even if -the number of arguments is different. - - The CLEAR command is used to "forget" matrices, arrays, operators -and scalar variables, returning their identifiers to the pristine state -to be used for other purposes. When CLEAR is applied to array elements, -the contents of the array element becomes the argument for CLEAR . -Thus, you get an error message if the element contains a number, or -some other expression that is not a legal argument to CLEAR . If the -element contains an identifier, it is cleared. When clear is applied -to matrix elements, an error message is returned if the element -evaluates to a number, otherwise there is no effect. Do not try to use -CLEAR to set array or matrix elements to 0. You will not be pleased -with the results. - - If you are trying to clear power or product substitution rules made -with either [*note LET::.] or [*note FORALL::.] ...LET , you must -reproduce the rule, exactly as you typed it with the same arguments, up -to but not including the equal sign, using the word CLEAR instead of -the word LET . This is shown in the last example. Any other type of LET -or FORALL ...LET substitution can be cleared with just the variable -or operator name. [*note MATCH::.] behaves the same as [*note LET::.] -in this situation. There is a more complicated example under [*note -FORALL::.] . - - -File: redhelp, Node: CLEARRULES, Next: DEFINE, Prev: CLEAR, Up: Declarations section - - CLEARRULES command - -syntax: - - CLEARRULES ,+ - - The operator CLEARRULES is used to remove previously defined [*note -RULE::.] lists from the system. can be an explicit rule list, -or evaluate to a rule list. - -examples: - - ____________________________________________________________ - - trig1 := {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2, - cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, - sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, - cos(~x)^2 => (1+cos(2*x))/2, - sin(~x)^2 => (1-cos(2*x))/2}$ - - let trig1; - cos(a)*cos(b); - - COS(A - B) + COS(A + B) - ----------------------- - 2 - - - clearrules trig1; - cos(a)*cos(b); - - COS(A)*COS(B) - - ____________________________________________________________ - - -File: redhelp, Node: DEFINE, Next: DEPEND, Prev: CLEARRULES, Up: Declarations section - - DEFINE command - - The command DEFINE allows you to supply a new name for an identifier -or replace it by any valid REDUCE expression. - -syntax: - - DEFINE = , = * - - is any valid REDUCE identifier, can be a -number, an identifier, an operator, a reserved word, or an expression. - -examples: - - ____________________________________________________________ - - - define is= :=, xx=y+z; - - - a is 10; - - A := 10 - - - - xx**2; - - 2 2 - Y + 2*Y*Z + Z - - - - xx := 10; - - Y + Z := 10 - - ____________________________________________________________ - The renaming is done at the input level, and therefore takes -precedence over any other replacement or substitution declared for the -same identifier. It remains in effect until the end of the REDUCE -session. Be careful with it, since you cannot easily undo it without -ending the session. - - -File: redhelp, Node: DEPEND, Next: EVEN, Prev: DEFINE, Up: Declarations section - - DEPEND declaration - - DEPEND declares that its first argument depends on the rest of its -arguments. - -syntax: - - DEPEND , + - - must be a legal variable name or a prefix operator (see -[*note KERNEL::.] ). - -examples: - - ____________________________________________________________ - - - depend y,x; - - - df(y**2,x); - - 2*DF(Y,X)*Y - - - - depend z,cos(x),y; - - - df(sin(z),cos(x)); - - COS(Z)*DF(Z,COS(X)) - - - - df(z**2,x); - - 2*DF(Z,X)*Z - - - - nodepend z,y; - - - df(z**2,x); - - 2*DF(Z,X)*Z - - - - cc := df(y**2,x); - - CC := 2*DF(Y,X)*Y - - - - y := tan x; - - Y := TAN(X); - - - - cc; - - 2 - 2*TAN(X)*(TAN(X) + 1) - - ____________________________________________________________ - Dependencies can be removed by using the declaration [*note -NODEPEND::.] . The differentiation operator uses this information, as -shown in the examples above. Linear operators also use knowledge of -dependencies (see [*note LINEAR::.] ). Note that dependencies can be -nested: Having declared y to depend on x, and z to depend on y, we see -that the chain rule was applied to the derivative of a function of z -with respect to x. If the explicit function of the dependency is later -entered into the system, terms with DF(Y,X) , for example, are expanded -when they are displayed again, as shown in the last example. The -boolean operator [*note FREEOF::.] allows you to check the dependency -between two algebraic objects. - - -File: redhelp, Node: EVEN, Next: FACTOR declaration, Prev: DEPEND, Up: Declarations section - - EVEN declaration - -syntax: - - EVEN ,* - - This declaration is used to declare an operator even in its first -argument. Expressions involving an operator declared in this manner are -transformed if the first argument contains a minus sign. Any other -arguments are not affected. - -examples: - - ____________________________________________________________ - - even f; - - f(-a) - - F(A) - - - f(-a,-b) - - F(A,-B) - - ____________________________________________________________ - - -File: redhelp, Node: FACTOR declaration, Next: FORALL, Prev: EVEN, Up: Declarations section - - FACTOR declaration - - When a kernel is declared by FACTOR , all terms involving fixed -powers of that kernel are printed as a product of the fixed powers and -the rest of the terms. - -syntax: - - FACTOR , * - - must be a [*note KERNEL::.] or a [*note LIST::.] of KERNEL -s. - -examples: - - ____________________________________________________________ - - a := (x + y + z)**2; - - 2 2 2 - A := X + 2*X*Y + 2*X*Z + Y + 2*Y*Z + Z - - - factor y; - - a; - - 2 2 2 - Y + 2*Y*(X + Z) + X + 2*X*Z + Z - - - factor sin(x); - - c := df(sin(x)**4*x**2*z,x); - - 4 3 2 - C := 2*SIN(X) *X*Z + 4*SIN(X) *COS(X)*X *Z - - - remfac sin(x); - - c; - - 3 - 2*SIN(X) *X*Z*(2*COS(X)*X + SIN(X)) - - ____________________________________________________________ - Use the FACTOR declaration to display variables of interest so that -you can see their powers more clearly, as shown in the example. Remove -this special treatment with the declaration [*note REMFAC::.] . The -FACTOR declaration is only effective when the switch [*note PRI::.] is -on. - - The FACTOR declaration is not a factoring command; to factor -expressions use the [*note FACTOR::.] switch or the [*note -FACTORIZE::.] command. - - The FACTOR declaration is helpful in such cases as Taylor polynomials -where the explicit powers of the variable are expected at the top -level, not buried in various factored forms. - - -File: redhelp, Node: FORALL, Next: INFIX, Prev: FACTOR declaration, Up: Declarations section - - FORALL command - - The FORALL or (preferably) FOR ALL command is used as a modifier for -[*note LET::.] statements, indicating the universal applicability of -the rule, with possible qualifications. - -syntax: - - FOR ALL ,* LET - - or - - FOR ALL ,* SUCH THAT LET - - may be any valid REDUCE identifier, can -be an operator, a product or power, or a group or block statement. - must be a logical or comparison operator returning true or -false. - -examples: - - ____________________________________________________________ - - for all x let f(x) = sin(x**2); - - - - Declare F operator ? (Y or N) - - - y - - f(a); - - 2 - SIN(A ) - - - operator pos; - - for all x such that x>=0 let pos(x) = sqrt(x + 1); - - pos(5); - - SQRT(6) - - - pos(-5); - - POS(-5) - - - clear pos; - - pos(5); - - Declare POS operator ? (Y or N) - - - for all a such that numberp a let x**a = 1; - - x**4; - - 1 - - - clear x**a; - - *** X**A not found - - - for all a clear x**a; - - x**4; - - 1 - - - for all a such that numberp a clear x**a; - - x**4; - - 4 - X - - ____________________________________________________________ - Substitution rules defined by FOR ALL or FOR ALL ...SUCH THAT -commands that involve products or powers are cleared by reproducing the -command, with exactly the same variable names used, up to but not -including the equal sign, with [*note CLEAR::.] replacing LET , as -shown in the last example. Other substitutions involving variables or -operator names can be cleared with just the name, like any other -variable. - - The [*note MATCH::.] command can also be used in product and power -substitutions. The syntax of its use and clearing is exactly like LET -. A MATCH substitution only replaces the term if it is exactly like the -pattern, for example MATCH X**5 = 1 replaces only terms of X**5 and not -terms of higher powers. - - It is easier to declare your potential operator before defining the -FOR ALL rule, since the system will ask you to declare it an operator -anyway. Names of declared arrays or matrices or scalar variables are -invalid as operator names, to avoid ambiguity. Either FOR ALL ...LET -statements or procedures are often used to define operators. One -difference is that procedures implement "call by value" meaning that -assignments involving their formal parameters do not change the calling -variables that replace them. If you use assignment statements on the -formal parameters in a FOR ALL ...LET statement, the effects are seen -in the calling variables. Be careful not to redefine a system operator -unless you mean it: the statement FOR ALL X LET SIN(X)=0; has exactly -that effect, and the usual definition for sin(x) has been lost for the -remainder of the REDUCE session. - - -File: redhelp, Node: INFIX, Next: INTEGER, Prev: FORALL, Up: Declarations section - - INFIX declaration - - INFIX declares identifiers to be infix operators. - -syntax: - - INFIX ,* - - can be any valid REDUCE identifier, which has not -already been declared an operator, array or matrix, and is not reserved -by the system. - -examples: - - ____________________________________________________________ - - infix aa; - - for all x,y let aa(x,y) = cos(x)*cos(y) - sin(x)*sin(y); - - x aa y; - - COS(X)*COS(Y) - SIN(X)*SIN(Y) - - - pi/3 aa pi/2; - - SQRT(3) - - ------- - 2 - - - aa(pi,pi); - - 1 - - ____________________________________________________________ - A [*note LET::.] statement must be used to attach functionality to -the operator. Note that the operator is defined in prefix form in the -LET statement. After its definition, the operator may be used in -either prefix or infix mode. The above operator aa finds the cosine of -the sum of two angles by the formula - - cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y). - - Precedence may be attached to infix operators with the [*note -PRECEDENCE::.] declaration. - - User-defined infix operators may be used in prefix form. If they are -used in infix form, a space must be left on each side of the operator -to avoid ambiguity. Infix operators are always binary. - - -File: redhelp, Node: INTEGER, Next: KORDER, Prev: INFIX, Up: Declarations section - - INTEGER declaration - - The INTEGER declaration must be made immediately after a [*note -BEGIN::.] (or other variable declaration such as [*note REAL::.] and -[*note SCALAR::.] ) and declares local integer variables. They are -initialized to 0. - -syntax: - - INTEGER ,* - - may be any valid REDUCE identifier, except T or NIL . - - Integer variables remain local, and do not share values with -variables of the same name outside the [*note BEGIN::.] ...END block. -When the block is finished, the variables are removed. You may use the -words [*note REAL::.] or [*note SCALAR::.] in the place of INTEGER . -INTEGER does not indicate typechecking by the current REDUCE; it is -only for your own information. Declaration statements must immediately -follow the BEGIN , without a semicolon between BEGIN and the first -variable declaration. - - Any variables used inside BEGIN ...END blocks that were not -declared SCALAR , REAL or INTEGER are global, and any change made to -them inside the block affects their global value. Any [*note ARRAY::.] -or [*note MATRIX::.] declared inside a block is always global. - - -File: redhelp, Node: KORDER, Next: LET, Prev: INTEGER, Up: Declarations section - - KORDER declaration - - The KORDER declaration changes the internal canonical ordering of -kernels. - -syntax: - - KORDER , * - - must be a REDUCE [*note KERNEL::.] or a [*note LIST::.] of -KERNEL s. - - The declaration KORDER changes the internal ordering, but not the -print ordering, so the effects cannot be seen on output. However, in -some calculations, the order of the variables can have significant -effects on the time and space demands of a calculation. If you are -doing a demanding calculation with several kernels, you can experiment -with changing the canonical ordering to improve behavior. - - The first kernel in the argument list is given the highest priority, -the second gets the next highest, and so on. Kernels not named in a -KORDER ordering otherwise. A new KORDER declaration replaces the -previous one. To return to canonical ordering, use the command KORDER -NIL . - - To change the print ordering, use the declaration [*note ORDER::.] . - - -File: redhelp, Node: LET, Next: LINEAR, Prev: KORDER, Up: Declarations section - - LET command - - The LET command defines general or specific substitution rules. - -syntax: - - LET = , = * - - can be any valid REDUCE identifier except an array, and -in some cases can be an expression; can be any valid REDUCE -expression. - -examples: - - ____________________________________________________________ - - let a = sin(x); - - b := a; - - B := SIN X; - - - let c = a; - - exp(a); - - SIN(X) - E - - - a := x**2; - - 2 - A := X - - - exp(a); - - 2 - X - E - - - exp(b); - - SIN(X) - E - - - exp(c); - - 2 - X - E - - - let m + n = p; - - (m + n)**5; - - 5 - P - - - operator h; - - let h(u,v) = u - v; - - h(u,v); - - U - V - - - h(x,y); - - H(X,Y) - - - array q(10); - - let q(1) = 15; - - ***** Substitution for 0 not allowed - - ____________________________________________________________ - The LET command is also used to activate a RULE SETS . - -syntax: - - LET ,+ - - can be an explicit [*note RULE::.] LIST , or evaluate to a -rule list. - -examples: - - ____________________________________________________________ - - trig1 := {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2, - cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, - sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, - cos(~x)^2 => (1+cos(2*x))/2, - sin(~x)^2 => (1-cos(2*x))/2}$ - - let trig1; - cos(a)*cos(b); - - COS(A - B) + COS(A + B) - ------------------------ - 2 - - ____________________________________________________________ - A LET command returns no value, though the substitution rule is -entered. Assignment rules made by [*note assign::.] and LET rules are -at the same level, and cancel each other. There is a difference in their -operation, however, as shown in the first example: a LET assignment -tracks the changes in what it is assigned to, while a := assignment is -fixed at the value it originally had. - - The use of expressions as left-hand sides of LET statements is a -little complicated. The rules of operation are: - - (i) Expressions of the form A*B = C do not change A, B or C, but set -A*B to C. - - (ii) Expressions of the form A+B = C substitute C - B for A, but do -not change B or C. - - (iii) Expressions of the form A-B = C substitute B + C for A, but do -not change B or C. - - (iv) Expressions of the form A/B = C substitute B*C for A, but do -not change B or C. - - (v) Expressions of the form A**N = C substitute C for A**N in every -expression of a power of A to N or greater. An asymptotic command such -as A**N = 0 sets all terms involving A to powers greater than or equal -to N to 0. Finite fields may be generated by requiring modular -arithmetic (the [*note MODULAR::.] switch) and defining the primitive -polynomial via a LET statement. - - LET substitutions involving expressions are cleared by using the -[*note CLEAR::.] command with exactly the same expression. - - Note when a simple LET statement is used to assign functionality to -an operator, it is valid only for the exact identifiers used. For the -use of the LET command to attach more general functionality to an -operator, see [*note FORALL::.] . - - Arrays as a whole cannot be arguments to LET statements, but -matrices as a whole can be legal arguments, provided both arguments are -matrices. However, it is important to note that the two matrices are -then linked. Any change to an element of one matrix changes the -corresponding value in the other. Unless you want this behavior, you -should not use LET for matrices. The assignment operator [*note -assign::.] can be used for non-tracking assignments, avoiding the side -effects. Matrices are redimensioned as needed in LET statements. - - When array or matrix elements are used as the left-hand side of LET -statements, the contents of that element is used as the argument. When -the contents is a number or some other expression that is not a valid -left-hand side for LET , you get an error message. If the contents is an -identifier or simple expression, the LET rule is globally attached to -that identifier, and is in effect not only inside the array or matrix, -but everywhere. Because of such unwanted side effects, you should not -use LET with array or matrix elements. The assignment operator := can -be used to put values into array or matrix elements without the side -effects. - - Local variables declared inside BEGIN ...END blocks cannot be used -as the left-hand side of LET statements. However, [*note BEGIN::.] -...END blocks themselves can be used as the right-hand side of LET -statements. The construction: - -syntax: - - FOR ALL LET ()= - - is an alternative to the - -syntax: - - PROCEDURE (); - - construction. One important difference between the two constructions -is that the as formal parameters to a procedure have their -global values protected against change by the procedure, while the - of a LET statement are changed globally by its actions. - - Be careful in using a construction such as LET X = X + 1 except -inside a controlled loop statement. The process of resubstitution -continues until a stack overflow message is given. - - The LET statement may be used to make global changes to variables -from inside procedures. If X is a formal parameter to a procedure, the -command LET X = ... makes the change to the calling variable. For -example, if a procedure was defined by - ____________________________________________________________ - - procedure f(x,y); - let x = 15; - ____________________________________________________________ - - and the procedure was called as - ____________________________________________________________ - - f(a,b); - ____________________________________________________________ - - A would have its value changed to 15. Be careful when using LET -statements inside procedures to avoid unwanted side effects. - - It is also important to be careful when replacing LET statements with -other LET statements. The overlapping of these substitutions can be -unpredictable. Ordinarily the latest-entered rule is the first to be -applied. Sometimes the previous rule is superseded completely; other -times it stays around as a special case. The order of entering a set of -related LET expressions is very important to their eventual behavior. -The best approach is to assume that the rules will be applied in an -arbitrary order. - - -File: redhelp, Node: LINEAR, Next: LINELENGTH, Prev: LET, Up: Declarations section - - LINEAR declaration - - An operator can be declared linear in its first argument over powers -of its second argument by the declaration LINEAR. - -syntax: - - LINEAR , * - - must have been declared to be an operator. Be careful not -to use a system operator name, because this command may change its -definition. The operator being declared must have at least two -arguments, and the second one must be a [*note KERNEL::.] . - -examples: - - ____________________________________________________________ - - operator f; - - linear f; - - f(0,x); - - 0 - - - f(-y,x); - - - F(1,X)*Y - - - f(y+z,x); - - F(1,X)*(Y + Z) - - - f(y*z,x); - - F(1,X)*Y*Z - - - depend z,x; - - f(y*z,x); - - F(Z,X)*Y - - - f(y/z,x); - - 1 - F(-,X)*Y - Z - - - depend y,x; - - f(y/z,x); - - Y - F(-,X) - Z - - - nodepend z,x; - - f(y/z,x); - - F(Y,X) - ------ - Z - - - f(2*e**sin(x),x); - - SIN(X) - 2*F(E ,X) - - ____________________________________________________________ - Even when the operator has not had its functionality attached, it -exhibits linear properties as shown in the examples. Notice the -difference when dependencies are added. Dependencies are also in effect -when the operator's first argument contains its second, as in the last -line above. - - For a fully-developed example of the use of linear operators, refer -to the article in the , Vol. 14 -(1974), pp. 301-317, "Analytic Computation of Some Integrals in Fourth -Order Quantum Electrodynamics," by J.A. Fox and A.C. Hearn. The article -includes the complete listing of REDUCE procedures used for this work. - - -File: redhelp, Node: LINELENGTH, Next: LISP, Prev: LINEAR, Up: Declarations section - - LINELENGTH declaration - - The LINELENGTH declaration sets the length of the output line. -Default is 80. - -syntax: - - LINELENGTH - - To change the linelength, must evaluate to a positive -integer less than 128 (although this varies from system to system), and -should not be less than 20 or so for proper operation. - - LINELENGTH returns the previous linelength. If you want the current -linelength value, but not change it, say LINELENGTH NIL . - - -File: redhelp, Node: LISP, Next: LISTARGP, Prev: LINELENGTH, Up: Declarations section - - LISP command - - The LISP command changes REDUCE's mode of operation to symbolic. When -LISP is followed by an expression, that expression is evaluated in -symbolic mode, but REDUCE's mode is not changed. This command is -equivalent to [*note SYMBOLIC::.] . - -examples: - - ____________________________________________________________ - - lisp; - - NIL - - - car '(a b c d e); - - A - - - algebraic; - - c := (lisp car '(first second))**2; - - - - 2 - C := FIRST - - ____________________________________________________________ - - -File: redhelp, Node: LISTARGP, Next: NODEPEND, Prev: LISP, Up: Declarations section - - LISTARGP declaration - -syntax: - - LISTARGP , * - - If an operator other than those specifically defined for lists is -given a single argument that is a [*note LIST::.] , then the result of -this operation will be a list in which that operator is applied to each -element of the list. This process can be inhibited for a specific -operator, or list of operators, by using the declaration LISTARGP . - -examples: - - ____________________________________________________________ - - log {a,b,c}; - - LOG(A),LOG(B),LOG(C) - - - listargp log; - - log {a,b,c}; - - LOG(A,B,C) - - ____________________________________________________________ - It is possible to inhibit such distribution globally by turning on -the switch [*note LISTARGS::.] . In addition, if an operator has more -than one argument, no such distribution occurs, so LISTARGP has no -effect. - - -File: redhelp, Node: NODEPEND, Next: MATCH, Prev: LISTARGP, Up: Declarations section - - NODEPEND declaration - - The NODEPEND declaration removes the dependency declared with [*note -DEPEND::.] . - -syntax: - - NODEPEND ,+ - - must be a kernel that has had a dependency declared -upon the one or more other kernels that are its other arguments. - -examples: - - ____________________________________________________________ - - depend y,x,z; - - df(sin y,x); - - COS(Y)*DF(Y,X) - - - df(sin y,x,z); - - COS(Y)*DF(Y,X,Z) - DF(Y,X)*DF(Y,Z)*SIN(Y) - - - nodepend y,z; - - df(sin y,x); - - COS(Y)*DF(Y,X) - - - df(sin y,x,z); - - 0 - - ____________________________________________________________ - A warning message is printed if the dependency had not been declared -by DEPEND . - - -File: redhelp, Node: MATCH, Next: NONCOM, Prev: NODEPEND, Up: Declarations section - - MATCH command - - The MATCH command is similar to the [*note LET::.] command, except -that it matches only explicit powers in substitution. - -syntax: - - MATCH = , = * - - is generally a term involving powers, and is limited by the -rules for the [*note LET::.] command. may be any valid -REDUCE scalar expression. - -examples: - - ____________________________________________________________ - - match c**2*a**2 = d; - (a+c)**4; - - 4 3 3 4 - A + 4*A *C + 4*A*C + C + 6*D - - - match a+b = c; - - a + 2*b; - - B + C - - - (a + b + c)**2; - - 2 2 2 - A - B + 2*B*C + 3*C - - - clear a+b; - - (a + b + c)**2; - - 2 2 2 - A + 2*A*B + 2*A*C + B + 2*B*C + C - - - let p*r = s; - - match p*q = ss; - - (a + p*r)**2; - - 2 2 - A + 2*A*S + S - - - (a + p*q)**2; - - 2 2 2 - A + 2*A*SS + P *Q - - ____________________________________________________________ - Note in the last example that A + B has been explicitly matched -after the squaring was done, replacing each single power of A by C - B -. This kind of substitution, although following the rules, is confusing -and could lead to unrecognizable results. It is better to use MATCH -with explicit powers or products only. MATCH should not be used inside -procedures for the same reasons that LET should not be. - - Unlike [*note LET::.] substitutions, MATCH substitutions are executed -after all other operations are complete. The last example shows the -difference. MATCH commands can be cleared by using [*note CLEAR::.] , -with exactly the expression that the original MATCH took. MATCH -commands can also be done more generally with FOR ALL or [*note -FORALL::.] ...SUCH THAT commands. - - -File: redhelp, Node: NONCOM, Next: NONZERO, Prev: MATCH, Up: Declarations section - - NONCOM declaration - - NONCOM declares that already-declared operators are noncommutative -under multiplication. - -syntax: - - NONCOM ,* - - must have been declared an [*note OPERATOR::.] , or a -warning message is given. - -examples: - - ____________________________________________________________ - - operator f,h; - - noncom f; - - f(a)*f(b) - f(b)*f(a); - - F(A)*F(B) - F(B)*F(A) - - - h(a)*h(b) - h(b)*h(a); - - 0 - - - operator comm; - - for all x,y such that x neq y and ordp(x,y) - let f(x)*f(y) = f(y)*f(x) + comm(x,y); - - - f(1)*f(2); - - F(1)*F(2) - - - f(2)*f(1); - - COMM(2,1) + F(1)*F(2) - - ____________________________________________________________ - The last example introduces the commutator of f(x) and f(y) for all -x and y. The equality check is to prevent an infinite loop. The -operator f can have other functionality attached to it if desired, or it -can remain an indeterminate operator. - - -File: redhelp, Node: NONZERO, Next: ODD, Prev: NONCOM, Up: Declarations section - - NONZERO declaration - -syntax: - - NONZERO ,* - - If an [*note OPERATOR::.] F is declared [*note ODD::.] , then F(0) -is replaced by zero unless F is also declared non zero by the -declaration NONZERO . - -examples: - - ____________________________________________________________ - - odd f; - - f(0) - - 0 - - - nonzero f; - - f(0) - - F(0) - - ____________________________________________________________ - - -File: redhelp, Node: ODD, Next: OFF, Prev: NONZERO, Up: Declarations section - - ODD declaration - -syntax: - - ODD ,* - - This declaration is used to declare an operator odd in its first -argument. Expressions involving an operator declared in this manner are -transformed if the first argument contains a minus sign. Any other -arguments are not affected. - -examples: - - ____________________________________________________________ - - odd f; - - f(-a) - - -F(A) - - - f(-a,-b) - - -F(A,-B) - - - f(a,-b) - - F(A,-B) - - ____________________________________________________________ - - If say F is declared odd, then F(0) is replaced by zero unless F is -also declared non zero by the declaration [*note NONZERO::.] . - - -File: redhelp, Node: OFF, Next: ON, Prev: ODD, Up: Declarations section - - OFF command - - The OFF command is used to turn switches off. - -syntax: - - OFF ,* - - can be any SWITCH name. There is no problem if the switch -is already off. If the switch name is mistyped, an error message is -given. - - -File: redhelp, Node: ON, Next: OPERATOR, Prev: OFF, Up: Declarations section - - ON command - - The ON command is used to turn switches on. - -syntax: - - ON ,* - - can be any SWITCH name. There is no problem if the switch -is already on. If the switch name is mistyped, an error message is -given. - - -File: redhelp, Node: OPERATOR, Next: ORDER, Prev: ON, Up: Declarations section - - OPERATOR declaration - - Use the OPERATOR declaration to declare your own operators. - -syntax: - - OPERATOR ,* - - can be any valid REDUCE identifier, which is not the -name of a [*note MATRIX::.] , [*note ARRAY::.] , scalar variable or -previously-defined operator. - -examples: - - ____________________________________________________________ - - operator dis,fac; - - let dis(~x,~y) = sqrt(x^2 + y^2); - - dis(1,2); - - SQRT(5) - - - dis(a,10); - - 2 - SQRT(A + 100) - - - on rounded; - - dis(1.5,7.2); - - 7.35459040329 - - - let fac(~n) = if n=0 then 1 - else if not(fixp n and n>0) - then rederr "choose non-negative integer" - else for i := 1:n product i; - - - fac(5); - - 120 - - - fac(-2); - - ***** choose non-negative integer - - ____________________________________________________________ - The first operator is the Euclidean distance metric, the distance of -point (x,y) from the origin. The second operator is the factorial. - - Operators can have various properties assigned to them; they can be -declared [*note INFIX::.] , [*note LINEAR::.] , [*note SYMMETRIC::.] , -[*note ANTISYMMETRIC::.] , or [*note NONCOM::.] MUTATIVE . The default -operator is prefix, nonlinear, and commutative. Precedence can also be -assigned to operators using the declaration [*note PRECEDENCE::.] . - - Functionality is assigned to an operator by a [*note LET::.] -statement or a [*note FORALL::.] ...LET statement, (or possibly by a -procedure with the name of the operator). Be careful not to redefine a -system operator by accident. REDUCE permits you to redefine system -operators, giving you a warning message that the operator was already -defined. This flexibility allows you to add mathematical rules that do -what you want them to do, but can produce odd or erroneous behavior if -you are not careful. - - You can declare operators from inside [*note PROCEDURE::.] s, as -long as they are not local variables. Operators defined inside -procedures are global. A formal parameter may be declared as an -operator, and has the effect of declaring the calling variable as the -operator. - - -File: redhelp, Node: ORDER, Next: PRECEDENCE, Prev: OPERATOR, Up: Declarations section - - ORDER declaration - - The ORDER declaration changes the order of precedence of kernels for -display purposes only. - -syntax: - - ORDER ,* - - must be a valid [*note KERNEL::.] or [*note OPERATOR::.] -name complete with argument or a [*note LIST::.] of such objects. - -examples: - - ____________________________________________________________ - - x + y + z + cos(a); - - COS(A) + X + Y + Z - - - order z,y,x,cos(a); - - x + y + z + cos(a); - - Z + Y + X + COS(A) - - - (x + y)**2; - - 2 2 - Y + 2*Y*X + X - - - order nil; - - (z + cos(z))**2; - - 2 2 - COS(Z) + 2*COS(Z)*Z + Z - - ____________________________________________________________ - ORDER affects the printing order of the identifiers only; internal -order is unchanged. Change internal order of evaluation with the -declaration [*note KORDER::.] . You can use ORDER to feature variables -or functions you are particularly interested in. - - Declarations made with ORDER are cumulative: kernels in new order -declarations are ordered behind those in previous declarations, and -previous declarations retain their relative order. Of course, specific -kernels named in new declarations are removed from previous ones and -given the new priority. Return to the standard canonical printing order -with the statement ORDER NIL . - - The print order specified by ORDER commands is not in effect if the -switch [*note PRI::.] is off. - - -File: redhelp, Node: PRECEDENCE, Next: PRECISION, Prev: ORDER, Up: Declarations section - - PRECEDENCE declaration - - The PRECEDENCE declaration attaches a precedence to an infix -operator. - -syntax: - - PRECEDENCE , - - should have been declared an operator but may be a REDUCE -identifier that is not already an operator, array, or matrix. - must be a system infix operator or have had its -precedence already declared. - -examples: - - ____________________________________________________________ - - operator f,h; - - precedence f,+; - - precedence h,*; - - a + f(1,2)*c; - - (1 F 2)*C + A - - - a + h(1,2)*c; - - 1 H 2*C + A - - - a*1 f 2*c; - - A F 2*C - - - a*1 h 2*c; - - 1 H 2*A*C - - ____________________________________________________________ - The operator whose precedence is being declared is inserted into the -infix operator precedence list at the next higher place than -. - - Attaching a precedence to an operator has the side effect of -declaring the operator to be infix. If the identifier argument for -PRECEDENCE has not been declared to be an operator, an attempt to use -it causes an error message. After declaring it to be an operator, it -becomes an infix operator with the precedence previously given. Infix -operators may be used in prefix form; if they are used in infix form, a -space must be left on each side of the operator to avoid ambiguity. -Declared infix operators are always binary. - - To see the infix operator precedence list, enter symbolic mode and -type PRECLIS!*; . The lowest precedence operator is listed first. - - All prefix operators have precedence higher than infix operators. - - -File: redhelp, Node: PRECISION, Next: PRINT_PRECISION, Prev: PRECEDENCE, Up: Declarations section - - PRECISION declaration - - The PRECISION declaration sets the number of decimal places used when -[*note ROUNDED::.] is on. Default is system dependent, and normally -about 12. - -syntax: - - PRECISION () or PRECISION - - must be a positive integer. When is 0, the -current precision is displayed, but not changed. There is no upper -limit, but precision of greater than several hundred causes -unpleasantly slow operation on numeric calculations. - -examples: - - ____________________________________________________________ - - on rounded; - - 7/9; - - 0.777777777778 - - - precision 20; - - 20 - - - 7/9; - - 0.77777777777777777778 - - - sin(pi/4); - - 0.7071067811865475244 - - ____________________________________________________________ - Trailing zeroes are dropped, so sometimes fewer than 20 decimal -places are printed as in the last example. Turn on the switch [*note -FULLPREC::.] if you want to print all significant digits. The [*note -ROUNDED::.] mode carries calculations to two more places than given by -PRECISION , and rounds off. - - -File: redhelp, Node: PRINT_PRECISION, Next: REAL, Prev: PRECISION, Up: Declarations section - - PRINT_PRECISION declaration - -syntax: - - PRINT_PRECISION () or PRINT_PRECISION - - In [*note ROUNDED::.] mode, numbers are normally printed to the -specified precision. If the user wishes to print such numbers with less -precision, the printing precision can be set by the declaration -PRINT_PRECISION . - -examples: - - ____________________________________________________________ - - on rounded; - - 1/3; - - 0.333333333333 - - - print_precision 5; - - 1/3 - - 0.33333 - - ____________________________________________________________ - - -File: redhelp, Node: REAL, Next: REMFAC, Prev: PRINT_PRECISION, Up: Declarations section - - REAL declaration - - The REAL declaration must be made immediately after a [*note -BEGIN::.] (or other variable declaration such as [*note INTEGER::.] -and [*note SCALAR::.] ) and declares local integer variables. They are -initialized to zero. - -syntax: - - REAL ,* - - may be any valid REDUCE identifier, except T or NIL . - - Real variables remain local, and do not share values with variables -of the same name outside the [*note BEGIN::.] ...END block. When the -block is finished, the variables are removed. You may use the words -[*note INTEGER::.] or [*note SCALAR::.] in the place of REAL . REAL -does not indicate typechecking by the current REDUCE; it is only for -your own information. Declaration statements must immediately follow -the BEGIN , without a semicolon between BEGIN and the first variable -declaration. - - Any variables used inside a BEGIN ...END [*note block::.] that were -not declared SCALAR , REAL or INTEGER are global, and any change made -to them inside the block affects their global value. Any [*note -ARRAY::.] or [*note MATRIX::.] declared inside a block is always global. - - -File: redhelp, Node: REMFAC, Next: SCALAR, Prev: REAL, Up: Declarations section - - REMFAC declaration - - The REMFAC declaration removes the special factoring treatment of its -arguments that was declared with [*note FACTOR::.] . - -syntax: - - REMFAC ,+ - - must be a [*note KERNEL::.] or [*note OPERATOR::.] name that -was declared as special with the [*note FACTOR::.] declaration. - - -File: redhelp, Node: SCALAR, Next: SCIENTIFIC_NOTATION, Prev: REMFAC, Up: Declarations section - - SCALAR declaration - - The SCALAR declaration must be made immediately after a [*note -BEGIN::.] (or other variable declaration such as [*note INTEGER::.] -and [*note REAL::.] ) and declares local scalar variables. They are -initialized to 0. - -syntax: - - SCALAR ,* - - may be any valid REDUCE identifier, except T or NIL . - - Scalar variables remain local, and do not share values with -variables of the same name outside the [*note BEGIN::.] ...END [*note -block::.] . When the block is finished, the variables are removed. You -may use the words [*note REAL::.] or [*note INTEGER::.] in the place of -SCALAR . REAL and INTEGER do not indicate typechecking by the current -REDUCE; they are only for your own information. Declaration statements -must immediately follow the BEGIN , without a semicolon between BEGIN -and the first variable declaration. - - Any variables used inside BEGIN ...END blocks that were not -declared SCALAR , REAL or INTEGER are global, and any change made to -them inside the block affects their global value. Arrays declared -inside a block are always global. - - -File: redhelp, Node: SCIENTIFIC_NOTATION, Next: SHARE, Prev: SCALAR, Up: Declarations section - - SCIENTIFIC_NOTATION declaration - -syntax: - - SCIENTIFIC_NOTATION () or SCIENTIFIC_NOTATION (,) - - and are positive integers. SCIENTIFIC_NOTATION controls -the output format of floating point numbers. At the default settings, -any number with five or less digits before the decimal point is printed -in a fixed-point notation, e.g., 12345.6. Numbers with more than five -digits are printed in scientific notation, e.g., 1.234567E+5. -Similarly, by default, any number with eleven or more zeros after the -decimal point is printed in scientific notation. - - When SCIENTIFIC_NOTATION is called with the numerical argument m a -number with more than m digits before the decimal point, or m or more -zeros after the decimal point, is printed in scientific notation. When -SCIENTIFIC_NOTATION is called with a list ,, a number with more -than m digits before the decimal point, or n or more zeros after the -decimal point is printed in scientific notation. - -examples: - - ____________________________________________________________ - - - on rounded; - - - 12345.6; - - 12345.6 - - - - 123456.5; - - 1.234565e+5 - - - - 0.00000000000000012; - - 1.2e-16 - - - - scientific_notation 20; - - 5,11 - - - - 5: 123456.7; - - 123456.7 - - - - 0.00000000000000012; - - 0.00000000000000012 - - ____________________________________________________________ - - -File: redhelp, Node: SHARE, Next: SYMBOLIC, Prev: SCIENTIFIC_NOTATION, Up: Declarations section - - SHARE declaration - - The SHARE declaration allows access to its arguments by both -algebraic and symbolic modes. - -syntax: - - SHARE ,* - - can be any valid REDUCE identifier. - - Programming in [*note SYMBOLIC::.] as well as algebraic mode allows -you a wider range of techniques than just algebraic mode alone. -Expressions do not cross the boundary since they have different -representations, unless the SHARE declaration is used. For more -information on using symbolic mode, see the , and -the . - - You should be aware that a previously-declared array is destroyed by -the SHARE declaration. Scalar variables retain their values. You can -share a declared [*note MATRIX::.] that has not yet been dimensioned so -that it can be used by both modes. Values that are later put into the -matrix are accessible from symbolic mode too, but not by the usual -matrix reference mechanism. In symbolic mode, a matrix is stored as a -list whose first element is [*note MAT::.] , and whose next elements -are the rows of the matrix stored as lists of the individual elements. -Access in symbolic mode is by the operators [*note FIRST::.] , [*note -SECOND::.] , [*note THIRD::.] and [*note REST::.] . - - -File: redhelp, Node: SYMBOLIC, Next: SYMMETRIC, Prev: SHARE, Up: Declarations section - - SYMBOLIC command - - The SYMBOLIC command changes REDUCE's mode of operation to symbolic. -When SYMBOLIC is followed by an expression, that expression is -evaluated in symbolic mode, but REDUCE's mode is not changed. It is -equivalent to the [*note LISP::.] command. - -examples: - - ____________________________________________________________ - - symbolic; - - NIL - - - cdr '(a b c); - - (B C) - - - algebraic; - - x + symbolic car '(y z); - - X + Y - - ____________________________________________________________ - - -File: redhelp, Node: SYMMETRIC, Next: TR, Prev: SYMBOLIC, Up: Declarations section - - SYMMETRIC declaration - - When an operator is declared SYMMETRIC , its arguments are reordered -to conform to the internal ordering of the system. - -syntax: - - SYMMETRIC ,* - - is an identifier that has been declared an operator. - -examples: - - ____________________________________________________________ - - operator m,n; - - symmetric m,n; - - m(y,a,sin(x)); - - M(SIN(X),A,Y) - - - n(z,m(b,a,q)); - - N(M(A,B,Q),Z) - - ____________________________________________________________ - If has not been declared to be an operator, the flag -SYMMETRIC is still attached to it. When is subsequently -used as an operator, the message DECLARE OPERATOR ? (Y OR -N) is printed. If the user replies Y , the symmetric property of the -operator is used. - - -File: redhelp, Node: TR, Next: UNTR, Prev: SYMMETRIC, Up: Declarations section - - TR declaration - - The TR declaration is used to trace system or user-written -procedures. It is only useful to those with a good knowledge of both -Lisp and the internal formats used by REDUCE. - -syntax: - - TR ,* - - is the name of a REDUCE system procedure or one of your own -procedures. - -examples: - - ____________________________________________________________ - ____________________________________________________________ - The system procedure PREPSQ is traced, which prepares -REDUCE standard forms for printing by converting them to a Lisp prefix -form. - ____________________________________________________________ - - - tr prepsq; - - (PREPSQ) - - - x**2 + y; - - PREPSQ entry: - Arg 1: (((((X . 2) . 1) ((Y . 1) . 1)) . 1) - PREPSQ return value = (PLUS (EXPT X 2) Y) - PREPSQ entry: - Arg 1: (1 . 1) - PREPSQ return value = 1 - 2 - X + Y - - - untr prepsq; - - (PREPSQ) - - ____________________________________________________________ - - This example is for a PSL-based system; the above format will vary if -other Lisp systems are used. - - When a procedure is traced, the first lines show entry to the -procedure and the arguments it is given. The value returned by the -procedure is printed upon exit. If you are tracing several procedures, -with a call to one of them inside the other, the inner trace will be -indented showing procedure nesting. There are no trace options. -However, the format of the trace depends on the underlying Lisp system -used. The trace can be removed with the command [*note UNTR::.] . Note -that TRACE , below, is a matrix operator, while TR does procedure -tracing. - - -File: redhelp, Node: UNTR, Next: VARNAME, Prev: TR, Up: Declarations section - - UNTR declaration - - The UNTR declaration is used to remove a trace from system or -user-written procedures declared with [*note TR::.] . It is only useful -to those with a good knowledge of both Lisp and the internal formats -used by REDUCE. - -syntax: - - UNTR ,* - - is the name of a REDUCE system procedure or one of your own -procedures that has previously been the argument of a TR declaration. - - -File: redhelp, Node: VARNAME, Next: WEIGHT, Prev: UNTR, Up: Declarations section - - VARNAME declaration - - The declaration VARNAME instructs REDUCE to use its argument as the -default Fortran (when [*note FORT::.] is on) or [*note STRUCTR::.] -identifier and identifier stem, rather than using ANS . - -syntax: - - VARNAME - - can be any combination of one or more alphanumeric -characters. Try to avoid REDUCE reserved words. - -examples: - - ____________________________________________________________ - - varname ident; - - IDENT - - - on fort; - - x**2 + 1; - - IDENT=X**2+1. - - - off fort,exp; - - structr(((x+y)**2 + z)**3); - - 3 - IDENT2 - where - 2 - IDENT2 := IDENT1 + Z - IDENT1 := X + Y - - ____________________________________________________________ - [*note EXP::.] was turned off so that [*note STRUCTR::.] could show -the structure. If EXP had been on, the expression would have been -expanded into a polynomial. - - -File: redhelp, Node: WEIGHT, Next: WHERE, Prev: VARNAME, Up: Declarations section - - WEIGHT command - - The WEIGHT command is used to attach weights to kernels for -asymptotic constraints. - -syntax: - - WEIGHT = - - must be a REDUCE [*note KERNEL::.] , must be a -positive integer, not 0. - -examples: - - ____________________________________________________________ - - a := (x+y)**4; - - 4 3 2 2 3 4 - A := X + 4*X *Y + 6*X *Y + 4*X*Y + Y - - - weight x=2,y=3; - - wtlevel 8; - - a; - - 4 - X - - - wtlevel 10; - - a; - - 2 2 2 - X *(6*Y + 4*X*Y + X ) - - - int(x**2,x); - - ***** X invalid as KERNEL - - ____________________________________________________________ - Weights and [*note WTLEVEL::.] are used for asymptotic constraints, -where higher-order terms are considered insignificant. - - Weights are originally equivalent to 0 until set by a WEIGHT -command. To remove a weight from a kernel, use the [*note CLEAR::.] -command. Weights once assigned cannot be changed without clearing the -identifier. Once a weight is assigned to a kernel, it is no longer a -kernel and cannot be used in any REDUCE commands or operators that -require kernels, until the weight is cleared. Note that terms are -ordered by greatest weight. - - The weight level of the system is set by [*note WTLEVEL::.] , -initially at 2. Since no kernels have weights, no effect from WTLEVEL -can be seen. Once you assign weights to kernels, you must set WTLEVEL -correctly for the desired operation. When weighted variables appear in a -term, their weights are summed for the total weight of the term (powers -of variables multiply their weights). When a term exceeds the weight -level of the system, it is discarded from the result expression. - - -File: redhelp, Node: WHERE, Next: WHILE, Prev: WEIGHT, Up: Declarations section - - WHERE operator - - The WHERE operator provides an infix notation for one-time -substitutions for kernels in expressions. - -syntax: - - WHERE = , = -* - - can be any REDUCE scalar expression, must be a -[*note KERNEL::.] . Alternatively a [*note RULE::.] or a RULE LIST can -be a member of the right-hand part of a WHERE expression. - -examples: - - ____________________________________________________________ - - x**2 + 17*x*y + 4*y**2 where x=1,y=2; - - - 51 - - - for i := 1:5 collect x**i*q where q= for j := 1:i product j; - - - - 2 3 4 5 - {X,2*X ,6*X ,24*X ,120*X } - - - x**2 + y + z where z=y**3,y=3; - - 2 3 - X + Y + 3 - - ____________________________________________________________ - Substitution inside a WHERE expression has no effect upon the values -of the kernels outside the expression. The WHERE operator has the -lowest precedence of all the infix operators, which are lower than -prefix operators, so that the substitutions apply to the entire -expression preceding the WHERE operator. However, WHERE is applied -before command keywords such as THEN , REPEAT , or DO . - - A [*note RULE::.] or a RULE SET in the right-hand part of the WHERE -expression act as if the rules were activated by [*note LET::.] -immediately before the evaluation of the expression and deactivated by -[*note CLEARRULES::.] immediately afterwards. - - WHERE gives you a natural notation for auxiliary variables in -expressions. As the second example shows, the substitute expression can -be a command to be evaluated. The substitute assignments are made in -parallel, rather than sequentially, as the last example shows. The -expression resulting from the first round of substitutions is not -reexamined to see if any further such substitutions can be made. WHERE -can also be used to define auxiliary variables in [*note PROCEDURE::.] -definitions. - - -File: redhelp, Node: WHILE, Next: WTLEVEL, Prev: WHERE, Up: Declarations section - - WHILE command - - The WHILE command causes a statement to be repeatedly executed until -a given condition is true. If the condition is initially false, the -statement is not executed at all. - -syntax: - - WHILE DO - - is given by a logical operator, must be a -single REDUCE statement, or a [*note group::.] (<< ...>> ) or [*note -BEGIN::.] ...END [*note block::.] . - -examples: - - ____________________________________________________________ - - a := 10; - - A := 10 - - - while a <= 12 do <>; - - - - 10 - - - 11 - - 12 - - while a < 5 do <>; - - - - nothing is printed - - ____________________________________________________________ - - -File: redhelp, Node: WTLEVEL, Prev: WHILE, Up: Declarations section - - WTLEVEL command - - In conjunction with [*note WEIGHT::.] , WTLEVEL is used to implement -asymptotic constraints. Its default value is 2. - -syntax: - - WTLEVEL - - To change the weight level, must evaluate to a positive -integer that is the greatest weight term to be retained in expressions -involving kernels with weight assignments. WTLEVEL returns the new -weight level. If you want the current weight level, but not change it, -say WTLEVEL NIL . - -examples: - - ____________________________________________________________ - - (x+y)**4; - - - 4 3 2 2 3 4 - X + 4*X *Y + 6*X *Y + 4*X*Y + Y - - - weight x=2,y=3; - - wtlevel 8; - - (x+y)**4; - - 4 - X - - - wtlevel 10; - - (x+y)**4; - - 2 2 2 - X *(6*Y + 4*X*Y + X ) - - - int(x**2,x); - - ***** X invalid as KERNEL - - ____________________________________________________________ - WTLEVEL is used in conjunction with the command [*note WEIGHT::.] to -enable asymptotic constraints. Weight of a term is computed by -multiplying the weights of each variable in it by the power to which it -has been raised, and adding the resulting weights for each variable. If -the weight of the term is greater than WTLEVEL , the term is dropped -from the expression, and not used in any further computation involving -the expression. - - Once a weight has been attached to a [*note KERNEL::.] , it is no -longer recognized by the system as a kernel, though still a variable. -It cannot be used in REDUCE commands and operators that need kernels. -The weight attachment can be undone with a [*note CLEAR::.] command. -WTLEVEL can be changed as desired. - - -File: redhelp, Node: Declarations section, Next: Input and Output section, Prev: Algebraic Operators section, Up: Top - - Declarations section - -* Menu: - -* ALGEBRAIC:: command -* ANTISYMMETRIC:: declaration -* ARRAY:: declaration -* CLEAR:: command -* CLEARRULES:: command -* DEFINE:: command -* DEPEND:: declaration -* EVEN:: declaration -* FACTOR declaration:: declaration -* FORALL:: command -* INFIX:: declaration -* INTEGER:: declaration -* KORDER:: declaration -* LET:: command -* LINEAR:: declaration -* LINELENGTH:: declaration -* LISP:: command -* LISTARGP:: declaration -* NODEPEND:: declaration -* MATCH:: command -* NONCOM:: declaration -* NONZERO:: declaration -* ODD:: declaration -* OFF:: command -* ON:: command -* OPERATOR:: declaration -* ORDER:: declaration -* PRECEDENCE:: declaration -* PRECISION:: declaration -* PRINT_PRECISION:: declaration -* REAL:: declaration -* REMFAC:: declaration -* SCALAR:: declaration -* SCIENTIFIC_NOTATION:: declaration -* SHARE:: declaration -* SYMBOLIC:: command -* SYMMETRIC:: declaration -* TR:: declaration -* UNTR:: declaration -* VARNAME:: declaration -* WEIGHT:: command -* WHERE:: operator -* WHILE:: command -* WTLEVEL:: command - - -File: redhelp, Node: IN, Next: INPUT, Up: Input and Output section - - IN command - - The IN command takes a list of file names and inputs each file into -the system. - -syntax: - - IN ,* - - must be in the current directory, or be a valid pathname. -If the file name is not an identifier, double quote marks (" ) are -needed around the file name. - - A message is given if the file cannot be found, or has a mistake in -it. - - Ending the command with a semicolon causes the file to be echoed to -the screen; ending it with a dollar sign does not echo the file. If you -want some but not all of a file echoed, turn the switch [*note ECHO::.] -on or off in the file. - - An efficient way to develop procedures in REDUCE is to write them -into a file using a system editor of your choice, and then input the -files into an active REDUCE session. REDUCE reparses the procedure as -it takes information from the file, overwriting the previous procedure -definition. When it accepts the procedure, it echoes its name to the -screen. Data can also be input to the system from files. - - Files to be read in should always end in [*note END::.] ; to avoid -end-of-file problems. Note that this is an additional END; to any -ending procedures in the file. - - -File: redhelp, Node: INPUT, Next: OUT, Prev: IN, Up: Input and Output section - - INPUT command - - The INPUT command returns the input expression to the REDUCE numbered -prompt that is its argument. - -syntax: - - INPUT () or INPUT - - must be between 1 and the current REDUCE prompt number. - - An expression brought back by INPUT can be reexecuted with new -values or switch settings, or used as an argument in another expression. -The command [*note WS::.] brings back the results of a numbered REDUCE -statement. Two lists contain every input and every output statement -since the beginning of the session. If your session is very long, -storage space begins to fill up with these expressions, so it is a good -idea to end the session once in a while, saving needed expressions to -files with the [*note SAVEAS::.] and [*note OUT::.] commands. - - Switch settings and [*note LET::.] statements can also be reexecuted -by using INPUT . - - An error message is given if a number is called for that has not yet -been used. - - -File: redhelp, Node: OUT, Next: SHUT, Prev: INPUT, Up: Input and Output section - - OUT command - - The OUT command directs output to the filename that is its argument, -until another OUT changes the output file, or [*note SHUT::.] closes it. - -syntax: - - OUT or OUT " " or OUT T - - must be in the current directory, or be a valid complete -file description for your system. If the file name is not in the -current directory, quote marks are needed around the file name. If the -file already exists, a message is printed allowing you to decide -whether to supersede the contents of the file with new material. - - To restore output to the terminal, type OUT T , or [*note SHUT::.] -the file. When you use OUT T , the file remains available, and if you -open it again (with another OUT ), new material is appended rather than -overwriting. - - To write a file using OUT that can be input at a later time, the -switch [*note NAT::.] must be turned off, so that the standard linear -form is saved that can be read in by [*note IN::.] . If NAT is on, -exponents are printed on the line above the expression, which causes -trouble when REDUCE tries to read the file. - - There is a slight complication if you are using the OUT command from -inside a file to create another file. The [*note ECHO::.] switch is -normally off at the top-level and on while reading files (so you can -see what is being read in). If you create a file using OUT at the -top-level, the result lines are printed into the file as you want them. -But if you create such a file from inside a file, the ECHO switch is -on, and every line is echoed, first as you typed it, then as REDUCE -parsed it, and then once more for the file. Therefore, when you create -a file from a file, you need to turn ECHO off explicitly before the OUT -command, and turn it back on when you SHUT the created file, so your -executing file echoes as it should. This behavior also means that as you -watch the file execute, you cannot see the lines that are being put into -the OUT file. As soon as you turn ECHO on, you can see output again. - - -File: redhelp, Node: SHUT, Prev: OUT, Up: Input and Output section - - SHUT command - - The SHUT command closes output files. - -syntax: - - SHUT ,* - - must have been a file opened by [*note OUT::.] . - - A file that has been opened by [*note OUT::.] must be SHUT before it -is brought in by [*note IN::.] . Files that have been opened by OUT -should always be SHUT before the end of the REDUCE session, to avoid -either loss of information or the printing of extraneous information -into the file. In most systems, terminating a session by [*note -BYE::.] closes all open output files. - - -File: redhelp, Node: Input and Output section, Next: Elementary Functions section, Prev: Declarations section, Up: Top - - Input and Output section - -* Menu: - -* IN:: command -* INPUT:: command -* OUT:: command -* SHUT:: command - - -File: redhelp, Node: ACOS, Next: ACOSH, Up: Elementary Functions section - - ACOS operator - - The ACOS operator returns the arccosine of its argument. - -syntax: - - ACOS () or ACOS - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - acos(ab); - - ACOS(AB) - - - acos 15; - - ACOS(15) - - - df(acos(x*y),x); - - 2 2 - SQRT( - X *Y + 1)*Y - -------------------- - 2 2 - X *Y - 1 - - - on rounded; - - res := acos(sqrt(2)/2); - - RES := 0.785398163397 - - - res-pi/4; - - 0 - - ____________________________________________________________ - An explicit numeric value is not given unless the switch [*note -ROUNDED::.] is on and the argument has an absolute numeric value less -than or equal to 1. - - -File: redhelp, Node: ACOSH, Next: ACOT, Prev: ACOS, Up: Elementary Functions section - - ACOSH operator - - ACOSH represents the hyperbolic arccosine of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of ACOSH -is known to the system. Numerical values may also be found by turning -on the switch [*note ROUNDED::.] . - -syntax: - - ACOSH () or ACOSH - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - acosh a; - - ACOSH(A) - - - acosh(0); - - ACOSH(0) - - - df(acosh(a**2),a); - - 4 - 2*SQRT(A - 1)*A - ---------------- - 4 - A - 1 - - - int(acosh(x),x); - - INT(ACOSH(X),X) - - ____________________________________________________________ - You may attach functionality by defining ACOSH to be the inverse of -COSH . This is done by the commands - ____________________________________________________________ - - put('cosh,'inverse,'acosh); - put('acosh,'inverse,'cosh); - ____________________________________________________________ - - You can write a procedure to attach integrals or other functions to -ACOSH . You may wish to add a check to see that its argument is -properly restricted. - - -File: redhelp, Node: ACOT, Next: ACOTH, Prev: ACOSH, Up: Elementary Functions section - - ACOT operator - - ACOT represents the arccotangent of its argument. It takes an -arbitrary scalar expression as its argument. The derivative of ACOT is -known to the system. Numerical values may also be found by turning on -the switch [*note ROUNDED::.] . - -syntax: - - ACOT () or ACOT - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. You can add -functionality yourself with LET and procedures. - - -File: redhelp, Node: ACOTH, Next: ACSC, Prev: ACOT, Up: Elementary Functions section - - ACOTH operator - - ACOTH represents the inverse hyperbolic cotangent of its argument. -It takes an arbitrary scalar expression as its argument. The derivative -of ACOTH is known to the system. Numerical values may also be found by -turning on the switch [*note ROUNDED::.] . - -syntax: - - ACOTH () or ACOTH - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. You can add -functionality yourself with LET and procedures. - - -File: redhelp, Node: ACSC, Next: ACSCH, Prev: ACOTH, Up: Elementary Functions section - - ACSC operator - - The ACSC operator returns the arccosecant of its argument. - -syntax: - - ACSC () or ACSC - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - acsc(ab); - - ACSC(AB) - - - acsc 15; - - ACSC(15) - - - df(acsc(x*y),x); - - 2 2 - -SQRT(X *Y - 1) - ---------------- - 2 2 - X*(X *Y - 1) - - - on rounded; - - res := acsc(2/sqrt(3)); - - RES := 1.0471975512 - - - res-pi/3; - - 0 - - ____________________________________________________________ - An explicit numeric value is not given unless the switch ROUNDED is -on and the argument has an absolute numeric value less than or equal to -1. - - -File: redhelp, Node: ACSCH, Next: ASEC, Prev: ACSC, Up: Elementary Functions section - - ACSCH operator - - The ACSCH operator returns the hyperbolic arccosecant of its -argument. - -syntax: - - ACSCH () or ACSCH - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - acsch(ab); - - ACSCH(AB) - - - acsch 15; - - ACSCH(15) - - - df(acsch(x*y),x); - - 2 2 - -SQRT(X *Y + 1) - ---------------- - 2 2 - X*(X *Y + 1) - - - on rounded; - - res := acsch(3); - - RES := 0.327450150237 - - ____________________________________________________________ - An explicit numeric value is not given unless the switch ROUNDED is -on and the argument has an absolute numeric value less than or equal to -1. - - -File: redhelp, Node: ASEC, Next: ASECH, Prev: ACSCH, Up: Elementary Functions section - - ASEC operator - - The ASEC operator returns the arccosecant of its argument. - -syntax: - - ASEC () or ASEC - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - asec(ab); - - ASEC(AB) - - - asec 15; - - ASEC(15) - - - df(asec(x*y),x); - - 2 2 - SQRT(X *Y - 1) - --------------- - 2 2 - X*(X *Y - 1) - - - on rounded; - - res := asec sqrt(2); - - RES := 0.785398163397 - - - res-pi/4; - - 0 - - ____________________________________________________________ - An explicit numeric value is not given unless the switch ROUNDED is -on and the argument has an absolute numeric value greater or equal to 1. - - -File: redhelp, Node: ASECH, Next: ASIN, Prev: ASEC, Up: Elementary Functions section - - ASECH operator - - ASECH represents the hyperbolic arccosecant of its argument. It -takes an arbitrary scalar expression as its argument. The derivative of -ASECH is known to the system. Numerical values may also be found by -turning on the switch [*note ROUNDED::.] . - -syntax: - - ASECH () or ASECH - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - asech a; - - ASECH(A) - - - asech(1); - - 0 - - - df(acosh(a**2),a); - - 4 - 2*SQRT(- A + 1) - ---------------- - 4 - A*(A - 1) - - - int(asech(x),x); - - INT(ASECH(X),X) - - ____________________________________________________________ - You may attach functionality by defining ASECH to be the inverse of -SECH . This is done by the commands - ____________________________________________________________ - - put('sech,'inverse,'asech); - put('asech,'inverse,'sech); - ____________________________________________________________ - - You can write a procedure to attach integrals or other functions to -ASECH . You may wish to add a check to see that its argument is -properly restricted. - - -File: redhelp, Node: ASIN, Next: ASINH, Prev: ASECH, Up: Elementary Functions section - - ASIN operator - - The ASIN operator returns the arcsine of its argument. - -syntax: - - ASIN () or ASIN - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - asin(givenangle); - - ASIN(GIVENANGLE) - - - asin(5); - - ASIN(5) - - - df(asin(2*x),x); - - 2 - 2*SQRT( - 4*X + 1)) - - -------------------- - 2 - 4*X - 1 - - - on rounded; - - asin .5; - - 0.523598775598 - - - asin(sqrt(3)); - - ASIN(1.73205080757) - - - asin(sqrt(3)/2); - - 1.04719755120 - - ____________________________________________________________ - A numeric value is not returned by ASIN unless the switch ROUNDED -is on and its argument has an absolute value less than or equal to 1. - - -File: redhelp, Node: ASINH, Next: ATAN, Prev: ASIN, Up: Elementary Functions section - - ASINH operator - - The ASINH operator returns the hyperbolic arcsine of its argument. -The derivative of ASINH and some simple transformations are known to -the system. - -syntax: - - ASINH () or ASINH - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - asinh d; - - ASINH(D) - - - asinh(1); - - ASINH(1) - - - df(asinh(2*x),x); - - 2 - 2*SQRT(4*X + 1)) - ----------------- - 2 - 4*X + 1 - - ____________________________________________________________ - You may attach further functionality by defining ASINH to be the -inverse of SINH . This is done by the commands - ____________________________________________________________ - - put('sinh,'inverse,'asinh); - put('asinh,'inverse,'sinh); - ____________________________________________________________ - - A numeric value is not returned by ASINH unless the switch ROUNDED -is on and its argument evaluates to a number. - - -File: redhelp, Node: ATAN, Next: ATANH, Prev: ASINH, Up: Elementary Functions section - - ATAN operator - - The ATAN operator returns the arctangent of its argument. - -syntax: - - ATAN () or ATAN - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - atan(middle); - - ATAN(MIDDLE) - - - on rounded; - - atan 45; - - 1.54857776147 - - - off rounded; - - int(atan(x),x); - - 2 - 2*ATAN(X)*X - LOG(X + 1) - ------------------------- - 2 - - - df(atan(y**2),y); - - 2*Y - ------- - 4 - Y + 1 - - ____________________________________________________________ - A numeric value is not returned by ATAN unless the switch [*note -ROUNDED::.] is on and its argument evaluates to a number. - - -File: redhelp, Node: ATANH, Next: ATAN2, Prev: ATAN, Up: Elementary Functions section - - ATANH operator - - The ATANH operator returns the hyperbolic arctangent of its argument. -The derivative of ASINH and some simple transformations are known to -the system. - -syntax: - - ATANH () or ATANH - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - atanh aa; - - ATANH(AA) - - - atanh(1); - - ATANH(1) - - - df(atanh(x*y),y); - - - X - ---------- - 2 2 - X *Y - 1 - - ____________________________________________________________ - A numeric value is not returned by ASINH unless the switch ROUNDED -is on and its argument evaluates to a number. You may attach -additional functionality by defining ATANH to be the inverse of TANH . -This is done by the commands - - ____________________________________________________________ - - put('tanh,'inverse,'atanh); - put('atanh,'inverse,'tanh); - ____________________________________________________________ - - -File: redhelp, Node: ATAN2, Next: COS, Prev: ATANH, Up: Elementary Functions section - - ATAN2 operator - -syntax: - - ATAN2 (,) - - is any valid scalar REDUCE expression. In [*note -ROUNDED::.] mode, if a numerical value exists, ATAN2 returns the -principal value of the arc tangent of the second argument divided by -the first in the range [-pi,+pi] radians, using the signs of both -arguments to determine the quadrant of the return value. An expression -in terms of ATAN2 is returned in other cases. - -examples: - - ____________________________________________________________ - - atan2(3,2); - - ATAN2(3,2); - - - on rounded; - - atan2(3,2); - - 0.982793723247 - - - atan2(a,b); - - ATAN2(A,B); - - - atan2(1,0); - - 1.57079632679 - - ____________________________________________________________ - ATAN2 returns a numeric value only if [*note ROUNDED::.] is on. Then -ATAN2 is calculated to the current degree of floating point precision. - - -File: redhelp, Node: COS, Next: COSH, Prev: ATAN2, Up: Elementary Functions section - - COS operator - - The COS operator returns the cosine of its argument. - -syntax: - - COS () or COS - - is any valid scalar REDUCE expression, - is a single identifier or begins with a prefix -operator name. - -examples: - - ____________________________________________________________ - - - - cos abc; - - COS(ABC) - - - - cos(pi); - - -1 - - - - cos 4; - - COS(4) - - - - on rounded; - - - cos(4); - - - 0.653643620864 - - - - cos log 5; - - - 0.0386319699339 - - ____________________________________________________________ - COS returns a numeric value only if [*note ROUNDED::.] is on. Then -the cosine is calculated to the current degree of floating point -precision. - - -File: redhelp, Node: COSH, Next: COT, Prev: COS, Up: Elementary Functions section - - COSH operator - - The COSH operator returns the hyperbolic cosine of its argument. -The derivative of COSH and some simple transformations are known to the -system. - -syntax: - - COSH () or COSH - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - - cosh b; - - COSH(B) - - - - cosh(0); - - 1 - - - - df(cosh(x*y),x); - - SINH(X*Y)*Y - - - - int(cosh(x),x); - - SINH(X) - - ____________________________________________________________ - You may attach further functionality by defining its inverse (see -[*note ACOSH::.] ). A numeric value is not returned by COSH unless the -switch [*note ROUNDED::.] is on and its argument evaluates to a number. - - -File: redhelp, Node: COT, Next: COTH, Prev: COSH, Up: Elementary Functions section - - COT operator - - COT represents the cotangent of its argument. It takes an arbitrary -scalar expression as its argument. The derivative of ACOT and some -simple properties are known to the system. - -syntax: - - COT () or COT - - may be any scalar REDUCE expression. -must be a single identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - cot(a)*tan(a); - - COT(A)*TAN(A)) - - - cot(1); - - COT(1) - - - df(cot(2*x),x); - - 2 - - 2*(COT(2*X) + 1) - - ____________________________________________________________ - Numerical values of expressions involving COT may be found by -turning on the switch [*note ROUNDED::.] . - - -File: redhelp, Node: COTH, Next: CSC, Prev: COT, Up: Elementary Functions section - - COTH operator - - The COTH operator returns the hyperbolic cotangent of its argument. -The derivative of COTH and some simple transformations are known to the -system. - -syntax: - - COTH () or COTH - - may be any scalar REDUCE expression. -must be a single identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - df(coth(x*y),x); - - 2 - - Y*(COTH(X*Y) - 1) - - - - coth acoth z; - - Z - - ____________________________________________________________ - You can write [*note LET::.] statements and procedures to add further -functionality to COTH if you wish. Numerical values of expressions -involving COTH may also be found by turning on the switch [*note -ROUNDED::.] . - - -File: redhelp, Node: CSC, Next: CSCH, Prev: COTH, Up: Elementary Functions section - - CSC operator - - The CSC operator returns the cosecant of its argument. The -derivative of CSC and some simple transformations are known to the -system. - -syntax: - - CSC () or CSC - - may be any scalar REDUCE expression. -must be a single identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - - csc(q)*sin(q); - - CSC(Q)*SIN(Q) - - - - df(csc(x*y),x); - - -COT(X*Y)*CSC(X*Y)*Y - - ____________________________________________________________ - You can write [*note LET::.] statements and procedures to add further -functionality to CSC if you wish. Numerical values of expressions -involving CSC may also be found by turning on the switch [*note -ROUNDED::.] . - - -File: redhelp, Node: CSCH, Next: ERF, Prev: CSC, Up: Elementary Functions section - - CSCH operator - - The COSH operator returns the hyperbolic cosecant of its argument. -The derivative of CSCH and some simple transformations are known to the -system. - -syntax: - - CSCH () or CSCH - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - - csch b; - - CSCH(B) - - - - csch(0); - - 0 - - - - df(csch(x*y),x); - - - COTH(X*Y)*CSCH(X*Y)*Y - - - - int(csch(x),x); - - INT(CSCH(X),X) - - ____________________________________________________________ - A numeric value is not returned by CSCH unless the switch [*note -ROUNDED::.] is on and its argument evaluates to a number. - - -File: redhelp, Node: ERF, Next: EXP, Prev: CSCH, Up: Elementary Functions section - - ERF operator - - The ERF operator represents the error function, defined by - - erf(x) = (2/sqrt(pi))*int(e^(-x^2),x) - - A limited number of its properties are known to the system, -including the fact that it is an odd function. Its derivative is known, -and from this, some integrals may be computed. However, a complete -integration procedure for this operator is not currently included. - -examples: - - ____________________________________________________________ - - erf(0); - - 0 - - - erf(-a); - - - ERF(A) - - - df(erf(x**2),x); - - 4*SQRT(PI)*X - ------------ - 4 - X - E *PI - - - - int(erf(x),x); - - 2 - X - E *ERF(X)*PI*X + SQRT(PI) - --------------------------- - 2 - X - E *PI - - ____________________________________________________________ - - -File: redhelp, Node: EXP, Next: SEC, Prev: ERF, Up: Elementary Functions section - - EXP operator - - The EXP operator returns E raised to the power of its argument. - -syntax: - - EXP () or EXP - - can be any valid REDUCE scalar expression. - must be a single identifier or begin with a prefix -operator. - -examples: - - ____________________________________________________________ - - exp(sin(x)); - - SIN X - E - - - exp(11); - - 11 - E - - - on rounded; - - exp sin(pi/3); - - 2.37744267524 - - ____________________________________________________________ - Numeric values are returned only when ROUNDED is on. The single -letter E with the exponential operator ^ or ** may be substituted for -EXP without change of function. - - -File: redhelp, Node: SEC, Next: SECH, Prev: EXP, Up: Elementary Functions section - - SEC operator - - The SEC operator returns the secant of its argument. - -syntax: - - SEC () or SEC - - is any valid scalar REDUCE expression, - is a single identifier or begins with a prefix -operator name. - -examples: - - ____________________________________________________________ - - - - sec abc; - - SEC(ABC) - - - - sec(pi); - - -1 - - - - sec 4; - - SEC(4) - - - - on rounded; - - - sec(4); - - - 1.52988565647 - - - - sec log 5; - - - 25.8852966005 - - ____________________________________________________________ - SEC returns a numeric value only if [*note ROUNDED::.] is on. Then -the secant is calculated to the current degree of floating point -precision. - - -File: redhelp, Node: SECH, Next: SIN, Prev: SEC, Up: Elementary Functions section - - SECH operator - - The SECH operator returns the hyperbolic secant of its argument. - -syntax: - - SECH () or SECH - - is any valid scalar REDUCE expression, - is a single identifier or begins with a prefix -operator name. - -examples: - - ____________________________________________________________ - - sech abc; - - SECH(ABC) - - - - sech(0); - - 1 - - - - sech 4; - - SECH(4) - - - - on rounded; - - - sech(4); - - 0.0366189934737 - - - - sech log 5; - - 0.384615384615 - - ____________________________________________________________ - SECH returns a numeric value only if [*note ROUNDED::.] is on. Then -the expression is calculated to the current degree of floating point -precision. - - -File: redhelp, Node: SIN, Next: SINH, Prev: SECH, Up: Elementary Functions section - - SIN operator - - The SIN operator returns the sine of its argument. - -syntax: - - SIN () or SIN - - is any valid scalar REDUCE expression, - is a single identifier or begins with a prefix -operator name. - -examples: - - ____________________________________________________________ - - sin aa; - - SIN(AA) - - - sin(pi/2); - - 1 - - - on rounded; - - sin 3; - - 0.14112000806 - - - sin(pi/2); - - 1.0 - - ____________________________________________________________ - SIN returns a numeric value only if ROUNDED is on. Then the sine is -calculated to the current degree of floating point precision. The -argument in this case is assumed to be in radians. - - -File: redhelp, Node: SINH, Next: TAN, Prev: SIN, Up: Elementary Functions section - - SINH operator - - The SINH operator returns the hyperbolic sine of its argument. The -derivative of SINH and some simple transformations are known to the -system. - -syntax: - - SINH () or SINH - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - - sinh b; - - SINH(B) - - - - sinh(0); - - 0 - - - df(sinh(x**2),x); - - 2 - 2*COSH(X )*X - - - int(sinh(4*x),x); - - COSH(4*X) - --------- - 4 - - - on rounded; - - sinh 4; - - 27.2899171971 - - ____________________________________________________________ - You may attach further functionality by defining its inverse (see -[*note ASINH::.] ). A numeric value is not returned by SINH unless the -switch [*note ROUNDED::.] is on and its argument evaluates to a number. - - -File: redhelp, Node: TAN, Next: TANH, Prev: SINH, Up: Elementary Functions section - - TAN operator - - The TAN operator returns the tangent of its argument. - -syntax: - - TAN () or TAN - - is any valid scalar REDUCE expression, - is a single identifier or begins with a prefix -operator name. - -examples: - - ____________________________________________________________ - - tan a; - - TAN(A) - - - tan(pi/5); - - PI - TAN(--) - 5 - - - on rounded; - tan(pi/5); - - 0.726542528005 - - ____________________________________________________________ - TAN returns a numeric value only if ROUNDED is on. Then the tangent -is calculated to the current degree of floating point accuracy. - - When [*note ROUNDED::.] is on, no check is made to see if the -argument of TAN is a multiple of pi/2, for which the tangent goes to -positive or negative infinity. (Of course, since REDUCE uses a -fixed-point representation of pi/2, it produces a large but not -infinite number.) You need to make a check for multiples of pi/2 in any -program you use that might possibly ask for the tangent of such a -quantity. - - -File: redhelp, Node: TANH, Prev: TAN, Up: Elementary Functions section - - TANH operator - - The TANH operator returns the hyperbolic tangent of its argument. -The derivative of TANH and some simple transformations are known to the -system. - -syntax: - - TANH () or TANH - - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. - -examples: - - ____________________________________________________________ - - tanh b; - - TANH(B) - - - tanh(0); - - 0 - - - df(tanh(x*y),x); - - 2 - Y*( - TANH(X*Y) + 1) - - - int(tanh(x),x); - - 2*X - LOG(E + 1) - X - - - on rounded; tanh 2; - - 0.964027580076 - - ____________________________________________________________ - You may attach further functionality by defining its inverse (see -[*note ATANH::.] ). A numeric value is not returned by TANH unless the -switch [*note ROUNDED::.] is on and its argument evaluates to a number. - - -File: redhelp, Node: Elementary Functions section, Next: General Switches section, Prev: Input and Output section, Up: Top - - Elementary Functions section - -* Menu: - -* ACOS:: operator -* ACOSH:: operator -* ACOT:: operator -* ACOTH:: operator -* ACSC:: operator -* ACSCH:: operator -* ASEC:: operator -* ASECH:: operator -* ASIN:: operator -* ASINH:: operator -* ATAN:: operator -* ATANH:: operator -* ATAN2:: operator -* COS:: operator -* COSH:: operator -* COT:: operator -* COTH:: operator -* CSC:: operator -* CSCH:: operator -* ERF:: operator -* EXP:: operator -* SEC:: operator -* SECH:: operator -* SIN:: operator -* SINH:: operator -* TAN:: operator -* TANH:: operator - - -File: redhelp, Node: SWITCHES, Next: ALGINT, Up: General Switches section - - SWITCHES introduction - - Switches are set on or off using the commands [*note ON::.] or -[*note OFF::.] , respectively. The default setting of the switches -described in this section is [*note OFF::.] unless stated otherwise. - - -File: redhelp, Node: ALGINT, Next: ALLBRANCH, Prev: SWITCHES, Up: General Switches section - - ALGINT switch - - When the ALGINT switch is on, the algebraic integration module (which -must be loaded from the REDUCE library) is used for integration. - - Loading ALGINT from the library automatically turns on the ALGINT -switch. An error message will be given if ALGINT is turned on when the -ALGINT has not been loaded from the library. - - -File: redhelp, Node: ALLBRANCH, Next: ALLFAC, Prev: ALGINT, Up: General Switches section - - ALLBRANCH switch - - When ALLBRANCH is on, the operator [*note SOLVE::.] selects all -branches of solutions. When ALLBRANCH is off, it selects only the -principal branches. Default is ON . - -examples: - - ____________________________________________________________ - - - solve(log(sin(x+3)),x); - - {X=2*ARBINT(1)*PI - ASIN(1) - 3, - X=2*ARBINT(1)*PI + ASIN(1) + PI - 3} - - - off allbranch; - - solve(log(sin(x+3)),x); - - X=ASIN(1) - 3 - - ____________________________________________________________ - [*note ARBINT::.] (1) indicates an arbitrary integer, which is given -a unique identifier by REDUCE, showing that there are infinitely many -solutions of this type. When ALLBRANCH is off, the single canonical -solution is given. - - -File: redhelp, Node: ALLFAC, Next: ARBVARS, Prev: ALLBRANCH, Up: General Switches section - - ALLFAC switch - - The ALLFAC switch, when on, causes REDUCE to factor out automatically -common products in the output of expressions. Default is ON . - -examples: - - ____________________________________________________________ - - x + x*y**3 + x**2*cos(z); - - 3 - X*(COS(Z)*X + Y + 1) - - - off allfac; - - x + x*y**3 + x**2*cos(z); - - 2 3 - COS(Z)*X + X*Y + X - - ____________________________________________________________ - The ALLFAC switch has no effect when PRI is off. Although the switch -setting stays as it was, printing behavior is as if it were off. - - -File: redhelp, Node: ARBVARS, Next: BALANCED_MOD, Prev: ALLFAC, Up: General Switches section - - ARBVARS switch - - When ARBVARS is on, the solutions of singular or underdetermined -systems of equations are presented in terms of arbitrary complex -variables (see [*note ARBCOMPLEX::.] ). Otherwise, the solution is -parametrized in terms of some of the input variables. Default is ON . - -examples: - - ____________________________________________________________ - - solve({2x + y,4x + 2y},{x,y}); - - arbcomplex(1) - {{x= - -------------,y=arbcomplex(1)}} - 2 - - - solve({sqrt(x)+ y**3-1},{x,y}); - - - 6 3 - {{y=arbcomplex(2),x=y - 2*y + 1}} - - - off arbvars; - - solve({2x + y,4x + 2y},{x,y}); - - y - {{x= - -}} - 2 - - - solve({sqrt(x)+ y**3-1},{x,y}); - - - 6 3 - {{x=y - 2*y + 1}} - - ____________________________________________________________ - With ARBVARS off, the return value [[]] means that the equations -given to [*note SOLVE::.] imply no relation among the input variables. - - -File: redhelp, Node: BALANCED_MOD, Next: BFSPACE, Prev: ARBVARS, Up: General Switches section - - BALANCED_MOD switch - - [*note MODULAR::.] numbers are normally produced in the range -[0,...), where is the current modulus. With BALANCED_MOD on, the -range [-/2,/2] is used instead. - -examples: - - ____________________________________________________________ - - setmod 7; - - 1 - - - on modular; - - 4; - - 4 - - - on balanced_mod; - - 4; - - -3 - - ____________________________________________________________ - - -File: redhelp, Node: BFSPACE, Next: COMBINEEXPT, Prev: BALANCED_MOD, Up: General Switches section - - BFSPACE switch - - Floating point numbers are normally printed in a compact notation -(either fixed point or in scientific notation if [*note -SCIENTIFIC_NOTATION::.] has been used). In some (but not all) cases, it -helps comprehensibility if spaces are inserted in the number at regular -intervals. The switch BFSPACE , if on, will cause a blank to be -inserted in the number after every five characters. - -examples: - - ____________________________________________________________ - - on rounded; - - 1.2345678; - - 1.2345678 - - - on bfspace; - - 1.2345678; - - 1.234 5678 - - ____________________________________________________________ - - BFSPACE is normally off. - - -File: redhelp, Node: COMBINEEXPT, Next: COMBINELOGS, Prev: BFSPACE, Up: General Switches section - - COMBINEEXPT switch - - REDUCE is in general poor at surd simplification. However, when the -switch COMBINEEXPT is on, the system attempts to combine exponentials -whenever possible. - -examples: - - ____________________________________________________________ - - 3^(1/2)*3^(1/3)*3^(1/6); - - 1/3 1/6 - SQRT(3)*3 *3 - - - on combineexpt; - - ws; - - 1 - - ____________________________________________________________ - - -File: redhelp, Node: COMBINELOGS, Next: COMP, Prev: COMBINEEXPT, Up: General Switches section - - COMBINELOGS switch - - In many cases it is desirable to expand product arguments of -logarithms, or collect a sum of logarithms into a single logarithm. -Since these are inverse operations, it is not possible to provide rules -for doing both at the same time and preserve the REDUCE concept of -idempotent evaluation. As an alternative, REDUCE provides two switches -[*note EXPANDLOGS::.] and COMBINELOGS to carry out these operations. - -examples: - - ____________________________________________________________ - - on expandlogs; - - log(x*y); - - LOG(X) + LOG(Y) - - - on combinelogs; - - ws; - - LOG(X*Y) - - ____________________________________________________________ - - At the present time, it is possible to have both switches on at once, -which could lead to infinite recursion. However, an expression is -switched from one form to the other in this case. Users should not rely -on this behavior, since it may change in the next release. - - -File: redhelp, Node: COMP, Next: COMPLEX, Prev: COMBINELOGS, Up: General Switches section - - COMP switch - - When COMP is on, any succeeding function definitions are compiled -into a faster-running form. Default is OFF . - -examples: - - ____________________________________________________________ - ____________________________________________________________ - The following procedure finds Fibonacci numbers recursively. Create -a new file "refib" in your current directory with the following lines -in it: - ____________________________________________________________ - - - procedure refib(n); - if fixp n and n >= 0 then - if n <= 1 then 1 - else refib(n-1) + refib(n-2) - else rederr "nonnegative integer only"; - - end; - - ____________________________________________________________ - Now load REDUCE and run the following: - ____________________________________________________________ - - - on time; - - Time: 100 ms - - - - in "refib"$ - - Time: 0 ms - - - - - - REFIB - - - - - - Time: 260 ms - - - - - - Time: 20 ms - - - - refib(80); - - 37889062373143906 - - - - - - Time: 14840 ms - - - - on comp; - - Time: 80 ms - - - - in "refib"$ - - Time: 20 ms - - - - - - REFIB - - - - - - Time: 640 ms - - - - refib(80); - - 37889062373143906 - - - - - - Time: 10940 ms - - ____________________________________________________________ - - Note that the compiled procedure runs faster. Your time messages will -differ depending upon which system you have. Compiled functions remain -so for the duration of the REDUCE session, and are then lost. They must -be recompiled if wanted in another session. With the switch [*note -TIME::.] on as shown above, the CPU time used in executing the command -is returned in milliseconds. Be careful not to leave COMP on unless you -want it, as it makes the processing of procedures much slower. - - -File: redhelp, Node: COMPLEX, Next: CREF, Prev: COMP, Up: General Switches section - - COMPLEX switch - - When the COMPLEX switch is on, full complex arithmetic is used in -simplification, function evaluation, and factorization. Default is OFF . - -examples: - - ____________________________________________________________ - - - factorize(a**2 + b**2); - - 2 2 - {A + B } - - - on complex; - - - factorize(a**2 + b**2); - - {A - I*B,A + I*B} - - - - (x**2 + y**2)/(x + i*y); - - X - I*Y - - - - on rounded; - - *** Domain mode COMPLEX changed to COMPLEX_FLOAT - - - - sqrt(-17); - - 4.12310562562*I - - - - log(7*i); - - 1.94591014906 + 1.57079632679*I - - ____________________________________________________________ - Complex floating-point can be done by turning on [*note ROUNDED::.] -in addition to COMPLEX . With COMPLEX off however, REDUCE knows that i -is the square root of -1 but will not carry out more complicated -complex operations. If you want complex denominators cleared by -multiplication by their conjugates, turn on the switch [*note -RATIONALIZE::.] . - - -File: redhelp, Node: CREF, Next: CRAMER, Prev: COMPLEX, Up: General Switches section - - CREF switch - - The switch CREF invokes the CREF cross-reference program that -processes a set of procedure definitions to produce a summary of their -entry points, undefined procedures, non-local variables and so on. The -program will also check that procedures are called with a consistent -number of arguments, and print a diagnostic message otherwise. - - The output is alphabetized on the first seven characters of each -function name. - - To invoke the cross-reference program, CREF is first turned on. -This causes the program to load and the cross-referencing process to -begin. After all the required definitions are loaded, turning CREF off -will cause a cross-reference listing to be produced. - - Algebraic procedures in REDUCE are treated as if they were symbolic, -so that algebraic constructs will actually appear as calls to symbolic -functions, such as AEVAL . - - -File: redhelp, Node: CRAMER, Next: DEFN, Prev: CREF, Up: General Switches section - - CRAMER switch - - When the CRAMER switch is on, [*note MATRIX::.] inversion and linear -equation solving (operator [*note SOLVE::.] ) is done by Cramer's rule, -through exterior multiplication. Default is OFF . - -examples: - - ____________________________________________________________ - - on time; - - Time: 80 ms - - - off output; - - Time: 100 ms - - - mm := mat((a,b,c,d,f),(a,a,c,f,b),(b,c,a,c,d), (c,c,a,b,f), - (d,a,d,e,f)); - - - Time: 300 ms - - - inverse := 1/mm; - - Time: 18460 ms - - - on cramer; - - Time: 80 ms - - - cramersinv := 1/mm; - - Time: 9260 ms - - ____________________________________________________________ - Your time readings will vary depending on the REDUCE version you use. -After you invert the matrix, turn on [*note OUTPUT::.] and ask for one -of the elements of the inverse matrix, such as CRAMERSINV(3,2) , so that -you can see the size of the expressions produced. - - Inversion of matrices and the solution of linear equations with dense -symbolic entries in many variables is generally considerably faster with -CRAMER on. However, inversion of numeric-valued matrices is slower. -Consider the matrices you're inverting before deciding whether to turn -CRAMER on or off. A substantial portion of the time in matrix inversion -is given to formatting the results for printing. To save this time, -turn OUTPUT off, as shown in this example or terminate the expression -with a dollar sign instead of a semicolon. The results are still -available to you in the workspace associated with your prompt number, -or you can assign them to an identifier for further use. - - -File: redhelp, Node: DEFN, Next: DEMO, Prev: CRAMER, Up: General Switches section - - DEFN switch - - When the switch DEFN is on, the Standard Lisp equivalent of the -input statement or procedure is printed, but not evaluated. Default is -OFF . - -examples: - - ____________________________________________________________ - - - on defn; - - - 17/3; - - (AEVAL (LIST 'QUOTIENT 17 3)) - - - - df(sin(x),x,2); - - - (AEVAL (LIST 'DF (LIST 'SIN 'X) 'X 2)) - - - procedure coshval(a); - begin scalar g; - g := (exp(a) + exp(-a))/2; - return g - end; - - - (AEVAL - (PROGN - (FLAG '(COSHVAL) 'OPFN) - (DE COSHVAL (A) - (PROG (G) - (SETQ G - (AEVAL - (LIST - 'QUOTIENT - (LIST - 'PLUS - (LIST 'EXP A) - (LIST 'EXP (LIST 'MINUS A))) - 2))) - (RETURN G)))) ) - - - - coshval(1); - - (AEVAL (LIST 'COSHVAL 1)) - - - - off defn; - - - coshval(1); - - Declare COSHVAL operator? (Y or N) - - - - n - - procedure coshval(a); - begin scalar g; - g := (exp(a) + exp(-a))/2; - return g - end; - - - COSHVAL - - - - on rounded; - - - coshval(1); - - 1.54308063482 - - ____________________________________________________________ - The above function COSHVAL finds the hyperbolic cosine (cosh) of its -argument. When DEFN is on, you can see the Standard Lisp equivalent of -the function, but it is not entered into the system as shown by the -message DECLARE COSHVAL OPERATOR? . It must be reentered with DEFN off -to be recognized. This procedure is used as an example; a more -efficient procedure would eliminate the unnecessary local variable with - ____________________________________________________________ - - procedure coshval(a); - (exp(a) + exp(-a))/2; - ____________________________________________________________ - - -File: redhelp, Node: DEMO, Next: DFPRINT, Prev: DEFN, Up: General Switches section - - DEMO switch - - The DEMO switch is used for interactive files, causing the system to -pause after each command in the file until you type a RETURN . Default -is OFF . - - The switch DEMO has no effect on top level interactive statements. -Use it when you want to slow down operations in a file so you can see -what is happening. - - You can either include the ON DEMO command in the file, or enter it -from the top level before bringing in any file. Unlike the [*note -PAUSE::.] command, ON DEMO does not permit you to interrupt the file -for questions of your own. - - -File: redhelp, Node: DFPRINT, Next: DIV, Prev: DEMO, Up: General Switches section - - DFPRINT switch - - When DFPRINT is on, expressions in the differentiation operator -[*note DF::.] are printed in a more "natural" notation, with the -differentiation variables appearing as subscripts. In addition, if the -switch [*note NOARG::.] is on (the default), the arguments of the -differentiated operator are suppressed. - -examples: - - ____________________________________________________________ - - operator f; - - df(f x,x); - - DF(F(X),X); - - - on dfprint; - - ws; - - F - X - - - df(f(x,y),x,y); - - F - Y - - - off noarg; - - ws; - - F(X,Y) - X - - ____________________________________________________________ - - -File: redhelp, Node: DIV, Next: ECHO, Prev: DFPRINT, Up: General Switches section - - DIV switch - - When DIV is on, the system divides any simple factors found in the -denominator of an expression into the numerator. Default is OFF . - -examples: - - ____________________________________________________________ - - - on div; - - - a := x**2/y**2; - - 2 -2 - A := X *Y - - - - b := a/(3*z); - - 1 2 -2 -1 - B := -*X *Y *Z - 3 - - - - off div; - - - a; - - 2 - X - --- - 2 - Y - - - - b; - - 2 - X - ------- - 2 - 3*Y *Z - - ____________________________________________________________ - The DIV switch only has effect when the [*note PRI::.] switch is on. -When PRI is off, regardless of the setting of DIV , the printing -behavior is as if DIV were off. - - -File: redhelp, Node: ECHO, Next: ERRCONT, Prev: DIV, Up: General Switches section - - ECHO switch - - The ECHO switch is normally off for top-level entry, and on when -files are brought in. If ECHO is turned on at the top level, your input -statements are echoed to the screen (thus appearing twice). Default OFF -(but note default ON for files). - - If you want to display certain portions of a file and not others, -use the commands OFF ECHO and ON ECHO inside the file. If you want no -display of the file, use the input command - - IN filename$ - - rather than using the semicolon delimiter. - - Be careful when you use commands within a file to generate another -file. Since ECHO is on for files, the output file echoes input -statements (unlike its behavior from the top level). You should -explicitly turn off ECHO when writing output, and turn it back on when -you're done. - - -File: redhelp, Node: ERRCONT, Next: EVALLHSEQP, Prev: ECHO, Up: General Switches section - - ERRCONT switch - - When the ERRCONT switch is on, error conditions do not stop file -execution. Error messages will be printed whether ERRCONT is on or off. - - Default is OFF . - - The following describes what happens when an error occurs in a file -under each setting of ERRCONT and INT : - - Both off: Message is printed and parsing continues, but no further -statements are executed; no commands from keyboard accepted except bye -or end; - - ERRCONT off, INT on: Message is printed, and you are asked if you -wish to continue. (This is the default behavior); - - ERRCONT on, INT off: Message is printed, and file continues to -execute without pause; - - Both on: Message is printed, and file continues to execute without -pause. - - -File: redhelp, Node: EVALLHSEQP, Next: EXP switch, Prev: ERRCONT, Up: General Switches section - - EVALLHSEQP switch - - Under normal circumstances, the right-hand-side of an [*note -EQUATION::.] is evaluated but not the left-hand-side. This also applies -to any substitutions made by the [*note SUB::.] operator. If both sides -are to be evaluated, the switch EVALLHSEQP should be turned on. - - -File: redhelp, Node: EXP switch, Next: EXPANDLOGS, Prev: EVALLHSEQP, Up: General Switches section - - EXP switch - - When the EXP switch is on, powers and products of expressions are -expanded. Default is ON . - -examples: - - ____________________________________________________________ - - (x+1)**3; - - 3 2 - X + 3*X + 3*X + 1 - - - (a + b*i)*(c + d*i); - - A*C + A*D*I + B*C*I - B*D - - - off exp; - - (x+1)**3; - - 3 - (X + 1) - - - (a + b*i)*(c + d*i); - - (A + B*I)*(C + D*I) - - - length((x+1)**2/(y+1)); - - 2 - - ____________________________________________________________ - Note that REDUCE knows that i^2 = -1. When EXP is off, equivalent -expressions may not simplify to the same form, although zero -expressions still simplify to zero. Several operators that expect a -polynomial argument behave differently when EXP is off, such as [*note -LENGTH::.] . Be cautious about leaving EXP off. - - -File: redhelp, Node: EXPANDLOGS, Next: EZGCD, Prev: EXP switch, Up: General Switches section - - EXPANDLOGS switch - - In many cases it is desirable to expand product arguments of -logarithms, or collect a sum of logarithms into a single logarithm. -Since these are inverse operations, it is not possible to provide rules -for doing both at the same time and preserve the REDUCE concept of -idempotent evaluation. As an alternative, REDUCE provides two switches -EXPANDLOGS and [*note COMBINELOGS::.] to carry out these operations. -Both are off by default. - -examples: - - ____________________________________________________________ - - on expandlogs; - - log(x*y); - - LOG(X) + LOG(Y) - - - on combinelogs; - - ws; - - LOG(X*Y) - - ____________________________________________________________ - - At the present time, it is possible to have both switches on at once, -which could lead to infinite recursion. However, an expression is -switched from one form to the other in this case. Users should not rely -on this behavior, since it may change in the next release. - - -File: redhelp, Node: EZGCD, Next: FACTOR, Prev: EXPANDLOGS, Up: General Switches section - - EZGCD switch - - When EZGCD and [*note GCD::.] are on, greatest common divisors are -computed using the EZ GCD algorithm that uses modular arithmetic (and is -usually faster). Default is OFF . - - As a side effect of the gcd calculation, the expressions involved are -factored, though not the heavy-duty factoring of [*note FACTORIZE::.] . -The EZ GCD algorithm was introduced in a paper by J. Moses and D.Y.Y. -Yun in , 1973, pp. 159-166. - - Note that the [*note GCD::.] switch must also be on for EZGCD to have -effect. - - -File: redhelp, Node: FACTOR, Next: FAILHARD, Prev: EZGCD, Up: General Switches section - - FACTOR switch - - When the FACTOR switch is on, input expressions and results are -automatically factored. - -examples: - - ____________________________________________________________ - - - on factor; - - - aa := 3*x**3*a + 6*x**2*y*a + 3*x**3*b + 6*x**2*y*b - - + x*y*a + 2*y**2*a + x*y*b + 2*y**2*b; - - - - 2 - AA := (A + B)*(3*X + Y)*(X + 2*Y) - - - off factor; - - aa; - - 3 2 2 3 2 - 3*A*X + 6*A*X *Y + A*X*Y + 2*A*Y + 3*B*X + 6*B*X *Y - - - + B*X*Y + 2*B*Y^{2} - - on factor; - - ab := x**2 - 2; - - 2 - AB := X - 2 - - ____________________________________________________________ - REDUCE factors univariate and multivariate polynomials with integer -coefficients, finding any factors that also have integer coefficients. -The factoring is done by reducing multivariate problems to univariate -ones with symbolic coefficients, and then solving the univariate ones -modulo small primes. The results of these calculations are merged to -determine the factors of the original polynomial. The factorizer -normally selects evaluation points and primes using a random number -generator. Thus, the detailed factoring behavior may be different each -time any particular problem is tackled. - - When the FACTOR switch is turned on, the [*note EXP::.] switch is -turned off, and when the FACTOR switch is turned off, the [*note -EXP::.] switch is turned on, whether it was on previously or not. - - When the switch [*note TRFAC::.] is on, informative messages are -generated at each call to the factorizer. The [*note TRALLFAC::.] -switch causes the production of a more verbose trace message. It takes -precedence over TRFAC if they are both on. - - To factor a polynomial explicitly and store the results, use the -operator [*note FACTORIZE::.] . - - -File: redhelp, Node: FAILHARD, Next: FORT, Prev: FACTOR, Up: General Switches section - - FAILHARD switch - - When the FAILHARD switch is on, the integration operator [*note -INT::.] terminates with an error message if the integral cannot be done -in closed terms. Default is off. - - Use the FAILHARD switch when you are dealing with complicated -integrals and want to know immediately if REDUCE was unable to handle -them. The integration operator sometimes returns a formal integration -form that is more complicated than the original expression, when it is -unable to complete the integration. - - -File: redhelp, Node: FORT, Next: FORTUPPER, Prev: FAILHARD, Up: General Switches section - - FORT switch - - When FORT is on, output is given Fortran-compatible syntax. Default -is OFF . - -examples: - - ____________________________________________________________ - - on fort; - - df(sin(7*x + y),x); - - ANS=7.*COS(7*X+Y) - - - on rounded; - - b := log(sin(pi/5 + n*pi)); - - B=LOG(SIN(3.14159265359*N+0.628318530718)) - - ____________________________________________________________ - REDUCE results can be written to a file (using [*note OUT::.] ) and -used as data by Fortran programs when FORT is in effect. FORT knows -about correct statement length, continuation characters, defining a -symbol when it is first used, and other Fortran details. - - The [*note GENTRAN::.] package offers many more possibilities than -the FORT switch. It produces Fortran (or C or Ratfor) code from REDUCE -procedures or structured specifications, including facilities for -producing double precision output. - - -File: redhelp, Node: FORTUPPER, Next: FULLPREC, Prev: FORT, Up: General Switches section - - FORTUPPER switch - - When FORTUPPER is on, any Fortran-style output appears in upper case. -Default is OFF . - -examples: - - ____________________________________________________________ - - on fort; - - df(sin(7*x + y),x); - - ans=7.*cos(7*x+y) - - - on fortupper; - - df(sin(7*x + y),x); - - ANS=7.*COS(7*X+Y) - - ____________________________________________________________ - - -File: redhelp, Node: FULLPREC, Next: FULLROOTS, Prev: FORTUPPER, Up: General Switches section - - FULLPREC switch - - Trailing zeroes of rounded numbers to the full system precision are -normally not printed. If this information is needed, for example to get -a more understandable indication of the accuracy of certain data, the -switch FULLPREC can be turned on. - -examples: - - ____________________________________________________________ - - on rounded; - - 1/2; - - 0.5 - - - on fullprec; - - ws; - - 0.500000000000 - - ____________________________________________________________ - This is just an output options which neither influences the accuracy -of the computation nor does it give additional information about the -precision of the results. See also [*note SCIENTIFIC_NOTATION::.] . - - -File: redhelp, Node: FULLROOTS, Next: GC, Prev: FULLPREC, Up: General Switches section - - FULLROOTS switch - - Since roots of cubic and quartic polynomials can often be very -messy, a switch FULLROOTS controls the production of results in closed -form. [*note SOLVE::.] will apply the formulas for explicit forms for -degrees 3 and 4 only if FULLROOTS is ON . Otherwise the result forms -are built using [*note ROOT_OF::.] . Default is OFF . - - -File: redhelp, Node: GC, Next: GCD switch, Prev: FULLROOTS, Up: General Switches section - - GC switch - - With the GC switch, you can turn the garbage collection messages on -or off. The form of the message depends on the particular Lisp used for -the REDUCE implementation. - - See [*note RECLAIM::.] for an explanation of garbage collection. -REDUCE does garbage collection when needed even if you have turned the -notices off. - - -File: redhelp, Node: GCD switch, Next: HORNER, Prev: GC, Up: General Switches section - - GCD switch - - When GCD is on, common factors in numerators and denominators of -expressions are canceled. Default is OFF . - -examples: - - ____________________________________________________________ - - - (2*(f*h)**2 - f**2*g*h - (f*g)**2 - f*h**3 + f*h*g**2 - - h**4 + g*h**3)/(f**2*h - f**2*g - f*h**2 + 2*f*g*h - - f*g**2 - g*h**2 + g**2*h); - - - 2 2 2 2 2 2 3 3 4 - F *G + F *G*H - 2*F *H - F*G *H + F*H - G*H + H - ---------------------------------------------------- - 2 2 2 2 2 2 - F *G - F *H + F*G - 2*F*G*H + F*H - G *H + G*H - - - on gcd; - - ws; - - 2 - F*G + 2*F*H + H - ---------------- - F + G - - - e2 := a*c + a*d + b*c + b*d; - - E2 := A*C + A*D + B*C + B*D - - - off exp; - - e2; - - (A + B)*(C + D) - - ____________________________________________________________ - Even with GCD off, a check is automatically made for common variable -and numerical products in the numerators and denominators of expression, -and the appropriate cancellations made. Thus the example demonstrating -the use of GCD is somewhat complicated. Note when [*note EXP::.] is off, -GCD has the side effect of factoring the expression. - - -File: redhelp, Node: HORNER, Next: IFACTOR, Prev: GCD switch, Up: General Switches section - - HORNER switch - - When the HORNER switch is on, polynomial expressions are printed in -Horner's form for faster and safer numerical evaluation. Default is OFF -. The leading variable of the expression is selected as Horner -variable. To select the Horner variable explicitly use the [*note -KORDER::.] declaration. - -examples: - - ____________________________________________________________ - - on horner; - - (13p-4q)^3; - - 3 2 - ( - 64)*q + p*(624*q + p*(( - 2028)*q + p*2197)) - - - korder q; - - ws; - - 3 2 - 2197*p + q*(( - 2028)*p + q*(624*p + q*(-64))) - - ____________________________________________________________ - - -File: redhelp, Node: IFACTOR, Next: INT switch, Prev: HORNER, Up: General Switches section - - IFACTOR switch - - When the IFACTOR switch is on, any integer terms appearing as a -result of the [*note FACTORIZE::.] command are factored themselves into -primes. Default is OFF . If the argument of FACTORIZE is an integer, -IFACTOR has no effect, since the integer is always factored. - -examples: - - ____________________________________________________________ - - factorize(4*x**2 + 28*x + 48); - - {4,X + 3,X + 4} - - - factorize(22587); - - {3,7529} - - - on ifactor; - - factorize(4*x**2 + 28*x + 48); - - {2,2,X + 4,X + 3} - - - factorize(22587); - - {3,7529} - - ____________________________________________________________ - Constant terms that appear within nonconstant polynomial factors are -not factored. - - The IFACTOR switch affects only factoring done specifically with -[*note FACTORIZE::.] , not on factoring done automatically when the -[*note FACTOR::.] switch is on. - - -File: redhelp, Node: INT switch, Next: INTSTR, Prev: IFACTOR, Up: General Switches section - - INT switch - - The INT switch specifies an interactive mode of operation. Default -ON . - - There is no reason to turn INT off during interactive calculations, -since there are no benefits to be gained. If you do have INT off while -inputting a file, and REDUCE finds an error, it prints the message -"Continuing with parsing only." In this state, REDUCE accepts only -[*note END::.] ; or [*note BYE::.] ; from the keyboard; everything -else is ignored, even the command ON INT . - - -File: redhelp, Node: INTSTR, Next: LCM, Prev: INT switch, Up: General Switches section - - INTSTR switch - - If INTSTR (for "internal structure") is on, arguments of an operator -are printed in a more structured form. - -examples: - - ____________________________________________________________ - - operator f; - - f(2x+2y); - - F(2*X + 2*Y) - - - on intstr; - - ws; - - F(2*(X + Y)) - - ____________________________________________________________ - - -File: redhelp, Node: LCM, Next: LESSSPACE, Prev: INTSTR, Up: General Switches section - - LCM switch - - The LCM switch instructs REDUCE to compute the least common multiple -of denominators whenever rational expressions occur. Default is ON . - -examples: - - ____________________________________________________________ - - off lcm; - - z := 1/(x**2 - y**2) + 1/(x-y)**2; - - - - 2*X*(X - Y) - Z := ------------------------- - 4 3 3 4 - X - 2*X *Y + 2*X*Y - Y - - - on lcm; - - z; - - 2*X*(X - Y) - ------------------------- - 4 3 3 4 - X - 2*X *Y + 2*X*Y - Y - - - zz := 1/(x**2 - y**2) + 1/(x-y)**2; - - - - 2*X - ZZ := --------------------- - 3 2 2 3 - X - X *Y - X*Y + Y - - - on gcd; - - z; - - 2*X - ---------------------- - 3 2 2 3 - X - X *Y - X*Y + Y - - ____________________________________________________________ - Note that LCM has effect only when rational expressions are first -combined. It does not examine existing structures for simplifications on -display. That is shown above when z is entered with LCM off. It -remains unsimplified even after LCM is turned back on. However, a new -variable containing the same expression is simplified on entry. The -switch [*note GCD::.] does examine existing structures, as shown in the -last example line above. - - Full greatest common divisor calculations become expensive if work -with large rational expressions is required. A considerable savings of -time can be had if a full gcd check is made only when denominators are -combined, and only a partial check for numerators. This is the effect -of the LCM switch. - - -File: redhelp, Node: LESSSPACE, Next: LIMITEDFACTORS, Prev: LCM, Up: General Switches section - - LESSSPACE switch - - You can turn on the switch LESSSPACE if you want fewer blank lines -in your output. - - -File: redhelp, Node: LIMITEDFACTORS, Next: LIST switch, Prev: LESSSPACE, Up: General Switches section - - LIMITEDFACTORS switch - - To get limited factorization in cases where it is too expensive to -use full multivariate polynomial factorization, the switch -LIMITEDFACTORS can be turned on. In that case, only "inexpensive" -factoring operations, such as square-free factorization, will be used -when [*note FACTORIZE::.] is called. - -examples: - - ____________________________________________________________ - - a := (y-x)^2*(y^3+2x*y+5)*(y^2-3x*y+7)$ - - factorize a; - - { - X + Y, - X - Y, - 3 - 2*X*Y + Y + 5, - 2 - 3*X*Y - Y - 7} - - - on limitedfactors; - - factorize a; - - { - X + Y, - X - Y, - 2 2 4 3 5 3 2 - 6*X *Y + 3*X*Y - 2*X*Y + X*Y - Y - 7*Y - 5*Y - 35} - - ____________________________________________________________ - - -File: redhelp, Node: LIST switch, Next: LISTARGS, Prev: LIMITEDFACTORS, Up: General Switches section - - LIST switch - - The LIST switch causes REDUCE to print each term in any sum on -separate lines. - -examples: - - ____________________________________________________________ - - x**2*(y**2 + 2*y) + x*(y**2 + z)/(2*a); - - - - 2 2 - X*(2*A*X*Y + 4*A*X*Y + Y +Z) - ------------------------------ - 2*A - - - on list; - - ws; - - 2 - (X*(2*A*X*Y - + 4*A*X*Y - 2 - + Y - + Z))/(2*A) - - ____________________________________________________________ - - -File: redhelp, Node: LISTARGS, Next: MCD, Prev: LIST switch, Up: General Switches section - - LISTARGS switch - - If an operator other than those specifically defined for lists is -given a single argument that is a list, then the result of this -operation will be a list in which that operator is applied to each -element of the list. This process can be inhibited globally by turning -on the switch LISTARGS . - -examples: - - ____________________________________________________________ - - log {a,b,c}; - - LOG(A),LOG(B),LOG(C) - - - on listargs; - - log {a,b,c}; - - LOG(A,B,C) - - ____________________________________________________________ - It is possible to inhibit such distribution for a specific operator -by using the declaration [*note LISTARGP::.] . In addition, if an -operator has more than one argument, no such distribution occurs, so -LISTARGS has no effect. - - -File: redhelp, Node: MCD, Next: MODULAR, Prev: LISTARGS, Up: General Switches section - - MCD switch - - When MCD is on, sums and differences of rational expressions are put -on a common denominator. Default is ON . - -examples: - - ____________________________________________________________ - - a/(x+1) + b/5; - - 5*A + B*X + B - ------------- - 5*(X + 1) - - - off mcd; - - a/(x+1) + b/5; - - -1 - (X + 1) *A + 1/5*B - - - 1/6 + 1/7; - - 13/42 - - ____________________________________________________________ - Even with MCD off, rational expressions involving only numbers are -still put over a common denominator. - - Turning MCD off is useful when explicit negative powers are needed, -or if no greatest common divisor calculations are desired, or when -differentiating complicated rational expressions. Results when MCD is -off are no longer in canonical form, and expressions equivalent to zero -may not simplify to 0. Some operations, such as factoring cannot be done -while MCD is off. This option should therefore be used with some -caution. Turning MCD off is most valuable in intermediate parts of a -complicated calculation, and should be turned back on for the last -stage. - - -File: redhelp, Node: MODULAR, Next: MSG, Prev: MCD, Up: General Switches section - - MODULAR switch - - When MODULAR is on, polynomial coefficients are reduced by the -modulus set by [*note SETMOD::.] . If no modulus has been set, MODULAR -has no effect. - -examples: - - ____________________________________________________________ - - setmod 2; - - 1 - - - on modular; - - (x+y)**2; - - 2 2 - X + Y - - - 145*x**2 + 20*x**3 + 17 + 15*x*y; - - - - 2 - X + X*Y + 1 - - ____________________________________________________________ - Modular operations are only conducted on the coefficients, not the -exponents. The modulus is not restricted to being prime. When the -modulus is prime, division by a number not relatively prime to the -modulus results in a error message. When the modulus is -a composite number, division by a power of the modulus results in an -error message, but division by an integer which is a factor of the -modulus does not. The representation of modular number can be -influenced by [*note BALANCED_MOD::.] . - - -File: redhelp, Node: MSG, Next: MULTIPLICITIES, Prev: MODULAR, Up: General Switches section - - MSG switch - - When MSG is off, the printing of warning messages is suppressed. -Error messages are still printed. - - Warning messages include those about redimensioning an [*note -ARRAY::.] or declaring an [*note OPERATOR::.] where one is expected. - - -File: redhelp, Node: MULTIPLICITIES, Next: NAT, Prev: MSG, Up: General Switches section - - MULTIPLICITIES switch - - When [*note SOLVE::.] is applied to a set of equations with multiple -roots, solution multiplicities are normally stored in the global -variable [*note ROOT_MULTIPLICITIES::.] rather than the solution list. -If you want the multiplicities explicitly displayed, the switch -MULTIPLICITIES should be turned on. In this case, ROOT_MULTIPLICITIES -has no value. - -examples: - - ____________________________________________________________ - - solve(x^2=2x-1,x); - - X=1 - - - root_multiplicities; - - 2 - - - on multiplicities; - - solve(x^2=2x-1,x); - - X=1,X=1 - - - root_multiplicities; - - ____________________________________________________________ - - -File: redhelp, Node: NAT, Next: NERO, Prev: MULTIPLICITIES, Up: General Switches section - - NAT switch - - When NAT is on, output is printed to the screen in natural form, with -raised exponents. NAT should be turned off when outputting expressions -to a file for future input. Default is ON . - -examples: - - ____________________________________________________________ - - (x + y)**3; - - 3 2 2 3 - X + 3*X *Y + 3*X*Y + Y - - - off nat; - - (x + y)**3; - - X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3$ - - - on fort; - - (x + y)**3; - - ANS=X**3+3.*X**2*Y+3.*X*Y**2+Y**3 - - ____________________________________________________________ - With NAT off, a dollar sign is printed at the end of each expression. -An output file written with NAT off is ready to be read into REDUCE -using the command [*note IN::.] . - - -File: redhelp, Node: NERO, Next: NOARG, Prev: NAT, Up: General Switches section - - NERO switch - - When NERO is on, zero assignments (such as matrix elements) are not -printed. - -examples: - - ____________________________________________________________ - - matrix a; - a := mat((1,0),(0,1)); - - A(1,1) := 1 - A(1,2) := 0 - A(2,1) := 0 - A(2,2) := 1 - - - on nero; - - a; - - MAT(1,1) := 1 - MAT(2,2) := 1 - - - a(1,2); - ____________________________________________________________ - nothing is printed. - ____________________________________________________________ - - - - b := 0; - ____________________________________________________________ - nothing is printed. - ____________________________________________________________ - - - - off nero; - - b := 0; - - B := 0 - - ____________________________________________________________ - - NERO is often used when dealing with large sparse matrices, to avoid -being overloaded with zero assignments. - - -File: redhelp, Node: NOARG, Next: NOLNR, Prev: NERO, Up: General Switches section - - NOARG switch - - When [*note DFPRINT::.] is on, expressions in the differentiation -operator [*note DF::.] are printed in a more "natural" notation, with -the differentiation variables appearing as subscripts. When NOARG is on -(the default), the arguments of the differentiated operator are also -suppressed. - -examples: - - ____________________________________________________________ - - operator f; - - df(f x,x); - - DF(F(X),X); - - - on dfprint; - - ws; - - F - X - - - off noarg; - - ws; - - F(X) - X - - ____________________________________________________________ - - -File: redhelp, Node: NOLNR, Next: NOSPLIT, Prev: NOARG, Up: General Switches section - - NOLNR switch - - When NOLNR is on, the linear properties of the integration operator -[*note INT::.] are suppressed if the integral cannot be found in -closed terms. - - REDUCE uses the linear properties of integration to attempt to break -down an integral into manageable pieces. If an integral cannot be found -in closed terms, these pieces are returned. When the NOLNR switch is -off, as many of the pieces as possible are integrated. When it is on, -if any piece fails, the rest of them remain unevaluated. - - -File: redhelp, Node: NOSPLIT, Next: NUMVAL, Prev: NOLNR, Up: General Switches section - - NOSPLIT switch - - Under normal circumstances, the printing routines try to break an -expression across lines at a natural point. This is a fairly expensive -process. If you are not overly concerned about where the end-of-line -breaks come, you can speed up the printing of expressions by turning -off the switch NOSPLIT . This switch is normally on. - - -File: redhelp, Node: NUMVAL, Next: OUTPUT, Prev: NOSPLIT, Up: General Switches section - - NUMVAL switch - - With [*note ROUNDED::.] on, elementary functions with numerical -arguments will return a numerical answer where appropriate. If you wish -to inhibit this evaluation, NUMVAL should be turned off. It is normally -on. - -examples: - - ____________________________________________________________ - - on rounded; - - cos 3.4; - - - 0.966798192579 - - - off numval; - - cos 3.4; - - COS(3.4) - - ____________________________________________________________ - - -File: redhelp, Node: OUTPUT, Next: OVERVIEW, Prev: NUMVAL, Up: General Switches section - - OUTPUT switch - - When OUTPUT is off, no output is printed from any REDUCE calculation. -The calculations have their usual effects other than printing. Default -is ON . - - Turn output OFF if you do not wish to see output when executing -large files, or to save the time REDUCE spends formatting large -expressions for display. Results are still available with [*note WS::.] -, or in their assigned variables. - - -File: redhelp, Node: OVERVIEW, Next: PERIOD, Prev: OUTPUT, Up: General Switches section - - OVERVIEW switch - - When OVERVIEW is on, the amount of detail reported by the factorizer -switches [*note TRFAC::.] and [*note TRALLFAC::.] is reduced. - - -File: redhelp, Node: PERIOD, Next: PRECISE, Prev: OVERVIEW, Up: General Switches section - - PERIOD switch - - When PERIOD is on, periods are added after integers in -Fortran-compatible output (when [*note FORT::.] is on). There is no -effect when FORT is off. Default is ON . - - -File: redhelp, Node: PRECISE, Next: PRET, Prev: PERIOD, Up: General Switches section - - PRECISE switch - - When the PRECISE switch is on, simplification of roots of even -powers returns absolute values, a more precise answer mathematically. -Default is ON . - -examples: - - ____________________________________________________________ - - sqrt(x**2); - - X - - - (x**2)**(1/4); - - SQRT(X) - - - on precise; - - sqrt(x**2); - - ABS(X) - - - (x**2)**(1/4); - - SQRT(ABS(X)) - - ____________________________________________________________ - In many types of mathematical work, simplification of powers and -surds can proceed by the fastest means of simplifying the exponents -arithmetically. When it is important to you that the positive root be -returned, turn PRECISE on. One situation where this is important is -when graphing square-root expressions such as sqrt(x^2+y^2) to avoid a -spike caused by REDUCE simplifying sqrt(y^2) to y when x is zero. - - -File: redhelp, Node: PRET, Next: PRI, Prev: PRECISE, Up: General Switches section - - PRET switch - - When PRET is on, input is printed in standard REDUCE format and then -evaluated. - -examples: - - ____________________________________________________________ - - on pret; - - (x+1)^3; - - (x + 1)**3; - 3 2 - X + 3*X + 3*X + 1 - - - - procedure fac(n); - if not (fixp(n) and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n-1 product i+1; - - - procedure fac n; - if not (fixp n and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n - 1 product i + 1; - FAC - - - - fac 5; - - fac 5; - 120 - - ____________________________________________________________ - Note that all input is converted to lower case except strings (which -keep the same case) all operators with a single argument have had the -parentheses removed, and all infix operators have had a space added on -each side. In addition, syntactical constructs like IF ...THEN ...ELSE -are printed in a standard format. - - -File: redhelp, Node: PRI, Next: RAISE, Prev: PRET, Up: General Switches section - - PRI switch - - When PRI is on, the declarations [*note ORDER::.] and [*note -FACTOR::.] can be used, and the switches [*note ALLFAC::.] , [*note -DIV::.] , [*note RAT::.] , and [*note REVPRI::.] take effect when they -are on. Default is ON . - - Printing of expressions is faster with PRI off. The expressions are -then returned in one standard form, without any of the display options -that can be used to feature or display various parts of the expression. -You can also gain insight into REDUCE's representation of expressions -with PRI off. - - -File: redhelp, Node: RAISE, Next: RAT, Prev: PRI, Up: General Switches section - - RAISE switch - - When RAISE is on, lower case letters are automatically converted to -upper case on input. RAISE is normally on. - - This conversion affects the internal representation of the letter, -and is independent of the case with which a letter is printed, which is -normally lower case. - - -File: redhelp, Node: RAT, Next: RATARG, Prev: RAISE, Up: General Switches section - - RAT switch - - When the RAT switch is on, and kernels have been selected to display -with the [*note FACTOR::.] declaration, the denominator is printed with -each term rather than one common denominator at the end of an -expression. - -examples: - - ____________________________________________________________ - - (x+1)/x + x**2/sin y; - - - 3 - SIN(Y)*X + SIN(Y) + X - ---------------------- factor x; - SIN(Y)*X - - - (x+1)/x + x**2/sin y; - - - 3 - X + X*SIN(Y) + SIN(Y) - ---------------------- on rat; - X*SIN(Y) - - - (x+1)/x + x**2/sin y; - - - 2 - X -1 - ------ + 1 + X - SIN(Y) - - ____________________________________________________________ - The RAT switch only has effect when the [*note PRI::.] switch is on. -When PRI is off, regardless of the setting of RAT , the printing -behavior is as if RAT were off. RAT only has effect upon the display of -expressions, not their internal form. - - -File: redhelp, Node: RATARG, Next: RATIONAL, Prev: RAT, Up: General Switches section - - RATARG switch - - When RATARG is on, rational expressions can be given to operators -such as [*note COEFF::.] and [*note LTERM::.] that normally require -polynomials in one of their arguments. When RATARG is off, rational -expressions cause an error message. - -examples: - - ____________________________________________________________ - - aa := x/y**2 + 1/x + y/x**2; - - - 3 2 3 - X + X*Y + Y - AA := -------------- - 2 2 - X *Y - - - coeff(aa,x); - - 3 2 3 - X + X*Y + Y - ***** -------------- invalid as POLYNOMIAL - 2 2 - X *Y - - - on ratarg; - - coeff(aa,x); - - - Y 1 1 - {--,--,0,-----} - 2 2 2 2 - X X X *Y - - ____________________________________________________________ - - -File: redhelp, Node: RATIONAL, Next: RATIONALIZE, Prev: RATARG, Up: General Switches section - - RATIONAL switch - - When RATIONAL is on, polynomial expressions with rational -coefficients are produced. - -examples: - - ____________________________________________________________ - - x/2 + 3*y/4; - - 2*X + 3*Y - --------- - 4 - - - (x**2 + 5*x + 17)/2; - - 2 - X + 5*X + 17 - ------------- - 2 - - - on rational; - - x/2 + 3y/4; - - 1 3 - -*(X + -*Y) - 2 2 - - - (x**2 + 5*x + 17)/2; - - 1 2 - -*(X + 5*X + 17) - 2 - - ____________________________________________________________ - By using RATIONAL , polynomial expressions with rational -coefficients can be used in some commands that expect polynomials. With -RATIONAL off, such a polynomial becomes a rational expression, with -denominator the least common multiple of the denominators of the -rational number coefficients. - - -File: redhelp, Node: RATIONALIZE, Next: RATPRI, Prev: RATIONAL, Up: General Switches section - - RATIONALIZE switch - - When the RATIONALIZE switch is on, denominators of rational -expressions that contain complex numbers or root expressions are -simplified by multiplication by their conjugates. - -examples: - - ____________________________________________________________ - - qq := (1+sqrt(3))/(sqrt(3)-7); - - SQRT(3) + 1 - QQ := ----------- - SQRT(3) - 7 - - - on rationalize; - - qq; - - - 4*SQRT(3) - 5 - --------------- - 23 - - - 2/(4 + 6**(1/3)); - - 2/3 1/3 - 6 - 4*6 + 16 - ------------------ - 35 - - - (i-1)/(i+3); - - 2*I - 1 - ------- - 5 - - - off rationalize; - - (i-1)/(i+3); - - I - 1 - ------ - I + 3 - - ____________________________________________________________ - - -File: redhelp, Node: RATPRI, Next: REVPRI, Prev: RATIONALIZE, Up: General Switches section - - RATPRI switch - - When the RATPRI switch is on, rational expressions and fractions are -printed as two lines separated by a fraction bar, rather than in a -linear style. Default is ON . - -examples: - - ____________________________________________________________ - - 3/17; - - 3 - -- - 17 - - - 2/b + 3/y; - - 3*B + 2*Y - --------- - B*Y - - - off ratpri; - - 3/17; - - 3/17 - - - 2/b + 3/y; - - (3*B + 2*Y)/(B*Y) - - ____________________________________________________________ - - -File: redhelp, Node: REVPRI, Next: RLISP88, Prev: RATPRI, Up: General Switches section - - REVPRI switch - - When the REVPRI switch is on, terms are printed in reverse order from -the normal printing order. - -examples: - - ____________________________________________________________ - - x**5 + x**2 + 18 + sqrt(y); - - 5 2 - SQRT(Y) + X + X + 18 - - - a + b + c + w; - - A + B + C + W - - - on revpri; - - x**5 + x**2 + 18 + sqrt(y); - - 2 5 - 17 + X + X + SQRT(Y) - - - a + b + c + w; - - W + C + B + A - - ____________________________________________________________ - Turn REVPRI on when you want to display a polynomial in ascending -rather than descending order. - - -File: redhelp, Node: RLISP88, Next: ROUNDALL, Prev: REVPRI, Up: General Switches section - - RLISP88 switch - - Rlisp '88 is a superset of the Rlisp that has been traditionally -used for the support of REDUCE. It is fully documented in the book -Marti, J.B., "RLISP '88: An Evolutionary Approach to Program Design and -Reuse", World Scientific, Singapore (1993). It supports different -looping constructs from the traditional Rlisp, and treats "-" as a -letter unless separated by spaces. Turning on the switch RLISP88 -converts to Rlisp '88 parsing conventions in symbolic mode, and enables -the use of Rlisp '88 extensions. Turning off the switch reverts to the -traditional Rlisp and the previous mode ( ([*note SYMBOLIC::.] or -[*note ALGEBRAIC::.] ) in force before RLISP88 was turned on. - - -File: redhelp, Node: ROUNDALL, Next: ROUNDBF, Prev: RLISP88, Up: General Switches section - - ROUNDALL switch - - In [*note ROUNDED::.] mode, rational numbers are normally converted -to a floating point representation. If ROUNDALL is off, this conversion -does not occur. ROUNDALL is normally ON . - -examples: - - ____________________________________________________________ - - on rounded; - - 1/2; - - 0.5 - - - off roundall; - ____________________________________________________________ - - -File: redhelp, Node: ROUNDBF, Next: ROUNDED, Prev: ROUNDALL, Up: General Switches section - - ROUNDBF switch - - When [*note ROUNDED::.] is on, the normal defaults cause underflows -to be converted to zero. If you really want the small number that -results in such cases, ROUNDBF can be turned on. - -examples: - - ____________________________________________________________ - - on rounded; - - exp(-100000.1^2); - - 0 - - - on roundbf; - - exp(-100000.1^2); - - 1.18441281937E-4342953505 - - ____________________________________________________________ - If a polynomial is input in [*note ROUNDED::.] mode at the default -precision into any [*note ROOTS::.] function, and it is not possible to -represent any of the coefficients of the polynomial precisely in the -system floating point representation, the switch ROUNDBF will be -automatically turned on. All rounded computation will use the internal -bigfloat representation until the user subsequently turns ROUNDBF off. -(A message is output to indicate that this condition is in effect.) - - -File: redhelp, Node: ROUNDED, Next: SAVESTRUCTR, Prev: ROUNDBF, Up: General Switches section - - ROUNDED switch - - When ROUNDED is on, floating-point arithmetic is enabled, with -precision initially at a system default value, which is usually 12 -digits. The precise number can be found by the command [*note -PRECISION::.] (0). - -examples: - - ____________________________________________________________ - - pi; - - PI - - - 35/217; - - 5 - -- - 31 - - - on rounded; - - pi; - - 3.14159265359 - - - 35/217; - - 0.161 - - - sqrt(3); - - 1.73205080756 - - ____________________________________________________________ - - If more than the default number of decimal places are required, use -the [*note PRECISION::.] command to set the required number. - - -File: redhelp, Node: SAVESTRUCTR, Next: SOLVESINGULAR, Prev: ROUNDED, Up: General Switches section - - SAVESTRUCTR switch - - When SAVESTRUCTR is on, results of the [*note STRUCTR::.] command are -returned as a list whose first element is the representation for the -expression and the remaining elements are equations showing the -relationships of the generated variables. - -examples: - - ____________________________________________________________ - - off exp; - - structr((x+y)^3 + sin(x)^2); - - ANS3 - where - 3 2 - ANS3 := ANS1 + ANS2 - ANS2 := SIN(X) - ANS1 := X + Y - - - ans3; - - ANS3 - - - on savestructr; - - structr((x+y)^{3} + sin(x)^{2}); - - 3 2 - ANS3,ANS3=ANS1 + ANS2 ,ANS2=SIN(X),ANS1=X + Y - - - ans3 where rest ws; - - 3 2 - (X + Y) + SIN(X) - - ____________________________________________________________ - In normal operation, [*note STRUCTR::.] is only a display command. -With SAVESTRUCTR on, you can access the various parts of the expression -produced by STRUCTR . - - The generic system names use the stem ANS . You can change this to -your own stem by the command [*note VARNAME::.] . REDUCE adds integers -to this stem to make unique identifiers. - - -File: redhelp, Node: SOLVESINGULAR, Next: TIME, Prev: SAVESTRUCTR, Up: General Switches section - - SOLVESINGULAR switch - - When SOLVESINGULAR is on, singular or underdetermined systems of -linear equations are solved, using arbitrary real, complex or integer -variables in the answer. Default is ON . - -examples: - - ____________________________________________________________ - - solve({2x + y,4x + 2y},{x,y}); - - ARBCOMPLEX(1) - {{X= - -------------,Y=ARBCOMPLEX(1)}} - 2 - - - solve({7x + 15y - z,x - y - z},{x,y,z}); - - - 8*ARBCOMPLEX(3) - {{X=---------------- - 11 - 3*ARBCOMPLEX(3) - Y= - ---------------- - 11 - Z=ARBCOMPLEX(3)}} - - - off solvesingular; - - solve({2x + y,4x + 2y},{x,y}); - - ***** SOLVE given singular equations - - - solve({7x + 15y - z,x - y - z},{x,y,z}); - - - ***** SOLVE given singular equations - - ____________________________________________________________ - The integer following the identifier [*note ARBCOMPLEX::.] above is -assigned by the system, and serves to identify the variable uniquely. -It has no other significance. - - -File: redhelp, Node: TIME, Next: TRALLFAC, Prev: SOLVESINGULAR, Up: General Switches section - - TIME switch - - When TIME is on, the system time used in executing each REDUCE -statement is printed after the answer is printed. - -examples: - - ____________________________________________________________ - - on time; - - Time: 4940 ms - - - df(sin(x**2 + y),y); - - 2 - COS(X + Y ) - Time: 180 ms - - - solve(x**2 - 6*y,x); - - {X= - SQRT(Y)*SQRT(6), - X=SQRT(Y)*SQRT(6)} - Time: 320 ms - - ____________________________________________________________ - When TIME is first turned on, the time since the beginning of the -REDUCE session is printed. After that, the time used in computation, -(usually in milliseconds, though this is system dependent) is printed -after the results of each command. Idle time or time spent typing in -commands is not counted. If TIME is turned off, the first reading after -it is turned on again gives the time elapsed since it was turned off. -The time printed is CPU or wall clock time, depending on the system. - - -File: redhelp, Node: TRALLFAC, Next: TRFAC, Prev: TIME, Up: General Switches section - - TRALLFAC switch - - When TRALLFAC is on, a more detailed trace of factorizer calls is -generated. - - The TRALLFAC switch takes precedence over [*note TRFAC::.] if they -are both on. TRFAC gives a factorization trace with less detail in it. -When the [*note FACTOR::.] switch is on also, all input polynomials are -sent to the factorizer automatically and trace information is -generated. The [*note OUT::.] command saves the results of the -factoring, but not the trace. - - -File: redhelp, Node: TRFAC, Next: TRIGFORM, Prev: TRALLFAC, Up: General Switches section - - TRFAC switch - - When TRFAC is on, a narrative trace of any calls to the factorizer is -generated. Default is OFF . - - When the switch [*note FACTOR::.] is on, and TRFAC is on, every input -polynomial is sent to the factorizer, and a trace generated. With -FACTOR off, only polynomials that are explicitly factored with the -command [*note FACTORIZE::.] generate trace information. - - The [*note OUT::.] command saves the results of the factoring, but -not the trace. The [*note TRALLFAC::.] switch gives trace information -to a greater level of detail. - - -File: redhelp, Node: TRIGFORM, Next: TRINT, Prev: TRFAC, Up: General Switches section - - TRIGFORM switch - - When [*note FULLROOTS::.] is on, [*note SOLVE::.] will compute the -roots of a cubic or quartic polynomial is closed form. When TRIGFORM -is on, the roots will be expressed by trigonometric forms. Otherwise -nested surds are used. Default is ON . - - -File: redhelp, Node: TRINT, Next: TRNONLNR, Prev: TRIGFORM, Up: General Switches section - - TRINT switch - - When TRINT is on, a narrative tracing various steps in the -integration process is produced. - - The [*note OUT::.] command saves the results of the integration, but -not the trace. - - -File: redhelp, Node: TRNONLNR, Next: VAROPT, Prev: TRINT, Up: General Switches section - - TRNONLNR switch - - When TRNONLNR is on, a narrative tracing various steps in the -process for solving non-linear equations is produced. - - TRNONLNR can only be used after the solve package has been loaded -(e.g., by an explicit call of [*note LOAD_PACKAGE::.] ). The [*note -OUT::.] command saves the results of the equation solving, but not the -trace. - - -File: redhelp, Node: VAROPT, Prev: TRNONLNR, Up: General Switches section - - VAROPT switch - - When VAROPT is on, the sequence of variables is optimized by [*note -SOLVE::.] with respect to execution speed. Otherwise, the sequence -given in the call to [*note SOLVE::.] is preserved. Default is ON . - - In combination with the switch [*note ARBVARS::.] , VAROPT can be -used to control variable elimination. - -examples: - - ____________________________________________________________ - - off arbvars; - - solve({x+2z,x-3y},{x,y,z}); - - x x - {{y=-,z= - -}} - 3 2 - - - solve({x*y=1,z=x},{x,y,z}); - - 1 - {{z=x,y=-}} - x - - - off varopt; - - solve({x+2z,x-3y},{x,y,z}); - - 2*z - {{x= - 2*z,y= - ---}} - 3 - - - solve({x*y=1,z=x},{x,y,z}); - - 1 - {{y=-,x=z}} - z - - ____________________________________________________________ - - -File: redhelp, Node: General Switches section, Next: Matrix Operations section, Prev: Elementary Functions section, Up: Top - - General Switches section - -* Menu: - -* SWITCHES:: introduction -* ALGINT:: switch -* ALLBRANCH:: switch -* ALLFAC:: switch -* ARBVARS:: switch -* BALANCED_MOD:: switch -* BFSPACE:: switch -* COMBINEEXPT:: switch -* COMBINELOGS:: switch -* COMP:: switch -* COMPLEX:: switch -* CREF:: switch -* CRAMER:: switch -* DEFN:: switch -* DEMO:: switch -* DFPRINT:: switch -* DIV:: switch -* ECHO:: switch -* ERRCONT:: switch -* EVALLHSEQP:: switch -* EXP switch:: switch -* EXPANDLOGS:: switch -* EZGCD:: switch -* FACTOR:: switch -* FAILHARD:: switch -* FORT:: switch -* FORTUPPER:: switch -* FULLPREC:: switch -* FULLROOTS:: switch -* GC:: switch -* GCD switch:: switch -* HORNER:: switch -* IFACTOR:: switch -* INT switch:: switch -* INTSTR:: switch -* LCM:: switch -* LESSSPACE:: switch -* LIMITEDFACTORS:: switch -* LIST switch:: switch -* LISTARGS:: switch -* MCD:: switch -* MODULAR:: switch -* MSG:: switch -* MULTIPLICITIES:: switch -* NAT:: switch -* NERO:: switch -* NOARG:: switch -* NOLNR:: switch -* NOSPLIT:: switch -* NUMVAL:: switch -* OUTPUT:: switch -* OVERVIEW:: switch -* PERIOD:: switch -* PRECISE:: switch -* PRET:: switch -* PRI:: switch -* RAISE:: switch -* RAT:: switch -* RATARG:: switch -* RATIONAL:: switch -* RATIONALIZE:: switch -* RATPRI:: switch -* REVPRI:: switch -* RLISP88:: switch -* ROUNDALL:: switch -* ROUNDBF:: switch -* ROUNDED:: switch -* SAVESTRUCTR:: switch -* SOLVESINGULAR:: switch -* TIME:: switch -* TRALLFAC:: switch -* TRFAC:: switch -* TRIGFORM:: switch -* TRINT:: switch -* TRNONLNR:: switch -* VAROPT:: switch - - -File: redhelp, Node: COFACTOR, Next: DET, Up: Matrix Operations section - - COFACTOR operator - - The operator COFACTOR returns the cofactor of the element in row - and column of a [*note MATRIX::.] . Errors occur if - or do not evaluate to integer expressions or if the -matrix is not square. - -syntax: - - COFACTOR (,,) - -examples: - - ____________________________________________________________ - - cofactor(mat((a,b,c),(d,e,f),(p,q,r)),2,2); - - - A*R - C*P - - - cofactor(mat((a,b,c),(d,e,f)),1,1); - - - ***** non-square matrix - - ____________________________________________________________ - - -File: redhelp, Node: DET, Next: MAT, Prev: COFACTOR, Up: Matrix Operations section - - DET operator - - The DET operator returns the determinant of its (square [*note -MATRIX::.] ) argument. - -syntax: - - DET () or DET - - must evaluate to a square matrix. - -examples: - - ____________________________________________________________ - - - matrix m,n; - - - m := mat((a,b),(c,d)); - - M(1,1) := A - M(1,2) := B - M(2,1) := C - M(2,2) := D - - - - det m; - - A*D - B*C - - - n := mat((1,2),(1,2)); - - N(1,1) := 1 - N(1,2) := 2 - N(2,1) := 1 - N(2,2) := 2 - - - - - det(n); - - 0 - - - - det(5); - - 5 - - ____________________________________________________________ - Given a numerical argument, DET returns the number. However, given a -variable name that has not been declared of type matrix, or a non-square -matrix, DET returns an error message. - - -File: redhelp, Node: MAT, Next: MATEIGEN, Prev: DET, Up: Matrix Operations section - - MAT operator - - The MAT operator is used to represent a two-dimensional [*note -MATRIX::.] . - -syntax: - - MAT ((,*) (, *)*) - - may be any valid REDUCE scalar expression. - -examples: - - ____________________________________________________________ - - mat((1,2),(3,4)); - - MAT(1,1) := 1 - MAT(2,3) := 2 - MAT(2,1) := 3 - MAT(2,2) := 4 - - - mat(2,1); - - ***** Matrix mismatch - Cont? (Y or N) - - - matrix qt; - - qt := ws; - - QT(1,1) := 1 - QT(1,2) := 2 - QT(2,1) := 3 - QT(2,2) := 4 - - - matrix a,b; - - a := mat((x),(y),(z)); - - A(1,1) := X - A(2,1) := Y - A(3,1) := Z - - - b := mat((sin x,cos x,1)); - - B(1,1) := SIN(X) - B(1,2) := COS(X) - B(1,3) := 1 - - ____________________________________________________________ - Matrices need not have a size declared (unlike arrays). MAT -redimensions a matrix variable as needed. It is necessary, of course, -that all rows be the same length. An anonymous matrix, as shown in the -first example, must be named before it can be referenced (note error -message). When using MAT to fill a 1 x n matrix, the row of values must -be inside a second set of parentheses, to eliminate ambiguity. - - -File: redhelp, Node: MATEIGEN, Next: MATRIX, Prev: MAT, Up: Matrix Operations section - - MATEIGEN operator - - The MATEIGEN operator calculates the eigenvalue equation and the -corresponding eigenvectors of a [*note MATRIX::.] . - -syntax: - - MATEIGEN (,) - - must be a declared matrix of values, and must be -a legal REDUCE identifier. - -examples: - - ____________________________________________________________ - - aa := mat((2,5),(1,0))$ - - mateigen(aa,alpha); - - 2 - {{ALPHA - 2*ALPHA - 5, - 1, - 5*ARBCOMPLEX(1) - MAT(1,1) := ---------------, - ALPHA - 2 - @ - MAT(2,1) := ARBCOMPLEX(1) - }} - - charpoly := first first ws; - - 2 - CHARPOLY := ALPHA - 2*ALPHA - 5 - - - bb := mat((1,0,1),(1,1,0),(0,0,1))$ - - mateigen(bb,lamb); - - {{LAMB - 1,3, - [ 0 ] - [ARBCOMPLEX(2)] - [ 0 ] - }} - - ____________________________________________________________ - The MATEIGEN operator returns a list of lists of three elements. The -first element is a square free factor of the characteristic polynomial; -the second element is its multiplicity; and the third element is the -corresponding eigenvector. If the characteristic polynomial can be -completely factored, the product of the first elements of all the -sublists will produce the minimal polynomial. You can access the -various parts of the answer with the usual list access operators. - - If the matrix is degenerate, more than one eigenvector can be -produced for the same eigenvalue, as shown by more than one arbitrary -variable in the eigenvector. The identification numbers of the -arbitrary complex variables shown in the examples above may not be the -same as yours. Note that since LAMBDA is a reserved word in REDUCE, -you cannot use it as a tag-id for this operator. - - -File: redhelp, Node: MATRIX, Next: NULLSPACE, Prev: MATEIGEN, Up: Matrix Operations section - - MATRIX declaration - - Identifiers are declared to be of type MATRIX . - -syntax: - - MATRIX option (,) - - , option (,)* - - must not be an already-defined operator or array or the -name of a scalar variable. Dimensions are optional, and if used appear -inside parentheses. must be a positive integer. - -examples: - - ____________________________________________________________ - - matrix a,b(1,4),c(4,4); - - b(1,1); - - 0 - - - a(1,1); - - ***** Matrix A not set - - - a := mat((x0,y0),(x1,y1)); - - A(1,1) := X0 - A(1,2) := Y0 - A(2,1) := X0 - A(2,2) := X1 - - - length a; - - {2,2} - - - b := a**2; - - 2 - B(1,1) := X0 + X1*Y0 - B(1,2) := Y0*(X0 + Y1) - B(2,1) := X1*(X0 + Y1) - 2 - B(2,2) := X1*Y0 + Y1 - - ____________________________________________________________ - When a matrix variable has not been dimensioned, matrix elements -cannot be referenced until the matrix is set by the [*note MAT::.] -operator. When a matrix is dimensioned in its declaration, matrix -elements are set to 0. Matrix elements cannot stand for themselves. -When you use [*note LET::.] on a matrix element, there is no effect -unless the element contains a constant, in which case an error message -is returned. The same behavior occurs with [*note CLEAR::.] . Do -use [*note CLEAR::.] to try to set a matrix element to 0. [*note -LET::.] statements can be applied to matrices as a whole, if the -right-hand side of the expression is a matrix expression, and the -left-hand side identifier has been declared to be a matrix. - - Arithmetical operators apply to matrices of the correct dimensions. -The operators + and - can be used with matrices of the same dimensions. -The operator * can be used to multiply m x n matrices by n x p -matrices. Matrix multiplication is non-commutative. Scalars can also be -multiplied with matrices, with the result that each element of the -matrix is multiplied by the scalar. The operator / applied to two -matrices computes the first matrix multiplied by the inverse of the -second, if the inverse exists, and produces an error message otherwise. -Matrices can be divided by scalars, which results in dividing each -element of the matrix. Scalars can also be divided by matrices when the -matrices are invertible, and the result is the multiplication of the -scalar by the inverse of the matrix. Matrix inverses can by found by -1/A or /A , where A is a matrix. Square matrices can be raised to -positive integer powers, and also to negative integer powers if they are -nonsingular. - - When a matrix variable is assigned to the results of a calculation, -the matrix is redimensioned if necessary. - - -File: redhelp, Node: NULLSPACE, Next: RANK, Prev: MATRIX, Up: Matrix Operations section - - NULLSPACE operator - -syntax: - - NULLSPACE () - - calculates for its [*note MATRIX::.] argument, A , a -list of linear independent vectors (a basis) whose linear combinations -satisfy the equation a x = 0. The basis is provided in a form such that -as many upper components as possible are isolated. - -examples: - - ____________________________________________________________ - - nullspace mat((1,2,3,4),(5,6,7,8)); - - - { - [ 1 ] - [ ] - [ 0 ] - [ ] - [ - 3] - [ ] - [ 2 ] - , - [ 0 ] - [ ] - [ 1 ] - [ ] - [ - 2] - [ ] - [ 1 ] - } - - ____________________________________________________________ - Note that with B := NULLSPACE A , the expression LENGTH B is the -nullity/ of A, and that SECOND LENGTH A - LENGTH B calculates the rank/ -of A. The rank of a matrix expression can also be found more directly -by the [*note RANK::.] operator. - - In addition to the REDUCE matrix form, NULLSPACE accepts as input a -matrix given as a [*note LIST::.] of lists, that is interpreted as a -row matrix. If that form of input is chosen, the vectors in the result -will be represented by lists as well. This additional input syntax -facilitates the use of NULLSPACE in applications different from -classical linear algebra. - - -File: redhelp, Node: RANK, Next: TP, Prev: NULLSPACE, Up: Matrix Operations section - - RANK operator - -syntax: - - RANK () - - RANK calculates the rank of its matrix argument. - -examples: - - ____________________________________________________________ - - rank mat((a,b,c),(d,e,f)); - - 2 - - ____________________________________________________________ - The argument to RANK can also be a [*note LIST::.] of lists, -interpreted either as a row matrix or a set of equations. If that form -of input is chosen, the vectors in the result will be represented by -lists as well. This additional input syntax facilitates the use of -RANK in applications different from classical linear algebra. - - -File: redhelp, Node: TP, Next: TRACE, Prev: RANK, Up: Matrix Operations section - - TP operator - - The TP operator returns the transpose of its [*note MATRIX::.] -argument. - -syntax: - - TP or TP () - - must be a matrix, which either has had its dimensions -set in its declaration, or has had values put into it by MAT . - -examples: - - ____________________________________________________________ - - matrix m,n; - - m := mat((1,2,3),(4,5,6))$ - - n := tp m; - - N(1,1) := 1 - N(1,2) := 4 - N(2,1) := 2 - N(2,2) := 5 - N(3,1) := 3 - N(3,2) := 6 - - ____________________________________________________________ - In an assignment statement involving TP , the matrix identifier on -the left-hand side is redimensioned to the correct size for the -transpose. - - -File: redhelp, Node: TRACE, Prev: TP, Up: Matrix Operations section - - TRACE operator - - The TRACE operator finds the trace of its [*note MATRIX::.] argument. - -syntax: - - TRACE () or TRACE - - or must evaluate to a square matrix. - -examples: - - ____________________________________________________________ - - matrix a; - - a := mat((x1,y1),(x2,y2))$ - - trace a; - - X1 + Y2 - - ____________________________________________________________ - The trace is the sum of the entries along the diagonal of a square -matrix. Given a non-matrix expression, or a non-square matrix, TRACE -returns an error message. - - -File: redhelp, Node: Matrix Operations section, Next: Groebner package section, Prev: General Switches section, Up: Top - - Matrix Operations section - -* Menu: - -* COFACTOR:: operator -* DET:: operator -* MAT:: operator -* MATEIGEN:: operator -* MATRIX:: declaration -* NULLSPACE:: operator -* RANK:: operator -* TP:: operator -* TRACE:: operator - - -File: redhelp, Node: Groebner bases, Next: Ideal Parameters, Up: Groebner package section - - GROEBNER BASES introduction - - The GROEBNER package calculates GROEBNER BASES using the -BUCHBERGER ALGORITHM and provides related algorithms for arithmetic -with ideal bases, such as ideal quotients, Hilbert polynomials ( -HOLLMANN ALGORITHM ), basis conversion ( FAUGERE-GIANNI-LAZARD-MORA -ALGORITHM ), independent variable set ( KREDEL-WEISPFENNING ALGORITHM ). - - Some routines of the Groebner package are used by [*note SOLVE::.] - -in that context the package is loaded automatically. However, if you -want to use the package by explicit calls you must load it by - ____________________________________________________________ - - load_package groebner; - ____________________________________________________________ - - For the common parameter setting of most operators in this package -see [*note Ideal Parameters::.] . - - -File: redhelp, Node: Ideal Parameters, Next: Term order section, Prev: Groebner bases, Up: Groebner package section - - IDEAL PARAMETERS - - Most operators of the GROEBNER package compute expressions in a -polynomial ring which given as [,,...] where is the -current REDUCE coefficient domain. All algebraically exact domains of -REDUCE are supported. The package can operate over rings and fields. -The operation mode is distinguished automatically. In general the ring -mode is a bit faster than the field mode. The factoring variant can be -applied only over domains which allow you factoring of multivariate -polynomials. - - The variable sequence is either declared explicitly as argument -in form of a [*note LIST::.] in [*note TORDER::.] , or it is extracted -automatically from the expressions. In the second case the current -REDUCE system order is used (see [*note KORDER::.] ) for arranging the -variables. If some kernels should play the role of formal parameters -(the ground domain then is the polynomial ring over these), the -variable sequences must be given explicitly. - - All REDUCE [*note KERNEL::.] s can be used as variables. But please -note, that all variables are considered as independent. E.g. when using -SIN(A) and COS(A) as variables, the basic relation -SIN(A)^2+COS(A)^2-1=0 must be explicitly added to an equation set -because the Groebner operators don't include such knowledge -automatically. - - The terms (monomials) in polynomials are arranged according to the -current [*note Term order::.] . Note that the algebraic properties of -the computed results only are valid as long as neither the ordering nor -the variable sequence changes. - - The input expressions can be polynomials

, rational -functions / or equations = built from polynomials or -rational functions. Apart from the TRACING algorithms [*note -groebnert::.] and [*note preducet::.] , where the equations have a -specific meaning, equations are converted to simple expressions by -taking the difference of the left-hand and right-hand sides --=>

. Rational functions are converted to polynomials by -converting the expression to a common denominator form first, and then -using the numerator only =>

. So eventual zeros of the -denominators are ignored. - - A basis on input or output of an algorithm is coded as [*note -LIST::.] of expressions ,,... . - - -File: redhelp, Node: Term order, Next: TORDER, Up: Term order section - - TERM ORDER introduction - - For all GROEBNER operations the polynomials are represented in -distributive form: a sum of terms (monomials). The terms are ordered -corresponding to the actual TERM ORDER which is set by the [*note -TORDER::.] operator, and to the actual variable sequence which is -either given as explicit parameter or by the system [*note KERNEL::.] -order. - - -File: redhelp, Node: TORDER, Next: torder_compile, Prev: Term order, Up: Term order section - - TORDER operator - - The operator TORDER sets the actual variable sequence and term order. - - 1. simple term order: - -syntax: - - TORDER (, ) - - where is a [*note LIST::.] of variables ([*note KERNEL::.] s) -and is the name of a simple [*note Term order::.] mode [*note lex -term order::.] , [*note gradlex term order::.] , [*note revgradlex term -order::.] or another implemented parameterless mode. - - 2. stepped term order: - -syntax: - - TORDER (,,) - - where is the name of a two step term order, one of [*note -gradlexgradlex term order::.] , [*note gradlexrevgradlex term order::.] -, [*note lexgradlex term order::.] or [*note lexrevgradlex term -order::.] , and is a positive integer. - - 3. weighted term order - -syntax: - - TORDER (, WEIGHTED , ,,...); - - where the are positive integers, see [*note weighted term -order::.] . - - 4. matrix term order - -syntax: - - TORDER (, MATRIX , ); - - where is a matrix with integer elements, see [*note -torder_compile::.] . - - 5. compiled term order - -syntax: - - TORDER (, CO ); - - where is the name of a routine generated by [*note -torder_compile::.] . - - TORDER sets the variable sequence and the term order mode. If the an -empty list is used as variable sequence, the automatic variable -extraction is activated. The defaults are the empty variable list an the -[*note lex term order::.] . The previous setting is returned as a list. - - Alternatively to the above syntax the arguments of TORDER may be -collected in a [*note LIST::.] and passed as one argument to TORDER . - - -File: redhelp, Node: torder_compile, Next: lex term order, Prev: TORDER, Up: Term order section - - TORDER_COMPILE operator - - A matrix can be converted into a compilable LISP program for faster -execution by using - -syntax: - - TORDER_COMPILE (,) - - where is an identifier for the new term order and is an -integer matrix to be used as [*note matrix term order::.] . Afterwards -the term order can be activated by using in a [*note TORDER::.] -expression. The resulting program is compiled if the switch [*note -COMP::.] is on, or if the TORDER_COMPILE expression is part of a -compiled module. - - -File: redhelp, Node: lex term order, Next: gradlex term order, Prev: torder_compile, Up: Term order section - - LEX TERM ORDER - - The terms are ordered lexicographically: two terms t1 t2 are -compared for their degrees along the fixed variable sequence: t1 is -higher than t2 if the first different degree is higher in t1. This -order has the ELIMINATION PROPERTY for GROEBNER BASIS calculations. If -the ideal has a univariate polynomial in the last variable the groebner -basis will contain such polynomial. LEX is best suited for solving of -polynomial equation systems. - - -File: redhelp, Node: gradlex term order, Next: revgradlex term order, Prev: lex term order, Up: Term order section - - GRADLEX TERM ORDER - - The terms are ordered first with their total degree, and if the -total degree is identical the comparison is [*note lex term order::.] . -With GROEBNER basis calculations this term order produces polynomials -of lowest degree. - - -File: redhelp, Node: revgradlex term order, Next: gradlexgradlex term order, Prev: gradlex term order, Up: Term order section - - REVGRADLEX TERM ORDER - - The terms are ordered first with their total degree (degree sum), -and if the total degree is identical the comparison is the inverse of -[*note lex term order::.] . With [*note GROEBNER::.] and [*note -groebnerf::.] calculations this term order is similar to [*note gradlex -term order::.] ; it is known as most efficient ordering with respect to -computing time. - - -File: redhelp, Node: gradlexgradlex term order, Next: gradlexrevgradlex term order, Prev: revgradlex term order, Up: Term order section - - GRADLEXGRADLEX TERM ORDER - - The terms are separated into two groups where the second parameter -of the [*note TORDER::.] call determines the length of the first group. -For a comparison first the total degrees of both variable groups are -compared. If both are equal [*note gradlex term order::.] comparison -is applied to the first group, and if that does not decide [*note -gradlex term order::.] is applied for the second group. This order has -the elimination property for the variable groups. It can be used e.g. -for separating variables from parameters. - - -File: redhelp, Node: gradlexrevgradlex term order, Next: lexgradlex term order, Prev: gradlexgradlex term order, Up: Term order section - - GRADLEXREVGRADLEX TERM ORDER - - Similar to [*note gradlexgradlex term order::.] , but using [*note -revgradlex term order::.] for the second group. - - -File: redhelp, Node: lexgradlex term order, Next: lexrevgradlex term order, Prev: gradlexrevgradlex term order, Up: Term order section - - LEXGRADLEX TERM ORDER - - Similar to [*note gradlexgradlex term order::.] , but using [*note -lex term order::.] for the first group. - - -File: redhelp, Node: lexrevgradlex term order, Next: weighted term order, Prev: lexgradlex term order, Up: Term order section - - LEXREVGRADLEX TERM ORDER - - Similar to [*note gradlexgradlex term order::.] , but using [*note -lex term order::.] for the first group [*note revgradlex term -order::.] for the second group. - - -File: redhelp, Node: weighted term order, Next: graded term order, Prev: lexrevgradlex term order, Up: Term order section - - WEIGHTED TERM ORDER - - establishes a graduated ordering similar to [*note gradlex term -order::.] , where the exponents first are multiplied by the given -weights. If there are less weight values than variables, the weight -list is extended by ones. If the weighted degree comparison is not -decidable, the [*note lex term order::.] is used. - - -File: redhelp, Node: graded term order, Next: matrix term order, Prev: weighted term order, Up: Term order section - - GRADED TERM ORDER - - establishes a cascaded term ordering: first a graduated ordering -similar to [*note gradlex term order::.] is used, where the exponents -first are multiplied by the given weights. If there are less weight -values than variables, the weight list is extended by ones. If the -weighted degree comparison is not decidable, the term ordering -described in the following parameters of the [*note TORDER::.] command -is used. - - -File: redhelp, Node: matrix term order, Prev: graded term order, Up: Term order section - - MATRIX TERM ORDER - - Any arbitrary term order mode can be installed by a matrix with -integer elements where the row length corresponds to the variable -number. The matrix must have at least as many rows as columns. It must -have full rank, and the top nonzero element of each column must be -positive. - - The matrix TERM ORDER MODE defines a term order where the exponent -vectors of the monomials are first multiplied by the matrix and the -resulting vectors are compared lexicographically. - - If the switch [*note COMP::.] is on, the matrix is converted into a -compiled LISP program for faster execution. A matrix can also be -compiled explicitly, see [*note torder_compile::.] . - - -File: redhelp, Node: Term order section, Next: Basic Groebner operators section, Prev: Ideal Parameters, Up: Groebner package section - - Term order section - -* Menu: - -* Term order:: introduction -* TORDER:: operator -* torder_compile:: operator -* lex term order:: concept -* gradlex term order:: concept -* revgradlex term order:: concept -* gradlexgradlex term order::concept -* gradlexrevgradlex term order::concept -* lexgradlex term order:: concept -* lexrevgradlex term order::concept -* weighted term order:: concept -* graded term order:: concept -* matrix term order:: concept - - -File: redhelp, Node: GVARS, Next: GROEBNER, Up: Basic Groebner operators section - - GVARS operator - -syntax: - - GVARS (,,... ) - - where are expressions or [*note EQUATION::.] s. - - GVARS extracts from the expressions the [*note KERNEL::.] S which can -play the role of variables for a [*note GROEBNER::.] or [*note -groebnerf::.] calculation. - - -File: redhelp, Node: GROEBNER, Next: groebopt, Prev: GVARS, Up: Basic Groebner operators section - - GROEBNER operator - -syntax: - - GROEBNER (EXP , ...) - - where EXP , ... is a list of expressions or equations. - - The operator GROEBNER implements the Buchberger algorithm for -computing Groebner bases for a given set of expressions with respect to -the given set of variables in the order given. As a side effect, the -sequence of variables is stored as a REDUCE list in the shared variable -[*note gvarslast::.] - this is important in cases where the algorithm -rearranges the variable sequence because [*note groebopt::.] is ON . - -examples: - - ____________________________________________________________ - - groebner({x**2+y**2-1,x-y}) - - {X - Y,2*Y**2 -1} - - ____________________________________________________________ - -related: - - [*note groebnerf::.] operator - - [*note gvarslast::.] variable - - [*note groebopt::.] switch - - [*note groebprereduce::.] switch - - [*note groebfullreduction::.] switch - - [*note gltbasis::.] switch - - [*note gltb::.] variable - - [*note glterms::.] variable - - [*note groebstat::.] switch - - [*note trgroeb::.] switch - - [*note trgroebs::.] switch - - [*note groebprot::.] switch - - [*note groebprotfile::.] variable - - [*note groebnert::.] operator - - -File: redhelp, Node: groebopt, Next: gvarslast, Prev: GROEBNER, Up: Basic Groebner operators section - - GROEBOPT switch - - If GROEBOPT is set ON, the sequence of variables is optimized with -respect to execution speed of GROEBNER calculations; note that the -final list of variables is available in [*note gvarslast::.] . By -default GROEBOPT is off, conserving the original variable sequence. - - An explicitly declared dependency using the [*note DEPEND::.] -declaration supersedes the variable optimization. - -examples: - - ____________________________________________________________ - ____________________________________________________________ - - guarantees that a will be placed in front of x and y. - - -File: redhelp, Node: gvarslast, Next: groebprereduce, Prev: groebopt, Up: Basic Groebner operators section - - GVARSLAST variable - - After a [*note GROEBNER::.] or [*note groebnerf::.] calculation the -actual variable sequence is stored in the variable GVARSLAST . If -[*note groebopt::.] is ON GVARSLAST shows the variable sequence after -reordering. - - -File: redhelp, Node: groebprereduce, Next: groebfullreduction, Prev: gvarslast, Up: Basic Groebner operators section - - GROEBPREREDUCE switch - - If GROEBPREREDUCE set ON, [*note GROEBNER::.] and [*note -groebnerf::.] try to simplify the input expressions: if the head term -of an input expression is a multiple of the head term of another -expression, it can be reduced; these reductions are done cyclicly as -long as possible in order to shorten the main part of the algorithm. - - By default GROEBPREREDUCE is off. - - -File: redhelp, Node: groebfullreduction, Next: gltbasis, Prev: groebprereduce, Up: Basic Groebner operators section - - GROEBFULLREDUCTION switch - - If GROEBFULLREDUCTION set off, the polynomial reduction steps during -[*note GROEBNER::.] and [*note groebnerf::.] are limited to the pure -head term reduction; subsequent terms are reduced otherwise. - - By default GROEBFULLREDUCTION is on. - - -File: redhelp, Node: gltbasis, Next: gltb, Prev: groebfullreduction, Up: Basic Groebner operators section - - GLTBASIS switch - - If GLTBASIS set on, the leading terms of the result basis of a -[*note GROEBNER::.] or [*note groebnerf::.] calculation are extracted. -They are collected as a basis of monomials, which is available as value -of the global variable [*note gltb::.] . - - -File: redhelp, Node: gltb, Next: glterms, Prev: gltbasis, Up: Basic Groebner operators section - - GLTB variable - - See [*note gltbasis::.] - - -File: redhelp, Node: glterms, Next: groebstat, Prev: gltb, Up: Basic Groebner operators section - - GLTERMS variable - - If the expressions in a [*note GROEBNER::.] or [*note groebnerf::.] -call contain parameters (symbols which are not member of the variable -list), the share variable GLTERMS is set to a list of expression which -during the calculation were assumed to be nonzero. The calculated bases -are valid only under the assumption that all these expressions do not -vanish. - - -File: redhelp, Node: groebstat, Next: trgroeb, Prev: glterms, Up: Basic Groebner operators section - - GROEBSTAT switch - - if GROEBSTAT is on, a summary of the [*note GROEBNER::.] or [*note -groebnerf::.] computation is printed at the end including the computing -time, the number of intermediate H polynomials and the counters for the -criteria hits. - - -File: redhelp, Node: trgroeb, Next: trgroebs, Prev: groebstat, Up: Basic Groebner operators section - - TRGROEB switch - - if TRGROEB is on, intermediate H polynomials are printed during a -[*note GROEBNER::.] or [*note groebnerf::.] calculation. - - -File: redhelp, Node: trgroebs, Next: gzerodim?, Prev: trgroeb, Up: Basic Groebner operators section - - TRGROEBS switch - - if TRGROEBS is on, intermediate H and S polynomials are printed -during a [*note GROEBNER::.] or [*note groebnerf::.] calculation. - - -File: redhelp, Node: gzerodim?, Next: gdimension, Prev: trgroebs, Up: Basic Groebner operators section - - GZERODIM? operator - -syntax: - - GZERODIM!? () - - where is a Groebner basis in the current [*note Term order::.] -with the actual setting (see [*note Ideal Parameters::.] ). - - GZERODIM!? tests whether the ideal spanned by the given basis has -dimension zero. If yes, the number of zeros is returned, [*note NIL::.] -otherwise. - - -File: redhelp, Node: gdimension, Next: gindependent_sets, Prev: gzerodim?, Up: Basic Groebner operators section - - GDIMENSION operator - -syntax: - - GDIMENSION () - - where is a [*note GROEBNER::.] basis in the current term order -(see [*note Ideal Parameters::.] ). GDIMENSION computes the dimension -of the ideal spanned by the given basis and returns the dimension as an -integer number. The Kredel-Weispfenning algorithm is used: the dimension -is the length of the longest independent variable set, see [*note -gindependent_sets::.] - - -File: redhelp, Node: gindependent_sets, Next: dd_groebner, Prev: gdimension, Up: Basic Groebner operators section - - GINDEPENDENT_SETS operator - -syntax: - - GINDEPENDENT_SETS () - - where is a [*note GROEBNER::.] basis in any TERM ORDER (which -must be the current TERM ORDER ) with the specified variables (see -[*note Ideal Parameters::.] ). - - GINDEPENDENT_SETS computes the maximal left independent variable -sets of the ideal, that are the variable sets which play the role of -free parameters in the current ideal basis. Each set is a list which is -a subset of the variable list. The result is a list of these sets. For -an ideal with dimension zero the list is empty. The -Kredel-Weispfenning algorithm is used. - - -File: redhelp, Node: dd_groebner, Next: glexconvert, Prev: gindependent_sets, Up: Basic Groebner operators section - - DD_GROEBNER operator - - For a homogeneous system of polynomials under [*note graded term -order::.] , [*note gradlex term order::.] , [*note revgradlex term -order::.] or [*note weighted term order::.] a Groebner Base can be -computed with limiting the grade of the intermediate S polynomials: - -syntax: - - DD_GROEBNER (,,) - - where is a non negative integer and is an integer or -"infinity". A pair of polynomials is considered only if the grade of -the lcm of their head terms is between and . For the term -orders GRADED or WEIGHTED the (first) weight vector is used for the -grade computation. Otherwise the total degree of a term is used. - - -File: redhelp, Node: glexconvert, Next: greduce, Prev: dd_groebner, Up: Basic Groebner operators section - - GLEXCONVERT operator - -syntax: - - GLEXCONVERT ([,][,MAXDEG=] [,NEWVARS=]) - - where is a [*note GROEBNER::.] basis in the current term -order, (optional) is a positive integer and (optional) is a -list of variables (see [*note Ideal Parameters::.] ). - - The operator GLEXCONVERT converts the basis of a zero-dimensional -ideal (finite number of isolated solutions) from arbitrary ordering -into a basis under [*note lex term order::.] . - - The parameter defines the new variable sequence. If -omitted, the original variable sequence is used. If only a subset of -variables is specified here, the partial ideal basis is evaluated. - - If is a list with one element, the minimal UNIVARIATE -POLYNOMIAL is computed. - - is an upper limit for the degrees. The algorithm stops with -an error message, if this limit is reached. - - A warning occurs, if the ideal is not zero dimensional. - - During the call the TERM ORDER of the input basis must be active. - - -File: redhelp, Node: greduce, Next: preduce, Prev: glexconvert, Up: Basic Groebner operators section - - GREDUCE operator - -syntax: - - GREDUCE (exp, exp1, exp2, ... , expm) - - where exp is an expression, and exp1, exp2, ... , expm is a list of -expressions or equations. - - GREDUCE is functionally equivalent with a call to [*note -GROEBNER::.] and then a call to [*note preduce::.] . - - -File: redhelp, Node: preduce, Next: idealquotient, Prev: greduce, Up: Basic Groebner operators section - - PREDUCE operator - -syntax: - - PREDUCE (

, , ... ) - - where

is an expression, and , ... is a list of expressions -or equations. - - PREDUCE computes the remainder of EXP modulo the given set of -polynomials resp. equations. This result is unique (canonical) only if -the given set is a GROEBNER basis under the current [*note Term -order::.] - - see also: [*note preducet::.] operator. - - -File: redhelp, Node: idealquotient, Next: hilbertpolynomial, Prev: preduce, Up: Basic Groebner operators section - - IDEALQUOTIENT operator - -syntax: - - IDEALQUOTIENT (, ..., ) - - where ,... is a list of expressions or equations, is a -single expression or equation. - - IDEALQUOTIENT computes the ideal quotient: ideal spanned by the -expressions ,... divided by the single polynomial/expression . -The result is the [*note GROEBNER::.] basis of the quotient ideal. - - -File: redhelp, Node: hilbertpolynomial, Prev: idealquotient, Up: Basic Groebner operators section - - HILBERTPOLYNOMIAL operator - -syntax: - - hilbertpolynomial() - - where is a [*note GROEBNER::.] basis in the current [*note -Term order::.] . - - The degree of the HILBERT POLYNOMIAL is the dimension of the ideal -spanned by the basis. For an ideal of dimension zero the Hilbert -polynomial is a constant which is the number of common zeros of the -ideal (including eventual multiplicities). The HOLLMANN ALGORITHM is -used. - - -File: redhelp, Node: Basic Groebner operators section, Next: Factorizing Groebner bases section, Prev: Term order section, Up: Groebner package section - - Basic Groebner operators section - -* Menu: - -* GVARS:: operator -* GROEBNER:: operator -* groebopt:: switch -* gvarslast:: variable -* groebprereduce:: switch -* groebfullreduction:: switch -* gltbasis:: switch -* gltb:: variable -* glterms:: variable -* groebstat:: switch -* trgroeb:: switch -* trgroebs:: switch -* gzerodim?:: operator -* gdimension:: operator -* gindependent_sets:: operator -* dd_groebner:: operator -* glexconvert:: operator -* greduce:: operator -* preduce:: operator -* idealquotient:: operator -* hilbertpolynomial:: operator - - -File: redhelp, Node: groebnerf, Next: groebmonfac, Up: Factorizing Groebner bases section - - GROEBNERF operator - -syntax: - - GROEBNERF (, ...[,,, ... ]); - - where , ... is a list of expressions or equations, and ,... -is an optional list of polynomials to be considered as non zero for -this calculation. An empty list must be passed as second argument if -the non-zero list is specified. - - GROEBNERF tries to separate polynomials into individual factors and -to branch the computation in a recursive manner (factorization tree). -The result is a list of partial Groebner bases. Multiplicities (one -factor with a higher power, the same partial basis twice) are deleted -as early as possible in order to speed up the calculation. - - The third parameter of GROEBNERF declares some polynomials nonzero. -If any of these is found in a branch of the calculation the branch is -canceled. - -example: - - ____________________________________________________________ - - groebnerf({ 3*x**2*y+2*x*y+y+9*x**2+5*x = 3, - 2*x**3*y-x*y-y+6*x**3-2*x**2-3*x = -3, - x**3*y+x**2*y+3*x**3+2*x**2 }, {y,x}); - - {{Y - 3,X}, - - 2 - {2*Y + 2*X - 1,2*X - 5*X - 5}} - ____________________________________________________________ - -related: - - [*note groebresmax::.] variable - - [*note groebmonfac::.] variable - - [*note groebrestriction::.] variable - - [*note GROEBNER::.] operator - - [*note gvarslast::.] variable - - [*note groebopt::.] switch - - [*note groebprereduce::.] switch - - [*note groebfullreduction::.] switch - - [*note gltbasis::.] switch - - [*note gltb::.] variable - - [*note glterms::.] variable - - [*note groebstat::.] switch - - [*note trgroeb::.] switch - - [*note trgroebs::.] switch - - [*note groebnert::.] operator - - -File: redhelp, Node: groebmonfac, Next: groebresmax, Prev: groebnerf, Up: Factorizing Groebner bases section - - GROEBMONFAC variable - - The variable GROEBMONFAC is connected to the handling of monomial -factors. A monomial factor is a product of variable powers as a factor, -e.g. x**2*y in x**3*y - 2*x**2*y**2. A monomial factor represents a -solution of the type x = 0 or y = 0 with a certain multiplicity. With -[*note groebnerf::.] the multiplicity of monomial factors is lowered -to the value of the shared variable GROEBMONFAC which by default is 1 -(= monomial factors remain present, but their multiplicity is brought -down). With GROEBMONFAC := 0 the monomial factors are suppressed -completely. - - -File: redhelp, Node: groebresmax, Next: groebrestriction, Prev: groebmonfac, Up: Factorizing Groebner bases section - - GROEBRESMAX variable - - The variable GROEBRESMAX controls during [*note groebnerf::.] -calculations the number of partial results. Its default value is 300. If -more partial results are calculated, the calculation is terminated. - - -File: redhelp, Node: groebrestriction, Prev: groebresmax, Up: Factorizing Groebner bases section - - GROEBRESTRICTION variable - - During [*note groebnerf::.] calculations irrelevant branches can be -excluded by setting the variable GROEBRESTRICTION . The following -restrictions are implemented: - -syntax: - - GROEBRESTRICTION := NONNEGATIVE - - GROEBRESTRICTION := POSITIVE - - GROEBRESTRICTION := ZEROPOINT - - With NONNEGATIVE branches are excluded where one polynomial has no -nonnegative real zeros; with POSITIVE the restriction is sharpened to -positive zeros only. The restriction ZEROPOINT excludes all branches -which do not have the origin (0,0,...0) in their solution set. - - -File: redhelp, Node: Factorizing Groebner bases section, Next: Tracing Groebner bases section, Prev: Basic Groebner operators section, Up: Groebner package section - - Factorizing Groebner bases section - -* Menu: - -* groebnerf:: operator -* groebmonfac:: variable -* groebresmax:: variable -* groebrestriction:: variable - - -File: redhelp, Node: groebprot, Next: groebprotfile, Up: Tracing Groebner bases section - - GROEBPROT switch - - If GROEBPROT is ON the computation steps during [*note preduce::.] , -[*note greduce::.] and [*note GROEBNER::.] are collected in a list -which is assigned to the variable [*note groebprotfile::.] . - - -File: redhelp, Node: groebprotfile, Next: groebnert, Prev: groebprot, Up: Tracing Groebner bases section - - GROEBPROTFILE variable - - See [*note groebprot::.] switch. - - -File: redhelp, Node: groebnert, Next: preducet, Prev: groebprotfile, Up: Tracing Groebner bases section - - GROEBNERT operator - -syntax: - - GROEBNERT (=,...) - - where are [*note KERNEL::.] S (simple or indexed variables), - are polynomials. - - GROEBNERT is functionally equivalent to a [*note GROEBNER::.] call -for ,..., but the result is a set of equations where the left-hand -sides are the basis elements while the right-hand sides are the same -values expressed as combinations of the input formulas, expressed in -terms of the names - -example: - - ____________________________________________________________ - - groebnert({p1=2*x**2+4*y**2-100,p2=2*x-y+1}); - - GB1 := {2*X - Y + 1=P2, - - 2 - 9*Y - 2*Y - 199= - 2*X*P2 - Y*P2 + 2*P1 + P2} - ____________________________________________________________ - - -File: redhelp, Node: preducet, Prev: groebnert, Up: Tracing Groebner bases section - - PREDUCET operator - -syntax: - - PREDUCE (

,=...) - - where

is an expression, are kernels (simple or indexed -variables), EXP are polynomials. - - PREDUCET computes the remainder of

modulo ,... similar to -[*note preduce::.] , but the result is an equation which expresses the -remainder as combination of the polynomials. - -example: - - ____________________________________________________________ - - - GB2 := {G1=2*X - Y + 1,G2=9*Y**2 - 2*Y - 199} - preducet(q=x**2,gb2); - - - 16*Y + 208= - 18*X*G1 - 9*Y*G1 + 36*Q + 9*G1 - G2 - ____________________________________________________________ - - -File: redhelp, Node: Tracing Groebner bases section, Next: Groebner Bases for Modules section, Prev: Factorizing Groebner bases section, Up: Groebner package section - - Tracing Groebner bases section - -* Menu: - -* groebprot:: switch -* groebprotfile:: variable -* groebnert:: operator -* preducet:: operator - - -File: redhelp, Node: Module, Next: gmodule, Up: Groebner Bases for Modules section - - MODULE - - Given a polynomial ring, e.g. R=Z[x,y,...] and an integer n>1. The -vectors with n elements of R form a free MODULE under elementwise -addition and multiplication with elements of R. - - For a submodule given by a finite basis a Groebner basis can be -computed, and the facilities of the GROEBNER package are available -except the operators [*note groebnerf::.] and GROESOLVE . The vectors -are encoded using auxiliary variables which represent the unit vectors -in the module. These are declared in the share variable [*note -gmodule::.] . - - -File: redhelp, Node: gmodule, Prev: Module, Up: Groebner Bases for Modules section - - GMODULE variable - - The vectors of a free [*note Module::.] over a polynomial ring R are -encoded as linear combinations with unit vectors of M which are -represented by auxiliary variables. These must be collected in the -variable GMODULE before any call to an operator of the Groebner package. - - ____________________________________________________________ - - torder({x,y,v1,v2,v3})$ - gmodule := {v1,v2,v3}$ - g:=groebner({x^2*v1 + y*v2,x*y*v1 - v3,2y*v1 + y*v3}); - ____________________________________________________________ - compute the Groebner basis of the submodule - - ____________________________________________________________ - - ([x^2,y,0],[xy,0,-1],[0,2y,y]) - ____________________________________________________________ - The members of the list GMODULE are automatically appended to the -end of the variable list, if they are not yet members there. They take -part in the actual term ordering. - - -File: redhelp, Node: Groebner Bases for Modules section, Next: Computing with distributive polynomials section, Prev: Tracing Groebner bases section, Up: Groebner package section - - Groebner Bases for Modules section - -* Menu: - -* Module:: concept -* gmodule:: variable - - -File: redhelp, Node: gsort, Next: gsplit, Up: Computing with distributive polynomials section - - GSORT operator - -syntax: - - GSORT (

) - - where

is a polynomial or a list of polynomials. - - The polynomials are reordered and sorted corresponding to the -current [*note Term order::.] . - -examples: - - ____________________________________________________________ - - - torder lex; - - gsort(x**2+2x*y+y**2,{y,x}); - - y**2+2y*x+x**2 - - ____________________________________________________________ - - -File: redhelp, Node: gsplit, Next: gspoly, Prev: gsort, Up: Computing with distributive polynomials section - - GSPLIT operator - -syntax: - - GSPLIT (

[,]); - - where

is a polynomial or a list of polynomials. - - The polynomial is reordered corresponding to the the current [*note -Term order::.] and then separated into leading term and reductum. -Result is a list with the leading term as first and the reductum as -second element. - -examples: - - ____________________________________________________________ - - - torder lex; - - gsplit(x**2+2x*y+y**2,{y,x}); - - {y**2,2y*x+x**2} - - ____________________________________________________________ - - -File: redhelp, Node: gspoly, Prev: gsplit, Up: Computing with distributive polynomials section - - GSPOLY operator - -syntax: - - GSPOLY (,); - - where and are polynomials. - - The SUBTRACTION polynomial of p1 and p2 is computed corresponding to -the method of the Buchberger algorithm for computing GROEBNER BASES : -p1 and p2 are multiplied with terms such that when subtracting them the -leading terms cancel each other. - - -File: redhelp, Node: Computing with distributive polynomials section, Prev: Groebner Bases for Modules section, Up: Groebner package section - - Computing with distributive polynomials section - -* Menu: - -* gsort:: operator -* gsplit:: operator -* gspoly:: operator - - -File: redhelp, Node: Groebner package section, Next: High Energy Physics section, Prev: Matrix Operations section, Up: Top - - Groebner package section - -* Menu: - -* Groebner bases:: introduction -* Ideal Parameters:: concept -* Term order section:: -* Basic Groebner operators section:: -* Factorizing Groebner bases section:: -* Tracing Groebner bases section:: -* Groebner Bases for Modules section:: -* Computing with distributive polynomials section:: - - -File: redhelp, Node: HEPHYS, Next: HE-dot, Up: High Energy Physics section - - HEPHYS introduction - - The High-energy Physics package is historic for REDUCE, since REDUCE -originated as a program to aid in computations with Dirac expressions. -The commutation algebra of the gamma matrices is independent of their -representation, and is a natural subject for symbolic mathematics. Dirac -theory is applied to beta decay and the computation of cross-sections -and scattering. The high-energy physics operators are available in the -REDUCE main program, rather than as a module which must be loaded. - - -File: redhelp, Node: HE-dot, Next: EPS, Prev: HEPHYS, Up: High Energy Physics section - - . HE-DOT operator - - The . operator is used to denote the scalar product of two Lorentz -four-vectors. - -syntax: - - . - - must be an identifier declared to be of type VECTOR to have -the scalar product definition. When applied to arguments that are not -vectors, the [*note CONS::.] operator is used, whose symbol is also -"dot." - -examples: - - ____________________________________________________________ - - vector aa,bb,cc; - - let aa.bb = 0; - - aa.bb; - - 0 - - - aa.cc; - - AA.CC - - - q := aa.cc; - - Q := AA.CC - - - q; - - AA.CC - - ____________________________________________________________ - Since vectors are special high-energy physics entities that do not -contain values, the . product will not return a true scalar product. -You can assign a scalar identifier to the result of a . operation, or -assign a . operation to have the value of the scalar you supply, as -shown above. Note that the result of a . operation is a scalar, not a -vector. - - The metric tensor g(u,v) can be represented by U.V . If contraction -over the indices is required, U and V should be declared to be of type -[*note INDEX::.] . - - The dot operator has the highest precedence of the infix operators, -so expressions involving . and other operators have the scalar product -evaluated first before other operations are done. - - -File: redhelp, Node: EPS, Next: G, Prev: HE-dot, Up: High Energy Physics section - - EPS operator - - The EPS operator denotes the completely antisymmetric tensor of -order 4 and its contraction with Lorentz four-vectors, as used in -high-energy physics calculations. - -syntax: - - EPS (,,, ) - - must be a valid vector expression, and may be an index. - -examples: - - ____________________________________________________________ - - vector g0,g1,g2,g3; - - eps(g1,g0,g2,g3); - - - EPS(G0,G1,G2,G3); - - - eps(g1,g2,g0,g3); - - EPS(G0,G1,G2,G3); - - - eps(g1,g2,g3,g1); - - 0 - - ____________________________________________________________ - Vector identifiers are ordered alphabetically by REDUCE. When an odd -number of transpositions is required to restore the canonical order to -the four arguments of EPS , the term is ordered and carries a minus -sign. When an even number of transpositions is required, the term is -returned ordered and positive. When one of the arguments is repeated, -the value 0 is returned. A contraction of the form eps(_i j mu nu p_mu -q_nu) is represented by EPS(I,J,P,Q) when I and J have been declared to -be of type [*note INDEX::.] . - - -File: redhelp, Node: G, Next: INDEX, Prev: EPS, Up: High Energy Physics section - - G operator - - G is an n-ary operator used to denote a product of gamma matrices -contracted with Lorentz four-vectors, in high-energy physics. - -syntax: - - G (, ,*) - - is a scalar identifier representing a fermion line -identifier, can be any valid vector expression, -representing a vector or a gamma matrix. - -examples: - - ____________________________________________________________ - - vector aa,bb,cc; - - vector a; - - g(line1,aa,bb); - - AA.BB - - - g(line2,aa,a); - - 0 - - - g(id,aa,bb,cc); - - 0 - - - g(li1,aa,bb) + k; - - AA.BB + K - - - let aa.bb = m*k; - - g(ln1,aa)*g(ln1,bb); - - K*M - - - g(ln1,aa)*g(ln2,bb); - - 0 - - ____________________________________________________________ - The vector A is reserved in arguments of G to denote the special -gamma matrix gamma_5. It must be declared to be a vector before you use -it. - - Gamma matrix expressions are associated with fermion lines in a -Feynman diagram. If more than one line occurs in an expression, the -gamma matrices involved are separate (operating in independent spin -space), as shown in the last two example lines above. A product of -gamma matrices associated with a single line can be entered either as a -single G command with several vector arguments, or as products of -separate G commands each with a single argument. - - While the product of vectors is not defined, the product, sum and -difference of several gamma expressions are defined, as is the product -of a gamma expression with a scalar. If an expression involving gamma -matrices includes a scalar, the scalar is treated as if it were the -product of itself with a unit 4 x 4 matrix. - - Dirac expressions are evaluated by computing the trace of the -expression using the commutation algebra of gamma matrices. The -algorithms used are described in articles by J. S. R. Chisholm in Vol. 30, p. 426, 1963, and J. Kahane, , Vol. 9, p. 1732, 1968. The trace is then divided -by 4 to distinguish between the trace of a scalar and the trace of an -expression that is the product of a scalar with a unit 4 x 4 matrix. - - Trace calculations may be prevented over any line identifier by -declaring it to be [*note NOSPUR::.] . If it is later desired to -evaluate these traces, the declaration can be undone with the [*note -SPUR::.] declaration. - - The notation of Bjorken and Drell, -1964, is assumed in all operations involving gamma matrices. For an -example of the use of G in a calculation, see the . - - -File: redhelp, Node: INDEX, Next: MASS, Prev: G, Up: High Energy Physics section - - INDEX declaration - - The declaration INDEX flags a four-vector as an index for subsequent -high-energy physics calculations. - -syntax: - - INDEX ,* - - must have been declared of type VECTOR . - -examples: - - ____________________________________________________________ - - vector aa,bb,cc; - - index uu; - - let aa.bb = 0; - - (aa.uu)*(bb.uu); - - 0 - - - (aa.uu)*(cc.uu); - - AA.CC - - ____________________________________________________________ - Index variables are used to represent contraction over components of -vectors when scalar products are taken by the . operator, as well as -indicating contraction for the [*note EPS::.] operator or metric tensor. - - The special status of a vector as an index can be revoked with the -declaration [*note REMIND::.] . The object remains a vector, however. - - -File: redhelp, Node: MASS, Next: MSHELL, Prev: INDEX, Up: High Energy Physics section - - MASS command - - The MASS command associates a scalar variable as a mass with the -corresponding vector variable, in high-energy physics calculations. - -syntax: - - MASS = ,= * - - can be a declared vector variable; MASS will declare it -to be of type VECTOR if it is not. This may override an existing matrix -variable by that name. must be a scalar variable. - -examples: - - ____________________________________________________________ - - vector bb,cc; - - mass cc=m; - - mshell cc; - - cc.cc; - - 2 - M - - ____________________________________________________________ - Once a mass has been attached to a vector with a MASS declaration, -the [*note MSHELL::.] declaration puts the associated particle "on the -mass shell." Subsequent scalar (.) products of the vector with itself -will be replaced by the square of the mass expression. - - -File: redhelp, Node: MSHELL, Next: NOSPUR, Prev: MASS, Up: High Energy Physics section - - MSHELL command - - The MSHELL command puts particles on the mass shell in high-energy -physics calculations. - -syntax: - - MSHELL ,* - - must have had a mass attached to it by a [*note MASS::.] -declaration. - -examples: - - ____________________________________________________________ - - vector v1,v2; - - mass v1=m,v2=q; - - mshell v1; - - v1.v1; - - 2 - M - - - v2.v2; - - V2.V2 - - - mshell v2; - - v1.v1*v2.v2; - - 2 2 - M *Q - - ____________________________________________________________ - Even though a mass is attached to a vector variable representing a -particle, the replacement does not take place until the MSHELL -declaration is given for that vector variable. - - -File: redhelp, Node: NOSPUR, Next: REMIND, Prev: MSHELL, Up: High Energy Physics section - - NOSPUR declaration - - The NOSPUR declaration prevents the trace calculation over the given -line identifiers in high-energy physics calculations. - -syntax: - - NOSPUR ,* - - is a scalar identifier that will be used as a line -identifier. - -examples: - - ____________________________________________________________ - - vector a1,b1,c1; - - g(line1,a1,b1)*g(line2,b1,c1); - - A1.B1*B1.C1 - - - nospur line2; - - g(line1,a1,b1)*g(line2,b1,c1); - - A1.B1*G(LINE2,B1,C1) - - ____________________________________________________________ - Nospur declarations can be removed by making the declaration [*note -SPUR::.] . - - -File: redhelp, Node: REMIND, Next: SPUR, Prev: NOSPUR, Up: High Energy Physics section - - REMIND declaration - - The REMIND declaration removes the special status of its arguments -as indices, which was set in the [*note INDEX::.] declaration, in -high-energy physics calculations. - -syntax: - - REMIND ,* - - must have been declared to be of type [*note INDEX::.] . - - -File: redhelp, Node: SPUR, Next: VECDIM, Prev: REMIND, Up: High Energy Physics section - - SPUR declaration - - The SPUR declaration removes the special exemption from trace -calculations that was declared by [*note NOSPUR::.] , in high-energy -physics calculations. - -syntax: - - SPUR ,* - - must be a line-identifier that has previously been declared -NOSPUR . - - -File: redhelp, Node: VECDIM, Next: VECTOR, Prev: SPUR, Up: High Energy Physics section - - VECDIM command - - The command VECDIM changes the vector dimension from 4 to an -arbitrary integer or symbol. Used in high-energy physics calculations. - -syntax: - - VECDIM - - must be either an integer or a valid scalar identifier -that does not have a floating-point value. - - The [*note EPS::.] operator and the gamma_5 symbol (A ) are not -properly defined in anything except four dimensions and will print an -error message if you use them that way. The other high-energy physics -operators should work without problem. - - -File: redhelp, Node: VECTOR, Prev: VECDIM, Up: High Energy Physics section - - VECTOR declaration - - The VECTOR declaration declares that its arguments are of type -VECTOR . - -syntax: - - VECTOR ,* - - must be a valid REDUCE identifier. It may have already -been used for a matrix, array, operator or scalar variable. After an -identifier has been declared to be a vector, it may not be used as a -scalar variable. - - Vectors are special entities for high-energy physics calculations. -You cannot put values into their coordinates; they do not have -coordinates. They are legal arguments for the high-energy physics -operators [*note EPS::.] , [*note G::.] and . (dot). Vector variables -are used to represent gamma matrices and gamma matrices contracted with -Lorentz 4-vectors, since there are no Dirac variables per se in the -system. Vectors do follow the usual vector rules for arithmetic -operations: + and - operate upon two or more vectors, producing a -vector; * and / cannot be used between vectors; the scalar product is -represented by the . operator; and the product of a scalar and vector -expression is well defined, and is a vector. - - You can represent components of vectors by including representations -of unit vectors in your system. For instance, letting E0 represent the -unit vector (1,0,0,0), the command - - V1.E0 := 0; would set up the substitution of zero for the first -component of the vector V1 . - - Identifiers that are declared by the INDEX and MASS declarations are -automatically declared to be vectors. - - The following errors can occur in calculations using the high energy -physics package: - - A REPRESENTS ONLY GAMMA5 IN VECTOR EXPRESSIONS You have tried to use -A in some way other than gamma5 in a high-energy physics expression. - - GAMMA5 NOT ALLOWED UNLESS VECDIM IS 4 You have used gamma_5 in a -high-energy physics computation involving a vector dimension other than -4. - - HAS NO MASS - - One of the arguments to [*note MSHELL::.] has had no mass assigned -to it, in high-energy physics calculations. - - MISSING ARGUMENTS FOR G OPERATOR A line symbol is missing in a gamma -matrix expression in high-energy physics calculations. - - UNMATCHED INDEX - - The parser has found unmatched indices during the evaluation of a -gamma matrix expression in high-energy physics calculations. - - -File: redhelp, Node: High Energy Physics section, Next: Numeric Package section, Prev: Groebner package section, Up: Top - - High Energy Physics section - -* Menu: - -* HEPHYS:: introduction -* HE-dot:: . operator -* EPS:: operator -* G:: operator -* INDEX:: declaration -* MASS:: command -* MSHELL:: command -* NOSPUR:: declaration -* REMIND:: declaration -* SPUR:: declaration -* VECDIM:: command -* VECTOR:: declaration - - -File: redhelp, Node: Numeric Package, Next: Interval, Up: Numeric Package section - - NUMERIC PACKAGE introduction - - The numeric package supplies algorithms based on approximation -techniques of numerical mathematics. The algorithms use the [*note -ROUNDED::.] mode arithmetic of REDUCE, including the variable precision -feature which is exploited in some algorithms in an adaptive manner in -order to reach the desired accuracy. - - -File: redhelp, Node: Interval, Next: numeric accuracy, Prev: Numeric Package, Up: Numeric Package section - - INTERVAL type - - Intervals are generally coded as lower bound and upper bound -connected by the operator .. , usually associated to a variable in an -equation. - -syntax: - - = ( .. ) - - where is a [*note KERNEL::.] and , are numbers or -expression which evaluate to numbers with <=. - -examples: - - ____________________________________________________________ - ____________________________________________________________ - means that the variable x is taken in the range from 2.5 up to 3.5. - - -File: redhelp, Node: numeric accuracy, Next: TRNUMERIC, Prev: Interval, Up: Numeric Package section - - NUMERIC ACCURACY - - The keyword parameters ACCURACY=A and ITERATIONS=I , where A and I -must be positive integer numbers, control the iterative algorithms: the -iteration is continued until the local error is below 10**-a; if that -is impossible within I steps, the iteration is terminated with an error -message. The values reached so far are then returned as the result. - - -File: redhelp, Node: TRNUMERIC, Next: num_min, Prev: numeric accuracy, Up: Numeric Package section - - TRNUMERIC switch - - Normally the algorithms produce only a minimum of printed output -during their operation. In cases of an unsuccessful or unexpected long -operation a TRACE OF THE ITERATION can be printed by setting TRNUMERIC -ON . - - -File: redhelp, Node: num_min, Next: num_solve, Prev: TRNUMERIC, Up: Numeric Package section - - NUM_MIN operator - - The Fletcher Reeves version of the STEEPEST DESCENT algorithms is -used to find the MINIMUM of a function of one or more variables. The -function must have continuous partial derivatives with respect to all -variables. The starting point of the search can be specified; if not, -random values are taken instead. The steepest descent algorithms in -general find only local minima. - -syntax: - - NUM_MIN (, [=] [,[=] ... [,accuracy=] -[,iterations=]) - - or - - NUM_MIN (exp, [=] [,[=] ...] [,accuracy=] -[,iterations=]) - - where is a function expression, are the variables in - and are the (optional) start values. For and see -[*note numeric accuracy::.] . - - NUM_MIN tries to find the next local minimum along the descending -path starting at the given point. The result is a [*note LIST::.] with -the minimum function value as first element followed by a list of -[*note EQUATION::.] S , where the variables are equated to the -coordinates of the result point. - -examples: - - ____________________________________________________________ - - num_min(sin(x)+x/5, x) - - {4.9489585606,{X=29.643767785}} - - - num_min(sin(x)+x/5, x=0) - - { - 1.3342267466,{X= - 1.7721582671}} - - ____________________________________________________________ - - -File: redhelp, Node: num_solve, Next: num_int, Prev: num_min, Up: Numeric Package section - - NUM_SOLVE operator - - An adaptively damped Newton iteration is used to find an -approximative root of a function (function vector) or the solution of -an [*note EQUATION::.] (equation system). The expressions must have -continuous derivatives for all variables. A starting point for the -iteration can be given. If not given random values are taken instead. -When the number of forms is not equal to the number of variables, the -Newton method cannot be applied. Then the minimum of the sum of -absolute squares is located instead. - - With [*note COMPLEX::.] on, solutions with imaginary parts can be -found, if either the expression(s) or the starting point contain a -nonzero imaginary part. - -syntax: - - NUM_SOLVE (, [=][,accuracy=][,iterations=]) - - or - - NUM_SOLVE (,...,, [=],...,[=] -[,accuracy=][,iterations=]) - - or - - NUM_SOLVE (,...,, [=],...,[=] -[,accuracy=][,iterations=]) - - where are function expressions, are the variables, - are optional start values. For and see [*note numeric -accuracy::.] . - - NUM_SOLVE tries to find a zero/solution of the expression(s). -Result is a list of equations, where the variables are equated to the -coordinates of the result point. - - The JACOBIAN MATRIX is stored as side effect the shared variable -JACOBIAN . - -examples: - - ____________________________________________________________ - - num_solve({sin x=cos y, x + y = 1},{x=1,y=2}); - - - {X= - 1.8561957251,Y=2.856195584} - - - jacobian; - - [COS(X) SIN(Y)] - [ ] - [ 1 1 ] - - ____________________________________________________________ - - -File: redhelp, Node: num_int, Next: num_odesolve, Prev: num_solve, Up: Numeric Package section - - NUM_INT operator - - For the numerical evaluation of univariate integrals over a finite -interval the following strategy is used: If [*note INT::.] finds a -formal antiderivative which is bounded in the integration interval, -this is evaluated and the end points and the difference is returned. -Otherwise a [*note Chebyshev fit::.] is computed, starting with order -20, eventually up to order 80. If that is recognized as sufficiently -convergent it is used for computing the integral by directly -integrating the coefficient sequence. If none of these methods is -successful, an adaptive multilevel quadrature algorithm is used. - - For multivariate integrals only the adaptive quadrature is used. -This algorithm tolerates isolated singularities. The value ITERATIONS -here limits the number of local interval intersection levels. is a -measure for the relative total discretization error (comparison of -order 1 and order 2 approximations). - -syntax: - - NUM_INT (,=( .. ) [,=( .. ),...] -[,accuracy=][,iterations=]) - - where is the function to be integrated, are the -integration variables, are the lower bounds, are the upper -bounds. - - Result is the value of the integral. - -examples: - - ____________________________________________________________ - - num_int(sin x,x=(0 .. 3.1415926)); - - 2.0000010334 - - ____________________________________________________________ - - -File: redhelp, Node: num_odesolve, Next: bounds, Prev: num_int, Up: Numeric Package section - - NUM_ODESOLVE operator - - The RUNGE-KUTTA method of order 3 finds an approximate graph for the -solution of real ODE INITIAL VALUE PROBLEM . - -syntax: - - NUM_ODESOLVE (,=, =( .. ) -[,accuracy=][,iterations=]) - - or - - NUM_ODESOLVE (,,..., =,=,... -=( .. ) [,accuracy=][,iterations=]) - - where and specify the dependent variable(s) and the -starting point value (vector), , and specify the -independent variable and the integration interval (starting point and -end point), are equations or expressions which contain the first -derivative of the independent variable with respect to the dependent -variable. - - The ODEs are converted to an explicit form, which then is used for a -Runge Kutta iteration over the given range. The number of steps is -controlled by the value of (default: 20). If the steps are too -coarse to reach the desired accuracy in the neighborhood of the -starting point, the number is increased automatically. - - Result is a list of pairs, each representing a point of the -approximate solution of the ODE problem. - -examples: - - ____________________________________________________________ - - depend(y,x); - - num_odesolve(df(y,x)=y,y=1,x=(0 .. 1), iterations=5); - - - ,{0.2,1.2214},{0.4,1.49181796},{0.6,1.8221064563}, - {0.8,2.2255208258},{1.0,2.7182511366}} - - ____________________________________________________________ - In most cases you must declare the dependency relation between the -variables explicitly using [*note DEPEND::.] ; otherwise the formal -derivative might be converted to zero. - - The operator [*note SOLVE::.] is used to convert the form into an -explicit ODE. If that process fails or if it has no unique result, the -evaluation is stopped with an error message. - - -File: redhelp, Node: bounds, Next: Chebyshev fit, Prev: num_odesolve, Up: Numeric Package section - - BOUNDS operator - - Upper and lower bounds of a real valued function over an [*note -Interval::.] or a rectangular multivariate domain are computed by the -operator BOUNDS . The algorithmic basis is the computation with -inequalities: starting from the interval(s) of the variables, the -bounds are propagated in the expression using the rules for inequality -computation. Some knowledge about the behavior of special functions -like ABS, SIN, COS, EXP, LOG, fractional exponentials etc. is -integrated and can be evaluated if the operator BOUNDS is called with -rounded mode on (otherwise only algebraic evaluation rules are -available). - - If BOUNDS finds a singularity within an interval, the evaluation is -stopped with an error message indicating the problem part of the -expression. - -syntax: - - BOUNDS (,=( .. ) [,=( .. ) ...]) - - or - - BOUNDS (,=( .. ) [,=( .. ) ...]) - - where is the function to be investigated, are the -variables of , and specify the area as set of [*note -Interval::.] S . - - BOUNDS computes upper and lower bounds for the expression in the -given area. An [*note Interval::.] is returned. - -examples: - - ____________________________________________________________ - - bounds(sin x,x=(1 .. 2)); - - -1 .. 1 - - - on rounded; - - bounds(sin x,x=(1 .. 2)); - - 0.84147098481 .. 1 - - - bounds(x**2+x,x=(-0.5 .. 0.5)); - - - 0.25 .. 0.75 - - ____________________________________________________________ - - -File: redhelp, Node: Chebyshev fit, Next: num_fit, Prev: bounds, Up: Numeric Package section - - CHEBYSHEV FIT - - The operator family CHEBYSHEV_... implements approximation and -evaluation of functions by the Chebyshev method. Let T(N,A,B,X) be the -Chebyshev polynomial of order N transformed to the interval (A,B) . -Then a function F(X) can be approximated in (A,B) by a series - - ____________________________________________________________ - - for i := 0:n sum c(i)*T(i,a,b,x) - ____________________________________________________________ - The operator CHEBYSHEV_FIT computes this approximation and returns a -list, which has as first element the sum expressed as a polynomial and -as second element the sequence of Chebyshev coefficients. CHEBYSHEV_DF -and CHEBYSHEV_INT transform a Chebyshev coefficient list into the -coefficients of the corresponding derivative or integral respectively. -For evaluating a Chebyshev approximation at a given point in the basic -interval the operator CHEBYSHEV_EVAL can be used. CHEBYSHEV_EVAL is -based on a recurrence relation which is in general more stable than a -direct evaluation of the complete polynomial. - -syntax: - - CHEBYSHEV_FIT (,=( .. ),) - - CHEBYSHEV_EVAL (,=( .. ), =) - - CHEBYSHEV_DF (,=( .. )) - - CHEBYSHEV_INT (,=( .. )) - - where is an algebraic expression (the target function), -is the variable of , and are numerical real values which -describe an [*note Interval::.] < , the integer is the -approximation order (set to 20 if missing), is a number in the -interval and is a series of Chebyshev coefficients. - -examples: - - ____________________________________________________________ - - - on rounded; - - - w:=chebyshev_fit(sin x/x,x=(1 .. 3),5); - - - 3 2 - w := {0.03824*x - 0.2398*x + 0.06514*x + 0.9778, - {0.8991,-0.4066,-0.005198,0.009464,-0.00009511}} - - - chebyshev_eval(second w, x=(1 .. 3), x=2.1); - - - 0.4111 - - ____________________________________________________________ - - -File: redhelp, Node: num_fit, Prev: Chebyshev fit, Up: Numeric Package section - - NUM_FIT operator - - The operator NUM_FIT finds for a set of points the linear -combination of a given set of functions (function basis) which -approximates the points best under the objective of the LEAST SQUARES -criterion (minimum of the sum of the squares of the deviation). The -solution is found as zero of the gradient vector of the sum of squared -errors. - -syntax: - - NUM_FIT (,,=) - - where is a list of numeric values, is a variable used -for the approximation, is a list of coordinate values which -correspond to , is a set of functions varying in VAR which -is used for the approximation. - - The result is a list containing as first element the function which -approximates the given values, and as second element a list of -coefficients which were used to build this function from the basis. - -examples: - - ____________________________________________________________ - - - pts:=for i:=1 step 1 until 5 collect i$ - - vals:=for i:=1 step 1 until 5 collect - - for j:=1:i product j$ - - num_fit(vals,{1,x,x**2},x=pts); - - 2 - {14.571428571*X - 61.428571429*X + 54.6,{54.6, - - 61.428571429,14.571428571}} - - ____________________________________________________________ - - -File: redhelp, Node: Numeric Package section, Next: Roots Package section, Prev: High Energy Physics section, Up: Top - - Numeric Package section - -* Menu: - -* Numeric Package:: introduction -* Interval:: type -* numeric accuracy:: concept -* TRNUMERIC:: switch -* num_min:: operator -* num_solve:: operator -* num_int:: operator -* num_odesolve:: operator -* bounds:: operator -* Chebyshev fit:: concept -* num_fit:: operator - - -File: redhelp, Node: Roots Package, Next: MKPOLY, Up: Roots Package section - - ROOTS PACKAGE introduction - - The root finding package is designed so that it can be used to find -some or all of the roots of univariate polynomials with real or complex -coefficients, to the accuracy specified by the user. - - Not all operators of ROOTS PACKAGE are described here. For using the -operators - - ISOLATER (intervals isolating real roots) - - RLROOTNO (number of real roots in an interval) - - ROOTSAT-PREC (roots at system precision) - - ROOTVAL (result in equation form) - - FIRSTROOT (computing only one root) - - GETROOT (selecting roots from a collection) - - please consult the full documentation of the package. - - -File: redhelp, Node: MKPOLY, Next: NEARESTROOT, Prev: Roots Package, Up: Roots Package section - - MKPOLY operator - - Given a roots list as returned by [*note ROOTS::.] , the operator -MKPOLY constructs a polynomial which has these numbers as roots. - -syntax: - - MKPOLY - - where is a [*note LIST::.] with equations, which all have the -same [*note KERNEL::.] on their left-hand sides and numbers as -right-hand sides. - -examples: - - ____________________________________________________________ - - mkpoly{x=1,x=-2,x=i,x=-i}; - - x**4 + x**3 - x**2 + x - 2 - - ____________________________________________________________ - Note that this polynomial is unique only up to a numeric factor. - - -File: redhelp, Node: NEARESTROOT, Next: REALROOTS, Prev: MKPOLY, Up: Roots Package section - - NEARESTROOT operator - - The operator NEARESTROOT finds one root of a polynomial with an -iteration using a given starting point. - -syntax: - - NEARESTROOT (

,) - - where

is a univariate polynomial and is a number. - -examples: - - ____________________________________________________________ - - nearestroot(x^2+2,2); - - {x=1.41421*i} - - ____________________________________________________________ - The minimal accuracy of the result values is controlled by [*note -ROOTACC::.] . - - -File: redhelp, Node: REALROOTS, Next: ROOTACC, Prev: NEARESTROOT, Up: Roots Package section - - REALROOTS operator - - The operator REALROOTS finds that real roots of a polynomial to an -accuracy that is sufficient to separate them and which is a minimum of -6 decimal places. - -syntax: - - REALROOTS (

) or - - REALROOTS (

,,) - - where

is a univariate polynomial. The optional parameters - and classify an interval: if given, exactly the real roots -in this interval will be returned. and can also take the -values INFINITY or -INFINITY . If omitted all real roots will be -returned. Result is a [*note LIST::.] of equations which represent the -roots of the polynomial at the given accuracy. - -examples: - - ____________________________________________________________ - - realroots(x^5-2); - - {x=1.1487} - - - realroots(x^3-104*x^2+403*x-300,2,infinity); - - - {x=3.0,x=100.0} - - - realroots(x^3-104*x^2+403*x-300,-infinity,2); - - - {x=1} - - ____________________________________________________________ - The minimal accuracy of the result values is controlled by [*note -ROOTACC::.] . - - -File: redhelp, Node: ROOTACC, Next: ROOTS, Prev: REALROOTS, Up: Roots Package section - - ROOTACC operator - - The operator ROOTACC allows you to set the accuracy up to which the -roots package computes its results. - -syntax: - - ROOTACC () - - Here is an integer value. The internal accuracy of the ROOTS -package is adjusted to a value of MAX(6,N) . The default value is 6 . - - -File: redhelp, Node: ROOTS, Next: ROOT_VAL, Prev: ROOTACC, Up: Roots Package section - - ROOTS operator - - The operator ROOTS is the main top level function of the roots -package. It will find all roots, real and complex, of the polynomial p -to an accuracy that is sufficient to separate them and which is a -minimum of 6 decimal places. - -syntax: - - ROOTS (

) - - where

is a univariate polynomial. Result is a [*note LIST::.] of -equations which represent the roots of the polynomial at the given -accuracy. In addition, ROOTS stores separate lists of real roots and -complex roots in the global variables [*note ROOTSREAL::.] and [*note -ROOTSCOMPLEX::.] . - -examples: - - ____________________________________________________________ - - roots(x^5-2); - - {x=-0.929316 + 0.675188*i, - x=-0.929316 - 0.675188*i, - x=0.354967 + 1.09248*i, - x=0.354967 - 1.09248*i, - x=1.1487} - - ____________________________________________________________ - The minimal accuracy of the result values is controlled by [*note -ROOTACC::.] . - - -File: redhelp, Node: ROOT_VAL, Next: ROOTSCOMPLEX, Prev: ROOTS, Up: Roots Package section - - ROOT_VAL operator - - The operator ROOT_VAL computes the roots of a univariate polynomial -at system precision (or greater if required for root separation) and -presents its result as a list of numbers. - -syntax: - - ROOTS (

) - - where

is a univariate polynomial. - -examples: - - ____________________________________________________________ - - root_val(x^5-2); - - {-0.929316490603 + 0.6751879524*i, - -0.929316490603 - 0.6751879524*i, - 0.354967313105 + 1.09247705578*i, - 0.354967313105 - 1.09247705578*i, - 1.148698355} - - ____________________________________________________________ - - -File: redhelp, Node: ROOTSCOMPLEX, Next: ROOTSREAL, Prev: ROOT_VAL, Up: Roots Package section - - ROOTSCOMPLEX variable - - When the operator [*note ROOTS::.] is called the complex roots are -collected in the global variable ROOTSCOMPLEX as [*note LIST::.] . - - -File: redhelp, Node: ROOTSREAL, Prev: ROOTSCOMPLEX, Up: Roots Package section - - ROOTSREAL variable - - When the operator [*note ROOTS::.] is called the real roots are -collected in the global variable ROOTREAL as [*note LIST::.] . - - -File: redhelp, Node: Roots Package section, Next: Special Functions section, Prev: Numeric Package section, Up: Top - - Roots Package section - -* Menu: - -* Roots Package:: introduction -* MKPOLY:: operator -* NEARESTROOT:: operator -* REALROOTS:: operator -* ROOTACC:: operator -* ROOTS:: operator -* ROOT_VAL:: operator -* ROOTSCOMPLEX:: variable -* ROOTSREAL:: variable - - -File: redhelp, Node: Special Function Package, Next: Constants, Up: Special Functions section - - SPECIAL FUNCTION PACKAGE introduction - - The REDUCE SPECIAL FUNCTION PACKAGE supplies extended algebraic and -numeric support for a wide class of objects. This package was released -together with REDUCE 3.5 (October 1993) for the first time, a major -update is released with REDUCE 3.6. - - The functions included in this package are in most cases (unless -otherwise stated) defined and named like in the book by Abramowitz and -Stegun: Handbook of Mathematical Functions, Dover Publications. - - The aim is to collect as much information on the special functions -and simplification capabilities as possible, i.e. algebraic -simplifications and numeric (rounded mode) code, limits of the -functions together with the definitions of the functions, which are in -most cases a power series, a (definite) integral and/or a differential -equation. - - What can be found: Some famous constants, a variety of Bessel -functions, special polynomials, the Gamma function, the (Riemann) Zeta -function, Elliptic Functions, Elliptic Integrals, 3J symbols -(Clebsch-Gordan coefficients) and integral functions. - - What is missing: Mathieu functions, LerchPhi, etc.. The information -about the special functions which solve certain differential equation -is very limited. In several cases numerical approximation is -restricted to real arguments or is missing completely. - - The implementation of this package uses REDUCE rule sets to a large -extent, which guarantees a high 'readability' of the functions -definitions in the source file directory. It makes extensions to the -special functions code easy in most cases too. To look at these rules -it may be convenient to use the showrules operator e.g. - - [*note SHOWRULES::.] Besseli; - - . - - Some evaluations are improved if the special function package is -loaded, e.g. some (infinite) sums and products leading to expressions -including special functions are known in this case. - - Note: The special function package has to be loaded explicitly by -calling - ____________________________________________________________ - - load_package specfn; - ____________________________________________________________ - - The functions [*note MeijerG::.] and [*note HYPERGEOMETRIC::.] -require additionally - ____________________________________________________________ - - load_package specfn2; - ____________________________________________________________ - - -File: redhelp, Node: Constants, Next: Bernoulli Euler Zeta section, Prev: Special Function Package, Up: Special Functions section - - CONSTANTS - - There are a few constants known to the special function package, -namely - - EULER CONSTANT (which can be computed as -[*note PSI::.] (1)) and - - KHINCHIN CONSTANT (which is defined in Khinchin's book - - "Continued Fractions") and - - GOLDEN_RATIO (which can be computed as (1 + sqrt 5)/2) and - - CATALAN CONSTANT (which is known as an infinite sum of reciprocal -powers) - -examples: - - ____________________________________________________________ - - on rounded; - Euler_Gamma; - - 0.577215664902 - - - Khinchin; - - 2.68545200107 - - - Catalan - - 0.915965594177 - - - Golden_Ratio - - 1.61803398875 - - ____________________________________________________________ - - -File: redhelp, Node: BERNOULLI, Next: BERNOULLIP, Up: Bernoulli Euler Zeta section - - BERNOULLI operator - - The BERNOULLI operator returns the nth Bernoulli number. - -syntax: - - BERNOULLI () - -examples: - - ____________________________________________________________ - - bernoulli 20; - - - 174611 / 330 - - - bernoulli 17; - - 0 - - ____________________________________________________________ - All Bernoulli numbers with odd indices except for 1 are zero. - - -File: redhelp, Node: BERNOULLIP, Next: EULER, Prev: BERNOULLI, Up: Bernoulli Euler Zeta section - - BERNOULLIP operator - - The BERNOULLIP operator returns the nth Bernoulli Polynomial -evaluated at x. - -syntax: - - BERNOULLIP (,) - -examples: - - ____________________________________________________________ - - BernoulliP(3,z); - - 2 - z*(2*z - 3*z + 1)/2 - - - - BernoulliP(10,3); - - 338585 / 66 - - ____________________________________________________________ - The value of the nth Bernoulli Polynomial at 0 is the nth Bernoulli -number. - - -File: redhelp, Node: EULER, Next: EULERP, Prev: BERNOULLIP, Up: Bernoulli Euler Zeta section - - EULER operator - - The EULER operator returns the nth Euler number. - -syntax: - - EULER () - -examples: - - ____________________________________________________________ - - Euler 20; - - 370371188237525 - - - Euler 0; - - 1 - - ____________________________________________________________ - The EULER numbers are evaluated by a recursive algorithm which makes -it hard to compute Euler numbers above say 200. - - Euler numbers appear in the coefficients of the power series -representation of 1/cos(z). - - -File: redhelp, Node: EULERP, Next: ZETA, Prev: EULER, Up: Bernoulli Euler Zeta section - - EULERP operator - - The EULERP operator returns the nth Euler Polynomial. - -syntax: - - EULERP (,) - -examples: - - ____________________________________________________________ - - EulerP(2,xx); - - xx*(xx - 1) - - - EulerP(10,3); - - 2046 - - ____________________________________________________________ - The Euler numbers are the values of the Euler Polynomials at 1/2 -multiplied by 2**n. - - -File: redhelp, Node: ZETA, Prev: EULERP, Up: Bernoulli Euler Zeta section - - ZETA operator - - The ZETA operator returns Riemann's Zeta function, - - Zeta (z) := sum(1/(k**z),k,1,infinity) - -syntax: - - ZETA () - -examples: - - ____________________________________________________________ - - Zeta(2); - - 2 - pi / 6 - - - on rounded; - - Zeta 1.01; - - 100.577943338 - - ____________________________________________________________ - Numerical computation for the Zeta function for arguments close to 1 -are tedious, because the series is converging very slowly. In this case -a formula (e.g. found in Bender/Orzag: Advanced Mathematical Methods for -Scientists and Engineers, McGraw-Hill) is used. - - No numerical approximation for complex arguments is done. - - -File: redhelp, Node: Bernoulli Euler Zeta section, Next: Bessel Functions section, Prev: Constants, Up: Special Functions section - - Bernoulli Euler Zeta section - -* Menu: - -* BERNOULLI:: operator -* BERNOULLIP:: operator -* EULER:: operator -* EULERP:: operator -* ZETA:: operator - - -File: redhelp, Node: BESSELJ, Next: BESSELY, Up: Bessel Functions section - - BESSELJ operator - - The BESSELJ operator returns the Bessel function of the first kind. - -syntax: - - BESSELJ (,) - -examples: - - ____________________________________________________________ - - BesselJ(1/2,pi); - - 0 - - - on rounded; - - BesselJ(0,1); - - 0.765197686558 - - ____________________________________________________________ - - -File: redhelp, Node: BESSELY, Next: HANKEL1, Prev: BESSELJ, Up: Bessel Functions section - - BESSELY operator - - The BESSELY operator returns the Bessel function of the second kind. - -syntax: - - BESSELY (,) - -examples: - - ____________________________________________________________ - - BesselY (1/2,pi); - - - sqrt(2) / pi - - - on rounded; - - BesselY (1,3); - - 0.324674424792 - - ____________________________________________________________ - The operator BESSELY is also called Weber's function. - - -File: redhelp, Node: HANKEL1, Next: HANKEL2, Prev: BESSELY, Up: Bessel Functions section - - HANKEL1 operator - - The HANKEL1 operator returns the Hankel function of the first kind. - -syntax: - - HANKEL1 (,) - -examples: - - ____________________________________________________________ - - on complex; - - Hankel1 (1/2,pi); - - - i * sqrt(2) / pi - - - Hankel1 (1,pi); - - besselj(1,pi) + i*bessely(1,pi) - - ____________________________________________________________ - The operator HANKEL1 is also called Bessel function of the third -kind. There is currently no numeric evaluation of Hankel functions. - - -File: redhelp, Node: HANKEL2, Next: BESSELI, Prev: HANKEL1, Up: Bessel Functions section - - HANKEL2 operator - - The HANKEL2 operator returns the Hankel function of the second kind. - -syntax: - - HANKEL2 (,) - -examples: - - ____________________________________________________________ - - on complex; - - Hankel2 (1/2,pi); - - - i * sqrt(2) / pi - - - Hankel2 (1,pi); - - besselj(1,pi) - i*bessely(1,pi) - - ____________________________________________________________ - The operator HANKEL2 is also called Bessel function of the third -kind. There is currently no numeric evaluation of Hankel functions. - - -File: redhelp, Node: BESSELI, Next: BESSELK, Prev: HANKEL2, Up: Bessel Functions section - - BESSELI operator - - The BESSELI operator returns the modified Bessel function I. - -syntax: - - BESSELI (,) - -examples: - - ____________________________________________________________ - - on rounded; - - Besseli (1,1); - - 0.565159103992 - - ____________________________________________________________ - The knowledge about the operator BESSELI is currently fairly limited. - - -File: redhelp, Node: BESSELK, Next: StruveH, Prev: BESSELI, Up: Bessel Functions section - - BESSELK operator - - The BESSELK operator returns the modified Bessel function K. - -syntax: - - BESSELK (,) - -examples: - - ____________________________________________________________ - - df(besselk(0,x),x); - - - besselk(1,x) - - ____________________________________________________________ - There is currently no numeric support for the operator BESSELK . - - -File: redhelp, Node: StruveH, Next: StruveL, Prev: BESSELK, Up: Bessel Functions section - - STRUVEH operator - - The STRUVEH operator returns Struve's H function. - -syntax: - - STRUVEH (,) - -examples: - - ____________________________________________________________ - - struveh(-3/2,x); - - - besselj(3/2,x) / i - - ____________________________________________________________ - - -File: redhelp, Node: StruveL, Next: KummerM, Prev: StruveH, Up: Bessel Functions section - - STRUVEL operator - - The STRUVEL operator returns the modified Struve L function . - -syntax: - - STRUVEL (,) - -examples: - - ____________________________________________________________ - - struvel(-3/2,x); - - besseli(3/2,x) - - ____________________________________________________________ - - -File: redhelp, Node: KummerM, Next: KummerU, Prev: StruveL, Up: Bessel Functions section - - KUMMERM operator - - The KUMMERM operator returns Kummer's M function. - -syntax: - - KUMMERM (,,) - -examples: - - ____________________________________________________________ - - kummerm(1,1,x); - - x - e - - - on rounded; - - kummerm(1,3,1.3); - - 1.62046942914 - - ____________________________________________________________ - Kummer's M function is one of the Confluent Hypergeometric functions. -For reference see the [*note HYPERGEOMETRIC::.] operator. - - -File: redhelp, Node: KummerU, Next: WhittakerW, Prev: KummerM, Up: Bessel Functions section - - KUMMERU operator - - The KUMMERU operator returns Kummer's U function. - -syntax: - - KUMMERU (,,) - -examples: - - ____________________________________________________________ - - df(kummeru(1,1,x),x) - - - kummeru(2,2,x) - - ____________________________________________________________ - Kummer's U function is one of the Confluent Hypergeometric functions. -For reference see the [*note HYPERGEOMETRIC::.] operator. - - -File: redhelp, Node: WhittakerW, Prev: KummerU, Up: Bessel Functions section - - WHITTAKERW operator - - The WHITTAKERW operator returns Whittaker's W function. - -syntax: - - WHITTAKERW (,,) - -examples: - - ____________________________________________________________ - - WhittakerW(2,2,2); - - 1 - 4*sqrt(2)*kummeru(-,5,2) - 2 - ------------------------- - e - - ____________________________________________________________ - Whittaker's W function is one of the Confluent Hypergeometric -functions. For reference see the [*note HYPERGEOMETRIC::.] operator. - - -File: redhelp, Node: Bessel Functions section, Next: Airy Functions section, Prev: Bernoulli Euler Zeta section, Up: Special Functions section - - Bessel Functions section - -* Menu: - -* BESSELJ:: operator -* BESSELY:: operator -* HANKEL1:: operator -* HANKEL2:: operator -* BESSELI:: operator -* BESSELK:: operator -* StruveH:: operator -* StruveL:: operator -* KummerM:: operator -* KummerU:: operator -* WhittakerW:: operator - - -File: redhelp, Node: Airy_Ai, Next: Airy_Bi, Up: Airy Functions section - - AIRY_AI operator - - The AIRY_AI operator returns the Airy Ai function for a given -argument. - -syntax: - - AIRY_AI () - -examples: - - ____________________________________________________________ - - on complex; - on rounded; - Airy_Ai(0); - - - 0.355028053888 - - - Airy_Ai(3.45 + 17.97i); - - - 5.5561528511e+9 - 8.80397899932e+9*i - - ____________________________________________________________ - - -File: redhelp, Node: Airy_Bi, Next: Airy_Aiprime, Prev: Airy_Ai, Up: Airy Functions section - - AIRY_BI operator - - The AIRY_BI operator returns the Airy Bi function for a given -argument. - -syntax: - - AIRY_BI () - -examples: - - ____________________________________________________________ - - Airy_Bi(0); - - 0.614926627446 - - - Airy_Bi(3.45 + 17.97i); - - 8.80397899932e+9 - 5.5561528511e+9*i - - ____________________________________________________________ - - -File: redhelp, Node: Airy_Aiprime, Next: Airy_Biprime, Prev: Airy_Bi, Up: Airy Functions section - - AIRY_AIPRIME operator - - The AIRY_AIPRIME operator returns the Airy Aiprime function for a -given argument. - -syntax: - - AIRY_AIPRIME () - -examples: - - ____________________________________________________________ - - Airy_Aiprime(0); - - - 0.258819403793 - - - Airy_Aiprime(3.45+17.97i); - - - 3.83386421824e+19 + 2.16608828136e+19*i - - ____________________________________________________________ - - -File: redhelp, Node: Airy_Biprime, Prev: Airy_Aiprime, Up: Airy Functions section - - AIRY_BIPRIME operator - - The AIRY_BIPRIME operator returns the Airy Biprime function for a -given argument. - -syntax: - - AIRY_BIPRIME () - -examples: - - ____________________________________________________________ - - Airy_Biprime(0); - - - Airy_Biprime(3.45 + 17.97i); - - 3.84251916792e+19 - 2.18006297399e+19*i - - ____________________________________________________________ - - -File: redhelp, Node: Airy Functions section, Next: Jacobi Elliptic Functions and Elliptic Integrals section, Prev: Bessel Functions section, Up: Special Functions section - - Airy Functions section - -* Menu: - -* Airy_Ai:: operator -* Airy_Bi:: operator -* Airy_Aiprime:: operator -* Airy_Biprime:: operator - - -File: redhelp, Node: JacobiSN, Next: JacobiCN, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBISN operator - - The JACOBISN operator returns the Jacobi Elliptic function sn. - -syntax: - - JACOBISN (,) - -examples: - - ____________________________________________________________ - - Jacobisn(0.672, 0.36) - - 0.609519691792 - - - Jacobisn(1,0.9) - - 0.770085724907881 - - ____________________________________________________________ - - -File: redhelp, Node: JacobiCN, Next: JacobiDN, Prev: JacobiSN, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBICN operator - - The JACOBICN operator returns the Jacobi Elliptic function cn. - -syntax: - - JACOBICN (,) - -examples: - - ____________________________________________________________ - - Jacobicn(7.2, 0.6) - - 0.837288298482018 - - - Jacobicn(0.11, 19) - - 0.994403862690043 - 1.6219006985556e-16*i - - ____________________________________________________________ - - -File: redhelp, Node: JacobiDN, Next: JacobiCD, Prev: JacobiCN, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBIDN operator - - The JACOBIDN operator returns the Jacobi Elliptic function dn. - -syntax: - - JACOBIDN (,) - -examples: - - ____________________________________________________________ - - Jacobidn(15, 0.683) - - 0.640574162024592 - - - Jacobidn(0,0) - - 1 - - ____________________________________________________________ - - -File: redhelp, Node: JacobiCD, Next: JacobiSD, Prev: JacobiDN, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBICD operator - - The JACOBICD operator returns the Jacobi Elliptic function cd. - -syntax: - - JACOBICD (,) - -examples: - - ____________________________________________________________ - - Jacobicd(1, 0.34) - - 0.657683337805273 - - - Jacobicd(0.8,0.8) - - 0.925587311582301 - - ____________________________________________________________ - - -File: redhelp, Node: JacobiSD, Next: JacobiND, Prev: JacobiCD, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBISD operator - - The JACOBISD operator returns the Jacobi Elliptic function sd. - -syntax: - - JACOBISD (,) - -examples: - - ____________________________________________________________ - - Jacobisd(12, 0.4) - - 0.357189729437272 - - - Jacobisd(0.35,1) - - - 1.17713873203043 - - ____________________________________________________________ - - -File: redhelp, Node: JacobiND, Next: JacobiDC, Prev: JacobiSD, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBIND operator - - The JACOBIND operator returns the Jacobi Elliptic function nd. - -syntax: - - JACOBIND (,) - -examples: - - ____________________________________________________________ - - Jacobind(0.2, 17) - - 1.46553203037507 + 0.0000000000334032759313703*i - - - Jacobind(30, 0.001) - - 1.00048958438 - - ____________________________________________________________ - - -File: redhelp, Node: JacobiDC, Next: JacobiNC, Prev: JacobiND, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBIDC operator - - The JACOBIDC operator returns the Jacobi Elliptic function dc. - -syntax: - - JACOBIDC (,) - -examples: - - ____________________________________________________________ - - Jacobidc(0.003,1) - - 1 - - - Jacobidc(2, 0.75) - - 6.43472885111 - - ____________________________________________________________ - - -File: redhelp, Node: JacobiNC, Next: JacobiSC, Prev: JacobiDC, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBINC operator - - The JACOBINC operator returns the Jacobi Elliptic function nc. - -syntax: - - JACOBINC (,) - -examples: - - ____________________________________________________________ - - Jacobinc(1,0) - - 1.85081571768093 - - - Jacobinc(56, 0.4387) - - 39.304842663512 - - ____________________________________________________________ - - -File: redhelp, Node: JacobiSC, Next: JacobiNS, Prev: JacobiNC, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBISC operator - - The JACOBISC operator returns the Jacobi Elliptic function sc. - -syntax: - - JACOBISC (,) - -examples: - - ____________________________________________________________ - - Jacobisc(9, 0.88) - - - 1.16417697982095 - - - Jacobisc(0.34, 7) - - 0.305851938390775 - 9.8768100944891e-12*i - - ____________________________________________________________ - - -File: redhelp, Node: JacobiNS, Next: JacobiDS, Prev: JacobiSC, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBINS operator - - The JACOBINS operator returns the Jacobi Elliptic function ns. - -syntax: - - JACOBINS (,) - -examples: - - ____________________________________________________________ - - Jacobins(3, 0.9) - - 1.00945801599785 - - - Jacobins(0.887, 15) - - 0.683578280513975 - 0.85023411082469*i - - ____________________________________________________________ - - -File: redhelp, Node: JacobiDS, Next: JacobiCS, Prev: JacobiNS, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBIDS operator - - The JACOBISN operator returns the Jacobi Elliptic function ds. - -syntax: - - JACOBIDS (,) - -examples: - - ____________________________________________________________ - - Jacobids(98,0.223) - - - 1.061253961477 - - - Jacobids(0.36,0.6) - - 2.76693172243692 - - ____________________________________________________________ - - -File: redhelp, Node: JacobiCS, Next: JacobiAMPLITUDE, Prev: JacobiDS, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBICS operator - - The JACOBICS operator returns the Jacobi Elliptic function cs. - -syntax: - - JACOBICS (,) - -examples: - - ____________________________________________________________ - - Jacobics(0, 0.767) - - infinity - - - Jacobics(1.43, 0) - - 0.141734127352112 - - ____________________________________________________________ - - -File: redhelp, Node: JacobiAMPLITUDE, Next: AGM_FUNCTION, Prev: JacobiCS, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBIAMPLITUDE operator - - The JACOBIAMPLITUDE operator returns the amplitude of u. - -syntax: - - JACOBIAMPLITUDE (,) - -examples: - - ____________________________________________________________ - - JacobiAmplitude(7.239, 0.427) - - 0.0520978301448978 - - - JacobiAmplitude(0,0.1) - - 0 - - ____________________________________________________________ - Amplitude u = asin(JACOBISN(U,M) ) - - -File: redhelp, Node: AGM_FUNCTION, Next: LANDENTRANS, Prev: JacobiAMPLITUDE, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - AGM_FUNCTION operator - - The AGM_FUNCTION operator returns a list of (N, AGM, list of -aNtoa0, list of bNtob0, list of cNtoc0) where a0, b0 and c0 are the -initial values; N is the index number of the last term used to generate -the AGM. AGM is the Arithmetic Geometric Mean. - -syntax: - - AGM_FUNCTION (,,) - -examples: - - ____________________________________________________________ - - AGM_function(1,1,1) - - 1,1,1,1,1,1,0,1 - - - AGM_function(1, 0.1, 1.3) - - {6, - 2.27985615996629, - {2.27985615996629, 2.27985615996629, - 2.2798561599706, 2.2798624278857, - 2.28742283656583, 2.55, 1}, - {2.27985615996629, 2.27985615996629, - 2.27985615996198, 2.2798498920555, - 2.27230201920557, 2.02484567313166, 4.1}, - {0, 4.30803136219904e-12, 0.0000062679151007581, - 0.00756040868012758, 0.262577163434171, - 1.55, 5.9}} - - ____________________________________________________________ - The other Jacobi functions use this function with initial values -a0=1, b0=sqrt(1-m), c0=sqrt(m). - - -File: redhelp, Node: LANDENTRANS, Next: EllipticF, Prev: AGM_FUNCTION, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - LANDENTRANS operator - - The LANDENTRANS operator generates the descending landen -transformation of the given imput values, returning a list of these -values; initial to final in each case. - -syntax: - - LANDENTRANS (,) - -examples: - - ____________________________________________________________ - - landentrans(0,0.1) - - {{0,0,0,0,0},{0.1,0.0025041751943776, - - - - - 0.00000156772498954046,6.1444078 9914461e-13,0}} - - ____________________________________________________________ - The first list ascends in value, and the second descends in value. - - -File: redhelp, Node: EllipticF, Next: EllipticK, Prev: LANDENTRANS, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - ELLIPTICF operator - - The ELLIPTICF operator returns the Elliptic Integral of the First -Kind. - -syntax: - - ELLITPICF (,) - -examples: - - ____________________________________________________________ - - EllipticF(0.3, 8.222) - - 0.3 - - - EllipticF(7.396, 0.1) - - 7.58123216114307 - - ____________________________________________________________ - The Complete Elliptic Integral of the First Kind can be found by -putting the first argument to pi/2 or by using ELLIPTICK and the second -argument. - - -File: redhelp, Node: EllipticK, Next: EllipticKprime, Prev: EllipticF, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - ELLIPTICK operator - - The ELLIPTICK operator returns the Elliptic value K. - -syntax: - - ELLIPTICK () - -examples: - - ____________________________________________________________ - - EllipticK(0.2) - - 1.65962359861053 - - - EllipticK(4.3) - - 0.808442364282734 - 1.05562492399206*i - - - EllipticK(0.000481) - - 1.57098526617635 - - ____________________________________________________________ - The ELLIPTICK function is the Complete Elliptic Integral of the -First Kind. - - -File: redhelp, Node: EllipticKprime, Next: EllipticE, Prev: EllipticK, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - ELLIPTICKPRIME operator - - The ELLIPTICK' operator returns the Elliptic value K(m). - -syntax: - - ELLIPTICKPRIME () - -examples: - - ____________________________________________________________ - - EllipticKprime(0.2) - - 2.25720532682085 - - - EllipticKprime(4.3) - - 1.05562492399206 - - - EllipticKprime(0.000481) - - 5.206621921966 - - ____________________________________________________________ - The ELLIPTICKPRIME function is the Complete Elliptic Integral of the -First Kind of (1-m). - - -File: redhelp, Node: EllipticE, Next: EllipticTHETA, Prev: EllipticKprime, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - ELLIPTICE operator - - The ELLIPTICE operator used with two arguments returns the Elliptic -Integral of the Second Kind. - -syntax: - - ELLIPTICE (,) - -examples: - - ____________________________________________________________ - - EllipticE(1.2,0.22) - - 1.15094019180949 - - - EllipticE(0,4.35) - - 0 - - - EllipticE(9,0.00719) - - 8.98312465929145 - - ____________________________________________________________ - The Complete Elliptic Integral of the Second Kind can be obtained by -using just the second argument, or by using pi/2 as the first argument. - - The ELLIPTICE operator used with one argument returns the Elliptic -value E. - -syntax: - - ELLIPTICE () - -examples: - - ____________________________________________________________ - - EllipticE(0.22) - - 1.48046637439519 - - - EllipticE(pi/2, 0.22) - - 1.48046637439519 - - ____________________________________________________________ - - -File: redhelp, Node: EllipticTHETA, Next: JacobiZETA, Prev: EllipticE, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - ELLIPTICTHETA operator - - The ELLIPTICTHETA operator returns one of the four Theta functions. -It cannot except any number other than 1,2,3 or 4 as its first argument. - -syntax: - - ELLIPTICTHETA (,,) - -examples: - - ____________________________________________________________ - - EllipticTheta(1, 1.4, 0.72) - - 0.91634775373 - - - EllipticTheta(2, 3.9, 6.1 ) - - -48.0202736969 + 20.9881034377 i - - - EllipticTheta(3, 0.67, 0.2) - - 1.0083077448 - - - EllipticTheta(4, 8, 0.75) - - 0.894963369304 - - - EllipticTheta(5, 1, 0.1) - - ***** In EllipticTheta(a,u,m); a = 1,2,3 or 4. - - ____________________________________________________________ - Theta functions are important because every one of the Jacobian -Elliptic functions can be expressed as the ratio of two theta functions. - - -File: redhelp, Node: JacobiZETA, Prev: EllipticTHETA, Up: Jacobi Elliptic Functions and Elliptic Integrals section - - JACOBIZETA operator - - The JACOBIZETA operator returns the Jacobian function Zeta. - -syntax: - - JACOBIZETA (,) - -examples: - - ____________________________________________________________ - - JacobiZeta(3.2, 0.8) - - - 0.254536403439 - - - JacobiZeta(0.2, 1.6) - - 0.171766095970451 - 0.0717028569800147*i - - ____________________________________________________________ - The Jacobian function Zeta is related to the Jacobian function Theta. -But it is significantly different from Riemann's Zeta Function [*note -ZETA::.] . - - -File: redhelp, Node: Jacobi Elliptic Functions and Elliptic Integrals section, Next: Gamma and Related Functions section, Prev: Airy Functions section, Up: Special Functions section - - Jacobi Elliptic Functions and Elliptic Integrals section - -* Menu: - -* JacobiSN:: operator -* JacobiCN:: operator -* JacobiDN:: operator -* JacobiCD:: operator -* JacobiSD:: operator -* JacobiND:: operator -* JacobiDC:: operator -* JacobiNC:: operator -* JacobiSC:: operator -* JacobiNS:: operator -* JacobiDS:: operator -* JacobiCS:: operator -* JacobiAMPLITUDE:: operator -* AGM_FUNCTION:: operator -* LANDENTRANS:: operator -* EllipticF:: operator -* EllipticK:: operator -* EllipticKprime:: operator -* EllipticE:: operator -* EllipticTHETA:: operator -* JacobiZETA:: operator - - -File: redhelp, Node: POCHHAMMER, Next: GAMMA, Up: Gamma and Related Functions section - - POCHHAMMER operator - - The POCHHAMMER operator implements the Pochhammer notation (shifted -factorial). - -syntax: - - POCHHAMMER (,) - -examples: - - ____________________________________________________________ - - pochhammer(17,4); - - 116280 - - - - pochhammer(1/2,z); - - factorial(2*z) - -------------------- - 2*z - (2 *factorial(z)) - - ____________________________________________________________ - A number of complex rules for POCHHAMMER are inactive, because they -cause a huge system load in algebraic mode. If one wants to use more -rules for the simplification of Pochhammer's notation, one can do: - - let special!*pochhammer!*rules; - - -File: redhelp, Node: GAMMA, Next: BETA, Prev: POCHHAMMER, Up: Gamma and Related Functions section - - GAMMA operator - - The GAMMA operator returns the Gamma function. - -syntax: - - GAMMA () - -examples: - - ____________________________________________________________ - - gamma(10); - - 362880 - - - gamma(1/2); - - sqrt(pi) - - ____________________________________________________________ - - -File: redhelp, Node: BETA, Next: PSI, Prev: GAMMA, Up: Gamma and Related Functions section - - BETA operator - - The BETA operator returns the Beta function defined by - - Beta (z,w) := defint(t**(z-1)* (1 - t)**(w-1),t,0,1) . - -syntax: - - BETA (,) - -examples: - - ____________________________________________________________ - - Beta(2,2); - - 1 / 6 - - - Beta(x,y); - - gamma(x)*gamma(y) / gamma(x + y) - - ____________________________________________________________ - The operator BETA is simplified towards the [*note GAMMA::.] -operator. - - -File: redhelp, Node: PSI, Next: POLYGAMMA, Prev: BETA, Up: Gamma and Related Functions section - - PSI operator - - The PSI operator returns the Psi (or DiGamma) function. - - Psi(x) := df(Gamma(z),z)/ Gamma (z) - -syntax: - - GAMMA () - -examples: - - ____________________________________________________________ - - Psi(3); - - (2*log(2) + psi(1/2) + psi(1) + 3)/2 - - - on rounded; - - - Psi(1); - - 0.577215664902 - - ____________________________________________________________ - Euler's constant can be found as - Psi(1). - - -File: redhelp, Node: POLYGAMMA, Prev: PSI, Up: Gamma and Related Functions section - - POLYGAMMA operator - - The POLYGAMMA operator returns the Polygamma function. - - Polygamma(n,x) := df(Psi(z),z,n); - -syntax: - - POLYGAMMA (,) - -examples: - - ____________________________________________________________ - - Polygamma(1,2); - - 2 - (pi - 6) / 6 - - - on rounded; - - Polygamma(1,2.35); - - 0.52849689109 - - ____________________________________________________________ - The Polygamma function is used for simplification of the [*note -ZETA::.] function for some arguments. - - -File: redhelp, Node: Gamma and Related Functions section, Next: Miscellaneous Functions section, Prev: Jacobi Elliptic Functions and Elliptic Integrals section, Up: Special Functions section - - Gamma and Related Functions section - -* Menu: - -* POCHHAMMER:: operator -* GAMMA:: operator -* BETA:: operator -* PSI:: operator -* POLYGAMMA:: operator - - -File: redhelp, Node: DILOG extended, Next: Lambert_W function, Up: Miscellaneous Functions section - - DILOG EXTENDED operator - - The package SPECFN supplies an extended support for the [*note -DILOG::.] operator which implements the DILOGARITHM FUNCTION . - - dilog(x) := - defint(log(t)/(t - 1),t,1,x); - -syntax: - - DILOG (,) - -examples: - - ____________________________________________________________ - - defint(log(t)/(t - 1),t,1,x); - - - dilog (x) - - - dilog 2; - - 2 - - pi /12 - - - - on rounded; - - Dilog 20; - - - 5.92783972438 - - ____________________________________________________________ - The operator DILOG is sometimes called Spence's Integral for n = 2. - - -File: redhelp, Node: Lambert_W function, Prev: DILOG extended, Up: Miscellaneous Functions section - - LAMBERT_W FUNCTION operator - - Lambert's W function is the inverse of the function w * e^w. It is -used in the [*note SOLVE::.] package for equations containing -exponentials and logarithms. - -syntax: - - LAMBERT_W () - -examples: - - ____________________________________________________________ - - Lambert_W(-1/e); - - -1 - - - solve(w + log(w),w); - - w=lambert_w(1) - - - on rounded; - - Lambert_W(-0.05); - - - 0.0527059835515 - - ____________________________________________________________ - The current implementation will compute the principal branch in -rounded mode only. - - -File: redhelp, Node: Miscellaneous Functions section, Next: Orthogonal Polynomials section, Prev: Gamma and Related Functions section, Up: Special Functions section - - Miscellaneous Functions section - -* Menu: - -* DILOG extended:: operator -* Lambert_W function:: operator - - -File: redhelp, Node: ChebyshevT, Next: ChebyshevU, Up: Orthogonal Polynomials section - - CHEBYSHEVT operator - - The CHEBYSHEVT operator computes the nth Chebyshev T Polynomial (of -the first kind). - -syntax: - - CHEBYSHEVT (,) - -examples: - - ____________________________________________________________ - - ChebyshevT(3,xx); - - 2 - xx*(4*xx - 3) - - - - ChebyshevT(3,4); - - 244 - - ____________________________________________________________ - Chebyshev's T polynomials are computed using the recurrence relation: - - ChebyshevT(n,x) := 2x*ChebyshevT(n-1,x) - ChebyshevT(n-2,x) with - - ChebyshevT(0,x) := 0 and ChebyshevT(1,x) := x - - -File: redhelp, Node: ChebyshevU, Next: HermiteP, Prev: ChebyshevT, Up: Orthogonal Polynomials section - - CHEBYSHEVU operator - - The CHEBYSHEVU operator returns the nth Chebyshev U Polynomial (of -the second kind). - -syntax: - - CHEBYSHEVU (,) - -examples: - - ____________________________________________________________ - - ChebyshevU(3,xx); - - 2 - 4*x*(2*x - 1) - - - - ChebyshevU(3,4); - - 496 - - ____________________________________________________________ - Chebyshev's U polynomials are computed using the recurrence relation: - - ChebyshevU(n,x) := 2x*ChebyshevU(n-1,x) - ChebyshevU(n-2,x) with - - ChebyshevU(0,x) := 0 and ChebyshevU(1,x) := 2x - - -File: redhelp, Node: HermiteP, Next: LaguerreP, Prev: ChebyshevU, Up: Orthogonal Polynomials section - - HERMITEP operator - - The HERMITEP operator returns the nth Hermite Polynomial. - -syntax: - - HERMITEP (,) - -examples: - - ____________________________________________________________ - - HermiteP(3,xx); - - 2 - 4*xx*(2*xx - 3) - - - HermiteP(3,4); - - 464 - - ____________________________________________________________ - Hermite polynomials are computed using the recurrence relation: - - HermiteP(n,x) := 2x*HermiteP(n-1,x) - 2*(n-1)*HermiteP(n-2,x) with - - HermiteP(0,x) := 1 and HermiteP(1,x) := 2x - - -File: redhelp, Node: LaguerreP, Next: LegendreP, Prev: HermiteP, Up: Orthogonal Polynomials section - - LAGUERREP operator - - The LAGUERREP operator computes the nth Laguerre Polynomial. The -two argument call of LaguerreP is a (common) abbreviation of -LaguerreP(n,0,x). - -syntax: - - LAGUERREP (,) or - - LAGUERREP (,,) - -examples: - - ____________________________________________________________ - - LaguerreP(3,xx); - - 3 2 - (- xx + 9*xx - 18*xx + 6)/6 - - - - LaguerreP(2,3,4); - - -2 - - ____________________________________________________________ - Laguerre polynomials are computed using the recurrence relation: - - LaguerreP(n,a,x) := (2n+a-1-x)/n*LaguerreP(n-1,a,x) - - (n+a-1) * LaguerreP(n-2,a,x) with - - LaguerreP(0,a,x) := 1 and LaguerreP(2,a,x) := -x+1+a - - -File: redhelp, Node: LegendreP, Next: JacobiP, Prev: LaguerreP, Up: Orthogonal Polynomials section - - LEGENDREP operator - - The binary LEGENDREP operator computes the nth Legendre Polynomial -which is a special case of the nth Jacobi Polynomial with - - LegendreP(n,x) := JacobiP(n,0,0,x) - - The ternary form returns the associated Legendre Polynomial (see -below). - -syntax: - - LEGENDREP (,) or - - LEGENDREP (,,) - -examples: - - ____________________________________________________________ - - LegendreP(3,xx); - - 2 - xx*(5*xx - 3) - ---------------- - 2 - - - - LegendreP(3,2,xx); - - 2 - 15*xx*( - xx + 1) - - ____________________________________________________________ - The ternary form of the operator LEGENDREP is the associated -Legendre Polynomial defined as - - P(n,m,x) = (-1)**m * (1-x**2)**(m/2) * df(LegendreP(n,x),x,m) - - -File: redhelp, Node: JacobiP, Next: GegenbauerP, Prev: LegendreP, Up: Orthogonal Polynomials section - - JACOBIP operator - - The JACOBIP operator computes the nth Jacobi Polynomial. - -syntax: - - JACOBIP (,,, ) - -examples: - - ____________________________________________________________ - - JacobiP(3,4,5,xx); - - 3 2 - 7*(65*xx - 13*xx - 13*xx + 1) - ---------------------------------- - 8 - - - - JacobiP(3,4,5,6); - - 94465/8 - - ____________________________________________________________ - - -File: redhelp, Node: GegenbauerP, Next: SolidHarmonicY, Prev: JacobiP, Up: Orthogonal Polynomials section - - GEGENBAUERP operator - - The GEGENBAUERP operator computes Gegenbauer's (ultraspherical) -polynomials. - -syntax: - - GEGENBAUERP (,,) - -examples: - - ____________________________________________________________ - - GegenbauerP(3,2,xx); - - 2 - 4*xx*(8*xx - 3) - - - - GegenbauerP(3,2,4); - - 2000 - - ____________________________________________________________ - - -File: redhelp, Node: SolidHarmonicY, Next: SphericalHarmonicY, Prev: GegenbauerP, Up: Orthogonal Polynomials section - - SOLIDHARMONICY operator - - The SOLIDHARMONICY operator computes Solid harmonic (Laplace) -polynomials. - -syntax: - - SOLIDHARMONICY (,, -,,,) - -examples: - - ____________________________________________________________ - - - SolidHarmonicY(3,-2,x,y,z,r2); - - 2 2 - sqrt(105)*z*(-2*i*x*y + x - y ) - --------------------------------- - 4*sqrt(pi)*sqrt(2) - - ____________________________________________________________ - - -File: redhelp, Node: SphericalHarmonicY, Prev: SolidHarmonicY, Up: Orthogonal Polynomials section - - SPHERICALHARMONICY operator - - The SPHERICALHARMONICY operator computes Spherical harmonic (Laplace) -polynomials. These are special cases of the solid harmonic polynomials, -[*note SolidHarmonicY::.] . - -syntax: - - SPHERICALHARMONICY (,, ,) - -examples: - - ____________________________________________________________ - - SphericalHarmonicY(3,2,theta,phi); - - - 2 2 2 - sqrt(105)*cos(theta)*sin(theta) *(cos(phi) +2*cos(phi)*sin(phi)*i-sin(phi) ) - ----------------------------------------------------------------------------- - 4*sqrt(pi)*sqrt(2) - - ____________________________________________________________ - - -File: redhelp, Node: Orthogonal Polynomials section, Next: Integral Functions section, Prev: Miscellaneous Functions section, Up: Special Functions section - - Orthogonal Polynomials section - -* Menu: - -* ChebyshevT:: operator -* ChebyshevU:: operator -* HermiteP:: operator -* LaguerreP:: operator -* LegendreP:: operator -* JacobiP:: operator -* GegenbauerP:: operator -* SolidHarmonicY:: operator -* SphericalHarmonicY:: operator - - -File: redhelp, Node: Si, Next: Shi, Up: Integral Functions section - - SI operator - - The SI operator returns the Sine Integral function. - -syntax: - - SI () - -examples: - - ____________________________________________________________ - - limit(Si(x),x,infinity); - - pi / 2 - - - on rounded; - - Si(0.35); - - 0.347626790989 - - ____________________________________________________________ - The numeric values for the operator SI are computed via the power -series representation, which limits the argument range. - - -File: redhelp, Node: Shi, Next: s_i, Prev: Si, Up: Integral Functions section - - SHI operator - - The SHI operator returns the hyperbolic Sine Integral function. - -syntax: - - SHI () - -examples: - - ____________________________________________________________ - - df(shi(x),x); - - sinh(x) / x - - - on rounded; - - Shi(0.35); - - 0.352390716351 - - ____________________________________________________________ - The numeric values for the operator SHI are computed via the power -series representation, which limits the argument range. - - -File: redhelp, Node: s_i, Next: Ci, Prev: Shi, Up: Integral Functions section - - S_I operator - - The S_I operator returns the Sine Integral function si. - -syntax: - - S_I () - -examples: - - ____________________________________________________________ - - s_i(xx); - - (2*Si(xx) - pi) / 2 - - - df(s_i(x),x); - - sin(x) / x - - ____________________________________________________________ - The operator name S_I is simplified towards [*note Si::.] . Since -REDUCE is not case sensitive by default the name "si" can't be used. - - -File: redhelp, Node: Ci, Next: Chi, Prev: s_i, Up: Integral Functions section - - CI operator - - The CI operator returns the Cosine Integral function. - -syntax: - - CI () - -examples: - - ____________________________________________________________ - - defint(cos(t)/t,t,x,infinity); - - - ci (x) - - - on rounded; - - Ci(0.35); - - - 0.50307556932 - - ____________________________________________________________ - The numeric values for the operator CI are computed via the power -series representation, which limits the argument range. - - -File: redhelp, Node: Chi, Next: ERF extended, Prev: Ci, Up: Integral Functions section - - CHI operator - - The CHI operator returns the Hyperbolic Cosine Integral function. - -syntax: - - CHI () - -examples: - - ____________________________________________________________ - - defint((cosh(t)-1)/t,t,0,x); - - - log(x) + psi(1) + chi(x) - - - on rounded; - - Chi(0.35); - - - 0.44182471827 - - ____________________________________________________________ - The numeric values for the operator CHI are computed via the power -series representation, which limits the argument range. - - -File: redhelp, Node: ERF extended, Next: erfc, Prev: Chi, Up: Integral Functions section - - ERF EXTENDED operator - - The special function package supplies an extended support for the -[*note ERF::.] operator which implements the ERROR FUNCTION - - defint(e**(-x**2),x,0,infinity) * 2/sqrt(pi) - - . - -syntax: - - ERF () - -examples: - - ____________________________________________________________ - - erf(-x); - - - erf(x) - - - on rounded; - - erf(0.35); - - 0.379382053562 - - ____________________________________________________________ - The numeric values for the operator ERF are computed via the power -series representation, which limits the argument range. - - -File: redhelp, Node: erfc, Next: Ei, Prev: ERF extended, Up: Integral Functions section - - ERFC operator - - The ERFC operator returns the complementary Error function - - 1 - defint(e**(-x**2),x,0,infinity) * 2/sqrt(pi) - - . - -syntax: - - ERFC () - -examples: - - ____________________________________________________________ - - erfc(xx); - - - erf(xx) + 1 - - ____________________________________________________________ - The operator ERFC is simplified towards the [*note ERF::.] operator. - - -File: redhelp, Node: Ei, Next: Fresnel_C, Prev: erfc, Up: Integral Functions section - - EI operator - - The EI operator returns the Exponential Integral function. - -syntax: - - EI () - -examples: - - ____________________________________________________________ - - df(ei(x),x); - - x - e - --- - x - - - on rounded; - - Ei(0.35); - - - 0.0894340019184 - - ____________________________________________________________ - The numeric values for the operator EI are computed via the power -series representation, which limits the argument range. - - -File: redhelp, Node: Fresnel_C, Next: Fresnel_S, Prev: Ei, Up: Integral Functions section - - FRESNEL_C operator - - The FRESNEL_C operator represents Fresnel's Cosine function. - -syntax: - - FRESNEL_C () - -examples: - - ____________________________________________________________ - - int(cos(t^2*pi/2),t,0,x); - - fresnel_c(x) - - - on rounded; - - fresnel_c(2.1); - - 0.581564135061 - - ____________________________________________________________ - The operator FRESNEL_C has a limited numeric evaluation of large -values of its argument. - - -File: redhelp, Node: Fresnel_S, Prev: Fresnel_C, Up: Integral Functions section - - FRESNEL_S operator - - The FRESNEL_S operator represents Fresnel's Sine Integral function. - -syntax: - - FRESNEL_S () - -examples: - - ____________________________________________________________ - - int(sin(t^2*pi/2),t,0,x); - - fresnel_s(x) - - - on rounded; - - fresnel_s(2.1); - - 0.374273359378 - - ____________________________________________________________ - The operator FRESNEL_S has a limited numeric evaluation of large -values of its argument. - - -File: redhelp, Node: Integral Functions section, Next: Combinatorial Operators section, Prev: Orthogonal Polynomials section, Up: Special Functions section - - Integral Functions section - -* Menu: - -* Si:: operator -* Shi:: operator -* s_i:: operator -* Ci:: operator -* Chi:: operator -* ERF extended:: operator -* erfc:: operator -* Ei:: operator -* Fresnel_C:: operator -* Fresnel_S:: operator - - -File: redhelp, Node: BINOMIAL, Next: STIRLING1, Up: Combinatorial Operators section - - BINOMIAL operator - - The BINOMIAL operator returns the Binomial coefficient if both -parameter are integer and expressions involving the Gamma function -otherwise. - -syntax: - - BINOMIAL (,) - -examples: - - ____________________________________________________________ - - Binomial(49,6); - - 13983816 - - - - Binomial(n,3); - - gamma(n + 1) - --------------- - 6*gamma(n - 2) - - ____________________________________________________________ - The operator BINOMIAL evaluates the Binomial coefficients from the -explicit form and therefore it is not the best algorithm if you want to -compute many binomial coefficients with big indices in which case a -recursive algorithm is preferable. - - -File: redhelp, Node: STIRLING1, Next: STIRLING2, Prev: BINOMIAL, Up: Combinatorial Operators section - - STIRLING1 operator - - The STIRLING1 operator returns the Stirling Numbers S(n,m) of the -first kind, i.e. the number of permutations of n symbols which have -exactly m cycles (divided by (-1)**(n-m)). - -syntax: - - STIRLING1 (,) - -examples: - - ____________________________________________________________ - - Stirling1 (17,4); - - -87077748875904 - - - Stirling1 (n,n-1); - - -gamma(n+1) - ------------- - 2*gamma(n-1) - - ____________________________________________________________ - The operator STIRLING1 evaluates the Stirling numbers of the first -kind by rulesets for special cases or by a computing the closed form, -which is a series involving the operators [*note BINOMIAL::.] and -[*note STIRLING2::.] . - - -File: redhelp, Node: STIRLING2, Prev: STIRLING1, Up: Combinatorial Operators section - - STIRLING2 operator - - The STIRLING1 operator returns the Stirling Numbers S(n,m) of the -second kind, i.e. the number of ways of partitioning a set of n elements -into m non-empty subsets. - -syntax: - - STIRLING2 (,) - -examples: - - ____________________________________________________________ - - Stirling2 (17,4); - - 694337290 - - - Stirling2 (n,n-1); - - gamma(n+1) - ------------- - 2*gamma(n-1) - - ____________________________________________________________ - The operator STIRLING2 evaluates the Stirling numbers of the second -kind by rulesets for special cases or by a computing the closed form. - - -File: redhelp, Node: Combinatorial Operators section, Next: 3j and 6j symbols section, Prev: Integral Functions section, Up: Special Functions section - - Combinatorial Operators section - -* Menu: - -* BINOMIAL:: operator -* STIRLING1:: operator -* STIRLING2:: operator - - -File: redhelp, Node: ThreejSymbol, Next: Clebsch_Gordan, Up: 3j and 6j symbols section - - THREEJSYMBOL operator - - The THREEJSYMBOL operator implements the 3j symbol. - -syntax: - - THREEJSYMBOL (,, ) - -examples: - - ____________________________________________________________ - - - ThreejSymbol({j+1,m},{j+1,-m},{1,0}); - - - j - ( - 1) *(abs(j - m + 1) - abs(j + m + 1)) - ------------------------------------------- - 3 2 m - 2*sqrt(2*j + 9*j + 13*j + 6)*( - 1) - - ____________________________________________________________ - - -File: redhelp, Node: Clebsch_Gordan, Next: SixjSymbol, Prev: ThreejSymbol, Up: 3j and 6j symbols section - - CLEBSCH_GORDAN operator - - The CLEBSCH_GORDAN operator implements the Clebsch_Gordan -coefficients. This is closely related to the [*note ThreejSymbol::.] . - -syntax: - - CLEBSCH_GORDAN (,, ) - -examples: - - ____________________________________________________________ - - Clebsch_Gordan({2,0},{2,0},{2,0}); - - - -2 - --------- - sqrt(14) - - ____________________________________________________________ - - -File: redhelp, Node: SixjSymbol, Prev: Clebsch_Gordan, Up: 3j and 6j symbols section - - SIXJSYMBOL operator - - The SIXJSYMBOL operator implements the 6j symbol. - -syntax: - - SIXJSYMBOL (,) - -examples: - - ____________________________________________________________ - - - SixjSymbol({7,6,3},{2,4,6}); - - 1 - ------------- - 14*sqrt(858) - - ____________________________________________________________ - The operator SIXJSYMBOL uses the [*note INEQ::.] package in order to -find minima and maxima for the summation index. - - -File: redhelp, Node: 3j and 6j symbols section, Next: Miscellaneous section, Prev: Combinatorial Operators section, Up: Special Functions section - - 3j and 6j symbols section - -* Menu: - -* ThreejSymbol:: operator -* Clebsch_Gordan:: operator -* SixjSymbol:: operator - - -File: redhelp, Node: HYPERGEOMETRIC, Next: MeijerG, Up: Miscellaneous section - - HYPERGEOMETRIC operator - - The HYPERGEOMETRIC operator provides simplifications for the -generalized hypergeometric functions. The HYPERGEOMETRIC operator is -included in the package specfn2. - -syntax: - - HYPERGEOMETRIC (,, -) - -examples: - - ____________________________________________________________ - - load specfn2; - - hypergeometric ({1/2,1},{3/2},-x^2); - - - atan(x) - -------- - x - - - hypergeometric ({},{},z); - - z - e - - ____________________________________________________________ - The special case where the length of the first list is equal to 2 and -the length of the second list is equal to 1 is often called "the -hypergeometric function" (notated as 2F1(a1,a2,b;x)). - - -File: redhelp, Node: MeijerG, Next: Heaviside, Prev: HYPERGEOMETRIC, Up: Miscellaneous section - - MEIJERG operator - - The MEIJERG operator provides simplifications for Meijer's G -function. The simplifications are performed towards polynomials, -elementary or special functions or (generalized) [*note -HYPERGEOMETRIC::.] functions. - - The MEIJERG operator is included in the package specfn2. - -syntax: - - MEIJERG (,, ) - - The first element of the lists has to be the list containing the -first group (mostly called "m" and "n") of parameters. This passes the -four parameters of a Meijer's G function implicitly via the length of -the lists. - -examples: - - ____________________________________________________________ - - load specfn2; - - MeijerG({{},1},{{0}},x); - - heaviside(-x+1) - - - MeijerG({{}},{{1+1/4},1-1/4},(x^2)/4) * sqrt pi; - - - - 2 - sqrt(2)*sin(x)*x - ------------------ - 4*sqrt(x) - - ____________________________________________________________ - Many well-known functions can be written as G functions, e.g. -exponentials, logarithms, trigonometric functions, Bessel functions and -hypergeometric functions. The formulae can be found e.g. in - - A.P.Prudnikov, Yu.A.Brychkov, O.I.Marichev: Integrals and Series, -Volume 3: More special functions, Gordon and Breach Science Publishers -(1990). - - -File: redhelp, Node: Heaviside, Next: erfi, Prev: MeijerG, Up: Miscellaneous section - - HEAVISIDE operator - - The HEAVISIDE operator returns the Heaviside function. - - Heaviside(~w) => if (w < 0) then 0 else 1 - - when numberp w; - -syntax: - - HEAVISIDE () - - This operator is often included in the result of the simplification -of a generalized [*note HYPERGEOMETRIC::.] function or a [*note -MeijerG::.] function. - - No simplification is done for this function. - - -File: redhelp, Node: erfi, Prev: Heaviside, Up: Miscellaneous section - - ERFI operator - - The ERFI operator returns the error function of an imaginary -argument. - - erfi(~x) => 2/sqrt(pi) * defint(e**(t**2),t,0,x); - -syntax: - - ERFI () - - This operator is sometimes included in the result of the -simplification of a generalized [*note HYPERGEOMETRIC::.] function or a -[*note MeijerG::.] function. - - No simplification is done for this function. - - -File: redhelp, Node: Miscellaneous section, Prev: 3j and 6j symbols section, Up: Special Functions section - - Miscellaneous section - -* Menu: - -* HYPERGEOMETRIC:: operator -* MeijerG:: operator -* Heaviside:: operator -* erfi:: operator - - -File: redhelp, Node: Special Functions section, Next: Taylor series section, Prev: Roots Package section, Up: Top - - Special Functions section - -* Menu: - -* Special Function Package::introduction -* Constants:: concept -* Bernoulli Euler Zeta section:: -* Bessel Functions section:: -* Airy Functions section:: -* Jacobi Elliptic Functions and Elliptic Integrals section:: -* Gamma and Related Functions section:: -* Miscellaneous Functions section:: -* Orthogonal Polynomials section:: -* Integral Functions section:: -* Combinatorial Operators section:: -* 3j and 6j symbols section:: -* Miscellaneous section:: - - -File: redhelp, Node: TAYLOR introduction, Next: taylor, Up: Taylor series section - - TAYLOR introduction - - This short note describes a package of REDUCE procedures that allow -Taylor expansion in one or more variables and efficient manipulation of -the resulting Taylor series. Capabilities include basic operations -(addition, subtraction, multiplication and division) and also -application of certain algebraic and transcendental functions. To a -certain extent, Laurent expansion can be performed as well. - - -File: redhelp, Node: taylor, Next: taylorautocombine, Prev: TAYLOR introduction, Up: Taylor series section - - TAYLOR operator - - The TAYLOR operator is used for expanding an expression into a -Taylor series. - -syntax: - - TAYLOR ( , , , - - , , , *) - - can be any valid REDUCE algebraic expression. -must be a [*note KERNEL::.] , and is the expansion variable. The - following it denotes the point about which the expansion -is to take place. must be a non-negative integer and denotes -the maximum expansion order. If more than one triple is specified -TAYLOR will expand its first argument independently with respect to -all the variables. Note that once the expansion has been done it is -not possible to calculate higher orders. - - Instead of a [*note KERNEL::.] , may also be a list of -kernels. In this case expansion will take place in a way so that the -sum/ of the degrees of the kernels does not exceed the maximum -expansion order. If the expansion point evaluates to the special -identifier INFINITY , TAYLOR tries to expand in a series in 1/. - - The expansion is performed variable per variable, i.e. in the -example above by first expanding exp(x^2+y^2) with respect to X and -then expanding every coefficient with respect to Y . - -examples: - - ____________________________________________________________ - - taylor(e^(x^2+y^2),x,0,2,y,0,2); - - - 2 2 2 2 2 2 - 1 + Y + X + Y *X + O(X ,Y ) - - - taylor(e^(x^2+y^2),{x,y},0,2); - - - 2 2 2 2 - 1 + Y + X + O({X ,Y }) - - ____________________________________________________________ - The following example shows the case of a non-analytical function. - ____________________________________________________________ - - - taylor(x*y/(x+y),x,0,2,y,0,2); - - - ***** Not a unit in argument to QUOTTAYLOR - - ____________________________________________________________ - - Note that it is not generally possible to apply the standard reduce -operators to a Taylor kernel. For example, [*note PART::.] , [*note -COEFF::.] , or [*note COEFFN::.] cannot be used. Instead, the -expression at hand has to be converted to standard form first using -the [*note taylortostandard::.] operator. - - Differentiation of a Taylor expression is possible. If you -differentiate with respect to one of the Taylor variables the order -will decrease by one. - - Substitution is a bit restricted: Taylor variables can only be -replaced by other kernels. There is one exception to this rule: you -can always substitute a Taylor variable by an expression that -evaluates to a constant. Note that REDUCE will not always be able to -determine that an expression is constant: an example is sin(acos(4)). - - Only simple taylor kernels can be integrated. More complicated -expressions that contain Taylor kernels as parts of themselves are -automatically converted into a standard representation by means of the -[*note taylortostandard::.] operator. In this case a suitable warning -is printed. - - -File: redhelp, Node: taylorautocombine, Next: taylorautoexpand, Prev: taylor, Up: Taylor series section - - TAYLORAUTOCOMBINE switch - - If you set TAYLORAUTOCOMBINE to ON , REDUCE automatically combines -Taylor expressions during the simplification process. This is -equivalent to applying [*note taylorcombine::.] to every expression -that contains Taylor kernels. Default is ON . - - -File: redhelp, Node: taylorautoexpand, Next: taylorcombine, Prev: taylorautocombine, Up: Taylor series section - - TAYLORAUTOEXPAND switch - - TAYLORAUTOEXPAND makes Taylor expressions "contagious" in the sense -that [*note taylorcombine::.] tries to Taylor expand all non-Taylor -subexpressions and to combine the result with the rest. Default is -OFF . - - -File: redhelp, Node: taylorcombine, Next: taylorkeeporiginal, Prev: taylorautoexpand, Up: Taylor series section - - TAYLORCOMBINE operator - - This operator tries to combine all Taylor kernels found in its -argument into one. Operations currently possible are: - - Addition, subtraction, multiplication, and division. - - Roots, exponentials, and logarithms. - - Trigonometric and hyperbolic functions and their inverses. - -examples: - - ____________________________________________________________ - - hugo := taylor(exp(x),x,0,2); - - 1 2 3 - HUGO := 1 + X + -*X + O(X ) - 2 - - - taylorcombine log hugo; - - 3 - X + O(X ) - - - taylorcombine(hugo + x); - - 1 2 3 - (1 + X + -*X + O(X )) + X - 2 - - - on taylorautoexpand; - - taylorcombine(hugo + x); - - 1 2 3 - 1 + 2*X + -*X + O(X ) - 2 - - ____________________________________________________________ - Application of unary operators like LOG and ATAN will nearly always -succeed. For binary operations their arguments have to be Taylor -kernels with the same template. This means that the expansion variable -and the expansion point must match. Expansion order is not so -important, different order usually means that one of them is truncated -before doing the operation. - - If [*note taylorkeeporiginal::.] is set to ON and if all Taylor -kernels in its argument have their original expressions kept -TAYLORCOMBINE will also combine these and store the result as the -original expression of the resulting Taylor kernel. There is also the -switch [*note taylorautoexpand::.] . - - There are a few restrictions to avoid mathematically undefined -expressions: it is not possible to take the logarithm of a Taylor -kernel which has no terms (i.e. is zero), or to divide by such a -beast. There are some provisions made to detect singularities during -expansion: poles that arise because the denominator has zeros at the -expansion point are detected and properly treated, i.e. the Taylor -kernel will start with a negative power. (This is accomplished by -expanding numerator and denominator separately and combining the -results.) Essential singularities of the known functions (see above) -are handled correctly. - - -File: redhelp, Node: taylorkeeporiginal, Next: taylororiginal, Prev: taylorcombine, Up: Taylor series section - - TAYLORKEEPORIGINAL switch - - TAYLORKEEPORIGINAL , if set to ON , forces the [*note taylor::.] -and all Taylor kernel manipulation operators to keep the original -expression, i.e. the expression that was Taylor expanded. All -operations performed on the Taylor kernels are also applied to this -expression which can be recovered using the operator [*note -taylororiginal::.] . Default is OFF . - - -File: redhelp, Node: taylororiginal, Next: taylorprintorder, Prev: taylorkeeporiginal, Up: Taylor series section - - TAYLORORIGINAL operator - - Recovers the original expression (the one that was expanded) from -the Taylor kernel that is given as its argument. - -syntax: - - TAYLORORIGINAL () or TAYLORORIGINAL - -examples: - - ____________________________________________________________ - - hugo := taylor(exp(x),x,0,2); - - 1 2 3 - HUGO := 1 + X + -*X + O(X ) - 2 - - - taylororiginal hugo; - - ***** Taylor kernel doesn't have an original part in TAYLORORIGINAL - - - on taylorkeeporiginal; - - hugo := taylor(exp(x),x,0,2); - - 1 2 3 - HUGO := 1 + X + -*X + O(X ) - 2 - - - taylororiginal hugo; - - X - E - - ____________________________________________________________ - An error is signalled if the argument is not a Taylor kernel or if -the original expression was not kept, i.e. if [*note -taylorkeeporiginal::.] was set OFF during expansion. - - -File: redhelp, Node: taylorprintorder, Next: taylorprintterms, Prev: taylororiginal, Up: Taylor series section - - TAYLORPRINTORDER switch - - TAYLORPRINTORDER , if set to ON , causes the remainder to be -printed in big-O notation. Otherwise, three dots are printed. Default -is ON . - - -File: redhelp, Node: taylorprintterms, Next: taylorrevert, Prev: taylorprintorder, Up: Taylor series section - - TAYLORPRINTTERMS variable - - Only a certain number of (non-zero) coefficients are printed. If -there are more, an expression of the form N TERMS is printed to -indicate how many non-zero terms have been suppressed. The number of -terms printed is given by the value of the shared algebraic variable -TAYLORPRINTTERMS . Allowed values are integers and the special -identifier ALL . The latter setting specifies that all terms are to be -printed. The default setting is 5. - -examples: - - ____________________________________________________________ - - taylor(e^(x^2+y^2),x,0,4,y,0,4); - - - 2 1 4 2 2 2 5 5 - 1 + Y + -*Y + X + Y *X + (4 terms) + O(X ,Y ) - 2 - - - taylorprintterms := all; - - TAYLORPRINTTERMS := ALL - - - taylor(e^(x^2+y^2),x,0,4,y,0,4); - - - 2 1 4 2 2 2 1 4 2 1 4 1 2 4 - 1 + Y + -*Y + X + Y *X + -*Y *X + -*X + -*Y *X - 2 2 2 2 - 1 4 4 5 5 - + -*Y *X + O(X ,Y ) - 4 - - - ____________________________________________________________ - - -File: redhelp, Node: taylorrevert, Next: taylorseriesp, Prev: taylorprintterms, Up: Taylor series section - - TAYLORREVERT operator - - TAYLORREVERT allows reversion of a Taylor series of a function f, -i.e., to compute the first terms of the expansion of the inverse of f -from the expansion of f. - -syntax: - - TAYLORREVERT (, , ) - - The first argument must evaluate to a Taylor kernel with the second -argument being one of its expansion variables. - -examples: - - ____________________________________________________________ - - taylor(u - u**2,u,0,5); - - 2 6 - U - U + O(U ) - - - taylorrevert (ws,u,x); - - 2 3 4 5 6 - X + X + 2*X + 5*X + 14*X + O(X ) - - ____________________________________________________________ - - -File: redhelp, Node: taylorseriesp, Next: taylortemplate, Prev: taylorrevert, Up: Taylor series section - - TAYLORSERIESP operator - - This operator may be used to determine if its argument is a Taylor -kernel. - -syntax: - - TAYLORSERIESP () or TAYLORSERIESP - -examples: - - ____________________________________________________________ - - hugo := taylor(exp(x),x,0,2); - - 1 2 3 - HUGO := 1 + X + -*X + O(X ) - 2 - - - if taylorseriesp hugo then OK; - - OK - - - if taylorseriesp(hugo + y) then OK else NO; - - - NO - - ____________________________________________________________ - Note that this operator is subject to the same restrictions as, -e.g., ORDP or NUMBERP , i.e. it may only be used in boolean -expressions in IF or LET statements. - - -File: redhelp, Node: taylortemplate, Next: taylortostandard, Prev: taylorseriesp, Up: Taylor series section - - TAYLORTEMPLATE operator - - The template of a Taylor kernel, i.e. the list of all variables -with respect to which expansion took place together with expansion -point and order can be extracted using - -syntax: - - TAYLORTEMPLATE () or TAYLORTEMPLATE - - This returns a list of lists with the three elements -(VAR,VAR0,ORDER). An error is signalled if the argument is not a -Taylor kernel. - -examples: - - ____________________________________________________________ - - hugo := taylor(exp(x),x,0,2); - - 1 2 3 - HUGO := 1 + X + -*X + O(X ) - 2 - - - taylortemplate hugo; - - {{X,0,2}} - - ____________________________________________________________ - - -File: redhelp, Node: taylortostandard, Prev: taylortemplate, Up: Taylor series section - - TAYLORTOSTANDARD operator - - This operator converts all Taylor kernels in its argument into -standard form and resimplifies the result. - -syntax: - - TAYLORTOSTANDARD () or TAYLORTOSTANDARD - - -examples: - - ____________________________________________________________ - - hugo := taylor(exp(x),x,0,2); - - 1 2 3 - HUGO := 1 + X + -*X + O(X ) - 2 - - - taylortostandard hugo; - - 2 - X + 2*X + 2 - ------------ - 2 - - ____________________________________________________________ - - -File: redhelp, Node: Taylor series section, Next: Gnuplot package section, Prev: Special Functions section, Up: Top - - Taylor series section - -* Menu: - -* TAYLOR introduction:: introduction -* taylor:: operator -* taylorautocombine:: switch -* taylorautoexpand:: switch -* taylorcombine:: operator -* taylorkeeporiginal:: switch -* taylororiginal:: operator -* taylorprintorder:: switch -* taylorprintterms:: variable -* taylorrevert:: operator -* taylorseriesp:: operator -* taylortemplate:: operator -* taylortostandard:: operator - - -File: redhelp, Node: GNUPLOT and REDUCE, Next: Axes names, Up: Gnuplot package section - - GNUPLOT AND REDUCE introduction - - The GNUPLOT system provides easy to use graphics output for curves -or surfaces which are defined by formulas and/or data sets. GNUPLOT -supports a great variety of output devices such as X-windows, VGA -screen, postscript, picTeX. The REDUCE GNUPLOT package lets one use -the GNUPLOT graphical output directly from inside REDUCE, either for -the interactive display of curves/surfaces or for the production of -pictures on paper. - - Note that this package may not be supported on all system platforms. - - For a detailed description you should read the GNUPLOT system -documentation, available together with the GNUPLOT installation -material from several servers by anonymous FTP. - - The REDUCE developers thank the GNUPLOT people for their permission -to distribute GNUPLOT together with REDUCE. - - -File: redhelp, Node: Axes names, Next: Pointset, Prev: GNUPLOT and REDUCE, Up: Gnuplot package section - - AXES NAMES - - Inside REDUCE the choice of variable names for a graph is completely -free. For referring to the GNUPLOT axes the names X and Y for 2 -dimensions, X,Y and Z for 3 dimensions are used in the usual schoolbook -sense independent from the variables of the REDUCE expression. - -examples: - - ____________________________________________________________ - ____________________________________________________________ - - -File: redhelp, Node: Pointset, Next: PLOT, Prev: Axes names, Up: Gnuplot package section - - POINTSET type - - A curve can be give as set of precomputed points (a polygon) in 2 or -3 dimensions. Such a point set is a [*note LIST::.] of points, where -each point is a [*note LIST::.] 2 (or 3) numbers. These numbers are -interpreted as (X,Y) (or X,Y,Z ) coordinates. All points of one set -must have the same dimension. - -examples: - - ____________________________________________________________ - ____________________________________________________________ - Also a surface in 3d can be given by precomputed points, but only on -a logically orthogonal mesh: the surface is defined by a list of curves -(in 3d) which must have a uniform length. GNUPLOT then will draw an -orthogonal mesh by first drawing the given lines, and second connecting -the 1st point of the 1st curve with the 1st point of the 2nd curve, -that one with the 1st point of the 3rd curve and so on for all curves -and for all indexes. - - -File: redhelp, Node: PLOT, Next: PLOTRESET, Prev: Pointset, Up: Gnuplot package section - - PLOT command - - The command PLOT is the main entry for drawing a picture from inside -REDUCE. - -syntax: - - PLOT (,,...) - - where is a , a or an ,,,) - - , :- matrices. - - , :- positive integers. - - COPY_INTO copies matrix into with (1,1) at -(,). - -examples: - - ____________________________________________________________ - - - G := mat((0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0)); - - - [0 0 0 0 0] - [ ] - [0 0 0 0 0] - [ ] - g := [0 0 0 0 0] - [ ] - [0 0 0 0 0] - [ ] - [0 0 0 0 0] - - - - copy_into(A,G,1,2); - - [0 1 2 3 0] - [ ] - [0 4 5 6 0] - [ ] - [0 7 8 9 0] - [ ] - [0 0 0 0 0] - [ ] - [0 0 0 0 0] - - ____________________________________________________________ - Related functions: [*note augment_columns::.] , [*note extend::.] , -[*note matrix_augment::.] , [*note matrix_stack::.] , [*note -stack_rows::.] , [*note sub_matrix::.] . - - -File: redhelp, Node: diagonal, Next: extend, Prev: copy_into, Up: Linear Algebra package section - - DIAGONAL operator - -syntax: - - DIAGONAL () - - (If you are feeling lazy then the braces can be omitted.) - - :- each can be either a scalar expression or a square -[*note MATRIX::.] . - - DIAGONAL creates a matrix that contains the input on the diagonal. - -examples: - - ____________________________________________________________ - - - H := mat((66,77),(88,99)); - - [66 77] - h := [ ] - [88 99] - - - - diagonal({A,x,H}); - - [1 2 3 0 0 0 ] - [ ] - [4 5 6 0 0 0 ] - [ ] - [7 8 9 0 0 0 ] - [ ] - [0 0 0 x 0 0 ] - [ ] - [0 0 0 0 66 77] - [ ] - [0 0 0 0 88 99] - - ____________________________________________________________ - Related functions: [*note jordan_block::.] . - - -File: redhelp, Node: extend, Next: find_companion, Prev: diagonal, Up: Linear Algebra package section - - EXTEND operator - -syntax: - - EXTEND (,,,) - - :- a [*note MATRIX::.] . - - , :- positive integers. - - :- algebraic expression or symbol. - - EXTEND returns a copy of that has been extended by rows -and columns. The new entries are made equal to . - -examples: - - ____________________________________________________________ - - - extend(A,1,2,x); - - [1 2 3 x x] - [ ] - [4 5 6 x x] - [ ] - [7 8 9 x x] - [ ] - [x x x x x] - - ____________________________________________________________ - Related functions: [*note copy_into::.] , [*note matrix_augment::.] -, [*note matrix_stack::.] , [*note remove_columns::.] , [*note -remove_rows::.] . - - -File: redhelp, Node: find_companion, Next: get_columns, Prev: extend, Up: Linear Algebra package section - - FIND_COMPANION operator - -syntax: - - FIND_COMPANION (,) - - :- a [*note MATRIX::.] . - - :- the variable. - - Given a companion matrix, FIND_COMPANION finds the polynomial from -which it was made. - -examples: - - ____________________________________________________________ - - - C := companion(x^4+17*x^3-9*x^2+11,x); - - - [0 0 0 -11] - [ ] - [1 0 0 0 ] - c := [ ] - [0 1 0 9 ] - [ ] - [0 0 1 -17] - - - - find_companion(C,x); - - 4 3 2 - x +17*x -9*x +11 - - ____________________________________________________________ - Related functions: [*note companion::.] . - - -File: redhelp, Node: get_columns, Next: get_rows, Prev: find_companion, Up: Linear Algebra package section - - GET_COLUMNS operator - - Get columns, get rows: - -syntax: - - GET_COLUMNS (,) - - :- a [*note MATRIX::.] . - - :- either a positive integer or a list of positive integers. - - GET_COLUMNS removes the columns of specified in - and returns them as a list of column matrices. - - GET_ROWS performs the same task on the rows of . - -examples: - - ____________________________________________________________ - - - get_columns(A,{1,3}); - - { - [1] - [ ] - [4] - [ ] - [7] - , - [3] - [ ] - [6] - [ ] - [9] - } - - - - get_rows(A,2); - - { - [4 5 6] - } - - ____________________________________________________________ - Related functions: [*note augment_columns::.] , [*note -stack_rows::.] , [*note sub_matrix::.] . - - -File: redhelp, Node: get_rows, Next: gram_schmidt, Prev: get_columns, Up: Linear Algebra package section - - GET_ROWS operator - - see: [*note get_columns::.] . - - -File: redhelp, Node: gram_schmidt, Next: hermitian_tp, Prev: get_rows, Up: Linear Algebra package section - - GRAM_SCHMIDT operator - -syntax: - - GRAM_SCHMIDT () - - (If you are feeling lazy then the braces can be omitted.) - - :- linearly independent vectors. Each vector must be -written as a list, eg:1,0,0. - - GRAM_SCHMIDT performs the gram_schmidt orthonormalization on the -input vectors. - - It returns a list of orthogonal normalized vectors. - -examples: - - ____________________________________________________________ - - - gram_schmidt({{1,0,0},{1,1,0},{1,1,1}}); - - - {{1,0,0},{0,1,0},{0,0,1}} - - - - gram_schmidt({{1,2},{3,4}}); - - - 1 2 2*sqrt(5) -sqrt(5) - {{ ------- , ------- },{ --------- , -------- }} - sqrt(5) sqrt(5) 5 5 - - ____________________________________________________________ - - -File: redhelp, Node: hermitian_tp, Next: hessian, Prev: gram_schmidt, Up: Linear Algebra package section - - HERMITIAN_TP operator - -syntax: - - HERMITIAN_TP () - - :- a [*note MATRIX::.] . - - HERMITIAN_TP computes the hermitian transpose of . - - This is a [*note MATRIX::.] in which the (i,j)'th entry is the -conjugate of the (j,i)'th entry of . - -examples: - - ____________________________________________________________ - - - J := mat((i+1,i+2,i+3),(4,5,2),(1,i,0)); - - - [i + 1 i + 2 i + 3] - [ ] - j := [ 4 5 2 ] - [ ] - [ 1 i 0 ] - - - - hermitian_tp(j); - - [ - i + 1 4 1 ] - [ ] - [ - i + 2 5 - i] - [ ] - [ - i + 3 2 0 ] - - ____________________________________________________________ - Related functions: [*note TP::.] . - - -File: redhelp, Node: hessian, Next: hilbert, Prev: hermitian_tp, Up: Linear Algebra package section - - HESSIAN operator - -syntax: - - HESSIAN (,) - - :- a scalar expression. - - :- either a single variable or a list of variables. - - HESSIAN computes the hessian matrix of w.r.t. the variables -in . - - This is an n by n matrix where n is the number of variables and the -(i,j)'th entry is [*note DF::.] (,(i), -(j)). - -examples: - - ____________________________________________________________ - - - hessian(x*y*z+x^2,{w,x,y,z}); - - [0 0 0 0] - [ ] - [0 2 z y] - [ ] - [0 z 0 x] - [ ] - [0 y x 0] - - ____________________________________________________________ - Related functions: [*note DF::.] . - - -File: redhelp, Node: hilbert, Next: jacobian, Prev: hessian, Up: Linear Algebra package section - - HILBERT operator - -syntax: - - HILBERT (,) - - :- a positive integer. - - :- an algebraic expression. - - HILBERT computes the square hilbert matrix of dimension -. - - This is the symmetric matrix in which the (i,j)'th entry is -1/(i+j-). - -examples: - - ____________________________________________________________ - - - hilbert(3,y+x); - - [ - 1 - 1 - 1 ] - [----------- ----------- -----------] - [ x + y - 2 x + y - 3 x + y - 4 ] - [ ] - [ - 1 - 1 - 1 ] - [----------- ----------- -----------] - [ x + y - 3 x + y - 4 x + y - 5 ] - [ ] - [ - 1 - 1 - 1 ] - [----------- ----------- -----------] - [ x + y - 4 x + y - 5 x + y - 6 ] - - ____________________________________________________________ - - -File: redhelp, Node: jacobian, Next: jordan_block, Prev: hilbert, Up: Linear Algebra package section - - JACOBIAN operator - -syntax: - - JACOBIAN (,) - - :- either a single algebraic expression or a list of -algebraic expressions. - - :- either a single variable or a list of variables. - - JACOBIAN computes the jacobian matrix of w.r.t. -. - - This is a matrix whose (i,j)'th entry is [*note DF::.] ( -(i),(j)). - - The matrix is n by m where n is the number of variables and m the -number of expressions. - -examples: - - ____________________________________________________________ - - - jacobian({x^4,x*y^2,x*y*z^3},{w,x,y,z}); - - - [ 3 ] - [0 4*x 0 0 ] - [ ] - [ 2 ] - [0 y 2*x*y 0 ] - [ ] - [ 3 3 2] - [0 y*z x*z 3*x*y*z ] - - ____________________________________________________________ - Related functions: [*note hessian::.] , [*note DF::.] . - - -File: redhelp, Node: jordan_block, Next: lu_decom, Prev: jacobian, Up: Linear Algebra package section - - JORDAN_BLOCK operator - -syntax: - - JORDAN_BLOCK (,) - - :- an algebraic expression or symbol. - - :- a positive integer. - - JORDAN_BLOCK computes the square jordan block matrix J of dimension -. - - The entries of J are: - - J(i,i) = for i=1 ... n, J(i,i+1) = 1 for i=1 ... n-1, and -all other entries are 0. - -examples: - - ____________________________________________________________ - - - jordan_block(x,5); - - [x 1 0 0 0] - [ ] - [0 x 1 0 0] - [ ] - [0 0 x 1 0] - [ ] - [0 0 0 x 1] - [ ] - [0 0 0 0 x] - - ____________________________________________________________ - Related functions: [*note diagonal::.] , [*note companion::.] . - - -File: redhelp, Node: lu_decom, Next: make_identity, Prev: jordan_block, Up: Linear Algebra package section - - LU_DECOM operator - -syntax: - - LU_DECOM () - - :- a [*note MATRIX::.] containing either numeric entries -or imaginary entries with numeric coefficients. - - LU_DECOM performs LU decomposition on , ie: it returns L,U -where L is a lower diagonal [*note MATRIX::.] , U an upper diagonal -[*note MATRIX::.] and A = LU. - - Caution: - - The algorithm used can swap the rows of during the -calculation. This means that LU does not equal but a row -equivalent of it. Due to this, LU_DECOM returns L,U,vec. The call -CONVERT(META[MATRIX ,vec) will return the matrix that has been -decomposed, i.e: LU = convert(,vec). - -examples: - - ____________________________________________________________ - - - K := mat((1,3,5),(-4,3,7),(8,6,4)); - - - [1 3 5] - [ ] - k := [-4 3 7] - [ ] - [8 6 4] - - - - on rounded; - - lu := lu_decom(K); - - lu := { - [8 0 0 ] - [ ] - [-4 6.0 0 ] - [ ] - [1 2.25 1.125] - , - [1 0.75 0.5] - [ ] - [0 1 1.5] - [ ] - [0 0 1 ] - , - [3 2 3]} - - - - first lu * second lu; - - [8 6.0 4.0] - [ ] - [-4 3.0 7.0] - [ ] - [1 3.0 5.0] - - - - convert(K,third lu); - - P := mat((i+1,i+2,i+3),(4,5,2),(1,i,0)); - [i + 1 i + 2 i + 3] - [ ] - p := [ 4 5 2 ] - [ ] - [ 1 i 0 ] - - - lu := lu_decom(P); - - lu := { - [ 1 0 0 ] - [ ] - [ 4 - 4*i + 5 0 ] - [ ] - [i + 1 3 0.414634146341*i + 2.26829268293] - , - [1 i 0 ] - [ ] - [0 1 0.19512195122*i + 0.243902439024] - [ ] - [0 0 1 ] - , - [3 2 3]} - - - - first lu * second lu; - - [ 1 i 0 ] - [ ] - [ 4 5 2.0 ] - [ ] - [i + 1 i + 2 i + 3.0] - - - - convert(P,third lu); - - [ 1 i 0 ] - [ ] - [ 4 5 2 ] - [ ] - [i + 1 i + 2 i + 3] - - ____________________________________________________________ - - Related functions: [*note cholesky::.] . - - -File: redhelp, Node: make_identity, Next: matrix_augment, Prev: lu_decom, Up: Linear Algebra package section - - MAKE_IDENTITY operator - -syntax: - - MAKE_IDENTITY () - - :- a positive integer. - - MAKE_IDENTITY creates the identity matrix of dimension . - -examples: - - ____________________________________________________________ - - - make_identity(4); - - [1 0 0 0] - [ ] - [0 1 0 0] - [ ] - [0 0 1 0] - [ ] - [0 0 0 1] - - ____________________________________________________________ - Related functions: [*note diagonal::.] . - - -File: redhelp, Node: matrix_augment, Next: matrixp, Prev: make_identity, Up: Linear Algebra package section - - MATRIX_AUGMENT operator - - Matrix augment, matrix stack: - -syntax: - - MATRIX_AUGMENT - - (If you are feeling lazy then the braces can be omitted.) - - :- matrices. - - MATRIX_AUGMENT sticks the matrices in together -horizontally. - - MATRIX_STACK sticks the matrices in together -vertically. - -examples: - - ____________________________________________________________ - - - matrix_augment({A,A}); - - [1 2 3 1 2 3] - [ ] - [4 5 6 4 5 6] - [ ] - [7 8 9 7 8 9] - - - - matrix_stack(A,A); - - [1 2 3] - [ ] - [4 5 6] - [ ] - [7 8 9] - [ ] - [1 2 3] - [ ] - [4 5 6] - [ ] - [7 8 9] - - ____________________________________________________________ - Related functions: [*note augment_columns::.] , [*note -stack_rows::.] , [*note sub_matrix::.] . - - -File: redhelp, Node: matrixp, Next: matrix_stack, Prev: matrix_augment, Up: Linear Algebra package section - - MATRIXP operator - -syntax: - - MATRIXP () - - :- anything you like. - - MATRIXP is a boolean function that returns t if the input is a -matrix and nil otherwise. - -examples: - - ____________________________________________________________ - - - matrixp A; - - t - - - matrixp(doodlesackbanana); - - nil - - ____________________________________________________________ - Related functions: [*note squarep::.] , [*note symmetricp::.] . - - -File: redhelp, Node: matrix_stack, Next: minor, Prev: matrixp, Up: Linear Algebra package section - - MATRIX_STACK operator - - see: [*note matrix_augment::.] . - - -File: redhelp, Node: minor, Next: mult_columns, Prev: matrix_stack, Up: Linear Algebra package section - - MINOR operator - -syntax: - - MINOR (,,) - - :- a [*note MATRIX::.] . , :- positive integers. - - MINOR computes the (,)'th minor of . This is created -by removing the 'th row and the 'th column from . - -examples: - - ____________________________________________________________ - - - minor(A,1,3); - - [4 5] - [ ] - [7 8] - - ____________________________________________________________ - Related functions: [*note remove_columns::.] , [*note -remove_rows::.] . - - -File: redhelp, Node: mult_columns, Next: mult_rows, Prev: minor, Up: Linear Algebra package section - - MULT_COLUMNS operator - - Mult columns, mult rows: - -syntax: - - MULT_COLUMNS (,,) - - :- a [*note MATRIX::.] . - - :- a positive integer or a list of positive integers. - - :- an algebraic expression. - - MULT_COLUMNS returns a copy of in which the columns -specified in have been multiplied by . - - MULT_ROWS performs the same task on the rows of . - -examples: - - ____________________________________________________________ - - - mult_columns(A,{1,3},x); - - [ x 2 3*x] - [ ] - [4*x 5 6*x] - [ ] - [7*x 8 9*x] - - - - mult_rows(A,2,10); - - [1 2 3 ] - [ ] - [40 50 60] - [ ] - [7 8 9 ] - - ____________________________________________________________ - Related functions: [*note add_to_columns::.] , [*note -add_to_rows::.] . - - -File: redhelp, Node: mult_rows, Next: pivot, Prev: mult_columns, Up: Linear Algebra package section - - MULT_ROWS operator - - see: [*note mult_columns::.] . - - -File: redhelp, Node: pivot, Next: pseudo_inverse, Prev: mult_rows, Up: Linear Algebra package section - - PIVOT operator - -syntax: - - PIVOT (,,) - - :- a matrix. - - , :- positive integers such that (, ) neq 0. - - PIVOT pivots about it's (,)'th entry. - - To do this, multiples of the 'th row are added to every other row -in the matrix. - - This means that the 'th column will be 0 except for the -(,)'th entry. - -examples: - - ____________________________________________________________ - - - pivot(A,2,3); - - [ - 1 ] - [-1 ------ 0] - [ 2 ] - [ ] - [4 5 6] - [ ] - [ 1 ] - [1 --- 0] - [ 2 ] - - ____________________________________________________________ - Related functions: [*note rows_pivot::.] . - - -File: redhelp, Node: pseudo_inverse, Next: random_matrix, Prev: pivot, Up: Linear Algebra package section - - PSEUDO_INVERSE operator - -syntax: - - PSEUDO_INVERSE () - - :- a [*note MATRIX::.] . - - PSEUDO_INVERSE , also known as the Moore-Penrose inverse, computes -the pseudo inverse of . - - Given the singular value decomposition of , i.e: A = -U*P*V^T, then the pseudo inverse A^-1 is defined by A^-1 = V^T*P^-1*U. - - Thus * pseudo_inverse(A) = Id. (Id is the identity matrix). - -examples: - - ____________________________________________________________ - - - R := mat((1,2,3,4),(9,8,7,6)); - - [1 2 3 4] - r := [ ] - [9 8 7 6] - - - - on rounded; - - pseudo_inverse(R); - - [ - 0.199999999996 0.100000000013 ] - [ ] - [ - 0.0499999999988 0.0500000000037 ] - [ ] - [ 0.0999999999982 - 5.57825497203e-12] - [ ] - [ 0.249999999995 - 0.0500000000148 ] - - ____________________________________________________________ - Related functions: [*note svd::.] . - - -File: redhelp, Node: random_matrix, Next: remove_columns, Prev: pseudo_inverse, Up: Linear Algebra package section - - RANDOM_MATRIX operator - -syntax: - - RANDOM_MATRIX (,,) - - ,, :- positive integers. - - RANDOM_MATRIX creates an by matrix with random entries in -the range -limit < entry < limit. - - Switches: - - IMAGINARY :- if on then matrix entries are x+i*y where -limit < x,y -< . - - NOT_NEGATIVE :- if on then 0 < entry < . In the imaginary -case we have 0 < x,y < . - - ONLY_INTEGER :- if on then each entry is an integer. In the imaginary - case x and y are integers. - - SYMMETRIC :- if on then the matrix is symmetric. - - UPPER_MATRIX :- if on then the matrix is upper triangular. - - LOWER_MATRIX :- if on then the matrix is lower triangular. - -examples: - - ____________________________________________________________ - - - on rounded; - - random_matrix(3,3,10); - - [ - 8.11911717343 - 5.71677292768 0.620580830035 ] - [ ] - [ - 0.032596262422 7.1655452861 5.86742633837 ] - [ ] - [ - 9.37155438255 - 7.55636708637 - 8.88618627557] - - - - on only_integer, not_negative, upper_matrix, imaginary; - - random_matrix(4,4,10); - - [70*i + 15 28*i + 8 2*i + 79 27*i + 44] - [ ] - [ 0 46*i + 95 9*i + 63 95*i + 50] - [ ] - [ 0 0 31*i + 75 14*i + 65] - [ ] - [ 0 0 0 5*i + 52 ] - - ____________________________________________________________ - - -File: redhelp, Node: remove_columns, Next: remove_rows, Prev: random_matrix, Up: Linear Algebra package section - - REMOVE_COLUMNS operator - - Remove columns, remove rows: - -syntax: - - REMOVE_COLUMNS (,) - - :- a [*note MATRIX::.] . :- either a -positive integer or a list of positive integers. - - REMOVE_COLUMNS removes the columns specified in from -. - - REMOVE_ROWS performs the same task on the rows of . - -examples: - - ____________________________________________________________ - - - remove_columns(A,2); - - [1 3] - [ ] - [4 6] - [ ] - [7 9] - - - - remove_rows(A,{1,3}); - - [4 5 6] - - ____________________________________________________________ - Related functions: [*note minor::.] . - - -File: redhelp, Node: remove_rows, Next: row_dim, Prev: remove_columns, Up: Linear Algebra package section - - REMOVE_ROWS operator - - see: [*note remove_columns::.] . - - -File: redhelp, Node: row_dim, Next: rows_pivot, Prev: remove_rows, Up: Linear Algebra package section - - ROW_DIM operator - - see: [*note column_dim::.] . - - -File: redhelp, Node: rows_pivot, Next: simplex, Prev: row_dim, Up: Linear Algebra package section - - ROWS_PIVOT operator - -syntax: - - ROWS_PIVOT (,,,) - - :- a namerefmatrix. - - , :- positive integers such that (, ) neq 0. - - :- positive integer or a list of positive integers. - - ROWS_PIVOT performs the same task as PIVOT but applies the pivot -only to the rows specified in . - -examples: - - ____________________________________________________________ - - - N := mat((1,2,3),(4,5,6),(7,8,9),(1,2,3),(4,5,6)); - - - [1 2 3] - [ ] - [4 5 6] - [ ] - n := [7 8 9] - [ ] - [1 2 3] - [ ] - [4 5 6] - - - - rows_pivot(N,2,3,{4,5}); - - [1 2 3] - [ ] - [4 5 6] - [ ] - [7 8 9] - [ ] - [ - 1 ] - [-1 ------ 0] - [ 2 ] - [ ] - [0 0 0] - - ____________________________________________________________ - Related functions: [*note pivot::.] . - - -File: redhelp, Node: simplex, Next: squarep, Prev: rows_pivot, Up: Linear Algebra package section - - SIMPLEX operator - -syntax: - - SIMPLEX (,, ) - - :- either max or min (signifying maximize and - minimize). - - :- the function you are maximizing or - minimizing. - - :- the constraint inequalities. Each one must -be of the form sum of variables ( <=,=,>=) number. - - SIMPLEX applies the revised simplex algorithm to find the -optimal(either maximum or minimum) value of the -under the linear inequality constraints. - - It returns optimal value, values of variables at this optimal. - - The algorithm implies that all the variables are non-negative. - -examples: - - ____________________________________________________________ - - - simplex(max,x+y,{x>=10,y>=20,x+y<=25}); - - - ***** Error in simplex: Problem has no feasible solution - - - - simplex(max,10x+5y+5.5z,{5x+3z<=200,x+0.1y+0.5z<=12, - 0.1x+0.2y+0.3z<=9, 30x+10y+50z<=1500}); - - - {525.0,{x=40.0,y=25.0,z=0}} - - ____________________________________________________________ - - -File: redhelp, Node: squarep, Next: stack_rows, Prev: simplex, Up: Linear Algebra package section - - SQUAREP operator - -syntax: - - SQUAREP () - - :- a [*note MATRIX::.] . - - SQUAREP is a predicate that returns t if the is square and -nil otherwise. - -examples: - - ____________________________________________________________ - - - squarep(mat((1,3,5))); - - nil - - - squarep(A); - t - ____________________________________________________________ - Related functions: [*note matrixp::.] , [*note symmetricp::.] . - - -File: redhelp, Node: stack_rows, Next: sub_matrix, Prev: squarep, Up: Linear Algebra package section - - STACK_ROWS operator - - see: [*note augment_columns::.] . - - -File: redhelp, Node: sub_matrix, Next: svd, Prev: stack_rows, Up: Linear Algebra package section - - SUB_MATRIX operator - -syntax: - - SUB_MATRIX (,,) - - :- a matrix. , :- either a -positive integer or a list of positive integers. - - namesub_matrix produces the matrix consisting of the intersection of -the rows specified in and the columns specified in -. - -examples: - - ____________________________________________________________ - - - sub_matrix(A,{1,3},{2,3}); - - [2 3] - [ ] - [8 9] - - ____________________________________________________________ - Related functions: [*note augment_columns::.] , [*note -stack_rows::.] . - - -File: redhelp, Node: svd, Next: swap_columns, Prev: sub_matrix, Up: Linear Algebra package section - - SVD operator - - Singular value decomposition: - -syntax: - - SVD () - - :- a [*note MATRIX::.] containing only numeric entries. - - SVD computes the singular value decomposition of . - - It returns - - U,P,V - - where A = U*P*V^T - - and P = diag(sigma(1) ... sigma(n)). - - sigma(i) for i= 1 ... n are the singular values of . - - n is the column dimension of . - - The singular values of are the non-negative square roots of -the eigenvalues of A^T*A. - - U and V are such that U*U^T = V*V^T = V^T*V = Id. Id is the -identity matrix. - -examples: - - ____________________________________________________________ - - - Q := mat((1,3),(-4,3)); - - [1 3] - q := [ ] - [-4 3] - - - - on rounded; - - svd(Q); - - { - [ 0.289784137735 0.957092029805] - [ ] - [ - 0.957092029805 0.289784137735] - , - [5.1491628629 0 ] - [ ] - [ 0 2.9130948854] - , - [ - 0.687215403194 0.726453707825 ] - [ ] - [ - 0.726453707825 - 0.687215403194] - } - - ____________________________________________________________ - - -File: redhelp, Node: swap_columns, Next: swap_entries, Prev: svd, Up: Linear Algebra package section - - SWAP_COLUMNS operator - - Swap columns, swap rows: - -syntax: - - SWAP_COLUMNS (,,) - - :- a [*note MATRIX::.] . - - , :- positive integers. - - SWAP_COLUMNS swaps column of with column . - - SWAP_ROWS performs the same task on two rows of . - -examples: - - ____________________________________________________________ - - - swap_columns(A,2,3); - - [1 3 2] - [ ] - [4 6 5] - [ ] - [7 9 8] - - - - swap_rows(A,1,3); - - [7 8 9] - [ ] - [4 5 6] - [ ] - [1 2 3] - - ____________________________________________________________ - Related functions: [*note swap_entries::.] . - - -File: redhelp, Node: swap_entries, Next: swap_rows, Prev: swap_columns, Up: Linear Algebra package section - - SWAP_ENTRIES operator - -syntax: - - SWAP_ENTRIES (,,,, ) - - :- a [*note MATRIX::.] . - - ,,, :- positive integers. - - SWAP_ENTRIES swaps (,) with (,). - -examples: - - ____________________________________________________________ - - - swap_entries(A,{1,1},{3,3}); - - [9 2 3] - [ ] - [4 5 6] - [ ] - [7 8 1] - - ____________________________________________________________ - Related functions: [*note swap_columns::.] , [*note swap_rows::.] . - - -File: redhelp, Node: swap_rows, Next: symmetricp, Prev: swap_entries, Up: Linear Algebra package section - - SWAP_ROWS operator - - see: [*note swap_columns::.] . - - -File: redhelp, Node: symmetricp, Next: toeplitz, Prev: swap_rows, Up: Linear Algebra package section - - SYMMETRICP operator - -syntax: - - SYMMETRICP () - - :- a [*note MATRIX::.] . - - SYMMETRICP is a predicate that returns t if the matrix is symmetric -and nil otherwise. - -examples: - - ____________________________________________________________ - - - symmetricp(make_identity(11)); - - t - - - symmetricp(A); - - nil - - ____________________________________________________________ - Related functions: [*note matrixp::.] , [*note squarep::.] . - - -File: redhelp, Node: toeplitz, Next: vandermonde, Prev: symmetricp, Up: Linear Algebra package section - - TOEPLITZ operator - -syntax: - - TOEPLITZ () - - (If you are feeling lazy then the braces can be omitted.) - - :- list of algebraic expressions. - - TOEPLITZ creates the toeplitz matrix from the . - - This is a square symmetric matrix in which the first expression is -placed on the diagonal and the i'th expression is placed on the (i-1)'th -sub and super diagonals. - - It has dimension n where n is the number of expressions. - -examples: - - ____________________________________________________________ - - - toeplitz({w,x,y,z}); - - [w x y z] - [ ] - [x w x y] - [ ] - [y x w x] - [ ] - [z y x w] - - ____________________________________________________________ - - -File: redhelp, Node: vandermonde, Prev: toeplitz, Up: Linear Algebra package section - - VANDERMONDE operator - -syntax: - - VANDERMONDE () - - (If you are feeling lazy then the braces can be omitted.) - - :- list of algebraic expressions. - - VANDERMONDE creates the vandermonde matrix from the . - - This is the square matrix in which the (i,j)'th entry is -(i)^(j-1). - - It has dimension n where n is the number of expressions. - -examples: - - ____________________________________________________________ - - vandermonde({x,2*y,3*z}); - - - [ 2 ] - [1 x x ] - [ ] - [ 2] - [1 2*y 4*y ] - [ ] - [ 2] - [1 3*z 9*z ] - - ____________________________________________________________ - - -File: redhelp, Node: Linear Algebra package section, Next: Matrix Normal Forms section, Prev: Gnuplot package section, Up: Top - - Linear Algebra package section - -* Menu: - -* Linear Algebra package:: introduction -* fast_la:: switch -* add_columns:: operator -* add_rows:: operator -* add_to_columns:: operator -* add_to_rows:: operator -* augment_columns:: operator -* band_matrix:: operator -* block_matrix:: operator -* char_matrix:: operator -* char_poly:: operator -* cholesky:: operator -* coeff_matrix:: operator -* column_dim:: operator -* companion:: operator -* copy_into:: operator -* diagonal:: operator -* extend:: operator -* find_companion:: operator -* get_columns:: operator -* get_rows:: operator -* gram_schmidt:: operator -* hermitian_tp:: operator -* hessian:: operator -* hilbert:: operator -* jacobian:: operator -* jordan_block:: operator -* lu_decom:: operator -* make_identity:: operator -* matrix_augment:: operator -* matrixp:: operator -* matrix_stack:: operator -* minor:: operator -* mult_columns:: operator -* mult_rows:: operator -* pivot:: operator -* pseudo_inverse:: operator -* random_matrix:: operator -* remove_columns:: operator -* remove_rows:: operator -* row_dim:: operator -* rows_pivot:: operator -* simplex:: operator -* squarep:: operator -* stack_rows:: operator -* sub_matrix:: operator -* svd:: operator -* swap_columns:: operator -* swap_entries:: operator -* swap_rows:: operator -* symmetricp:: operator -* toeplitz:: operator -* vandermonde:: operator - - -File: redhelp, Node: Smithex, Next: Smithex_int, Up: Matrix Normal Forms section - - SMITHEX operator - - The operator SMITHEX computes the Smith normal form S of a [*note -MATRIX::.] A (say). It returns S,P,P^-1 where P*S*P^-1 = A. - -syntax: - - SMITHEX (,) - - :- a rectangular [*note MATRIX::.] of univariate -polynomials in . :- the variable. - -examples: - - ____________________________________________________________ - - a := mat((x,x+1),(0,3*x^2)); - - [x x + 1] - [ ] - a := [ 2 ] - [0 3*x ] - - - - smithex(a,x); - - [1 0 ] [1 0] [x x + 1] - { [ ], [ ], [ ] } - [ 3] [ 2 ] [ ] - [0 x ] [3*x 1] [-3 -3 ] - - ____________________________________________________________ - - -File: redhelp, Node: Smithex_int, Next: Frobenius, Prev: Smithex, Up: Matrix Normal Forms section - - SMITHEX_INT operator - - The operator SMITHEX_INT performs the same task as SMITHEX but on -matrices containing only integer entries. Namely, SMITHEX_INT returns -S,P,P^-1 where S is the smith normal form of the input [*note -MATRIX::.] (A say), and P*S*P^-1 = A. - -syntax: - - SMITHEX_INT () - - :- a rectangular [*note MATRIX::.] of integer entries. - -examples: - - ____________________________________________________________ - - a := mat((9,-36,30),(-36,192,-180),(30,-180,180)); - - - [ 9 -36 30 ] - [ ] - a := [-36 192 -180] - [ ] - [30 -180 180 ] - - - - smithex_int(a); - - [3 0 0 ] [-17 -5 -4 ] [1 -24 30 ] - [ ] [ ] [ ] - { [0 12 0 ], [64 19 15 ], [-1 25 -30] } - [ ] [ ] [ ] - [0 0 60] [-50 -15 -12] [0 -1 1 ] - - ____________________________________________________________ - - -File: redhelp, Node: Frobenius, Next: Ratjordan, Prev: Smithex_int, Up: Matrix Normal Forms section - - FROBENIUS operator - - The operator FROBENIUS computes the FROBENIUS normal form F of a -[*note MATRIX::.] (A say). It returns F,P,P^-1 where P*F*P^-1 = A. - -syntax: - - FROBENIUS () - - :- a square [*note MATRIX::.] . - - Field Extensions: - - By default, calculations are performed in the rational numbers. To -extend this field the [*note ARNUM::.] package can be used. The package -must first be loaded by load_package arnum;. The field can now be -extended by using the defpoly command. For example, defpoly sqrt2**2-2; -will extend the field to include the square root of 2 (now defined by -sqrt2). - - Modular Arithmetic: - - FROBENIUS can also be calculated in a modular base. To do this first -type on modular;. Then setmod p; (where p is a prime) will set the -modular base of calculation to p. By further typing on balanced_mod the -answer will appear using a symmetric modular representation. See [*note -Ratjordan::.] for an example. - -examples: - - ____________________________________________________________ - - a := mat((x,x^2),(3,5*x)); - - [ 2 ] - [x x ] - a := [ ] - [3 5*x] - - - frobenius(a); - - [ 2] [1 x] [ - x ] - { [0 - 2*x ], [ ], [1 -----] } - [ ] [0 3] [ 3 ] - [1 6*x ] [ ] - [ 1 ] - [0 --- ] - [ 3 ] - - - load_package arnum; - - defpoly sqrt2**2-2; - - a := mat((sqrt2,5),(7*sqrt2,sqrt2)); - - - [ sqrt2 5 ] - a := [ ] - [7*sqrt2 sqrt2] - - - - frobenius(a); - - [0 35*sqrt2 - 2] [1 sqrt2 ] [ 1 ] - { [ ], [ ], [1 - --- ] } - [1 2*sqrt2 ] [1 7*sqrt2] [ 7 ] - [ ] - [ 1 ] - [0 ----*sqrt2] - [ 14 ] - - ____________________________________________________________ - - -File: redhelp, Node: Ratjordan, Next: Jordansymbolic, Prev: Frobenius, Up: Matrix Normal Forms section - - RATJORDAN operator - - The operator RATJORDAN computes the rational Jordan normal form R of -a [*note MATRIX::.] (A say). It returns R,P,P^-1 where P*R*P^-1 = A. - -syntax: - - RATJORDAN () - - :- a square [*note MATRIX::.] . - - Field Extensions: - - By default, calculations are performed in the rational numbers. To -extend this field the ARNUM package can be used. The package must first -be loaded by load_package arnum;. The field can now be extended by -using the defpoly command. For example, defpoly sqrt2**2-2; will extend -the field to include the square root of 2 (now defined by sqrt2). See -[*note Frobenius::.] for an example. - - Modular Arithmetic: - - RATJORDAN can also be calculated in a modular base. To do this first -type on modular;. Then setmod p; (where p is a prime) will set the -modular base of calculation to p. By further typing on balanced_mod the -answer will appear using a symmetric modular representation. - -examples: - - ____________________________________________________________ - - a := mat((5,4*x),(2,x^2)); - - [5 4*x] - [ ] - a := [ 2 ] - [2 x ] - - - - ratjordan(a); - - [0 x*( - 5*x + 8)] [1 5] [ -5 ] - { [ ], [ ], [1 -----] } - [ 2 ] [0 2] [ 2 ] - [1 x + 5 ] [ ] - [ 1 ] - [0 -----] - [ 2 ] - - - on modular; - - setmod 23; - - a := mat((12,34),(56,78)); - - [12 11] - a := [ ] - [10 9 ] - - - - ratjordan(a); - - [15 0] [16 8] [1 21] - { [ ], [ ], [ ] } - [0 6] [19 4] [1 4 ] - - - - on balanced_mod; - - ratjordan(a); - - [- 8 0] [ - 7 8] [1 - 2] - { [ ], [ ], [ ] } - [ 0 6] [ - 4 4] [1 4 ] - - ____________________________________________________________ - - -File: redhelp, Node: Jordansymbolic, Next: Jordan, Prev: Ratjordan, Up: Matrix Normal Forms section - - JORDANSYMBOLIC operator - - The operator JORDANSYMBOLIC computes the Jordan normal form J of a -[*note MATRIX::.] (A say). It returns J,L,P,P^-1 where P*J*P^-1 = A. L -= ll,mm where mm is a name and ll is a list of irreducible factors of -p(mm). - -syntax: - - JORDANSYMBOLIC () - - :- a square [*note MATRIX::.] . - - Field Extensions: - - By default, calculations are performed in the rational numbers. To -extend this field the [*note ARNUM::.] package can be used. The package -must first be loaded by load_package arnum;. The field can now be -extended by using the defpoly command. For example, defpoly sqrt2**2-2; -will extend the field to include the square root of 2 (now defined by -sqrt2). See [*note Frobenius::.] for an example. - - Modular Arithmetic: - - JORDANSYMBOLIC can also be calculated in a modular base. To do this -first type on modular;. Then setmod p; (where p is a prime) will set -the modular base of calculation to p. By further typing on balanced_mod -the answer will appear using a symmetric modular representation. See -[*note Ratjordan::.] for an example. - -examples: - - ____________________________________________________________ - - - a := mat((1,y),(2,5*y)); - - [1 y ] - a := [ ] - [2 5*y] - - - - jordansymbolic(a); - - { - [lambda11 0 ] - [ ] - [ 0 lambda12] - , - 2 - lambda - 5*lambda*y - lambda + 3*y,lambda, - [lambda11 - 5*y lambda12 - 5*y] - [ ] - [ 2 2 ] - , - [ 2*lambda11 - 5*y - 1 5*lambda11*y - lambda11 - y + 1 ] - [---------------------- ---------------------------------] - [ 2 2 ] - [ 25*y - 2*y + 1 2*(25*y - 2*y + 1) ] - [ ] - [ 2*lambda12 - 5*y - 1 5*lambda12*y - lambda12 - y + 1 ] - [---------------------- ---------------------------------] - [ 2 2 ] - [ 25*y - 2*y + 1 2*(25*y - 2*y + 1) ] - } - - ____________________________________________________________ - - -File: redhelp, Node: Jordan, Prev: Jordansymbolic, Up: Matrix Normal Forms section - - JORDAN operator - - The operator JORDAN computes the Jordan normal form J of a [*note -MATRIX::.] (A say). It returns J,P,P^-1 where P*J*P^-1 = A. - -syntax: - - JORDAN () - - :- a square [*note MATRIX::.] . - - Field Extensions: By default, calculations are performed in the -rational numbers. To extend this field the ARNUM package can be used. -The package must first be loaded by load_package arnum;. The field can -now be extended by using the defpoly command. For example, defpoly -sqrt2**2-2; will extend the field to include the square root of 2 (now -defined by sqrt2). See [*note Frobenius::.] for an example. - - Modular Arithmetic: JORDAN can also be calculated in a modular -base. To do this first type on modular;. Then setmod p; (where p is a -prime) will set the modular base of calculation to p. By further typing -on balanced_mod the answer will appear using a symmetric modular -representation. See [*note Ratjordan::.] for an example. - -examples: - - ____________________________________________________________ - - - a := mat((1,x),(0,x)); - - [1 x] - a := [ ] - [0 x] - - - - jordan(a); - - { - [1 0] - [ ] - [0 x] - , - [ 1 x ] - [------- --------------] - [ x - 1 2 ] - [ x - 2*x + 1 ] - [ ] - [ 1 ] - [ 0 ------- ] - [ x - 1 ] - , - [x - 1 - x ] - [ ] - [ 0 x - 1] - } - - ____________________________________________________________ - - -File: redhelp, Node: Matrix Normal Forms section, Next: Miscellaneous Packages section, Prev: Linear Algebra package section, Up: Top - - Matrix Normal Forms section - -* Menu: - -* Smithex:: operator -* Smithex_int:: operator -* Frobenius:: operator -* Ratjordan:: operator -* Jordansymbolic:: operator -* Jordan:: operator - - -File: redhelp, Node: Miscellaneous Packages, Next: ALGINT package, Up: Miscellaneous Packages section - - MISCELLANEOUS PACKAGES introduction - - REDUCE includes a large number of packages that have been -contributed by users from various fields. Some of these, together with -their relevant commands, switches and so on (e.g., the NUMERIC -package), have been described elsewhere. This section describes those -packages for which no separate help material exists. Each has its own -switches, commands, and operators, and some redefine special characters -to aid in their notation. However, the brief descriptions given here do -not include all such information. Readers are referred to the general -package documentation in this case, which can be found, along with the -source code, under the subdirectories DOC and SRC in the REDUCE -directory. The [*note LOAD_PACKAGE::.] command is used to load the -files you wish into your system. There will be a short delay while the -package is loaded. A package cannot be unloaded. Once it is in your -system, it stays there until you end the session. Each package also has -a test file, which you will find under its name in the $REDUCE/XMPL -directory. - - Finally, it should be mentioned that such user-contributed packages -are unsupported; any questions or problems should be directed to their -authors. - - -File: redhelp, Node: ALGINT package, Next: APPLYSYM, Prev: Miscellaneous Packages, Up: Miscellaneous Packages section - - ALGINT package - - Author: James H. Davenport - - The ALGINT package provides indefinite integration of square roots. -This package, which is an extension of the basic integration package -distributed with REDUCE, will analytically integrate a wide range of -expressions involving square roots. The [*note ALGINT::.] switch -provides for the use of the facilities given by the package, and is -automatically turned on when the package is loaded. If you want to -return to the standard integration algorithms, turn [*note ALGINT::.] -off. An error message is given if you try to turn the [*note ALGINT::.] -switch on when its package is not loaded. - - -File: redhelp, Node: APPLYSYM, Next: ARNUM, Prev: ALGINT package, Up: Miscellaneous Packages section - - APPLYSYM package - - Author: Thomas Wolf - - This package provides programs APPLYSYM, QUASILINPDE and DETRAFO for -computing with infinitesimal symmetries of differential equations. - - -File: redhelp, Node: ARNUM, Next: ASSIST, Prev: APPLYSYM, Up: Miscellaneous Packages section - - ARNUM package - - Author: Eberhard Schruefer - - This package provides facilities for handling algebraic numbers as -polynomial coefficients in REDUCE calculations. It includes facilities -for introducing indeterminates to represent algebraic numbers, for -calculating splitting fields, and for factoring and finding greatest -common divisors in such domains. - - -File: redhelp, Node: ASSIST, Next: AVECTOR, Prev: ARNUM, Up: Miscellaneous Packages section - - ASSIST package - - Author: Hubert Caprasse - - ASSIST contains a large number of additional general purpose -functions that allow a user to better adapt REDUCE to various -calculational strategies and to make the programming task more -straightforward and more efficient. - - -File: redhelp, Node: AVECTOR, Next: BOOLEAN, Prev: ASSIST, Up: Miscellaneous Packages section - - AVECTOR package - - Author: David Harper - - This package provides REDUCE with the ability to perform vector -algebra using the same notation as scalar algebra. The basic algebraic -operations are supported, as are differentiation and integration of -vectors with respect to scalar variables, cross product and dot -product, component manipulation and application of scalar functions -(e.g. cosine) to a vector to yield a vector result. - - -File: redhelp, Node: BOOLEAN, Next: CALI, Prev: AVECTOR, Up: Miscellaneous Packages section - - BOOLEAN package - - Author: Herbert Melenk - - This package supports the computation with boolean expressions in the -propositional calculus. The data objects are composed from algebraic -expressions connected by the infix boolean operators and, or, implies, -equiv, and the unary prefix operator not. Boolean allows you to -simplify expressions built from these operators, and to test properties -like equivalence, subset property etc. - - -File: redhelp, Node: CALI, Next: CAMAL, Prev: BOOLEAN, Up: Miscellaneous Packages section - - CALI package - - Author: Hans-Gert Gr"abe - - This package contains algorithms for computations in commutative -algebra closely related to the Groebner algorithm for ideals and -modules. Its heart is a new implementation of the Groebner algorithm -that also allows for the computation of syzygies. This implementation -is also applicable to submodules of free modules with generators -represented as rows of a matrix. - - -File: redhelp, Node: CAMAL, Next: CHANGEVR, Prev: CALI, Up: Miscellaneous Packages section - - CAMAL package - - Author: John P. Fitch - - This packages implements in REDUCE the Fourier transform procedures -of the CAMAL package for celestial mechanics. - - -File: redhelp, Node: CHANGEVR, Next: COMPACT, Prev: CAMAL, Up: Miscellaneous Packages section - - CHANGEVR package - - Author: G. Ucoluk - - This package provides facilities for changing the independent -variables in a differential equation. It is basically the application -of the chain rule. - - -File: redhelp, Node: COMPACT, Next: CONTFR, Prev: CHANGEVR, Up: Miscellaneous Packages section - - COMPACT package - - Author: Anthony C. Hearn - - COMPACT is a package of functions for the reduction of a polynomial -in the presence of side relations. COMPACT applies the side relations -to the polynomial so that an equivalent expression results with as few -terms as possible. For example, the evaluation of - - ____________________________________________________________ - - compact(s*(1-sin x^2)+c*(1-cos x^2)+sin x^2+cos x^2, - {cos x^2+sin x^2=1}); - - ____________________________________________________________ - yields the result - ____________________________________________________________ - - - 2 2 - SIN(X) *C + COS(X) *S + 1 - ____________________________________________________________ - - The first argument to the operator COMPACT is the expression and the -second is a list of side relations that can be equations or simple -expressions (implicitly equated to zero). The kernels in the side -relations may also be free variables with the same meaning as in rules, -e.g. - ____________________________________________________________ - - sin_cos_identity := {cos ~w^2+sin ~w^2=1}$ - compact(u,in_cos_identity); - ____________________________________________________________ - - Also the full rule syntax with the replacement operator is allowed -here. - - -File: redhelp, Node: CONTFR, Next: CRACK, Prev: COMPACT, Up: Miscellaneous Packages section - - CONTFR package - - Author: Herbert Melenk - - This package provides for the simultaneous approximation of a real -number by a continued fraction and a rational number with optional user -controlled precision (an upper bound for the denominator). - - To use this package, the MISC package should be loaded. One can then -use the operator CONTINUED_FRACTION to approximate the real number by a -continued fraction. This operator has one or two arguments, the number -to be converted and an optional precision. The result is a list of two -elements: the first is the rational value of the approximation and the -second the list of terms of the continued fraction that represent the -same value according to the definition t0 +1/(t1 + 1/(t2 + ...)). The -second optional parameter SIZE is an upper bound on the absolute value -of the result denominator. If omitted, the approximation is performed -up to the current system precision. - -examples: - - ____________________________________________________________ - - continued_fraction pi; - - 1146408 - {---------,{3,7,15,1,292,1,1,1,2,1}} - 364913 - - - - continued_fraction(pi,100); - - 22 - {----,{3,7}} - 7 - - ____________________________________________________________ - - -File: redhelp, Node: CRACK, Next: CVIT, Prev: CONTFR, Up: Miscellaneous Packages section - - CRACK package - - Authors: Andreas Brand, Thomas Wolf - - CRACK is a package for solving overdetermined systems of partial or -ordinary differential equations (PDEs, ODEs). Examples of programs which -make use of CRACK for investigating ODEs (finding symmetries, first -integrals, an equivalent Lagrangian or a "differential factorization") -are included. - - -File: redhelp, Node: CVIT, Next: DEFINT, Prev: CRACK, Up: Miscellaneous Packages section - - CVIT package - - Authors: V.Ilyin, A.Kryukov, A.Rodionov, A.Taranov - - This package provides an alternative method for computing traces of -Dirac gamma matrices, based on an algorithm by Cvitanovich that treats -gamma matrices as 3-j symbols. - - -File: redhelp, Node: DEFINT, Next: DESIR, Prev: CVIT, Up: Miscellaneous Packages section - - DEFINT package - - Authors: Kerry Gaskell, Stanley M. Kameny, Winfried Neun - - This package finds the definite integral of an expression in a stated -interval. It uses several techniques, including an innovative approach -based on the Meijer G-function, and contour integration. - - -File: redhelp, Node: DESIR, Next: DFPART, Prev: DEFINT, Up: Miscellaneous Packages section - - DESIR package - - Authors: C. Dicrescenzo, F. Richard-Jung, E. Tournier - - This package enables the basis of formal solutions to be computed -for an ordinary homogeneous differential equation with polynomial -coefficients over Q of any order, in the neighborhood of zero (regular -or irregular singular point, or ordinary point). - - -File: redhelp, Node: DFPART, Next: DUMMY, Prev: DESIR, Up: Miscellaneous Packages section - - DFPART package - - Author: Herbert Melenk - - This package supports computations with total and partial -derivatives of formal function objects. Such computations can be useful -in the context of differential equations or power series expansions. - - -File: redhelp, Node: DUMMY, Next: EXCALC, Prev: DFPART, Up: Miscellaneous Packages section - - DUMMY package - - Author: Alain Dresse - - This package allows a user to find the canonical form of expressions -involving dummy variables. In that way, the simplification of -polynomial expressions can be fully done. The indeterminates are general -operator objects endowed with as few properties as possible. In that way -the package may be used in a large spectrum of applications. - - -File: redhelp, Node: EXCALC, Next: FPS, Prev: DUMMY, Up: Miscellaneous Packages section - - EXCALC package - - Author: Eberhard Schruefer - - The EXCALC package is designed for easy use by all who are familiar -with the calculus of Modern Differential Geometry. The program is -currently able to handle scalar-valued exterior forms, vectors and -operations between them, as well as non-scalar valued forms (indexed -forms). It is thus an ideal tool for studying differential equations, -doing calculations in general relativity and field theories, or doing -simple things such as calculating the Laplacian of a tensor field for -an arbitrary given frame. - - -File: redhelp, Node: FPS, Next: FIDE, Prev: EXCALC, Up: Miscellaneous Packages section - - FPS package - - Authors: Wolfram Koepf, Winfried Neun - - This package can expand a specific class of functions into their -corresponding Laurent-Puiseux series. - - -File: redhelp, Node: FIDE, Next: GENTRAN, Prev: FPS, Up: Miscellaneous Packages section - - FIDE package - - Author: Richard Liska - - This package performs automation of the process of numerically -solving partial differential equations systems (PDES) by means of -computer algebra. For PDES solving, the finite difference method is -applied. The computer algebra system REDUCE and the numerical -programming language FORTRAN are used in the presented methodology. The -main aim of this methodology is to speed up the process of preparing -numerical programs for solving PDES. This process is quite often, -especially for complicated systems, a tedious and time consuming task. - - -File: redhelp, Node: GENTRAN, Next: IDEALS, Prev: FIDE, Up: Miscellaneous Packages section - - GENTRAN package - - Author: Barbara L. Gates - - This package is an automatic code GENerator and TRANslator. It -constructs complete numerical programs based on sets of algorithmic -specifications and symbolic expressions. Formatted FORTRAN, RATFOR or C -code can be generated through a series of interactive commands or under -the control of a template processing routine. Large expressions can be -automatically segmented into subexpressions of manageable size, and a -special file-handling mechanism maintains stacks of open I/O channels -to allow output to be sent to any number of files simultaneously and to -facilitate recursive invocation of the whole code generation process. - - -File: redhelp, Node: IDEALS, Next: INEQ, Prev: GENTRAN, Up: Miscellaneous Packages section - - IDEALS package - - Author: Herbert Melenk - - This package implements the basic arithmetic for polynomial ideals by -exploiting the Groebner bases package of REDUCE. In order to save -computing time all intermediate Groebner bases are stored internally -such that time consuming repetitions are inhibited. - - -File: redhelp, Node: INEQ, Next: INVBASE, Prev: IDEALS, Up: Miscellaneous Packages section - - INEQ package - - Author: Herbert Melenk - - This package supports the operator INEQ_SOLVE that tries to solves -single inequalities and sets of coupled inequalities. - - -File: redhelp, Node: INVBASE, Next: LAPLACE, Prev: INEQ, Up: Miscellaneous Packages section - - INVBASE package - - Authors: A.Yu. Zharkov and Yu.A. Blinkov - - Involutive bases are a new tool for solving problems in connection -with multivariate polynomials, such as solving systems of polynomial -equations and analyzing polynomial ideals. An involutive basis of -polynomial ideal is nothing but a special form of a redundant Groebner -basis. The construction of involutive bases reduces the problem of -solving polynomial systems to simple linear algebra. - - -File: redhelp, Node: LAPLACE, Next: LIE, Prev: INVBASE, Up: Miscellaneous Packages section - - LAPLACE package - - Authors: C. Kazasov, M. Spiridonova, V. Tomov - - This package can calculate ordinary and inverse Laplace transforms of -expressions. Documentation is in plain text. - - -File: redhelp, Node: LIE, Next: MODSR, Prev: LAPLACE, Up: Miscellaneous Packages section - - LIE package - - Authors: Carsten and Franziska Sch"obel - - LIE is a package of functions for the classification of real -n-dimensional Lie algebras. It consists of two modules: LIENDMC1 and -LIE1234 . With the help of the functions in the LIENDMCL module, real -n-dimensional Lie algebras L with a derived algebra L^(1) of dimension -1 can be classified. - - -File: redhelp, Node: MODSR, Next: NCPOLY, Prev: LIE, Up: Miscellaneous Packages section - - MODSR package - - Author: Herbert Melenk - - This package supports solve (M_SOLVE) and roots (M_ROOTS) operators -for modular polynomials and modular polynomial systems. The moduli need -not be primes. M_SOLVE requires a modulus to be set. M_ROOTS takes the -modulus as a second argument. For example: - - ____________________________________________________________ - - on modular; setmod 8; - m_solve(2x=4); -> {{X=2},{X=6}} - m_solve({x^2-y^3=3}); - -> {{X=0,Y=5}, {X=2,Y=1}, {X=4,Y=5}, {X=6,Y=1}} - m_solve({x=2,x^2-y^3=3}); -> {{X=2,Y=1}} - off modular; - m_roots(x^2-1,8); -> {1,3,5,7} - m_roots(x^3-x,7); -> {0,1,6} - ____________________________________________________________ - - -File: redhelp, Node: NCPOLY, Next: ORTHOVEC, Prev: MODSR, Up: Miscellaneous Packages section - - NCPOLY package - - Authors: Herbert Melenk, Joachim Apel - - This package allows the user to set up automatically a consistent -environment for computing in an algebra where the non-commutativity is -defined by Lie-bracket commutators. The package uses the REDUCE NONCOM -mechanism for elementary polynomial arithmetic; the commutator rules -are automatically computed from the Lie brackets. - - -File: redhelp, Node: ORTHOVEC, Next: PHYSOP, Prev: NCPOLY, Up: Miscellaneous Packages section - - ORTHOVEC package - - Author: James W. Eastwood - - ORTHOVEC is a collection of REDUCE procedures and operations which -provide a simple-to-use environment for the manipulation of scalars and -vectors. Operations include addition, subtraction, dot and cross -products, division, modulus, div, grad, curl, laplacian, -differentiation, integration, and Taylor expansion. - - -File: redhelp, Node: PHYSOP, Next: PM, Prev: ORTHOVEC, Up: Miscellaneous Packages section - - PHYSOP package - - Author: Mathias Warns - - This package has been designed to meet the requirements of -theoretical physicists looking for a computer algebra tool to perform -complicated calculations in quantum theory with expressions containing -operators. These operations consist mainly of the calculation of -commutators between operator expressions and in the evaluations of -operator matrix elements in some abstract space. - - -File: redhelp, Node: PM, Next: RANDPOLY, Prev: PHYSOP, Up: Miscellaneous Packages section - - PM package - - Author: Kevin McIsaac - - PM is a general pattern matcher similar in style to those found in -systems such as SMP and Mathematica, and is based on the pattern -matcher described in Kevin McIsaac, "Pattern Matching Algebraic -Identities", SIGSAM Bulletin, 19 (1985), 4-13. - - -File: redhelp, Node: RANDPOLY, Next: REACTEQN, Prev: PM, Up: Miscellaneous Packages section - - RANDPOLY package - - Author: Francis J. Wright - - This package is based on a port of the Maple random polynomial -generator together with some support facilities for the generation of -random numbers and anonymous procedures. - - -File: redhelp, Node: REACTEQN, Next: RESET, Prev: RANDPOLY, Up: Miscellaneous Packages section - - REACTEQN package - - Author: Herbert Melenk - - This package allows a user to transform chemical reaction systems -into ordinary differential equation systems (ODE) corresponding to the -laws of pure mass action. - - -File: redhelp, Node: RESET, Next: RESIDUE, Prev: REACTEQN, Up: Miscellaneous Packages section - - RESET package - - Author: John Fitch - - This package defines a command command RESETREDUCE that works -through the history of previous commands, and clears any values which -have been assigned, plus any rules, arrays and the like. It also sets -the various switches to their initial values. It is not complete, but -does work for most things that cause a gradual loss of space. It would -be relatively easy to make it interactive, so allowing for selective -resetting. - - -File: redhelp, Node: RESIDUE, Next: RLFI, Prev: RESET, Up: Miscellaneous Packages section - - RESIDUE package - - Author: Wolfram Koepf - - This package supports the calculation of residues of arbitrary -expressions. - - -File: redhelp, Node: RLFI, Next: SCOPE, Prev: RESIDUE, Up: Miscellaneous Packages section - - RLFI package - - Author: Richard Liska - - This package adds LaTeX syntax to REDUCE. Text generated by REDUCE -in this mode can be directly used in LaTeX source documents. Various -mathematical constructions are supported by the interface including -subscripts, superscripts, font changing, Greek letters, divide-bars, -integral and sum signs, derivatives, and so on. - - -File: redhelp, Node: SCOPE, Next: SETS, Prev: RLFI, Up: Miscellaneous Packages section - - SCOPE package - - Author: J.A. van Hulzen - - SCOPE is a package for the production of an optimized form of a set -of expressions. It applies an heuristic search for common -(sub)expressions to almost any set of proper REDUCE assignment -statements. The output is obtained as a sequence of assignment -statements. GENTRAN is used to facilitate expression output. - - -File: redhelp, Node: SETS, Next: SPDE, Prev: SCOPE, Up: Miscellaneous Packages section - - SETS package - - Author: Francis J. Wright - - The SETS package provides algebraic-mode support for set operations -on lists regarded as sets (or representing explicit sets) and on -implicit sets represented by identifiers. - - -File: redhelp, Node: SPDE, Next: SYMMETRY, Prev: SETS, Up: Miscellaneous Packages section - - SPDE package - - Author: Fritz Schwartz - - The package SPDE provides a set of functions which may be used to -determine the symmetry group of Lie- or point-symmetries of a given -system of partial differential equations. In many cases the determining -system is solved completely automatically. In other cases the user has -to provide additional input information for the solution algorithm to -terminate. - - -File: redhelp, Node: SYMMETRY, Next: TPS, Prev: SPDE, Up: Miscellaneous Packages section - - SYMMETRY package - - Author: Karin Gatermann - - This package computes symmetry-adapted bases and block diagonal -forms of matrices which have the symmetry of a group. The package is the -implementation of the theory of linear representations for small finite -groups such as the dihedral groups. - - -File: redhelp, Node: TPS, Next: TRI, Prev: SYMMETRY, Up: Miscellaneous Packages section - - TPS package - - Authors: Alan Barnes, Julian Padget - - This package implements formal Laurent series expansions in one -variable using the domain mechanism of REDUCE. This means that power -series objects can be added, multiplied, differentiated etc., like -other first class objects in the system. A lazy evaluation scheme is -used and thus terms of the series are not evaluated until they are -required for printing or for use in calculating terms in other power -series. The series are extendible giving the user the impression that -the full infinite series is being manipulated. The errors that can -sometimes occur using series that are truncated at some fixed depth -(for example when a term in the required series depends on terms of an -intermediate series beyond the truncation depth) are thus avoided. - - -File: redhelp, Node: TRI, Next: TRIGSIMP, Prev: TPS, Up: Miscellaneous Packages section - - TRI package - - Author: Werner Antweiler - - This package provides facilities written in REDUCE-Lisp for -typesetting REDUCE formulas using TeX. The TeX-REDUCE-Interface -incorporates three levels of TeX output: without line breaking, with -line breaking, and with line breaking plus indentation. - - -File: redhelp, Node: TRIGSIMP, Next: XCOLOR, Prev: TRI, Up: Miscellaneous Packages section - - TRIGSIMP package - - Author: Wolfram Koepf - - TRIGSIMP is a useful tool for all kinds of trigonometric and -hyperbolic simplification and factorization. There are three procedures -included in TRIGSIMP: TRIGSIMP , TRIGFACTORIZE and TRIGGCD . The first -is for finding simplifications of trigonometric or hyperbolic -expressions with many options, the second for factorizing them and the -third for finding the greatest common divisor of two trigonometric or -hyperbolic polynomials. - - -File: redhelp, Node: XCOLOR, Next: XIDEAL, Prev: TRIGSIMP, Up: Miscellaneous Packages section - - XCOLOR package - - Author: A. Kryukov - - This package calculates the color factor in non-abelian gauge field -theories using an algorithm due to Cvitanovich. - - -File: redhelp, Node: XIDEAL, Next: WU, Prev: XCOLOR, Up: Miscellaneous Packages section - - XIDEAL package - - Author: David Hartley - - XIDEAL constructs Groebner bases for solving the left ideal -membership problem: Groebner left ideal bases or GLIBs. For graded -ideals, where each form is homogeneous in degree, the distinction -between left and right ideals vanishes. Furthermore, if the generating -forms are all homogeneous, then the Groebner bases for the non-graded -and graded ideals are identical. In this case, XIDEAL is able to save -time by truncating the Groebner basis at some maximum degree if desired. - - -File: redhelp, Node: WU, Next: ZEILBERG, Prev: XIDEAL, Up: Miscellaneous Packages section - - WU package - - Author: Russell Bradford - - This is a simple implementation of the Wu algorithm implemented in -REDUCE working directly from "A Zero Structure Theorem for -Polynomial-Equations-Solving," Wu Wen-tsun, Institute of Systems -Science, Academia Sinica, Beijing. - - -File: redhelp, Node: ZEILBERG, Next: ZTRANS, Prev: WU, Up: Miscellaneous Packages section - - ZEILBERG package - - Authors: Gregor St"olting and Wolfram Koepf - - This package is a careful implementation of the Gosper and Zeilberger -algorithms for indefinite and definite summation of hypergeometric -terms, respectively. Extensions of these algorithms are also included -that are valid for ratios of products of powers, factorials, gamma -function terms, binomial coefficients, and shifted factorials that are -rational-linear in their arguments. - - -File: redhelp, Node: ZTRANS, Prev: ZEILBERG, Up: Miscellaneous Packages section - - ZTRANS package - - Authors: Wolfram Koepf, Lisa Temme - - This package is an implementation of the Z-transform of a sequence. -This is the discrete analogue of the Laplace Transform. - - -File: redhelp, Node: Miscellaneous Packages section, Next: Outmoded Operations section, Prev: Matrix Normal Forms section, Up: Top - - Miscellaneous Packages section - -* Menu: - -* Miscellaneous Packages:: introduction -* ALGINT package:: package -* APPLYSYM:: package -* ARNUM:: package -* ASSIST:: package -* AVECTOR:: package -* BOOLEAN:: package -* CALI:: package -* CAMAL:: package -* CHANGEVR:: package -* COMPACT:: package -* CONTFR:: package -* CRACK:: package -* CVIT:: package -* DEFINT:: package -* DESIR:: package -* DFPART:: package -* DUMMY:: package -* EXCALC:: package -* FPS:: package -* FIDE:: package -* GENTRAN:: package -* IDEALS:: package -* INEQ:: package -* INVBASE:: package -* LAPLACE:: package -* LIE:: package -* MODSR:: package -* NCPOLY:: package -* ORTHOVEC:: package -* PHYSOP:: package -* PM:: package -* RANDPOLY:: package -* REACTEQN:: package -* RESET:: package -* RESIDUE:: package -* RLFI:: package -* SCOPE:: package -* SETS:: package -* SPDE:: package -* SYMMETRY:: package -* TPS:: package -* TRI:: package -* TRIGSIMP:: package -* XCOLOR:: package -* XIDEAL:: package -* WU:: package -* ZEILBERG:: package -* ZTRANS:: package - - -File: redhelp, Node: ED, Next: EDITDEF, Up: Outmoded Operations section - - ED command - - The ED command invokes a simple line editor for REDUCE input -statements. - -syntax: - - ED or ED - - ED called with no argument edits the last input statement. If - is greater than or equal to the current line number, an error -message is printed. Reenter a proper ED command or return to the top -level with a semicolon. - - The editor formats REDUCE's version of the desired input statement, -dividing it into lines at semicolons and dollar signs. The statement is -printed at the beginning of the edit session. The editor works on one -line at a time, and has a pointer (shown by ^ ) to the current -character of that line. When the session begins, the pointer is at the -left hand side of the first line. The editing prompt is > . - - The following commands are available. They may be entered in either -upper or lower case. All commands are activated by the carriage return, -which also prints out the current line after changes. Several commands -can be placed on a single line, except that commands terminated by an -ESC must be the last command before the carriage return. - - b Move pointer to beginning of current line. - - d Delete current character and next (digit-1) characters. An -error message is printed if anything other than a single digit follows -d. If there are fewer than characters left on the line, all but -the final dollar sign or semicolon is removed. To delete a line -completely, use the k command. - - e End the current session, causing the edited expression to be -reparsed by REDUCE. - - f Find the next occurrence of the character to the -right of the pointer on the current line and move the pointer to it. If -the character is not found, an error message is printed and the pointer -remains in its original position. Other lines are not searched. The f -command is not case-sensitive. - - iESC Insert in front of pointer. The ESC key is your -delimiter for the input string. No other command may follow this one on -the same line. - - k Kill rest of the current line, including the semicolon or dollar -sign terminator. If there are characters remaining on the current line, -and it is the last line of the input statement, a semicolon is added to -the line as a terminator for REDUCE. If the current line is now empty, -one of the following actions is performed: If there is a following -line, it becomes the current line and the pointer is placed at its -first character. If the current line was the final line of the -statement, and there is a previous line, the previous line becomes the -current line. If the current line was the only line of the statement, -and it is empty, a single semicolon is inserted for REDUCE to parse. - - l Finish editing this line and move to the last previous line. An -error message is printed if there is no previous line. - - n Finish editing this line and move to the next line. An error -message is printed if there is no next line. - - p Print out all the lines of the statement. Then a dotted line is -printed, and the current line is reprinted, with the pointer under it. - - q Quit the editing session without saving the changes. If a -semicolon is entered after q, a new line prompt is given, otherwise -REDUCE prompts you for another command. Whatever you type in to the -prompt appearing after the q is entered is stored as the input for the -line number in which you called the edit. Thus if you enter a -semicolon, neither [*note INPUT::.] ED will find anything under the -current number. - - r Replace the character at the pointer by . - - sESC Search for the first occurrence of to the -right of the pointer on the current line and move the pointer to its -first character. The ESC key is your delimiter for the input string. -The s function does not search other lines of the statement. If the -string is not found, an error message is printed and the pointer -remains in its original position. The s command is not case-sensitive. -No other command may follow this one on the same line. - - x Move the pointer one character to the right. If the -pointer is already at the end of the line, an error message is printed. - - - <(minus)> Move the pointer one character to the left. If the -pointer is already at the beginning of the line, an error message is -printed. - - ? Display the Help menu, showing the commands and their actions. - -examples: - - ____________________________________________________________ - ____________________________________________________________ - (Line numbers are shown in the following examples) - ____________________________________________________________ - - - 2: >>x**2 + y; - - X^{2} + Y - - 3: >>ed 2; - - X**2 + Y; - - ^ - - For help, type '?' - - ?- (Enter three spaces and (Key){Return}) - - X**2 + Y; - - ^ - - ?- r5 - - X**5 + Y; - - ^ - - ?- fY - - X**5 + Y; - - ^ - - ?- iabc (Terminate with (Key){ESC} and (Key){Return}) - - X**5 + abcY; - - ^ - - ?- ---- - - X**5 + abcY; - - ^ - - ?- fbd2 - - X**5 + aY; - - ^ - - ?- b - - X**5 + aY; - - ^ - - ?- e - - AY + X^{5} - - 4: >>procedure dumb(a); - - >>write a; - - DUMB - - 5: >>dumb(17); - - 17 - - 6: >>ed 4; - - PROCEDURE DUMB (A); - - ^ - - WRITE A; - - ?- fArBn - - WRITE A; - - ^ - - ?- ibegin scalar a; a := b + 10; (Type a space, (Key){ESC}, and (Key){Return}) - - begin scalar a; a := b + 10; WRITE A; - - ?- f;i end (Key){ESC}, (Key){Return} - - begin scalar b; b := a + 10; WRITE A end; - - ^ - - ?- p - - PROCEDURE DUMB (B); - - begin scalar b; b := a + 10; WRITE A end; - - - - - - - - - - - - - - begin scalar b; b := a + 10; WRITE A end; - - ^ - - ?- e - - DUMB - - 7: >>dumb(17); - - 27 - - 8: >> - - ____________________________________________________________ - - Note that REDUCE reparsed the procedure DUMB and updated the -definition. - - Since REDUCE divides the expression to be edited into lines at -semicolons or dollar sign terminators, some lines may occupy more than -one line of screen space. If the pointer is directly beneath the last -line of text, it refers to the top line of text. If there is a blank -line between the last line of text and the pointer, it refers to the -second line of text, and likewise for cases of greater than two lines -of text. In other words, the entire REDUCE statement up to the next -terminator is printed, even if it runs to several lines, then the -pointer line is printed. - - You can insert new statements which contain semicolons of their own -into the current line. They are run into the current line where you -placed them until you edit the statement again. REDUCE will understand -the set of statements if the syntax is correct. - - If you leave out needed closing brackets when you exit the editor, a -message is printed allowing you to redo the edit (you can edit the -previous line number and return to where you were). If you leave out a -closing double-quotation mark, an error message is printed, and the -editing must be redone from the original version; the edited version -has been destroyed. Most syntax errors which you inadvertently leave -in an edited statement are caught as usual by the REDUCE parser, and -you will be able to re-edit the statement. - - When the editor processes a previous statement for your editing, -escape characters are removed. Most special characters that you may use -in identifiers are printed in legal fashion, prefixed by the exclamation -point. Be sure to treat the special character and its escape as a pair -in your editing. The characters ( ) # ; ' are different. Since they -have special meaning in Lisp, they are double-escaped in the editor. -It is unwise to use these characters inside identifiers anyway, due to -the probability of confusion. - - If you see a Lisp error message during editing, the edit has been -aborted. Enter a semicolon and you will see a new line prompt. - - Since the editor has no dependence on any window system, it can be -used if you are running REDUCE without windows. - - -File: redhelp, Node: EDITDEF, Prev: ED, Up: Outmoded Operations section - - EDITDEF command - - The interactive editor [*note ED::.] may be used to edit a -user-defined procedure that has not been compiled. - -syntax: - - EDITDEF (IDENTIFIER ) - - where IDENTIFIER is the name of the procedure. When EDITDEF is -invoked, the procedure definition will be displayed in editing mode, -and may then be edited and redefined on exiting from the editor using -standard [*note ED::.] commands. - - -File: redhelp, Node: Outmoded Operations section, Prev: Miscellaneous Packages section, Up: Top - - Outmoded Operations section - -* Menu: - -* ED:: command -* EDITDEF:: command - - -File: redhelp, Node: Top, Up: (dir) - - - - Top - -* Menu: - -* Concepts section:: -* Variables section:: -* Syntax section:: -* Arithmetic Operations section:: -* Boolean Operators section:: -* General Commands section:: -* Algebraic Operators section:: -* Declarations section:: -* Input and Output section:: -* Elementary Functions section:: -* General Switches section:: -* Matrix Operations section:: -* Groebner package section:: -* High Energy Physics section:: -* Numeric Package section:: -* Roots Package section:: -* Special Functions section:: -* Taylor series section:: -* Gnuplot package section:: -* Linear Algebra package section:: -* Matrix Normal Forms section:: -* Miscellaneous Packages section:: -* Outmoded Operations section:: - - - -Tag Table: -Node: IDENTIFIER89 -Node: KERNEL2324 -Node: STRING3392 -Node: Concepts section3968 -Node: assumptions4176 -Node: CARD_NO4826 -Node: E5867 -Node: EVAL_MODE6482 -Node: FORT_WIDTH6999 -Node: HIGH_POW7815 -Node: I8845 -Node: INFINITY9764 -Node: LOW_POW10098 -Node: NIL10948 -Node: PI11255 -Node: requirements11766 -Node: ROOT_MULTIPLICITIES12497 -Node: T12952 -Node: Variables section13268 -Node: semicolon13737 -Node: dollar15078 -Node: percent16681 -Node: dot17543 -Node: assign18702 -Node: equalsign21581 -Node: replace22690 -Node: plussign23164 -Node: minussign23972 -Node: asterisk24929 -Node: slash26263 -Node: power27815 -Node: caret29236 -Node: geqsign30594 -Node: greater31612 -Node: leqsign32756 -Node: less33720 -Node: tilde34705 -Node: group35013 -Node: AND36337 -Node: BEGIN37441 -Node: block39104 -Node: COMMENT39416 -Node: CONS40186 -Node: END41485 -Node: EQUATION42355 -Node: FIRST44096 -Node: FOR44791 -Node: FOREACH49527 -Node: GEQ50216 -Node: GOTO51311 -Node: GREATERP52299 -Node: IF53401 -Node: LIST55938 -Node: OR57485 -Node: PROCEDURE58633 -Node: REPEAT64274 -Node: REST65495 -Node: RETURN66354 -Node: REVERSE68550 -Node: RULE69700 -Node: Free Variable71388 -Node: Optional Free Variable71963 -Node: SECOND72895 -Node: SET73615 -Node: SETQ74805 -Node: THIRD76899 -Node: WHEN77657 -Node: Syntax section77939 -Node: ARITHMETIC_OPERATIONS79927 -Node: ABS80256 -Node: ADJPREC81171 -Node: ARG82070 -Node: CEILING82722 -Node: CHOOSE83550 -Node: DEG2DMS84254 -Node: DEG2RAD85082 -Node: DIFFERENCE85822 -Node: DILOG86780 -Node: DMS2DEG87560 -Node: DMS2RAD88337 -Node: FACTORIAL89119 -Node: FIX89844 -Node: FIXP90758 -Node: FLOOR91726 -Node: EXPT92537 -Node: GCD93564 -Node: LN94527 -Node: LOG95533 -Node: LOGB96404 -Node: MAX97393 -Node: MIN98071 -Node: MINUS98747 -Node: NEXTPRIME99421 -Node: NOCONVERT100086 -Node: NORM100684 -Node: PERM101514 -Node: PLUS102277 -Node: QUOTIENT103281 -Node: RAD2DEG105295 -Node: RAD2DMS106032 -Node: RECIP106813 -Node: REMAINDER107352 -Node: ROUND108400 -Node: SETMOD109013 -Node: SIGN110331 -Node: SQRT111038 -Node: TIMES112284 -Node: Arithmetic Operations section113292 -Node: boolean value114899 -Node: EQUAL115270 -Node: EVENP116460 -Node: false117495 -Node: FREEOF117929 -Node: LEQ119109 -Node: LESSP120184 -Node: MEMBER121267 -Node: NEQ122259 -Node: NOT123321 -Node: NUMBERP124252 -Node: ORDP125124 -Node: PRIMEP126108 -Node: TRUE126817 -Node: Boolean Operators section127468 -Node: BYE128155 -Node: CONT128479 -Node: DISPLAY129810 -Node: LOAD_PACKAGE130698 -Node: PAUSE131120 -Node: QUIT133498 -Node: RECLAIM133854 -Node: REDERR134624 -Node: RETRY136051 -Node: SAVEAS136864 -Node: SHOWTIME137941 -Node: WRITE138900 -Node: General Commands section140314 -Node: APPEND140915 -Node: ARBINT142112 -Node: ARBCOMPLEX142691 -Node: ARGLENGTH143426 -Node: COEFF144430 -Node: COEFFN146529 -Node: 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All procedures are entered into the system as operators, so -the name of a procedure may not be used as a matrix, array, or operator -identifier either. - \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # KERNEL} - -${\footnote \pard\plain \sl240 \fs20 $ KERNEL} - -+{\footnote \pard\plain \sl240 \fs20 + g2:0644} - - K{\footnote \pard\plain \sl240 \fs20 K KERNEL type;type} - -}{\b\f2 KERNEL}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par -\par - -A }{\f3 kernel} {\f2 is a form that cannot be modified further by the REDUCE -canonical simplifier. Scalar variables are always kernels. The -other important class of kernels are operators with their arguments. -Some examples should help clarify this concept: -\par -\par -\pard \tx3420 }{\f4 \par - Expression Kernel? \par - \par - x Yes \par - varname Yes \par - cos(a) Yes \par - log(sin(x**2)) Yes \par - a*b No \par - (x+y)**4 No \par - matrix-identifier No \par -\pard \sl240 }{\f2 Many REDUCE operators expect kernels among their arguments. Error messages -result from attempts to use non-kernel expressions for these arguments. - \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # STRING} - -${\footnote \pard\plain \sl240 \fs20 $ STRING} - -+{\footnote \pard\plain \sl240 \fs20 + g2:0645} - - K{\footnote \pard\plain \sl240 \fs20 K STRING type;type} - -}{\b\f2 STRING}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par -\par - -A }{\f3 string} {\f2 is any collection of characters enclosed in double quotation -marks (}{\f3 "} {\f2 ). It may be used as an argument for a variety of commands -and operators, such as }{\f3 in} {\f2 , }{\f3 rederr} {\f2 and }{\f3 write} {\f2 . - \par -examples: \par -\pard \tx3420 }{\f4 \par -write "this is a string"; \par - \par - this is a string \par - \par - \par -write a, " ", b, " ",c,"!"; \par - \par - A B C! \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g2} - -${\footnote \pard\plain \sl240 \fs20 $ Concepts} - -+{\footnote \pard\plain \sl240 \fs20 + index:0002} -}{\b\f2 Concepts}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb IDENTIFIER type} -{\v\f2 IDENTIFIER}{\f2 \par -}{\f2 \tab}{\f2\uldb KERNEL type} -{\v\f2 KERNEL}{\f2 \par -}{\f2 \tab}{\f2\uldb STRING type} -{\v\f2 STRING}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # assumptions} - -${\footnote \pard\plain \sl240 \fs20 $ assumptions} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0646} - - K{\footnote \pard\plain \sl240 \fs20 K solve;assumptions variable;variable} - -}{\b\f2 ASSUMPTIONS}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -After solving a linear or polynomial equation system -with parameters, the variable }{\f3 assumptions} {\f2 contains a list -of side relations for the parameters. The solution is valid only -as long as none of these expression is zero. - \par -examples: \par -\pard \tx3420 }{\f4 \par -solve(\{a*x-b*y+x,y-c\},\{x,y\}); \par - \par - b*c \par - \{\{x=-----,y=c\}\} \par - a + 1 \par - \par - \par -assumptions; \par - \par - \{a + 1\} \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CARD\_NO} - -${\footnote \pard\plain \sl240 \fs20 $ CARD_NO} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0647} - - K{\footnote \pard\plain \sl240 \fs20 K output;FORTRAN;CARD_NO variable;variable} - -}{\b\f2 CARD\_NO}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -}{\f3 card_no} {\f2 sets the total number of cards allowed in a Fortran -output statement when }{\f3 fort} {\f2 is on. Default is 20. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on fort; \par - \par -card_no := 4; \par - \par - CARD_NO=4. \par - \par - \par -z := (x + y)**15; \par - \par - ANS1=5005.*X**6*Y**9+3003.*X**5*Y**10+1365.*X**4*Y** \par - . 11+455.*X**3*Y**12+105.*X**2*Y**13+15.*X*Y**14+Y**15 \par - Z=X**15+15.*X**14*Y+105.*X**13*Y**2+455.*X**12*Y**3+ \par - . 1365.*X**11*Y**4+3003.*X**10*Y**5+5005.*X**9*Y**6+ \par - . 6435.*X**8*Y**7+6435.*X**7*Y**8+ANS1 \par - \par -\pard \sl240 }{\f2 Twenty total cards means 19 continuation cards. You may set it for more -if your Fortran system allows more. Expressions are broken apart in a -Fortran-compatible way if they extend for more than }{\f3 card_no} {\f2 -continuation cards. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # E} - -${\footnote \pard\plain \sl240 \fs20 $ E} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0648} - - K{\footnote \pard\plain \sl240 \fs20 K E constant;constant} - -}{\b\f2 E}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - -The constant }{\f3 e} {\f2 is reserved for use as the base of the natural -logarithm. Its value is approximately 2.71828284590, which REDUCE gives -to the current decimal precision when the switch } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on. -\par -\par -}{\f3 e} {\f2 may be used as an iterative variable in a } -{\f2\uldb for}{\v\f2 FOR} -{\f2 statement, -or as a local variable or a } -{\f2\uldb procedure}{\v\f2 PROCEDURE} -{\f2 . If }{\f3 e} {\f2 is defined -as a local -variable inside the procedure, the normal definition as the base of the -natural logarithm would be suspended inside the procedure. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EVAL\_MODE} - -${\footnote \pard\plain \sl240 \fs20 $ EVAL_MODE} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0649} - - K{\footnote \pard\plain \sl240 \fs20 K symbolic;algebraic;EVAL_MODE variable;variable} - -}{\b\f2 EVAL\_MODE}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -The system variable }{\f3 eval_mode} {\f2 contains the current mode, either -} -{\f2\uldb algebraic}{\v\f2 ALGEBRAIC} -{\f2 or } -{\f2\uldb symbolic}{\v\f2 SYMBOLIC} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -EVAL\_MODE; \par - \par - ALGEBRAIC \par - \par -\pard \sl240 }{\f2 Some commands do not behave the same way in algebraic and symbolic modes. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FORT\_WIDTH} - -${\footnote \pard\plain \sl240 \fs20 $ FORT_WIDTH} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0650} - - K{\footnote \pard\plain \sl240 \fs20 K FORTRAN;output;FORT_WIDTH variable;variable} - -}{\b\f2 FORT\_WIDTH}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -The }{\f3 fort_width} {\f2 variable sets the number of characters in a line of -Fortran-compatible output produced when the } -{\f2\uldb fort}{\v\f2 FORT} -{\f2 switch is on. -Default is 70. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -fort_width := 30; \par - \par - FORT_WIDTH := 30 \par - \par - \par -on fort; \par - \par -df(sin(x**3*y),x); \par - \par - ANS=3.*COS(X \par - . **3*Y)*X**2* \par - . Y \par - \par -\pard \sl240 }{\f2 }{\f3 fort_width} {\f2 includes the usually blank characters at the beginning -of the card. As you may notice above, it is conservative and makes the -lines even shorter than it was told. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # HIGH\_POW} - -${\footnote \pard\plain \sl240 \fs20 $ HIGH_POW} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0651} - - K{\footnote \pard\plain \sl240 \fs20 K degree;polynomial;HIGH_POW variable;variable} - -}{\b\f2 HIGH\_POW}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -The variable }{\f3 high_pow} {\f2 is set by } -{\f2\uldb coeff}{\v\f2 COEFF} -{\f2 to the highest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -coeff((x+1)^5*(x*(y+3)^2)^2,x); \par - \par - \{0, \par - 0, \par - 4 3 2 \par - Y + 12*Y + 54*Y + 108*Y + 81, \par - 4 3 2 \par - 5*(Y + 12*Y + 54*Y + 108*Y + 81), \par - 4 3 2 \par - 10*(Y + 12*Y + 54*Y + 108*Y + 81), \par - 4 3 2 \par - 10*(Y + 12*Y + 54*Y + 108*Y + 81), \par - 4 3 2 \par - 5*(Y + 12*Y + 54*Y + 108*Y + 81), \par - 4 3 2 \par - Y + 12*Y + 54*Y + 108*Y + 81\} \par - \par - \par -high_pow; \par - \par - 7 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # I} - -${\footnote \pard\plain \sl240 \fs20 $ I} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0652} - - K{\footnote \pard\plain \sl240 \fs20 K complex;I constant;constant} - -}{\b\f2 I}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - - \par -\par -REDUCE knows }{\f3 i} {\f2 is the square root of -1, - and that i^2 = -1. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -(a + b*i)*(c + d*i); \par - \par - A*C + A*D*I + B*C*I - B*D \par - \par - \par -i**2; \par - \par - -1 \par - \par -\pard \sl240 }{\f2 }{\f3 i} {\f2 cannot be used as an identifier. It is all right to use }{\f3 i} {\f2 -as an index variable in a }{\f3 for} {\f2 loop, or as a local (}{\f3 scalar} {\f2 ) -variable inside a }{\f3 begin...end} {\f2 block, but it loses its definition as -the square root of -1 inside the block in that case. -\par -\par -Only the simplest properties of i are known by REDUCE unless -the switch } -{\f2\uldb complex}{\v\f2 COMPLEX} -{\f2 is turned on, which implements full complex -arithmetic in factoring, simplification, and functional values. -}{\f3 complex} {\f2 is ordinarily off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INFINITY} - -${\footnote \pard\plain \sl240 \fs20 $ INFINITY} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0653} - - K{\footnote \pard\plain \sl240 \fs20 K INFINITY constant;constant} - -}{\b\f2 INFINITY}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - -The name }{\f3 infinity} {\f2 is used to represent the infinite positive number. -However, at the present time, arithmetic in terms of this operator reflects -finite arithmetic, rather than true operations on infinity. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LOW\_POW} - -${\footnote \pard\plain \sl240 \fs20 $ LOW_POW} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0654} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;degree;LOW_POW variable;variable} - -}{\b\f2 LOW\_POW}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -The variable }{\f3 low_pow} {\f2 is set by } -{\f2\uldb coeff}{\v\f2 COEFF} -{\f2 to the lowest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -coeff((x+2*y)**6,y); \par - \par - 6 \par - \{X , \par - 5 \par - 12*X , \par - 4 \par - 60*X , \par - 3 \par - 160*X , \par - 2 \par - 240*X , \par - 192*X, \par - 64\} \par - \par - \par -low_pow; \par - \par - 0 \par - \par - \par -coeff(x**2*(x*sin(y) + 1),x); \par - \par - \par - \par - \{0,0,1,SIN(Y)\} \par - \par - \par -low_pow; \par - \par - 2 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NIL} - -${\footnote \pard\plain \sl240 \fs20 $ NIL} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0655} - - K{\footnote \pard\plain \sl240 \fs20 K false;NIL constant;constant} - -}{\b\f2 NIL}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - - \par -\par -}{\f3 nil} {\f2 represents the truth value false in symbolic mode, and is -a synonym for 0 in algebraic mode. It cannot be used for any other -purpose, even inside procedures or } -{\f2\uldb for}{\v\f2 FOR} -{\f2 loops. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PI} - -${\footnote \pard\plain \sl240 \fs20 $ PI} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0656} - - K{\footnote \pard\plain \sl240 \fs20 K PI constant;constant} - -}{\b\f2 PI}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - -The identifier }{\f3 pi} {\f2 is reserved for use as the circular constant. -Its value is given by 3.14159265358..., which REDUCE gives to the current -decimal precision when REDUCE is in a floating-point mode. -\par -\par -}{\f3 pi} {\f2 may be used as a looping variable in a } -{\f2\uldb for}{\v\f2 FOR} -{\f2 statement, -or as a local variable in a } -{\f2\uldb procedure}{\v\f2 PROCEDURE} -{\f2 . Its value in such cases -will be taken from the local environment. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # requirements} - -${\footnote \pard\plain \sl240 \fs20 $ requirements} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0657} - - K{\footnote \pard\plain \sl240 \fs20 K solve;requirements variable;variable} - -}{\b\f2 REQUIREMENTS}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -After an attempt to solve an inconsistent equation system -with parameters, the variable }{\f3 requirements} {\f2 contains a list -of expressions. These expressions define a set of conditions implicitly -equated with zero. Any solution to this system defines a setting for -the parameters sufficient to make the original system consistent. - \par -examples: \par -\pard \tx3420 }{\f4 \par -solve(\{x-a,x-y,y-1\},\{x,y\}); \par - \par - \{\} \par - \par - \par -requirements; \par - \par - \{a - 1\} \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROOT\_MULTIPLICITIES} - -${\footnote \pard\plain \sl240 \fs20 $ ROOT_MULTIPLICITIES} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0658} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;solve;root;ROOT_MULTIPLICITIES variable;variable} - -}{\b\f2 ROOT\_MULTIPLICITIES}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -The }{\f3 root_multiplicities} {\f2 variable is set to the list of the -multiplicities of the roots of an equation by the } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 operator. -\par -\par -} -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 returns its solutions in a list. The multiplicities of -each solution are put in the corresponding locations of the list -}{\f3 root_multiplicities} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # T} - -${\footnote \pard\plain \sl240 \fs20 $ T} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0659} - - K{\footnote \pard\plain \sl240 \fs20 K T constant;constant} - -}{\b\f2 T}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - -The constant }{\f3 t} {\f2 stands for the truth value true. It cannot be used -as a scalar variable in a } -{\f2\uldb block}{\v\f2 block} -{\f2 , as a looping variable in a -} -{\f2\uldb for}{\v\f2 FOR} -{\f2 statement or as an } -{\f2\uldb operator}{\v\f2 OPERATOR} -{\f2 name. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g3} - -${\footnote \pard\plain \sl240 \fs20 $ Variables} - -+{\footnote \pard\plain \sl240 \fs20 + index:0003} -}{\b\f2 Variables}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb assumptions variable} -{\v\f2 assumptions}{\f2 \par -}{\f2 \tab}{\f2\uldb CARD\_NO variable} -{\v\f2 CARD\_NO}{\f2 \par -}{\f2 \tab}{\f2\uldb E constant} -{\v\f2 E}{\f2 \par -}{\f2 \tab}{\f2\uldb EVAL\_MODE variable} -{\v\f2 EVAL\_MODE}{\f2 \par -}{\f2 \tab}{\f2\uldb FORT\_WIDTH variable} -{\v\f2 FORT\_WIDTH}{\f2 \par -}{\f2 \tab}{\f2\uldb HIGH\_POW variable} -{\v\f2 HIGH\_POW}{\f2 \par -}{\f2 \tab}{\f2\uldb I constant} -{\v\f2 I}{\f2 \par -}{\f2 \tab}{\f2\uldb INFINITY constant} -{\v\f2 INFINITY}{\f2 \par -}{\f2 \tab}{\f2\uldb LOW\_POW variable} -{\v\f2 LOW\_POW}{\f2 \par -}{\f2 \tab}{\f2\uldb NIL constant} -{\v\f2 NIL}{\f2 \par -}{\f2 \tab}{\f2\uldb PI constant} -{\v\f2 PI}{\f2 \par -}{\f2 \tab}{\f2\uldb requirements variable} -{\v\f2 requirements}{\f2 \par -}{\f2 \tab}{\f2\uldb ROOT\_MULTIPLICITIES variable} -{\v\f2 ROOT\_MULTIPLICITIES}{\f2 \par -}{\f2 \tab}{\f2\uldb T constant} -{\v\f2 T}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # semicolon} - -${\footnote \pard\plain \sl240 \fs20 $ semicolon} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0660} - - K{\footnote \pard\plain \sl240 \fs20 K semicolon command;command} - -}{\b\f2 ;}{\f2 \tab }{\b\f2 SEMICOLON}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The semicolon is a statement delimiter, indicating results are to be printed -when used in interactive mode. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -(x+1)**2; \par - \par - 2 \par - X + 2*X + 1 \par - \par - \par -df(x**2 + 1,x); \par - \par - 2*X \par - \par -\pard \sl240 }{\f2 Entering a }{\f3 Return} {\f2 without a semicolon or dollar sign results in a -prompt on the following line. A semicolon or dollar sign can be -added at this point to execute the statement. In interactive mode, a -statement that is ended with a semicolon and }{\f3 Return} {\f2 has its results -printed on the screen. -\par -\par -Inside a group statement }{\f3 <<} {\f2 ...}{\f3 >>} {\f2 -or a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 block, a -semicolon or dollar sign separates individual REDUCE statements. Since -results are not printed from a block without a specific }{\f3 return} {\f2 -statement, there is no difference between using the semicolon or dollar -sign. In a group statement, the last value produced is the value -returned by the group statement. Thus, if a semicolon or dollar sign is -placed between the last statement and the ending brackets, the group -statement returns the value 0 or nil, rather than the value of the -last statement. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # dollar} - -${\footnote \pard\plain \sl240 \fs20 $ dollar} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0661} - - K{\footnote \pard\plain \sl240 \fs20 K dollar command;command} - -}{\b\f2 $}{\f2 \tab }{\b\f2 DOLLAR}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The dollar sign is a statement delimiter, indicating results are not to be -printed when used in interactive mode. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -(x+1)**2$ \pard \sl240 }{\f2 The workspace is set to }{\f4 x^2 + 2x + 1}{\f2 - but nothing shows on the screen}{\f4 \pard \tx3420 \par - \par - \par -ws; \par - \par - 2 \par - X + 2*X + 1 \par - \par -\pard \sl240 }{\f2 -\par -\par -Entering a }{\f3 Return} {\f2 without a semicolon or dollar sign results in a -prompt on the following line. A semicolon or dollar sign can -be added at this point to execute the statement. In interactive mode, a -statement that ends with a dollar sign }{\f3 $} {\f2 and a }{\f3 Return} {\f2 is -executed, but the results not printed. -\par -\par -Inside a } -{\f2\uldb group}{\v\f2 group} -{\f2 statement }{\f3 <<} {\f2 ...}{\f3 >>} {\f2 -or a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 , a -semicolon or dollar sign separates individual REDUCE statements. Since -results are not printed from a } -{\f2\uldb block}{\v\f2 block} -{\f2 without a specific -} -{\f2\uldb return}{\v\f2 RETURN} -{\f2 \par -\par -statement, there is no difference between using the semicolon or dollar -sign. -\par -\par -In a group statement, the last value produced is the value returned by the -group statement. Thus, if a semicolon or dollar sign is placed between the -last statement and the ending brackets, the group statement returns the -value 0 or nil, rather than the value of the last statement. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # percent} - -${\footnote \pard\plain \sl240 \fs20 $ percent} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0662} - - K{\footnote \pard\plain \sl240 \fs20 K percent command;command} - -}{\b\f2 %}{\f2 \tab }{\b\f2 PERCENT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The percent sign is used to precede comments; everything from a percent -to the end of the line is ignored. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -df(x**3 + y,x);\% This is a comment \key\{Return\} \par - \par - \par - 2 \par - 3*X \par - \par - \par -int(3*x**2,x) \%This is a comment; \key\{Return\} \par -\pard \sl240 }{\f2 A prompt is given, waiting for the semicolon that was not -detected in the comment}{\f4 \pard \tx3420 \pard \sl240 }{\f2 -\par -\par -Statement delimiters }{\f3 ;} {\f2 and }{\f3 $} {\f2 are not detected between a -percent sign and the end of the line. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # dot} - -${\footnote \pard\plain \sl240 \fs20 $ dot} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0663} - - K{\footnote \pard\plain \sl240 \fs20 K list;dot operator;operator} - -}{\b\f2 .}{\f2 \tab }{\b\f2 DOT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The . (dot) infix binary operator adds a new item to the beginning of an -existing } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . In high energy physics expressions, -it can also be used -to represent the scalar product of two Lorentz four-vectors. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 .} {\f4 -\par -\par -}{\f2 \par - can be any REDUCE scalar expression, including a list; - must be a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 to avoid producing an error message. -The dot operator is right associative. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -liss := a . \{\}; \par - \par - LISS := \{A\} \par - \par - \par -liss := b . liss; \par - \par - LISS := \{B,A\} \par - \par - \par -newliss := liss . liss; \par - \par - NEWLISS := \{\{B,A\},B,A\} \par - \par - \par -firstlis := a . b . \{c\}; \par - \par - FIRSTLIS := \{A,B,C\} \par - \par - \par -secondlis := x . y . \{z\}; \par - \par - SECONDLIS := \{X,Y,Z\} \par - \par - \par -for i := 1:3 sum part(firstlis,i)*part(secondlis,i); \par - \par - \par - \par - A*X + B*Y + C*Z \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # assign} - -${\footnote \pard\plain \sl240 \fs20 $ assign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0664} - - K{\footnote \pard\plain \sl240 \fs20 K assign;assign operator;operator} - -}{\b\f2 :=}{\f2 \tab }{\b\f2 ASSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 :=} {\f2 is the assignment operator, assigning the value on the right-hand -side to the identifier or other valid expression on the left-hand side. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 :=} {\f4 -\par -\par -}{\f2 \par - is ordinarily a single identifier, though simple -expressions may be used (see Comments below). is any -valid REDUCE expression. If is a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 -identifier, then - can be a matrix identifier (redimensioned if -necessary) which has each element set to the corresponding elements -of the identifier on the right-hand side. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := x**2 + 1; \par - \par - 2 \par - A := X + 1 \par - \par - \par -a; \par - \par - 2 \par - X + 1 \par - \par - \par -first := second := third; \par - \par - FIRST := SECOND := THIRD \par - \par - \par -first; \par - \par - THIRD \par - \par - \par -second; \par - \par - THIRD \par - \par - \par -b := for i := 1:5 product i; \par - \par - B := 120 \par - \par - \par -b; \par - \par - 120 \par - \par - \par -w + (c := x + 3) + z; \par - \par - W + X + Z + 3 \par - \par - \par -c; \par - \par - X + 3 \par - \par - \par -y + b := c; \par - \par - Y + B := C \par - \par - \par -y; \par - \par - - (B - C) \par - \par -\pard \sl240 }{\f2 The assignment operator is right associative, as shown in the second and -third examples. A string of such assignments has all but the last -item set to the value of the last item. Embedding an assignment statement -in another expression has the side effect of making the assignment, as well -as causing the given replacement in the expression. -\par -\par -Assignments of values to expressions rather than simple identifiers (such as in -the last example above) can also be done, subject to the following remarks: -\par -\par -\tab (i) -If the left-hand side is an identifier, an operator, or a power, the -substitution rule is added to the rule table. -\par -\par -\tab (ii) -If the operators }{\f3 - + /} {\f2 appear on the left-hand side, all but the first -term of the expression is moved to the right-hand side. -\par -\par -\tab (iii) -If the operator }{\f3 *} {\f2 appears on the left-hand side, any constant terms are -moved to the right-hand side, but the symbolic factors remain. -\par -\par -Assignment is valid for } -{\f2\uldb array}{\v\f2 ARRAY} -{\f2 elements, but not for entire arrays. -The assignment operator can also be used to attach functionality to operators. -\par -\par -A recursive construction such as }{\f3 a := a + b} {\f2 is allowed, but when -}{\f3 a} {\f2 is referenced again, the process of resubstitution continues -until the expression stack overflows (you get an error message). -Recursive assignments can be done safely inside controlled loop -expressions, such as } -{\f2\uldb for}{\v\f2 FOR} -{\f2 ... or } -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # equalsign} - -${\footnote \pard\plain \sl240 \fs20 $ equalsign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0665} - - K{\footnote \pard\plain \sl240 \fs20 K equalsign operator;operator} - -}{\b\f2 =}{\f2 \tab }{\b\f2 EQUALSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 =} {\f2 operator is a prefix or infix equality comparison operator. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 =} {\f4 (}{\f3 ,} {\f4 ) - or - }{\f3 =} {\f4 -\par -\par -}{\f2 \par - can be any REDUCE scalar expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 4; \par - \par - A := 4 \par - \par - \par -if =(a,10) then write "yes" else write "no"; \par - \par - \par - \par - no \par - \par - \par -b := c; \par - \par - B := C \par - \par - \par -if b = c then write "yes" else write "no"; \par - \par - \par - \par - yes \par - \par - \par -on rounded; \par - \par -if 4.0 = 4 then write "yes" else write "no"; \par - \par - \par - \par - yes \par - \par -\pard \sl240 }{\f2 This logical equality operator can only be used inside a conditional -statement, such as } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 -or } -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 . In other places the equal -sign establishes an algebraic object of type } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # replace} - -${\footnote \pard\plain \sl240 \fs20 $ replace} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0666} - - K{\footnote \pard\plain \sl240 \fs20 K replace operator;operator} - -}{\b\f2 =>}{\f2 \tab }{\b\f2 REPLACE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par -The }{\f3 =>} {\f2 operator is a binary operator used in } -{\f2\uldb rule}{\v\f2 RULE} -{\f2 lists to -denote replacements. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator f; \par - \par -let f(x) => x^2; \par - \par -f(x); \par - \par - 2 \par - x \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # plussign} - -${\footnote \pard\plain \sl240 \fs20 $ plussign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0667} - - K{\footnote \pard\plain \sl240 \fs20 K plussign operator;operator} - -}{\b\f2 +}{\f2 \tab }{\b\f2 PLUSSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 +} {\f2 operator is a prefix or infix n-ary addition operator. -\par -\par - \par -syntax: \par -}{\f4 \{}{\f3 +} {\f4 \}+ -\par -\par -or }{\f3 +} {\f4 ( \{,\}+) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x**4 + 4*x**2 + 17*x + 1; \par - \par - 4 2 \par - X + 4*X + 17*X + 1 \par - \par - \par -14 + 15 + x; \par - \par - X + 29 \par - \par - \par -+(1,2,3,4,5); \par - \par - 15 \par - \par -\pard \sl240 }{\f2 }{\f3 +} {\f2 is also valid as an addition operator for } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 variables -that are of the same dimensions and for } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # minussign} - -${\footnote \pard\plain \sl240 \fs20 $ minussign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0668} - - K{\footnote \pard\plain \sl240 \fs20 K minussign operator;operator} - -}{\b\f2 -}{\f2 \tab }{\b\f2 MINUSSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 -} {\f2 operator is a prefix or infix binary subtraction operator, as well -as the unary minus operator. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 -} {\f4 -or }{\f3 -} {\f4 (,) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -15 - 4; \par - \par - 11 \par - \par - \par -x*(-5); \par - \par - - 5*X \par - \par - \par -a - b - 15; \par - \par - A - B - 15 \par - \par - \par --(a,4); \par - \par - A - 4 \par - \par -\pard \sl240 }{\f2 The subtraction operator is left associative, so that a - b - c is equivalent -to (a - b) - c, as shown in the third example. The subtraction operator is -also valid with } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 expressions of the correct dimensions -and with } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # asterisk} - -${\footnote \pard\plain \sl240 \fs20 $ asterisk} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0669} - - K{\footnote \pard\plain \sl240 \fs20 K asterisk operator;operator} - -}{\b\f2 *}{\f2 \tab }{\b\f2 ASTERISK}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 *} {\f2 operator is a prefix or infix n-ary multiplication operator. -\par -\par - \par -syntax: \par -}{\f4 \{}{\f3 *} {\f4 \}+ -\par -\par -or }{\f3 *} {\f4 ( \{,\}+) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -15*3; \par - \par - 45 \par - \par - \par -24*x*yvalue*2; \par - \par - 48*X*YVALUE \par - \par - \par -*(6,x); \par - \par - 6*X \par - \par - \par -on rounded; \par - \par -3*1.5*x*x*x; \par - \par - 3 \par - 4.5*X \par - \par - \par -off rounded; \par - \par -2x**2; \par - \par - 2 \par - 2*X \par - \par -\pard \sl240 }{\f2 REDUCE assumes you are using an implicit multiplication operator when an -identifier is preceded by a number, as shown in the last line above. Since -no valid identifiers can begin with numbers, there is no ambiguity in -making this assumption. -\par -\par -The multiplication operator is also valid with } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 expressions -of the -proper dimensions: matrices A and B -can be multiplied if -A is n x m and B is -m x p. Matrices and } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s can also be -multiplied by scalars: the -result is as if each element was multiplied by the scalar. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # slash} - -${\footnote \pard\plain \sl240 \fs20 $ slash} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0670} - - K{\footnote \pard\plain \sl240 \fs20 K slash operator;operator} - -}{\b\f2 /}{\f2 \tab }{\b\f2 SLASH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 /} {\f2 operator is a prefix or infix binary division operator or -prefix unary } -{\f2\uldb recip}{\v\f2 RECIP} -{\f2 rocal operator. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 /} {\f4 or - }{\f3 /} {\f4 -\par -\par -or }{\f3 /} {\f4 (,) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -20/5; \par - \par - 4 \par - \par - \par -100/6; \par - \par - 50 \par - -- \par - 3 \par - \par - \par -16/2/x; \par - \par - 8 \par - - \par - X \par - \par - \par -/b; \par - \par - 1 \par - - \par - B \par - \par - \par -/(y,5); \par - \par - Y \par - - \par - 5 \par - \par - \par -on rounded; \par - \par -35/4; \par - \par - 8.75 \par - \par - \par -/20; \par - \par - 0.05 \par - \par -\pard \sl240 }{\f2 The division operator is left associative, so that }{\f3 a/b/c} {\f2 is equivalent -to }{\f3 (a/b)/c} {\f2 . The division operator is also valid with square -} -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 expressions of the same dimensions: With A and -B both n x n matrices and B -invertible, A/B is -given by A*B^-1. -Division of a matrix by a scalar is defined, with the results being the -division of each element of the matrix by the scalar. Division of a -scalar by a matrix is defined if the matrix is invertible, and has the -effect of multiplying the scalar by the inverse of the matrix. When -}{\f3 /} {\f2 is used as a reciprocal operator for a matrix, the inverse of -the matrix is returned if it exists. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # power} - -${\footnote \pard\plain \sl240 \fs20 $ power} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0671} - - K{\footnote \pard\plain \sl240 \fs20 K power operator;operator} - -}{\b\f2 **}{\f2 \tab }{\b\f2 POWER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 **} {\f2 operator is a prefix or infix binary exponentiation operator. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 **} {\f4 - or }{\f3 **} {\f4 (,) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x**15; \par - \par - 15 \par - X \par - \par - \par -x**y**z; \par - \par - Y*Z \par - X \par - \par - \par -x**(y**z); \par - \par - Z \par - Y \par - X \par - \par - \par - **(y,4); \par - \par - 4 \par - Y \par - \par - \par -on rounded; \par - \par -2**pi; \par - \par - 8.82497782708 \par - \par -\pard \sl240 }{\f2 The exponentiation operator is left associative, so that }{\f3 a**b**c} {\f2 is -equivalent to }{\f3 (a**b)**c} {\f2 , as shown in the second example. Note -that this is not }{\f3 a**(b**c)} {\f2 , which would be right associative. -\par -\par -When } -{\f2\uldb nat}{\v\f2 NAT} -{\f2 is on (the default), REDUCE output produces raised -exponents, as shown. The symbol }{\f3 ^} {\f2 , which is the upper-case 6 on -most keyboards, may be used in the place of }{\f3 **} {\f2 . -\par -\par -A square } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 may also be raised to positive and negative powers -with the exponentiation operator (negative powers require the matrix to be -invertible). Scalar expressions and } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s may be raised to -fractional and floating-point powers. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # caret} - -${\footnote \pard\plain \sl240 \fs20 $ caret} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0672} - - K{\footnote \pard\plain \sl240 \fs20 K caret operator;operator} - -}{\b\f2 ^}{\f2 \tab }{\b\f2 CARET}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 ^} {\f2 operator is a prefix or infix binary exponentiation operator. -It is equivalent to } -{\f2\uldb power}{\v\f2 power} -{\f2 or **. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 ^} {\f4 - or }{\f3 ^} {\f4 (,) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x^15; \par - \par - 15 \par - X \par - \par - \par -x^y^z; \par - \par - Y*Z \par - X \par - \par - \par -x^(y^z); \par - \par - Z \par - Y \par - X \par - \par - \par -^(y,4); \par - \par - 4 \par - Y \par - \par - \par -on rounded; \par - \par -2^pi; \par - \par - 8.82497782708 \par - \par -\pard \sl240 }{\f2 The exponentiation operator is left associative, so that }{\f3 a^b^c} {\f2 is -equivalent to }{\f3 (a^b)^c} {\f2 , as shown in the second example. Note -that this is }{\f3 a^(b^c)} {\f2 , which would be right associative. -\par -\par -When } -{\f2\uldb nat}{\v\f2 NAT} -{\f2 is on (the default), REDUCE output produces raised -exponents, as shown. -\par -\par -A square } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 may also be raised to positive -and negative powers with -the exponentiation operator (negative powers require the matrix to be -invertible). Scalar expressions and } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s -may be raised to fractional and floating-point powers. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # geqsign} - -${\footnote \pard\plain \sl240 \fs20 $ geqsign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0673} - - K{\footnote \pard\plain \sl240 \fs20 K geqsign operator;operator} - -}{\b\f2 >=}{\f2 \tab }{\b\f2 GEQSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -}{\f3 >=} {\f2 is an infix binary comparison operator, which returns true if -its first argument is greater than or equal to its second argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 >=} {\f4 -\par -\par -}{\f2 \par - must evaluate to an integer or floating-point number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -if (3 >= 2) then yes; \par - \par - yes \par - \par - \par -a := 15; \par - \par - A := 15 \par - \par - \par -if a >= 20 then big else small; \par - \par - \par - small \par - \par -\pard \sl240 }{\f2 The binary comparison operators can only be used for comparisons between -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 -or } -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 or } -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # greater} - -${\footnote \pard\plain \sl240 \fs20 $ greater} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0674} - - K{\footnote \pard\plain \sl240 \fs20 K greater operator;operator} - -}{\b\f2 >}{\f2 \tab }{\b\f2 GREATER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 >} {\f2 is an infix binary comparison operator that returns - true if its first argument is strictly greater than its second. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 >} {\f4 -\par -\par -}{\f2 \par - must evaluate to a number, e.g., integer, rational or -floating point number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -if 3.0 > 3 then write "different" else write "same"; \par - \par - \par - same \par - \par - \par -off rounded; \par - \par -a := 20; \par - \par - A := 20 \par - \par - \par -if a > 20 then write "bigger" else write "not bigger"; \par - \par - \par - not bigger \par - \par -\pard \sl240 }{\f2 The binary comparison operators can only be used for comparisons between -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 or -} -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 or } -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # leqsign} - -${\footnote \pard\plain \sl240 \fs20 $ leqsign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0675} - - K{\footnote \pard\plain \sl240 \fs20 K leqsign operator;operator} - -}{\b\f2 <=}{\f2 \tab }{\b\f2 LEQSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -}{\f3 <=} {\f2 is an infix binary comparison operator that returns - true if its first argument is less than or equal to its second argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 <=} {\f4 -\par -\par -}{\f2 \par - must evaluate to a number, e.g., integer, rational or -floating point number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 10; \par - \par - A := 10 \par - \par - \par -if a <= 10 then true; \par - \par - true \par - \par -\pard \sl240 }{\f2 The binary comparison operators can only be used for comparisons between -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 or -} -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 or } -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # less} - -${\footnote \pard\plain \sl240 \fs20 $ less} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0676} - - K{\footnote \pard\plain \sl240 \fs20 K less operator;operator} - -}{\b\f2 <}{\f2 \tab }{\b\f2 LESS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -}{\f3 <} {\f2 is an infix binary logical comparison operator that -returns true if its first argument is strictly less than its second -argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 <} {\f4 -\par -\par -}{\f2 \par - must evaluate to a number, e.g., integer, rational or -floating point number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -f := -3; \par - \par - F := -3 \par - \par - \par -if f < -3 then write "yes" else write "no"; \par - \par - \par - no \par - \par -\pard \sl240 }{\f2 The binary comparison operators can only be used for comparisons between -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 -or } -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 or -} -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # tilde} - -${\footnote \pard\plain \sl240 \fs20 $ tilde} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0677} - - K{\footnote \pard\plain \sl240 \fs20 K tilde operator;operator} - -}{\b\f2 ~}{\f2 \tab }{\b\f2 TILDE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 ~} {\f2 is used as a unary prefix operator in the left-hand -sides of } -{\f2\uldb rule}{\v\f2 RULE} -{\f2 s to mark } -{\f2\uldb free variable}{\v\f2 Free_Variable} -{\f2 s. A double tilde -marks an optional } -{\f2\uldb free variable}{\v\f2 Free_Variable} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # group} - -${\footnote \pard\plain \sl240 \fs20 $ group} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0678} - - K{\footnote \pard\plain \sl240 \fs20 K group command;command} - -}{\b\f2 <<}{\f2 \tab }{\b\f2 GROUP}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 <<} {\f2 ...}{\f3 >>} {\f2 command is a group statement, -used to group statements -together where REDUCE expects a single statement. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 <<} {\f4 \{; }{\f3 or} {\f4 - }{\f2 \}* }{\f3 >>} {\f2 -\par -\par -\par - may be any valid REDUCE statement or expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 2; \par - \par - A := 2 \par - \par - \par -if a < 5 then <>; \par - \par - \par - 12 \par - \par - \par -<>; \par - \par - \par - 2 \par - C + 90*C + 202 \par - ---------------- \par - 225 \par - \par -\pard \sl240 }{\f2 The value returned from a group statement is the value of the last -individual statement executed inside it. Note that when a semicolon is -placed between the last statement and the closing brackets, 0 or - nil is returned. Group statements are often used in the -consequence portions of } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 , -} -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 , and -} -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 -clauses. They may also be used in interactive -operation to execute several statements at one time. Statements inside -the group statement are separated by semicolons or dollar signs. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # AND} - -${\footnote \pard\plain \sl240 \fs20 $ AND} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0679} - - K{\footnote \pard\plain \sl240 \fs20 K AND operator;operator} - -}{\b\f2 AND}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 and} {\f2 binary logical operator returns true if both of its -arguments are true. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 and} {\f4 -\par -\par -}{\f2 \par - must evaluate to true or nil. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 12; \par - \par - A := 12 \par - \par - \par -if numberp a and a < 15 then write a**2 else write "no"; \par - \par - \par - \par - 144 \par - \par - \par -clear a; \par - \par -if numberp a and a < 15 then write a**2 else write "no"; \par - \par - \par - \par - no \par - \par -\pard \sl240 }{\f2 Logical operators can only be used inside conditional statements, such as -} -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 or -} -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 . }{\f3 and} {\f2 examines each of -its arguments in order, and quits, returning nil, on finding an -argument that is not true. An error results if it is used in other -contexts. -\par -\par -}{\f3 and} {\f2 is left associative: }{\f3 x and y and z} {\f2 is equivalent to -}{\f3 (x and y) and z} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BEGIN} - -${\footnote \pard\plain \sl240 \fs20 $ BEGIN} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0680} - - K{\footnote \pard\plain \sl240 \fs20 K BEGIN command;command} - -}{\b\f2 BEGIN}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -}{\f3 begin} {\f2 is used to start a } -{\f2\uldb block}{\v\f2 block} -{\f2 statement, which is closed with -}{\f3 end} {\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 begin} {\f4 \{}{\f3 ;} {\f4 \}* }{\f3 end} {\f4 -\par -\par -}{\f2 \par - is any valid REDUCE statement. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -begin for i := 1:3 do write i end; \par - \par - \par - 1 \par - 2 \par - 3 \par - \par - \par -begin scalar n;n:=1;b:=for i:=1:4 product(x-i);return n end; \par - \par - \par - \par - 1 \par - \par - \par -b; \par - \par - 4 3 2 \par - X - 10*X + 35*X - 50*X + 24 \par - \par -\pard \sl240 }{\f2 A }{\f3 begin} {\f2 ...}{\f3 end} {\f2 block can do actions (such as }{\f3 write} {\f2 ), but -does not -return a value unless instructed to by a } -{\f2\uldb return}{\v\f2 RETURN} -{\f2 statement, which must -be the last statement executed in the block. It is unnecessary to insert -a semicolon before the }{\f3 end} {\f2 . -\par -\par -Local variables, if any, are declared in the first statement immediately -after }{\f3 begin} {\f2 , and may be defined as }{\f3 scalar, integer,} {\f2 or -}{\f3 real} {\f2 . } -{\f2\uldb array}{\v\f2 ARRAY} -{\f2 variables declared -within a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 block -are global in every case, and } -{\f2\uldb let}{\v\f2 LET} -{\f2 statements have global -effects. A } -{\f2\uldb let}{\v\f2 LET} -{\f2 statement involving a formal parameter affects -the calling parameter that corresponds to it. } -{\f2\uldb let}{\v\f2 LET} -{\f2 statements -involving local variables make global assignments, overwriting outside -variables by the same name or creating them if they do not exist. You -can use this feature to affect global variables from procedures, but be -careful that you do not do it inadvertently. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # block} - -${\footnote \pard\plain \sl240 \fs20 $ block} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0681} - - K{\footnote \pard\plain \sl240 \fs20 K block command;command} - -}{\b\f2 BLOCK}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -A }{\f3 block} {\f2 is a sequence of statements enclosed by -commands } -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 and } -{\f2\uldb end}{\v\f2 END} -{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 begin} {\f4 \{}{\f3 ;} {\f4 \}* }{\f3 end} {\f4 -\par -\par -}{\f2 \par -For more details see } -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COMMENT} - -${\footnote \pard\plain \sl240 \fs20 $ COMMENT} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0682} - - K{\footnote \pard\plain \sl240 \fs20 K COMMENT command;command} - -}{\b\f2 COMMENT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -Beginning with the word }{\f3 comment} {\f2 , all text until the next statement -terminator (}{\f3 ;} {\f2 or }{\f3 $} {\f2 ) is ignored. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -x := a**2 comment--a is the velocity of the particle;; \par - \par - \par - \par - 2 \par - X := A \par - \par -\pard \sl240 }{\f2 Note that the first semicolon ends the comment and the second one -terminates the original REDUCE statement. -\par -\par -Multiple-line comments are often needed in interactive files. The -}{\f3 comment} {\f2 command allows a normal-looking text to accompany the -REDUCE statements in the file. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CONS} - -${\footnote \pard\plain \sl240 \fs20 $ CONS} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0683} - - K{\footnote \pard\plain \sl240 \fs20 K CONS operator;operator} - -}{\b\f2 CONS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 cons} {\f2 operator adds a new element to the beginning of a -} -{\f2\uldb list}{\v\f2 LIST} -{\f2 . Its -operation is identical to the symbol } -{\f2\uldb dot}{\v\f2 dot} -{\f2 (dot). It can be used -infix or prefix. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 cons} {\f4 (,) or }{\f3 cons} {\f4 -\par -\par -}{\f2 \par - can be any REDUCE scalar expression, including a list; -must be a list. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -liss := cons(a,\{b\}); \par - \par - \{A,B\} \par - \par - \par - \par -liss := c cons liss; \par - \par - \{C,A,B\} \par - \par - \par - \par -newliss := for each y in liss collect cons(y,list x); \par - \par - \par - \par - NEWLISS := \{\{C,X\},\{A,X\},\{B,X\}\} \par - \par - \par - \par -for each y in newliss sum (first y)*(second y); \par - \par - \par - \par - X*(A + B + C) \par - \par -\pard \sl240 }{\f2 If you want to use }{\f3 cons} {\f2 to put together two elements into a new list, -you must make the second one into a list with curly brackets or the }{\f3 list} {\f2 -command. You can also start with an empty list created by }{\f3 \{\}} {\f2 . -\par -\par -The }{\f3 cons} {\f2 operator is right associative: }{\f3 a cons b cons c} {\f2 is valid -if }{\f3 c} {\f2 is a list; }{\f3 b} {\f2 need not be a list. The list produced is -}{\f3 \{a,b,c\}} {\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # END} - -${\footnote \pard\plain \sl240 \fs20 $ END} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0684} - - K{\footnote \pard\plain \sl240 \fs20 K END command;command} - -}{\b\f2 END}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The command }{\f3 end} {\f2 has two main uses: -\par -\par -\tab (i) -as the ending of a } -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 ; and -\par -\tab (ii) -to end input from a file. -\par -\par -In a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 , there need not be a delimiter -(}{\f3 ;} {\f2 or }{\f3 $} {\f2 ) before the }{\f3 end} {\f2 , though there must be one -after it, or a right bracket matching an earlier left bracket. -\par -\par -Files to be read into REDUCE should end with }{\f3 end;} {\f2 , which must be -preceded by a semicolon (usually the last character of the previous line). -The additional semicolon avoids problems with mistakes in the files. If -you have suspended file operation by answering }{\f3 n} {\f2 to a }{\f3 pause} {\f2 -command, you are still, technically speaking, ``in" the file. Use -}{\f3 end} {\f2 to exit the file. -\par -\par -An }{\f3 end} {\f2 at the top level of a program is ignored. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EQUATION} - -${\footnote \pard\plain \sl240 \fs20 $ EQUATION} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0685} - - K{\footnote \pard\plain \sl240 \fs20 K =;arithmetic;equal;equation;EQUATION type;type} - -}{\b\f2 EQUATION}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par -\par - - \par -\par -An }{\f3 equation} {\f2 is an expression where two algebraic expressions -are connected by the (infix) operator } -{\f2\uldb equal}{\v\f2 EQUAL} -{\f2 or by }{\f3 =} {\f2 . -For access to the components of an }{\f3 equation} {\f2 the operators -} -{\f2\uldb lhs}{\v\f2 LHS} -{\f2 , } -{\f2\uldb rhs}{\v\f2 RHS} -{\f2 or } -{\f2\uldb part}{\v\f2 PART} -{\f2 can be used. The -evaluation of the left-hand side of an }{\f3 equation} {\f2 is controlled -by the switch } -{\f2\uldb evallhseqp}{\v\f2 EVALLHSEQP} -{\f2 , while the right-hand side is -evaluated unconditionally. When an }{\f3 equation} {\f2 is part of a -logical expression, e.g. in a } -{\f2\uldb if}{\v\f2 IF} -{\f2 or } -{\f2\uldb while}{\v\f2 WHILE} -{\f2 statement, -the equation is evaluated by subtracting both sides can comparing -the result with zero. -\par -\par -Equations occur in many contexts, e.g. as arguments of the } -{\f2\uldb sub}{\v\f2 SUB} -{\f2 -operator and in the arguments and the results -of the operator } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 . An equation can be member of a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 -and you may assign an equation to a variable. Elementary arithmetic is supported -for equations: if } -{\f2\uldb evallhseqp}{\v\f2 EVALLHSEQP} -{\f2 is on, you may add and subtract -equations, and you can combine an equation with a scalar expression by -addition, subtraction, multiplication, division and raise an equation -to a power. - \par -examples: \par -\pard \tx3420 }{\f4 \par -on evallhseqp; \par - \par -u:=x+y=1$ \par - \par -v:=2x-y=0$ \par - \par -2*u-v; \par - \par - - 3*y=-2 \par - \par - \par -ws/3; \par - \par - 2 \par - y=-- \par - 3 \par - \par -\pard \sl240 }{\f2 \par -\par -Important: the equation must occur in the leftmost term of such an expression. -For other operations, e.g. taking function values of both sides, use the -} -{\f2\uldb map}{\v\f2 MAP} -{\f2 operator. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FIRST} - -${\footnote \pard\plain \sl240 \fs20 $ FIRST} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0686} - - K{\footnote \pard\plain \sl240 \fs20 K decomposition;list;FIRST operator;operator} - -}{\b\f2 FIRST}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 first} {\f2 operator returns the first element of a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 first} {\f4 () or }{\f3 first} {\f4 -\par -\par -}{\f2 \par - must be a non-empty list to avoid an error message. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -alist := \{a,b,c,d\}; \par - \par - ALIST := \{A,B,C,D\} \par - \par - \par -first alist; \par - \par - A \par - \par - \par -blist := \{x,y,\{ww,aa,qq\},z\}; \par - \par - BLIST := \{X,Y,\{WW,AA,QQ\},Z\} \par - \par - \par -first third blist; \par - \par - WW \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FOR} - -${\footnote \pard\plain \sl240 \fs20 $ FOR} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0687} - - K{\footnote \pard\plain \sl240 \fs20 K loop;FOR command;command} - -}{\b\f2 FOR}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 for} {\f2 command is used for iterative loops. There are many -possible forms it can take. -\par -\par -\pard \tx3420 }{\f4 \par - / \ \par - / |STEP UNTIL| \ \par - |:=| || \par -FOR| | : | | \par - | \ / | \par - |EACH IN | \par - \ / \par - \par - where ::= DO|PRODUCT|SUM|COLLECT|JOIN. \par -\pard \sl240 }{\f2 can be any valid REDUCE identifier except }{\f3 t} {\f2 or -}{\f3 nil} {\f2 , , and can be any expression -that evaluates to a positive or negative integer. must be a -valid } -{\f2\uldb list}{\v\f2 LIST} -{\f2 structure. -The action taken must be one of the actions shown -above, each of which is followed by a single REDUCE expression, statement -or a } -{\f2\uldb group}{\v\f2 group} -{\f2 (}{\f3 <<} {\f2 ...}{\f3 >>} {\f2 ) or } -{\f2\uldb block}{\v\f2 block} -{\f2 -(} -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 ...} -{\f2\uldb end}{\v\f2 END} -{\f2 ) statement. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -for i := 1:10 sum i; \par - \par - \par - \par - 55 \par - \par - \par -for a := -2 step 3 until 6 product a; \par - \par - \par - \par - -8 \par - \par - \par -a := 3; \par - \par - A := 3 \par - \par - \par -for iter := 4:a do write iter; \par - \par -m := 0; \par - \par - M := 0 \par - \par - \par -for s := 10 step -1 until 3 do <>; \par - \par -m; \par - \par - 520 \par - \par - \par -for each x in \{q,r,s\} sum x**2; \par - \par - 2 2 2 \par - Q + R + S \par - \par - \par -for i := 1:4 collect 1/i; \par - \par - \par - \par - 1 1 1 \par - \{1,-,-,-\} \par - 2 3 4 \par - \par - \par -for i := 1:3 join list solve(x**2 + i*x + 1,x); \par - \par - \par - \par - SQRT(3)*I + 1 \par - \{\{X= --------------, \par - 2 \par - SQRT(3)*I - 1 \par - X= --------------\} \par - 2 \par - \{X=-1\}, \par - SQRT(5) + 3 SQRT(5) - 3 \par - \{X= - -----------,X=-----------\}\} \par - 2 2 \par - \par -\pard \sl240 }{\f2 The behavior of each of the five action words follows: -\par -\par -\pard \tx3420 }{\f4 \par - Action Word Behavior \par -Keyword Argument Type Action \par - do statement, command, group Evaluates its argument once \par - or block for each iteration of the loop, \par - not saving results \par -collect expression, statement, Evaluates its argument once for \par - command, group, block, list each iteration of the loop, \par - storing the results in a list \par - which is returned by the for \par - statement when done \par - join list or an operator which Evaluates its argument once for \par - produces a list each iteration of the loop, \par - appending the elements in each \par - individual result list onto the \par - overall result list \par -product expression, statement, Evaluates its argument once for \par - command, group or block each iteration of the loop, \par - multiplying the results together \par - and returning the overall product \par - sum expression, statement, Evaluates its argument once for \par - command, group or block each iteration of the loop, \par - adding the results together and \par - returning the overall sum \par -\pard \sl240 }{\f2 For number-driven }{\f3 for} {\f2 statements, if the ending limit is smaller -than the beginning limit (larger in the case of negative steps) the action -statement is not executed at all. The iterative variable is local to the -}{\f3 for} {\f2 statement, and does not affect the value of an identifier with -the same name. For list-driven }{\f3 for} {\f2 statements, if the list is -empty, the action statement is not executed, but no error occurs. -\par -\par -You can use nested }{\f3 for} {\f2 statements, with the inner }{\f3 for} {\f2 -statement after the action keyword. You must make sure that your inner -statement returns an expression that the outer statement can handle. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FOREACH} - -${\footnote \pard\plain \sl240 \fs20 $ FOREACH} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0688} - - K{\footnote \pard\plain \sl240 \fs20 K loop;FOREACH command;command} - -}{\b\f2 FOREACH}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -}{\f3 foreach} {\f2 is a synonym for the }{\f3 for each} {\f2 variant of the -} -{\f2\uldb for}{\v\f2 FOR} -{\f2 construct. It is designed to iterate down a list, and an -error will occur if a list is not used. The use of }{\f3 for each} {\f2 is -preferred to }{\f3 foreach} {\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 foreach} {\f4 in -\par -\par -where ::= }{\f3 do | product | sum | collect | join} {\f4 -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -foreach x in \{q,r,s\} sum x**2; \par - \par - 2 2 2 \par - Q + R + S \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GEQ} - -${\footnote \pard\plain \sl240 \fs20 $ GEQ} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0689} - - K{\footnote \pard\plain \sl240 \fs20 K GEQ operator;operator} - -}{\b\f2 GEQ}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 geq} {\f2 operator is a binary infix or prefix logical operator. It -returns true if its first argument is greater than or equal to its second -argument. As an infix operator it is identical with }{\f3 >=} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 geq} {\f4 (,) or -}{\f3 geq} {\f4 -\par -\par -}{\f2 \par - can be any valid REDUCE expression that evaluates to a -number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 20; \par - \par - A := 20 \par - \par - \par -if geq(a,25) then write "big" else write "small"; \par - \par - \par - \par - small \par - \par - \par -if a geq 20 then write "big" else write "small"; \par - \par - \par - \par - big \par - \par - \par -if (a geq 18) then write "big" else write "small"; \par - \par - \par - \par - big \par - \par -\pard \sl240 }{\f2 Logical operators can only be used in conditional statements such as -\par -\par -} -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 or -} -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GOTO} - -${\footnote \pard\plain \sl240 \fs20 $ GOTO} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0690} - - K{\footnote \pard\plain \sl240 \fs20 K GOTO command;command} - -}{\b\f2 GOTO}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -Inside a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 , }{\f3 goto} {\f2 , or -preferably, }{\f3 go to} {\f2 , transfers flow of control to a labeled statement. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 go to} {\f4 or }{\f3 goto} {\f4 -\par -\par -}{\f2 \par - is of the form >; \par - \par - \par - \par - 46 \par - \par - \par -max(-5,-10,-a); \par - \par - -5 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MIN} - -${\footnote \pard\plain \sl240 \fs20 $ MIN} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0730} - - K{\footnote \pard\plain \sl240 \fs20 K minimum;MIN operator;operator} - -}{\b\f2 MIN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 min} {\f2 is an n-ary prefix operator, which returns the -smallest value in its arguments. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 min} {\f4 (\{,\}*) -\par -\par -}{\f2 \par - must evaluate to a number. }{\f3 min} {\f2 of an empty list -returns 0. - \par -examples: \par -\pard \tx3420 }{\f4 \par -min(-3,0,17,2); \par - \par - -3 \par - \par - \par -<>; \par - \par - \par - \par - 16 \par - \par - \par -min(5,10,a); \par - \par - 5 \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MINUS} - -${\footnote \pard\plain \sl240 \fs20 $ MINUS} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0731} - - K{\footnote \pard\plain \sl240 \fs20 K MINUS operator;operator} - -}{\b\f2 MINUS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 minus} {\f2 operator is a unary minus, returning the negative of its -argument. It is equivalent to the unary }{\f3 -} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 minus} {\f4 () -\par -\par -\par -\par -}{\f2 may be any scalar REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -minus(a); \par - \par - - A \par - \par - \par -minus(-1); \par - \par - 1 \par - \par - \par -minus((x+1)**4); \par - \par - 4 3 2 \par - - (X + 4*X + 6*X + 4*X + 1) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NEXTPRIME} - -${\footnote \pard\plain \sl240 \fs20 $ NEXTPRIME} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0732} - - K{\footnote \pard\plain \sl240 \fs20 K prime number;NEXTPRIME operator;operator} - -}{\b\f2 NEXTPRIME}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 nextprime} {\f4 () -\par -\par -}{\f2 \par -If the argument of }{\f3 nextprime} {\f2 is an integer, the least prime greater -than that argument is returned. Otherwise, a type error results. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -nextprime 5001; \par - \par - 5003 \par - \par - \par -nextprime(10^30); \par - \par - 1000000000000000000000000000057 \par - \par - \par -nextprime a; \par - \par - ***** A invalid as integer \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NOCONVERT} - -${\footnote \pard\plain \sl240 \fs20 $ NOCONVERT} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0733} - - K{\footnote \pard\plain \sl240 \fs20 K NOCONVERT switch;switch} - -}{\b\f2 NOCONVERT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -Under normal circumstances when }{\f3 rounded} {\f2 is on, REDUCE converts the -number 1.0 to the integer 1. If this is not desired, the switch -}{\f3 noconvert} {\f2 can be turned on. - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -1.0000000000001; \par - \par - 1 \par - \par - \par -on noconvert; \par - \par -1.0000000000001; \par - \par - 1.0 \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NORM} - -${\footnote \pard\plain \sl240 \fs20 $ NORM} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0734} - - K{\footnote \pard\plain \sl240 \fs20 K complex;NORM operator;operator} - -}{\b\f2 NORM}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 norm} {\f4 () -\par -\par -}{\f2 \par -If }{\f3 rounded} {\f2 is on, and the argument is a real number, -returns its absolute value. If }{\f3 complex} {\f2 is also on, -returns the square root of the sum of squares of the real and imaginary -parts of the argument. In all other cases, a result is returned in -terms of the original operator. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -norm (-2); \par - \par - NORM(-2) \par - \par - \par -on rounded; \par - \par -ws; \par - \par - 2.0 \par - \par - \par -norm(3+4i); \par - \par - NORM(4*I+3) \par - \par - \par -on complex; \par - \par -ws; \par - \par - 5.0 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PERM} - -${\footnote \pard\plain \sl240 \fs20 $ PERM} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0735} - - K{\footnote \pard\plain \sl240 \fs20 K permutation;PERM operator;operator} - -}{\b\f2 PERM}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 perm(,) -\par -\par -}{\f2 \par -If and evaluate to positive integers, -}{\f3 perm} {\f2 returns the number of permutations possible in selecting - objects from objects. -In other cases, an expression in the original operator is returned. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -perm(1,1); \par - \par - 1 \par - \par - \par -perm(3,5); \par - \par - 60 \par - \par - \par -perm(-3,5); \par - \par - PERM(-3,5) \par - \par - \par -perm(a,b); \par - \par - PERM(A,B) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PLUS} - -${\footnote \pard\plain \sl240 \fs20 $ PLUS} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0736} - - K{\footnote \pard\plain \sl240 \fs20 K PLUS operator;operator} - -}{\b\f2 PLUS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 plus} {\f2 operator is both an infix and prefix n-ary addition -operator. It exists because of the way in which REDUCE handles such -operators internally, and is not recommended for use in algebraic mode -programming. } -{\f2\uldb plussign}{\v\f2 plussign} -{\f2 , which has the identical effect, should be -used instead. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 plus} {\f4 (,\{,\} -*) or -\par -\par - }{\f3 plus} {\f4 \{}{\f3 plus} {\f4 \}* -\par -\par -}{\f2 \par - can be any valid REDUCE expression, including matrix -expressions of the same dimensions. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a plus b plus c plus d; \par - \par - A + B + C + D \par - \par - \par -4.5 plus 10; \par - \par - 29 \par - -- \par - 2 \par - \par - \par - \par -plus(x**2,y**2); \par - \par - 2 2 \par - X + Y \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # QUOTIENT} - -${\footnote \pard\plain \sl240 \fs20 $ QUOTIENT} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0737} - - K{\footnote \pard\plain \sl240 \fs20 K QUOTIENT operator;operator} - -}{\b\f2 QUOTIENT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 quotient} {\f2 operator is both an infix and prefix binary operator that -returns the quotient of its first argument divided by its second. It is -also a unary } -{\f2\uldb recip}{\v\f2 RECIP} -{\f2 rocal operator. It is identical to }{\f3 /} {\f2 and -} -{\f2\uldb slash}{\v\f2 slash} -{\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 quotient} {\f4 (,) or - }{\f3 quotient} {\f4 or -}{\f3 quotient} {\f4 () or -}{\f3 quotient} {\f4 -\par -\par -}{\f2 \par - can be any valid REDUCE scalar expression. Matrix -expressions can also be used if the second expression is invertible and the -matrices are of the correct dimensions. - \par -examples: \par -\pard \tx3420 }{\f4 \par -quotient(a,x+1); \par - \par - A \par - ----- \par - X + 1 \par - \par - \par -7 quotient 17; \par - \par - 7 \par - -- \par - 17 \par - \par - \par -on rounded; \par - \par -4.5 quotient 2; \par - \par - 2.25 \par - \par - \par -quotient(x**2 + 3*x + 2,x+1); \par - \par - X + 2 \par - \par - \par -matrix m,inverse; \par - \par -m := mat((a,b),(c,d)); \par - \par - M(1,1) := A; \par - M(1,2) := B; \par - M(2,1) := C \par - M(2,2) := D \par - \par - \par - \par -inverse := quotient m; \par - \par - D \par - INVERSE(1,1) := ---------- \par - A*D - B*C \par - B \par - INVERSE(1,2) := - ---------- \par - A*D - B*C \par - C \par - INVERSE(2,1) := - ---------- \par - A*D - B*C \par - A \par - INVERSE(2,2) := ---------- \par - A*D - B*C \par - \par -\pard \sl240 }{\f2 \par -\par -The }{\f3 quotient} {\f2 operator is left associative: }{\f3 a quotient b quotient c} {\f2 -is equivalent to }{\f3 (a quotient b) quotient c} {\f2 . -\par -\par -If a matrix argument to the unary }{\f3 quotient} {\f2 is not invertible, or if the -second matrix argument to the binary quotient is not invertible, an error -message is given. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RAD2DEG} - -${\footnote \pard\plain \sl240 \fs20 $ RAD2DEG} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0738} - - K{\footnote \pard\plain \sl240 \fs20 K radians;degrees;RAD2DEG operator;operator} - -}{\b\f2 RAD2DEG}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 rad2deg} {\f4 () -\par -\par -}{\f2 \par -In } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 mode, if is a real number, the -operator }{\f3 rad2deg} {\f2 will interpret it as radians, and convert it to -the equivalent degrees. In all other cases, an expression in terms of the -original operator is returned. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -rad2deg 1; \par - \par - RAD2DEG(1) \par - \par - \par -on rounded; \par - \par -ws; \par - \par - 57.2957795131 \par - \par - \par -rad2deg a; \par - \par - RAD2DEG(A) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RAD2DMS} - -${\footnote \pard\plain \sl240 \fs20 $ RAD2DMS} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0739} - - K{\footnote \pard\plain \sl240 \fs20 K radians;degrees;RAD2DMS operator;operator} - -}{\b\f2 RAD2DMS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 rad2dms} {\f4 () -\par -\par -}{\f2 \par -In } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 mode, if is a real number, the -operator }{\f3 rad2dms} {\f2 will interpret it as radians, and convert it to a -list containing the equivalent degrees, minutes and seconds. In all other -cases, an expression in terms of the original operator is returned. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -rad2dms 1; \par - \par - RAD2DMS(1) \par - \par - \par -on rounded; \par - \par -ws; \par - \par - \{57,17,44.8062470964\} \par - \par - \par -rad2dms a; \par - \par - RAD2DMS(A) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RECIP} - -${\footnote \pard\plain \sl240 \fs20 $ RECIP} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0740} - - K{\footnote \pard\plain \sl240 \fs20 K RECIP operator;operator} - -}{\b\f2 RECIP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -}{\f3 recip} {\f2 is the alphabetical name for the division operator }{\f3 /} {\f2 -or } -{\f2\uldb slash}{\v\f2 slash} -{\f2 used as a unary operator. The use of }{\f3 /} {\f2 is preferred. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -recip a; \par - \par - 1 \par - - \par - A \par - \par - \par -recip 2; \par - \par - 1 \par - -- \par - 2 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # REMAINDER} - -${\footnote \pard\plain \sl240 \fs20 $ REMAINDER} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0741} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;REMAINDER operator;operator} - -}{\b\f2 REMAINDER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 remainder} {\f2 operator returns the remainder after its first -argument is divided by its second argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 remainder} {\f4 (,) -\par -\par -}{\f2 \par - can be any valid REDUCE polynomial, and is not limited -to numeric values. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -remainder(13,6); \par - \par - 1 \par - \par - \par -remainder(x**2 + 3*x + 2,x+1); \par - \par - 0 \par - \par - \par -remainder(x**3 + 12*x + 4,x**2 + 1); \par - \par - \par - 11*X + 4 \par - \par - \par -remainder(sin(2*x),x*y); \par - \par - SIN(2*X) \par - \par -\pard \sl240 }{\f2 In the default case, remainders are calculated over the integers. If you -need the remainder with respect to another domain, it must be declared -explicitly. -\par -\par -If the first argument to }{\f3 remainder} {\f2 contains a denominator not equal to -1, an error occurs. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROUND} - -${\footnote \pard\plain \sl240 \fs20 $ ROUND} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0742} - - K{\footnote \pard\plain \sl240 \fs20 K integer;ROUND operator;operator} - -}{\b\f2 ROUND}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 round} {\f4 () -\par -\par -}{\f2 \par -If its argument has a numerical value, }{\f3 round} {\f2 rounds it to the -nearest integer. For non-numeric arguments, the value is an expression in -the original operator. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -round 3.4; \par - \par - 3 \par - \par - \par -round 3.5; \par - \par - 4 \par - \par - \par -round a; \par - \par - ROUND(A) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SETMOD} - -${\footnote \pard\plain \sl240 \fs20 $ SETMOD} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0743} - - K{\footnote \pard\plain \sl240 \fs20 K modular;SETMOD command;command} - -}{\b\f2 SETMOD}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 setmod} {\f2 command sets the modulus value for subsequent } -{\f2\uldb modular}{\v\f2 MODULAR} -{\f2 -arithmetic. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 setmod} {\f4 -\par -\par -}{\f2 \par - must be positive, and greater than 1. It need not be a prime -number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -setmod 6; \par - \par - 1 \par - \par - \par -on modular; \par - \par -16; \par - \par - 4 \par - \par - \par -x^2 + 5x + 7; \par - \par - 2 \par - X + 5*X + 1 \par - \par - \par -x/3; \par - \par - X \par - - \par - 3 \par - \par - \par -setmod 2; \par - \par - 6 \par - \par - \par -(x+1)^4; \par - \par - 4 \par - X + 1 \par - \par - \par -x/3; \par - \par - X \par - \par -\pard \sl240 }{\f2 }{\f3 setmod} {\f2 returns the previous modulus, or 1 if none has been set -before. }{\f3 setmod} {\f2 only has effect when } -{\f2\uldb modular}{\v\f2 MODULAR} -{\f2 is on. -\par -\par -Modular operations are done only on numbers such as coefficients of -polynomials, not on the exponents. The modulus need not be prime. -Attempts to divide by a power of the modulus produces an error message, since the -operation is equivalent to dividing by 0. However, dividing by a factor -of a non-prime modulus does not produce an error message. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SIGN} - -${\footnote \pard\plain \sl240 \fs20 $ SIGN} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0744} - - K{\footnote \pard\plain \sl240 \fs20 K SIGN operator;operator} - -}{\b\f2 SIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 sign} {\f4 -\par -\par -}{\f2 \par -}{\f3 sign} {\f2 tries to evaluate the sign of its argument. If this -is possible }{\f3 sign} {\f2 returns one of 1, 0 or -1. Otherwise, the result -is the original form or a simplified variant. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - sign(-5) \par - \par - -1 \par - \par - \par - sign(-a^2*b) \par - \par - -SIGN(B) \par - \par -\pard \sl240 }{\f2 Even powers of formal expressions are assumed to be positive only as long -as the switch } -{\f2\uldb complex}{\v\f2 COMPLEX} -{\f2 is off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SQRT} - -${\footnote \pard\plain \sl240 \fs20 $ SQRT} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0745} - - K{\footnote \pard\plain \sl240 \fs20 K square root;SQRT operator;operator} - -}{\b\f2 SQRT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 sqrt} {\f2 operator returns the square root of its argument. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 sqrt} {\f4 () -\par -\par -}{\f2 \par - can be any REDUCE scalar expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -sqrt(16*a^3); \par - \par - 4*SQRT(A)*A \par - \par - \par -sqrt(17); \par - \par - SQRT(17) \par - \par - \par -on rounded; \par - \par -sqrt(17); \par - \par - 4.12310562562 \par - \par - \par -off rounded; \par - \par -sqrt(a*b*c^5*d^3*27); \par - \par - 2 \par - 3*SQRT(D)*SQRT(C)*SQRT(B)*SQRT(A)*SQRT(3)*C *D \par - \par -\pard \sl240 }{\f2 }{\f3 sqrt} {\f2 checks its argument for squared factors and removes them. -\par -\par -Numeric values for square roots that are not exact integers are given only -when } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on. -\par -\par -Please note that }{\f3 sqrt(a**2)} {\f2 is given as }{\f3 a} {\f2 , which may be -incorrect if }{\f3 a} {\f2 eventually has a negative value. If you are -programming a calculation in which this is a concern, you can turn on the -} -{\f2\uldb precise}{\v\f2 PRECISE} -{\f2 switch, which causes the absolute value of the square root -to be returned. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TIMES} - -${\footnote \pard\plain \sl240 \fs20 $ TIMES} - -+{\footnote \pard\plain \sl240 \fs20 + g5:0746} - - K{\footnote \pard\plain \sl240 \fs20 K TIMES operator;operator} - -}{\b\f2 TIMES}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 times} {\f2 operator is an infix or prefix n-ary multiplication -operator. It is identical to }{\f3 *} {\f2 . - \par -syntax: \par -}{\f4 \par -\par - }{\f3 times} {\f4 \{}{\f3 times} {\f4 \}* -\par -\par -or }{\f3 times} {\f4 (, \{,\}*) -\par -\par -}{\f2 \par - can be any valid REDUCE scalar or matrix expression. -Matrix expressions must be of the correct dimensions. Compatible scalar -and matrix expressions can be mixed. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -var1 times var2; \par - \par - VAR1*VAR2 \par - \par - \par -times(6,5); \par - \par - 30 \par - \par - \par -matrix aa,bb; \par - \par -aa := mat((1),(2),(x))\$ \par - \par -bb := mat((0,3,1))\$ \par - \par -aa times bb times 5; \par - \par - [0 15 5 ] \par - [ ] \par - [0 30 10 ] \par - [ ] \par - [0 15*X 5*X] \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g5} - -${\footnote \pard\plain \sl240 \fs20 $ Arithmetic Operations} - -+{\footnote \pard\plain \sl240 \fs20 + index:0005} -}{\b\f2 Arithmetic Operations}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb ARITHMETIC\_OPERATIONS introduction} -{\v\f2 ARITHMETIC\_OPERATIONS}{\f2 \par -}{\f2 \tab}{\f2\uldb ABS operator} -{\v\f2 ABS}{\f2 \par -}{\f2 \tab}{\f2\uldb ADJPREC switch} -{\v\f2 ADJPREC}{\f2 \par -}{\f2 \tab}{\f2\uldb ARG operator} -{\v\f2 ARG}{\f2 \par -}{\f2 \tab}{\f2\uldb CEILING operator} -{\v\f2 CEILING}{\f2 \par -}{\f2 \tab}{\f2\uldb CHOOSE operator} -{\v\f2 CHOOSE}{\f2 \par -}{\f2 \tab}{\f2\uldb DEG2DMS operator} -{\v\f2 DEG2DMS}{\f2 \par -}{\f2 \tab}{\f2\uldb DEG2RAD operator} -{\v\f2 DEG2RAD}{\f2 \par -}{\f2 \tab}{\f2\uldb DIFFERENCE operator} -{\v\f2 DIFFERENCE}{\f2 \par -}{\f2 \tab}{\f2\uldb DILOG operator} -{\v\f2 DILOG}{\f2 \par -}{\f2 \tab}{\f2\uldb DMS2DEG operator} -{\v\f2 DMS2DEG}{\f2 \par -}{\f2 \tab}{\f2\uldb DMS2RAD operator} -{\v\f2 DMS2RAD}{\f2 \par -}{\f2 \tab}{\f2\uldb FACTORIAL operator} -{\v\f2 FACTORIAL}{\f2 \par -}{\f2 \tab}{\f2\uldb FIX operator} -{\v\f2 FIX}{\f2 \par -}{\f2 \tab}{\f2\uldb FIXP operator} -{\v\f2 FIXP}{\f2 \par -}{\f2 \tab}{\f2\uldb FLOOR operator} -{\v\f2 FLOOR}{\f2 \par -}{\f2 \tab}{\f2\uldb EXPT operator} -{\v\f2 EXPT}{\f2 \par -}{\f2 \tab}{\f2\uldb GCD operator} -{\v\f2 GCD}{\f2 \par -}{\f2 \tab}{\f2\uldb LN operator} -{\v\f2 LN}{\f2 \par -}{\f2 \tab}{\f2\uldb LOG operator} -{\v\f2 LOG}{\f2 \par -}{\f2 \tab}{\f2\uldb LOGB operator} -{\v\f2 LOGB}{\f2 \par -}{\f2 \tab}{\f2\uldb MAX operator} -{\v\f2 MAX}{\f2 \par -}{\f2 \tab}{\f2\uldb MIN operator} -{\v\f2 MIN}{\f2 \par -}{\f2 \tab}{\f2\uldb MINUS operator} -{\v\f2 MINUS}{\f2 \par -}{\f2 \tab}{\f2\uldb NEXTPRIME operator} -{\v\f2 NEXTPRIME}{\f2 \par -}{\f2 \tab}{\f2\uldb NOCONVERT switch} -{\v\f2 NOCONVERT}{\f2 \par -}{\f2 \tab}{\f2\uldb NORM operator} -{\v\f2 NORM}{\f2 \par -}{\f2 \tab}{\f2\uldb PERM operator} -{\v\f2 PERM}{\f2 \par -}{\f2 \tab}{\f2\uldb PLUS operator} -{\v\f2 PLUS}{\f2 \par -}{\f2 \tab}{\f2\uldb QUOTIENT operator} -{\v\f2 QUOTIENT}{\f2 \par -}{\f2 \tab}{\f2\uldb RAD2DEG operator} -{\v\f2 RAD2DEG}{\f2 \par -}{\f2 \tab}{\f2\uldb RAD2DMS operator} -{\v\f2 RAD2DMS}{\f2 \par -}{\f2 \tab}{\f2\uldb RECIP operator} -{\v\f2 RECIP}{\f2 \par -}{\f2 \tab}{\f2\uldb REMAINDER operator} -{\v\f2 REMAINDER}{\f2 \par -}{\f2 \tab}{\f2\uldb ROUND operator} -{\v\f2 ROUND}{\f2 \par -}{\f2 \tab}{\f2\uldb SETMOD command} -{\v\f2 SETMOD}{\f2 \par -}{\f2 \tab}{\f2\uldb SIGN operator} -{\v\f2 SIGN}{\f2 \par -}{\f2 \tab}{\f2\uldb SQRT operator} -{\v\f2 SQRT}{\f2 \par -}{\f2 \tab}{\f2\uldb TIMES operator} -{\v\f2 TIMES}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # boolean_value} - -${\footnote \pard\plain \sl240 \fs20 $ boolean_value} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0747} - - K{\footnote \pard\plain \sl240 \fs20 K boolean value concept;concept} - -}{\b\f2 BOOLEAN VALUE}{\f2 \par -\par - -There are no extra symbols for the truth values true -and false. Instead, } -{\f2\uldb nil}{\v\f2 NIL} -{\f2 and the number zero -are interpreted as truth value false in algebraic -programs (see } -{\f2\uldb false}{\v\f2 false} -{\f2 ), while any different -value is considered as true (see } -{\f2\uldb true}{\v\f2 TRUE} -{\f2 ). -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EQUAL} - -${\footnote \pard\plain \sl240 \fs20 $ EQUAL} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0748} - - K{\footnote \pard\plain \sl240 \fs20 K equation;EQUAL operator;operator} - -}{\b\f2 EQUAL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 equal} {\f2 is an infix binary comparison -operator. It is identical with }{\f3 =} {\f2 . It returns } -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if its two -arguments are equal. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 equal} {\f4 -\par -\par -}{\f2 \par -Equality is given between floating point numbers and integers that have -the same value. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -a := 4; \par - \par - A := 4 \par - \par - \par -b := 4.0; \par - \par - B := 4.0 \par - \par - \par -if a equal b then write "true" else write "false"; \par - \par - \par - \par - true \par - \par - \par -if a equal 5 then write "true" else write "false"; \par - \par - \par - \par - false \par - \par - \par -if a equal sqrt(16) then write "true" else write "false"; \par - \par - \par - \par - true \par - \par -\pard \sl240 }{\f2 Comparison operators can only be used as conditions in conditional commands -such as }{\f3 if} {\f2 ...}{\f3 then} {\f2 and }{\f3 repeat} {\f2 ...}{\f3 until} {\f2 . - can also be used as a prefix operator. However, this use -is not encouraged. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EVENP} - -${\footnote \pard\plain \sl240 \fs20 $ EVENP} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0749} - - K{\footnote \pard\plain \sl240 \fs20 K EVENP operator;operator} - -}{\b\f2 EVENP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 evenp} {\f2 logical operator returns } -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if its argument is an -even integer, and } -{\f2\uldb nil}{\v\f2 NIL} -{\f2 if its argument is an odd integer. An error -message is returned if its argument is not an integer. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 evenp} {\f4 () or }{\f3 evenp} {\f4 -\par -\par -}{\f2 \par - must evaluate to an integer. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -aa := 1782; \par - \par - AA := 1782 \par - \par - \par -if evenp aa then yes else no; \par - \par - YES \par - \par - \par -if evenp(-3) then yes else no; \par - \par - NO \par - \par -\pard \sl240 }{\f2 Although you would not ordinarily enter an expression such as the last -example above, note that the negative term must be enclosed in parentheses -to be correctly parsed. The }{\f3 evenp} {\f2 operator can only be used in -conditional statements such as }{\f3 if} {\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 -or }{\f3 while} {\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # false} - -${\footnote \pard\plain \sl240 \fs20 $ false} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0750} - - K{\footnote \pard\plain \sl240 \fs20 K false concept;concept} - -}{\b\f2 FALSE}{\f2 \par -\par - -The symbol } -{\f2\uldb nil}{\v\f2 NIL} -{\f2 and the number zero are considered -as } -{\f2\uldb boolean value}{\v\f2 boolean_value} -{\f2 false if used in a place where -a boolean value is required. Most builtin operators return -} -{\f2\uldb nil}{\v\f2 NIL} -{\f2 as false value. Algebraic programs use better zero. -Note that }{\f3 nil} {\f2 is not printed when returned as result to -a top level evaluation. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FREEOF} - -${\footnote \pard\plain \sl240 \fs20 $ FREEOF} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0751} - - K{\footnote \pard\plain \sl240 \fs20 K FREEOF operator;operator} - -}{\b\f2 FREEOF}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 freeof} {\f2 logical operator returns -} -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if its first argument does -not contain its second argument anywhere in its structure. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 freeof} {\f4 (,) or - }{\f3 freeof} {\f4 -\par -\par -}{\f2 \par - can be any valid scalar REDUCE expression, must -be a kernel expression (see }{\f3 kernel} {\f2 ). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := x + sin(y)**2 + log sin z; \par - \par - \par - \par - 2 \par - A := LOG(SIN(Z)) + SIN(Y) + X \par - \par - \par -if freeof(a,sin(y)) then write "free" else write "not free"; \par - \par - \par - \par - not free \par - \par - \par -if freeof(a,sin(x)) then write "free" else write "not free"; \par - \par - \par - \par - free \par - \par - \par -if a freeof sin z then write "free" else write "not free"; \par - \par - \par - \par - not free \par - \par -\pard \sl240 }{\f2 Logical operators can only be used in conditional expressions such as -\par -\par -}{\f3 if} {\f2 ...}{\f3 then} {\f2 or }{\f3 while} {\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LEQ} - -${\footnote \pard\plain \sl240 \fs20 $ LEQ} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0752} - - K{\footnote \pard\plain \sl240 \fs20 K LEQ operator;operator} - -}{\b\f2 LEQ}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 leq} {\f2 operator is a binary infix or prefix logical operator. It -returns } -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if its first argument is less than or equal to its second -argument. As an infix operator it is identical with }{\f3 <=} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 leq} {\f4 (,) or -}{\f3 leq} {\f4 -\par -\par -\par -\par -}{\f2 can be any valid REDUCE expression that evaluates to a -number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 15; \par - \par - A := 15 \par - \par - \par -if leq(a,25) then write "yes" else write "no"; \par - \par - \par - \par - yes \par - \par - \par -if leq(a,15) then write "yes" else write "no"; \par - \par - \par - \par - yes \par - \par - \par -if leq(a,5) then write "yes" else write "no"; \par - \par - \par - \par - no \par - \par -\pard \sl240 }{\f2 Logical operators can only be used in conditional statements such as -\par -\par -}{\f3 if} {\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 or }{\f3 while} {\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LESSP} - -${\footnote \pard\plain \sl240 \fs20 $ LESSP} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0753} - - K{\footnote \pard\plain \sl240 \fs20 K LESSP operator;operator} - -}{\b\f2 LESSP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 lessp} {\f2 operator is a binary infix or prefix logical operator. It -returns } -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if its first argument is strictly less than its second -argument. As an infix operator it is identical with }{\f3 <} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 lessp} {\f4 (,) -or }{\f3 lessp} {\f4 -\par -\par -\par -\par -}{\f2 can be any valid REDUCE expression that evaluates to a -number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 15; \par - \par - A := 15 \par - \par - \par -if lessp(a,25) then write "yes" else write "no"; \par - \par - \par - \par - yes \par - \par - \par -if lessp(a,15) then write "yes" else write "no"; \par - \par - \par - \par - no \par - \par - \par -if lessp(a,5) then write "yes" else write "no"; \par - \par - \par - \par - no \par - \par -\pard \sl240 }{\f2 Logical operators can only be used in conditional statements such as -\par -\par -}{\f3 if} {\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 or }{\f3 while} {\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MEMBER} - -${\footnote \pard\plain \sl240 \fs20 $ MEMBER} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0754} - - K{\footnote \pard\plain \sl240 \fs20 K list;MEMBER operator;operator} - -}{\b\f2 MEMBER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 member} {\f4 -\par -\par -}{\f2 \par -}{\f3 member} {\f2 is an infix binary comparison operator that evaluates to -} -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if is } -{\f2\uldb equal}{\v\f2 EQUAL} -{\f2 to a member of -the } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -if a member \{a,b\} then 1 else 0; \par - \par - 1 \par - \par - \par -if 1 member(1,2,3) then a else b; \par - \par - a \par - \par - \par -if 1 member(1.0,2) then a else b; \par - \par - b \par - \par -\pard \sl240 }{\f2 Logical operators can only be used in conditional statements such as -\par -\par -}{\f3 if} {\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 or }{\f3 while} {\f2 ...}{\f3 do} {\f2 . - can also be used as a prefix operator. However, this use -is not encouraged. Finally, } -{\f2\uldb equal}{\v\f2 EQUAL} -{\f2 (}{\f3 =} {\f2 ) is used for the test -within the list, so expressions must be of the same type to match. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NEQ} - -${\footnote \pard\plain \sl240 \fs20 $ NEQ} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0755} - - K{\footnote \pard\plain \sl240 \fs20 K NEQ operator;operator} - -}{\b\f2 NEQ}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The operator }{\f3 neq} {\f2 is an infix binary comparison -operator. It returns } -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if its two -arguments are not } -{\f2\uldb equal}{\v\f2 EQUAL} -{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 neq} {\f4 -\par -\par -}{\f2 \par -An inequality is satisfied between floating point numbers and integers -that have the same value. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -a := 4; \par - \par - A := 4 \par - \par - \par -b := 4.0; \par - \par - B := 4.0 \par - \par - \par -if a neq b then write "true" else write "false"; \par - \par - \par - \par - false \par - \par - \par -if a neq 5 then write "true" else write "false"; \par - \par - \par - \par - true \par - \par -\pard \sl240 }{\f2 Comparison operators can only be used as conditions in conditional commands -such as }{\f3 if} {\f2 ...}{\f3 then} {\f2 and }{\f3 repeat} {\f2 ...}{\f3 until} {\f2 . - can also be used as a prefix operator. However, this use -is not encouraged. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NOT} - -${\footnote \pard\plain \sl240 \fs20 $ NOT} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0756} - - K{\footnote \pard\plain \sl240 \fs20 K NOT operator;operator} - -}{\b\f2 NOT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 not} {\f2 operator returns } -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if its argument evaluates to -} -{\f2\uldb nil}{\v\f2 NIL} -{\f2 , and }{\f3 nil} {\f2 if its argument is }{\f3 true} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 not} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -if not numberp(a) then write "indeterminate" else write a; \par - \par - \par - \par - indeterminate; \par - \par - \par -a := 10; \par - \par - A := 10 \par - \par - \par -if not numberp(a) then write "indeterminate" else write a; \par - \par - \par - \par - 10 \par - \par - \par -if not(numberp(a) and a < 0) then write "positive number"; \par - \par - \par - \par - positive number \par - \par -\pard \sl240 }{\f2 Logical operators can only be used in conditional statements such as -\par -\par -}{\f3 if} {\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 or }{\f3 while} {\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NUMBERP} - -${\footnote \pard\plain \sl240 \fs20 $ NUMBERP} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0757} - - K{\footnote \pard\plain \sl240 \fs20 K NUMBERP operator;operator} - -}{\b\f2 NUMBERP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 numberp} {\f2 operator returns } -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if its argument is a number, -and } -{\f2\uldb nil}{\v\f2 NIL} -{\f2 otherwise. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 numberp} {\f4 () or }{\f3 numberp} {\f4 -\par -\par -}{\f2 \par - can be any REDUCE scalar expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -cc := 15.3; \par - \par - CC := 15.3 \par - \par - \par -if numberp(cc) then write "number" else write "nonnumber"; \par - \par - \par - number \par - \par - \par -if numberp(cb) then write "number" else write "nonnumber"; \par - \par - \par - nonnumber \par - \par -\pard \sl240 }{\f2 Logical operators can only be used in conditional expressions, such as -\par -\par -}{\f3 if} {\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 and }{\f3 while} {\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ORDP} - -${\footnote \pard\plain \sl240 \fs20 $ ORDP} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0758} - - K{\footnote \pard\plain \sl240 \fs20 K order;ORDP operator;operator} - -}{\b\f2 ORDP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 ordp} {\f2 logical operator returns } -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if its first argument is -ordered ahead of its second argument in canonical internal ordering, or is -identical to it. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 ordp} {\f4 (,) -\par -\par -\par -\par -}{\f2 and can be any valid REDUCE scalar -expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -if ordp(x**2 + 1,x**3 + 3) then write "yes" else write "no"; \par - \par - \par - \par - no \par - \par - \par -if ordp(101,100) then write "yes" else write "no"; \par - \par - \par - \par - yes \par - \par - \par -if ordp(x,x) then write "yes" else write "no"; \par - \par - \par - \par - yes \par - \par -\pard \sl240 }{\f2 Logical operators can only be used in conditional expressions, such as -\par -\par -}{\f3 if} {\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 and }{\f3 while} {\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PRIMEP} - -${\footnote \pard\plain \sl240 \fs20 $ PRIMEP} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0759} - - K{\footnote \pard\plain \sl240 \fs20 K prime number;PRIMEP operator;operator} - -}{\b\f2 PRIMEP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 primep} {\f4 () or }{\f3 primep} {\f4 -\par -\par -}{\f2 \par -If evaluates to a integer, }{\f3 primep} {\f2 returns -} -{\f2\uldb true}{\v\f2 TRUE} -{\f2 \par -\par -if is a prime number and } -{\f2\uldb nil}{\v\f2 NIL} -{\f2 otherwise. -If does not have an integer value, a type error occurs. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -if primep 3 then write "yes" else write "no"; \par - \par - \par - YES \par - \par - \par -if primep a then 1; \par - \par - ***** A invalid as integer \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TRUE} - -${\footnote \pard\plain \sl240 \fs20 $ TRUE} - -+{\footnote \pard\plain \sl240 \fs20 + g6:0760} - - K{\footnote \pard\plain \sl240 \fs20 K false;TRUE concept;concept} - -}{\b\f2 TRUE}{\f2 \par -\par - -\par -\par -Any value of the boolean part of a logical expression which is neither -} -{\f2\uldb nil}{\v\f2 NIL} -{\f2 nor }{\f3 0} {\f2 is considered as }{\f3 true} {\f2 . Most -builtin test and compare functions return } -{\f2\uldb t}{\v\f2 T} -{\f2 for }{\f3 true} {\f2 -and } -{\f2\uldb nil}{\v\f2 NIL} -{\f2 for }{\f3 false} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -if member(3,\{1,2,3\}) then 1 else -1; \par - \par - \par - 1 \par - \par - \par -if floor(1.7) then 1 else -1; \par - \par - 1 \par - \par - \par -if floor(0.7) then 1 else -1; \par - \par - -1 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g6} - -${\footnote \pard\plain \sl240 \fs20 $ Boolean Operators} - -+{\footnote \pard\plain \sl240 \fs20 + index:0006} -}{\b\f2 Boolean Operators}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb boolean value concept} -{\v\f2 boolean_value}{\f2 \par -}{\f2 \tab}{\f2\uldb EQUAL operator} -{\v\f2 EQUAL}{\f2 \par -}{\f2 \tab}{\f2\uldb EVENP operator} -{\v\f2 EVENP}{\f2 \par -}{\f2 \tab}{\f2\uldb false concept} -{\v\f2 false}{\f2 \par -}{\f2 \tab}{\f2\uldb FREEOF operator} -{\v\f2 FREEOF}{\f2 \par -}{\f2 \tab}{\f2\uldb LEQ operator} -{\v\f2 LEQ}{\f2 \par -}{\f2 \tab}{\f2\uldb LESSP operator} -{\v\f2 LESSP}{\f2 \par -}{\f2 \tab}{\f2\uldb MEMBER operator} -{\v\f2 MEMBER}{\f2 \par -}{\f2 \tab}{\f2\uldb NEQ operator} -{\v\f2 NEQ}{\f2 \par -}{\f2 \tab}{\f2\uldb NOT operator} -{\v\f2 NOT}{\f2 \par -}{\f2 \tab}{\f2\uldb NUMBERP operator} -{\v\f2 NUMBERP}{\f2 \par -}{\f2 \tab}{\f2\uldb ORDP operator} -{\v\f2 ORDP}{\f2 \par -}{\f2 \tab}{\f2\uldb PRIMEP operator} -{\v\f2 PRIMEP}{\f2 \par -}{\f2 \tab}{\f2\uldb TRUE concept} -{\v\f2 TRUE}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BYE} - -${\footnote \pard\plain \sl240 \fs20 $ BYE} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0761} - - K{\footnote \pard\plain \sl240 \fs20 K BYE command;command} - -}{\b\f2 BYE}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 bye} {\f2 command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are at -the top level, the }{\f3 bye} {\f2 command exits REDUCE. }{\f3 quit} {\f2 is a -synonym for }{\f3 bye} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CONT} - -${\footnote \pard\plain \sl240 \fs20 $ CONT} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0762} - - K{\footnote \pard\plain \sl240 \fs20 K CONT command;command} - -}{\b\f2 CONT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The command }{\f3 cont} {\f2 returns control to an interactive file after a -} -{\f2\uldb pause}{\v\f2 PAUSE} -{\f2 command that has been answered with }{\f3 n} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 Suppose you are in the middle of an interactive file.}{\f4 \pard \tx3420 \par - \par - \par - \par - factorize(x**2 + 17*x + 60); \par - \par - \par - \par - \par - \{X + 5,X + 12\} \par - \par - \par - pause; \par - \par - Cont? (Y or N) \par - \par - \par -n \par - \par -saveas results; \par - \par -factor1 := first results; \par - \par - FACTOR1 := X + 5 \par - \par - \par -factor2 := second results; \par - \par - FACTOR2 := X + 12 \par - \par - \par -cont; \pard \sl240 }{\f2 the file resumes}{\f4 \pard \tx3420 \par - \par -\pard \sl240 }{\f2 -\par -\par -A } -{\f2\uldb pause}{\v\f2 PAUSE} -{\f2 allows you to enter your own REDUCE commands, change -switch values, inquire about results, or other such activities. When you -wish to resume operation of the interactive file, use }{\f3 cont} {\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DISPLAY} - -${\footnote \pard\plain \sl240 \fs20 $ DISPLAY} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0763} - - K{\footnote \pard\plain \sl240 \fs20 K interactive;history;DISPLAY command;command} - -}{\b\f2 DISPLAY}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -When given a numeric argument , }{\f3 display} {\f2 prints the -most recent input statements, identified by prompt numbers. If an empty -pair of parentheses is given, or if is greater than the current -number of statements, all the input statements since the beginning of -the session are printed. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 display} {\f4 () or }{\f3 display} {\f4 () -\par -\par -}{\f2 \par - should be a positive integer. However, if it is a real number, the -truncated integer value is used, and if a non-numeric argument is used, all -the input statements are printed. -\par -\par -The statements are displayed in upper case, with lines split at semicolons or -dollar signs, as they are in editing. If long files have been input during -the session, the }{\f3 display} {\f2 command is slow to format these for -printing. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LOAD\_PACKAGE} - -${\footnote \pard\plain \sl240 \fs20 $ LOAD_PACKAGE} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0764} - - K{\footnote \pard\plain \sl240 \fs20 K package;LOAD_PACKAGE command;command} - -}{\b\f2 LOAD\_PACKAGE}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 load_package} {\f2 command is used to load REDUCE packages, such as -}{\f3 gentran} {\f2 that are not automatically loaded by the system. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 load_package "} {\f4 }{\f3 "} {\f4 -\par -\par -}{\f2 \par -A package is only loaded once; subsequent calls of }{\f3 load_package} {\f2 -for the same package name are ignored. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PAUSE} - -${\footnote \pard\plain \sl240 \fs20 $ PAUSE} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0765} - - K{\footnote \pard\plain \sl240 \fs20 K interactive;PAUSE command;command} - -}{\b\f2 PAUSE}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 pause} {\f2 command, given in an interactive file, stops operation and -asks if you want to continue or not. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 An interactive file is running, and at some point you see the -question}{\f4 \pard \tx3420 \par - \par - Cont? (Y or N) \par -\pard \sl240 }{\f2 If you type}{\f4 \pard \tx3420 \par - \par -y\key\{Return\} \par -\pard \sl240 }{\f2 the file continues to run until the next pause or the end.}{\f4 \pard \tx3420 \par -\pard \sl240 }{\f2 If you type }{\f4 \pard \tx3420 \par - \par -n\key\{Return\} \par -\pard \sl240 }{\f2 you will get a numbered REDUCE prompt, and be allowed to -enter and execute any REDUCE statements. If you later wish to continue with -the file, type}{\f4 \pard \tx3420 \par - \par -cont; \par -\pard \sl240 }{\f2 and the file resumes.}{\f4 \pard \tx3420 \pard \sl240 }{\f2 -\par -\par -To use }{\f3 pause} {\f2 in your own interactive files, type -\par -\par -}{\f3 pause;} {\f2 in the file wherever you want it. -\par -\par -}{\f3 pause} {\f2 does not allow you to continue without typing either }{\f3 y} {\f2 -or }{\f3 n} {\f2 . Its use is to slow down scrolling of interactive files, or to -let you change parameters or switch settings for the calculations. -\par -\par -If you have stopped an interactive file at a }{\f3 pause,} {\f2 and do not wish to -resume the file, type }{\f3 end;} {\f2 . This does not end the REDUCE session, but -stops input from the file. A second }{\f3 end;} {\f2 ends the REDUCE session. -However, if you have pauses from more than one file stacked up, an }{\f3 end;} {\f2 -brings you back to the top level, not the file directly above. -\par -\par -A }{\f3 pause} {\f2 typed from the terminal has no effect. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # QUIT} - -${\footnote \pard\plain \sl240 \fs20 $ QUIT} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0766} - - K{\footnote \pard\plain \sl240 \fs20 K QUIT command;command} - -}{\b\f2 QUIT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 quit} {\f2 command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are at -the top level, the }{\f3 quit} {\f2 command exits REDUCE. } -{\f2\uldb bye}{\v\f2 BYE} -{\f2 is a -synonym for }{\f3 quit} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RECLAIM} - -${\footnote \pard\plain \sl240 \fs20 $ RECLAIM} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0767} - - K{\footnote \pard\plain \sl240 \fs20 K memory;RECLAIM operator;operator} - -}{\b\f2 RECLAIM}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -REDUCE's memory is in a storage structure called a heap. As REDUCE -statements execute, chunks of memory are used up. When these chunks are no -longer needed, they remain idle. When the memory is almost full, -the system executes a garbage collection, reclaiming space that is no -longer needed, and putting all the free space at one end. Depending on -the size of the image REDUCE is using, -garbage collection needs to be done more or less often. A -larger image means fewer but longer garbage collections. -Regardless of memory size, -if you ask REDUCE to do something ridiculous, like }{\f3 factorial(2000)} {\f2 , it may -garbage collect many times. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # REDERR} - -${\footnote \pard\plain \sl240 \fs20 $ REDERR} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0768} - - K{\footnote \pard\plain \sl240 \fs20 K error handling;REDERR command;command} - -}{\b\f2 REDERR}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 rederr} {\f2 command allows you to print an error message from inside -a } -{\f2\uldb procedure}{\v\f2 PROCEDURE} -{\f2 or a } -{\f2\uldb block}{\v\f2 block} -{\f2 statement. -The calculation is gracefully terminated. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 rederr} {\f4 -\par -\par -}{\f2 \par - is an error message, usually inside double quotation marks -(a } -{\f2\uldb string}{\v\f2 STRING} -{\f2 ). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -procedure fac(n); \par - if not (fixp(n) and n>=0) \par - then rederr "Choose nonneg. integer only" \par - else for i := 0:n-1 product i+1; \par -\pard \sl240 \par - \par - fac \par - \par - \par -fac a; \par - \par - ***** Choose nonneg. integer only \par - \par - \par -fac 5; \par - \par - 120 \par - \par -\pard \sl240 }{\f2 The above procedure finds the factorial of its argument. -If n is not a positive integer or 0, an error message is returned. -\par -\par -If your procedure is executed in a file, the usual error message is -printed, followed by }{\f3 Cont? (Y or N)} {\f2 , just as any other error does from -a file. Although the procedure is gracefully terminated, any switch settings or -variable assignments you made before the error occurred are not undone. If you -need to clean up such items before exiting, use a group statement, with the -}{\f3 rederr} {\f2 command as its last statement. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RETRY} - -${\footnote \pard\plain \sl240 \fs20 $ RETRY} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0769} - - K{\footnote \pard\plain \sl240 \fs20 K interactive;RETRY command;command} - -}{\b\f2 RETRY}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 retry} {\f2 command allows you to retry the latest statement that resulted -in an error message. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -matrix a; \par - \par -det a; \par - \par - ***** Matrix A not set \par - \par - \par -a := mat((1,2),(3,4)); \par - \par - A(1,1) := 1 \par - A(1,2) := 2 \par - A(2,1) := 3 \par - A(2,2) := 4 \par - \par - \par -retry; \par - \par - -2 \par - \par -\pard \sl240 }{\f2 }{\f3 retry} {\f2 remembers only the most recent statement that resulted in an -error message. It allows you to stop and fix something obvious, then -continue on your way without retyping the original command. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SAVEAS} - -${\footnote \pard\plain \sl240 \fs20 $ SAVEAS} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0770} - - K{\footnote \pard\plain \sl240 \fs20 K SAVEAS command;command} - -}{\b\f2 SAVEAS}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 saveas} {\f2 command saves the current workspace under the name of its -argument. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 saveas} {\f4 -\par -\par -}{\f2 \par - can be any valid REDUCE identifier. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 (The numbered prompts are shown below, unlike in most examples)}{\f4 \pard \tx3420 \par - \par -1: solve(x^2-3); \par - \par - \{x=sqrt(3),x= - sqrt(3)\} \par - \par - \par -2: saveas rts(0)\$ \par - \par -3: rts(0); \par - \par - \{x=sqrt(3),x= - sqrt(3)\} \par - \par -\pard \sl240 }{\f2 -\par -\par -}{\f3 saveas} {\f2 works only for the current workspace, the last algebraic -expression produced by REDUCE. This allows you to save a result that you -did not assign to an identifier when you originally typed the input. -For access to previous output use } -{\f2\uldb ws}{\v\f2 WS} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SHOWTIME} - -${\footnote \pard\plain \sl240 \fs20 $ SHOWTIME} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0771} - - K{\footnote \pard\plain \sl240 \fs20 K time;SHOWTIME command;command} - -}{\b\f2 SHOWTIME}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 showtime} {\f2 command prints the elapsed system time since the last -call of this command or since the beginning of the session, if it has not -been called before. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -showtime; \par - \par - Time: 1020 ms \par - \par - \par -factorize(x^4 - 8x^4 + 8x^2 - 136x - 153); \par - \par - \par - \par - 2 \par - \{X - 9,X + 17,X + 1\} \par - \par - \par -showtime; \par - \par - Time: 920 ms \par - \par -\pard \sl240 }{\f2 The time printed is either the elapsed cpu time or the elapsed wall clock -time, depending on your system. }{\f3 showtime} {\f2 allows you to see the -system time resources REDUCE uses in its calculations. Your time readings -will of course vary from this example according to the system you use. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # WRITE} - -${\footnote \pard\plain \sl240 \fs20 $ WRITE} - -+{\footnote \pard\plain \sl240 \fs20 + g7:0772} - - K{\footnote \pard\plain \sl240 \fs20 K output;WRITE command;command} - -}{\b\f2 WRITE}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 write} {\f2 command explicitly writes its arguments to the output device -(terminal or file). - \par -syntax: \par -}{\f4 \par -\par -}{\f3 write} {\f4 \{,\}* -\par -\par -}{\f2 \par - can be an expression, an assignment or a } -{\f2\uldb string}{\v\f2 STRING} -{\f2 -enclosed in double quotation marks (}{\f3 "} {\f2 ). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -write a, sin x, "this is a string"; \par - \par - \par - ASIN(X)this is a string \par - \par - \par -write a," ",sin x," this is a string"; \par - \par - \par - A SIN(X) this is a string \par - \par - \par -if not numberp(a) then write "the symbol ",a; \par - \par - \par - \par - the symbol A \par - \par - \par -array m(10); \par - \par -for i := 1:5 do write m(i) := 2*i; \par - \par - \par - M(1) := 2 \par - M(2) := 4 \par - M(3) := 6 \par - M(4) := 8 \par - M(5) := 10 \par - \par - \par -m(4); \par - \par - 8 \par - \par -\pard \sl240 }{\f2 The items specified by a single }{\f3 write} {\f2 statement print on a single line -unless they are too long. A printed line is always ended with a carriage -return, so the next item printed starts a new line. -\par -\par -When an assignment statement is printed, the assignment is also made. This -allows you to get feedback on filling slots in an array with a } -{\f2\uldb for}{\v\f2 FOR} -{\f2 - statement, as shown in the last example above. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g7} - -${\footnote \pard\plain \sl240 \fs20 $ General Commands} - -+{\footnote \pard\plain \sl240 \fs20 + index:0007} -}{\b\f2 General Commands}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb BYE command} -{\v\f2 BYE}{\f2 \par -}{\f2 \tab}{\f2\uldb CONT command} -{\v\f2 CONT}{\f2 \par -}{\f2 \tab}{\f2\uldb DISPLAY command} -{\v\f2 DISPLAY}{\f2 \par -}{\f2 \tab}{\f2\uldb LOAD\_PACKAGE command} -{\v\f2 LOAD\_PACKAGE}{\f2 \par -}{\f2 \tab}{\f2\uldb PAUSE command} -{\v\f2 PAUSE}{\f2 \par -}{\f2 \tab}{\f2\uldb QUIT command} -{\v\f2 QUIT}{\f2 \par -}{\f2 \tab}{\f2\uldb RECLAIM operator} -{\v\f2 RECLAIM}{\f2 \par -}{\f2 \tab}{\f2\uldb REDERR command} -{\v\f2 REDERR}{\f2 \par -}{\f2 \tab}{\f2\uldb RETRY command} -{\v\f2 RETRY}{\f2 \par -}{\f2 \tab}{\f2\uldb SAVEAS command} -{\v\f2 SAVEAS}{\f2 \par -}{\f2 \tab}{\f2\uldb SHOWTIME command} -{\v\f2 SHOWTIME}{\f2 \par -}{\f2 \tab}{\f2\uldb WRITE command} -{\v\f2 WRITE}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # APPEND} - -${\footnote \pard\plain \sl240 \fs20 $ APPEND} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0773} - - K{\footnote \pard\plain \sl240 \fs20 K list;APPEND operator;operator} - -}{\b\f2 APPEND}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 append} {\f2 operator constructs a new } -{\f2\uldb list}{\v\f2 LIST} -{\f2 -from the elements of its two arguments (which must be lists). -\par -\par - \par -syntax: \par -}{\f4 }{\f3 append} {\f4 (,) -\par -\par -}{\f2 \par - must be a list, though it may be the empty list (}{\f3 \{\}} {\f2 ). -Any arguments beyond the first two are ignored. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -alist := \{1,2,\{a,b\}\}; \par - \par - ALIST := \{1,2,\{A,B\}\} \par - \par - \par -blist := \{3,4,5,sin(y)\}; \par - \par - BLIST := \{3,4,5,SIN(Y)\} \par - \par - \par -append(alist,blist); \par - \par - \{1,2,\{A,B\},3,4,5,SIN(Y)\} \par - \par - \par -append(alist,\{\}); \par - \par - \{1,2,\{A,B\}\} \par - \par - \par -append(list z,blist); \par - \par - \{Z,3,4,5,SIN(Y)\} \par - \par -\pard \sl240 }{\f2 The new list consists of the elements of the second list appended to the -elements of the first list. You can }{\f3 append} {\f2 new elements to the -beginning or end of an existing list by putting the new element in a -list (use curly braces or the operator }{\f3 list} {\f2 ). This is -particularly helpful in an iterative loop. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ARBINT} - -${\footnote \pard\plain \sl240 \fs20 $ ARBINT} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0774} - - K{\footnote \pard\plain \sl240 \fs20 K arbitrary value;ARBINT operator;operator} - -}{\b\f2 ARBINT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 arbint} {\f2 is used to express arbitrary integer parts -of an expression, e.g. in the result of } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 when -} -{\f2\uldb allbranch}{\v\f2 ALLBRANCH} -{\f2 is on. - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -solve(log(sin(x+3)),x); \par - \par - \{X=2*ARBINT(1)*PI - ASIN(1) - 3, \par - X=2*ARBINT(1)*PI + ASIN(1) + PI - 3\} \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ARBCOMPLEX} - -${\footnote \pard\plain \sl240 \fs20 $ ARBCOMPLEX} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0775} - - K{\footnote \pard\plain \sl240 \fs20 K arbitrary value;ARBCOMPLEX operator;operator} - -}{\b\f2 ARBCOMPLEX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 arbcomplex} {\f2 is used to express arbitrary scalar parts -of an expression, e.g. in the result of } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 when -the solution is parametric in one of the variable. - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -solve(\{x+3=y-2z,y-3x=0\},\{x,y,z\}); \par - \par - \par - 2*ARBCOMPLEX(1) + 3 \par - \{X=-------------------, \par - 2 \par - 3*ARBCOMPLEX(1) + 3 \par - Y=-------------------, \par - 2 \par - Z=ARBCOMPLEX(1)\} \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ARGLENGTH} - -${\footnote \pard\plain \sl240 \fs20 $ ARGLENGTH} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0776} - - K{\footnote \pard\plain \sl240 \fs20 K argument;ARGLENGTH operator;operator} - -}{\b\f2 ARGLENGTH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 arglength} {\f2 returns the number of arguments of the top-level -operator in its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 arglength} {\f4 () -\par -\par -}{\f2 \par - can be any valid REDUCE algebraic expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -arglength(a + b + c + d); \par - \par - 4 \par - \par - \par -arglength(a/b/c); \par - \par - 2 \par - \par - \par -arglength(log(sin(df(r**3*x,x)))); \par - \par - \par - 1 \par - \par -\pard \sl240 }{\f2 In the first example, }{\f3 +} {\f2 is an n-ary operator, so the number of terms -is returned. In the second example, since }{\f3 /} {\f2 is a binary operator, the -argument is actually (a/b)/c, so there are two terms at the top level. In -the last example, no matter how deeply the operators are nested, there is -still only one argument at the top level. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COEFF} - -${\footnote \pard\plain \sl240 \fs20 $ COEFF} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0777} - - K{\footnote \pard\plain \sl240 \fs20 K coefficient;COEFF operator;operator} - -}{\b\f2 COEFF}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 coeff} {\f2 operator returns the coefficients of the powers of the -specified variable in the given expression, in a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 coeff} {\f4 (}{\f3 ,} {\f4 ) -\par -\par -}{\f2 \par - is expected to be a polynomial expression, not a rational -expression. Rational expressions are accepted when the switch -} -{\f2\uldb ratarg}{\v\f2 RATARG} -{\f2 is on. must be a kernel. The results are -returned in a list. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -coeff((x+y)**3,x); \par - \par - 3 2 \par - \{Y ,3*Y ,3*Y,1\} \par - \par - \par -coeff((x+2)**4 + sin(x),x); \par - \par - \{SIN(X) + 16,32,24,8,1\} \par - \par - \par -high_pow; \par - \par - 4 \par - \par - \par -low_pow; \par - \par - 0 \par - \par - \par -ab := x**9 + sin(x)*x**7 + sqrt(y); \par - \par - \par - \par - 7 9 \par - AB := SQRT(Y) + SIN(X)*X + X \par - \par - \par -coeff(ab,x); \par - \par - \{SQRT(Y),0,0,0,0,0,0,SIN(X),0,1\} \par - \par -\pard \sl240 }{\f2 The variables } -{\f2\uldb high_pow}{\v\f2 HIGH\_POW} -{\f2 and } -{\f2\uldb low_pow}{\v\f2 LOW\_POW} -{\f2 are set to the -highest and lowest powers of the variable, respectively, appearing in the -expression. -\par -\par -The coefficients are put into a list, with the coefficient of the lowest -(constant) term first. You can use the usual list access methods -(}{\f3 first} {\f2 , }{\f3 second} {\f2 , }{\f3 third} {\f2 , }{\f3 rest} {\f2 , }{\f3 length} {\f2 , and -}{\f3 part} {\f2 ) to extract them. If a power does not appear in the -expression, the corresponding element of the list is zero. Terms involving -functions of the specified variable but not including powers of it (for -example in the expression }{\f3 x**4 + 3*x**2 + tan(x)} {\f2 ) are placed in the -constant term. -\par -\par -Since the }{\f3 coeff} {\f2 command deals with the expanded form of the expression, -you may get unexpected results when } -{\f2\uldb exp}{\v\f2 EXP} -{\f2 is off, or when -} -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 or } -{\f2\uldb ifactor}{\v\f2 IFACTOR} -{\f2 are on. -\par -\par -If you want only a specific coefficient rather than all of them, use the -} -{\f2\uldb coeffn}{\v\f2 COEFFN} -{\f2 operator. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COEFFN} - -${\footnote \pard\plain \sl240 \fs20 $ COEFFN} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0778} - - K{\footnote \pard\plain \sl240 \fs20 K coefficient;COEFFN operator;operator} - -}{\b\f2 COEFFN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 coeffn} {\f2 operator takes three arguments: an expression, a kernel, and -a non-negative integer. It returns the coefficient of the kernel to that -integer power, appearing in the expression. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 coeffn} {\f4 (,,) -\par -\par -}{\f2 \par - must be a polynomial, unless } -{\f2\uldb ratarg}{\v\f2 RATARG} -{\f2 is on which -allows rational expressions. must be a kernel, and - must be a non-negative integer. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -ff := x**7 + sin(y)*x**5 + y**4 + x + 7; \par - \par - \par - 5 7 4 \par - FF := SIN(Y)*X + X + X + Y + 7 \par - \par - \par -coeffn(ff,x,5); \par - \par - SIN(Y) \par - \par - \par -coeffn(ff,z,3); \par - \par - 0 \par - \par - \par -coeffn(ff,y,0); \par - \par - 5 7 \par - SIN(Y)*X + X + X + 7 \par - \par - \par - \par -rr := 1/y**2+y**3+sin(y); \par - \par - 2 5 \par - SIN(Y)*Y + Y + 1 \par - RR := -------------------- \par - 2 \par - Y \par - \par - \par -on ratarg; \par - \par - \par -coeffn(rr,y,-2); \par - \par - ***** -2 invalid as COEFFN index \par - \par - \par - \par -coeffn(rr,y,5); \par - \par - 1 \par - --- \par - 2 \par - Y \par - \par -\pard \sl240 }{\f2 If the given power of the kernel does not appear in the expression, -}{\f3 coeffn} {\f2 returns 0. Negative powers are never detected, even if -they appear in the expression and } -{\f2\uldb ratarg}{\v\f2 RATARG} -{\f2 are on. }{\f3 coeffn} {\f2 -with an integer argument of 0 returns any terms in the expression that -do not contain the given kernel. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CONJ} - -${\footnote \pard\plain \sl240 \fs20 $ CONJ} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0779} - - K{\footnote \pard\plain \sl240 \fs20 K complex;conjugate;CONJ operator;operator} - -}{\b\f2 CONJ}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 conj} {\f4 () or }{\f3 conj} {\f4 -\par -\par -}{\f2 \par -This operator returns the complex conjugate of an expression, if that -argument has an numerical value. A non-numerical argument is returned as -an expression in the operators } -{\f2\uldb repart}{\v\f2 REPART} -{\f2 and } -{\f2\uldb impart}{\v\f2 IMPART} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -conj(1+i); \par - \par - 1-I \par - \par - \par -conj(a+i*b); \par - \par - REPART(A) - REPART(B)*I - IMPART(A)*I - IMPART(B) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CONTINUED_FRACTION} - -${\footnote \pard\plain \sl240 \fs20 $ CONTINUED_FRACTION} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0780} - - K{\footnote \pard\plain \sl240 \fs20 K rational numbers;approximation;CONTINUED_FRACTION operator;operator} - -}{\b\f2 CONTINUED_FRACTION}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 continued_fraction} {\f4 () -or }{\f3 continued_fraction} {\f4 ( ,) -\par -\par -}{\f2 \par -This operator approximates the real number -( } -{\f2\uldb rational}{\v\f2 RATIONAL} -{\f2 number, } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 number) -into a continued fraction. The result is a list of two elements: the -first one is the rational value of the approximation, the second one -is the list of terms of the continued fraction which represents the -same value according to the definition }{\f3 t0 +1/(t1 + 1/(t2 + ...))} {\f2 . -Precision: the second optional parameter is an upper bound -for the absolute value of the result denominator. If omitted, the -approximation is performed up to the current system precision. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -continued_fraction pi; \par - \par - \par - 1146408 \par - \{-------,\{3,7,15,1,292,1,1,1,2,1\}\} \par - 364913 \par - \par - \par -continued_fraction(pi,100); \par - \par - \par - 22 \par - \{--,\{3,7\}\} \par - 7 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DECOMPOSE} - -${\footnote \pard\plain \sl240 \fs20 $ DECOMPOSE} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0781} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;decomposition;DECOMPOSE operator;operator} - -}{\b\f2 DECOMPOSE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 decompose} {\f2 operator takes a multivariate polynomial as argument, -and returns an expression and a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of -} -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s from which the -original polynomial can be found by composition. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 decompose} {\f4 () or }{\f3 decompose} {\f4 - -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -decompose(x^8-88*x^7+2924*x^6-43912*x^5+263431*x^4- \par - 218900*x^3+65690*x^2-7700*x+234) \par -\pard \sl240 \par - \par - \par - 2 2 2 \par - U + 35*U + 234, U=V + 10*V, V=X - 22*X \par - \par - \par - decompose(u^2+v^2+2u*v+1) \par - \par - 2 \par - W + 1, W=U + V \par - \par -\pard \sl240 }{\f2 Unlike factorization, this decomposition is not unique. Further -details can be found in V.S. Alagar, M.Tanh, , Proc. EUROCAL 1985, pp 150-153 (Springer) and J. von zur -Gathen, -, J. -Symbolic Computation (1990) 9, 281-299. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DEG} - -${\footnote \pard\plain \sl240 \fs20 $ DEG} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0782} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;degree;DEG operator;operator} - -}{\b\f2 DEG}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 deg} {\f2 returns the highest degree of its variable argument -found in its expression argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 deg} {\f4 (,) -\par -\par -}{\f2 \par - is expected to be a polynomial expression, not a rational -expression. Rational expressions are accepted when the switch -} -{\f2\uldb ratarg}{\v\f2 RATARG} -{\f2 is on. must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 . The -results are returned in a list. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -deg((x+y)**5,x); \par - \par - 5 \par - \par - \par - \par -deg((a+b)*(c+2*d)**2,d); \par - \par - 2 \par - \par - \par - \par -deg(x**2 + cos(y),sin(x)); \par - \par - \par -deg((x**2 + sin(x))**5,sin(x)); \par - \par - 5 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DEN} - -${\footnote \pard\plain \sl240 \fs20 $ DEN} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0783} - - K{\footnote \pard\plain \sl240 \fs20 K rational expression;denominator;DEN operator;operator} - -}{\b\f2 DEN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 den} {\f2 operator returns the denominator of its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 den} {\f4 () -\par -\par -}{\f2 \par - is ordinarily a rational expression, but may be any valid -scalar REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -a := x**3 + 3*x**2 + 12*x; \par - \par - 2 \par - A := X*(X + 3*X + 12) \par - \par - \par - \par -b := 4*x*y + x*sin(x); \par - \par - B := X*(SIN(X) + 4*Y) \par - \par - \par - \par -den(a/b); \par - \par - SIN(X) + 4*Y \par - \par - \par - \par -den(aa/4 + bb/5); \par - \par - 20 \par - \par - \par - \par -den(100/6); \par - \par - 3 \par - \par - \par - \par -den(sin(x)); \par - \par - 1 \par - \par -\pard \sl240 }{\f2 }{\f3 den} {\f2 returns the denominator of the expression after it has been -simplified by REDUCE. As seen in the examples, this includes putting -sums of rational expressions over a common denominator, and reducing -common factors where possible. If the expression does not have any -other denominator, 1 is returned. -\par -\par -Switch settings, such as } -{\f2\uldb mcd}{\v\f2 MCD} -{\f2 or } -{\f2\uldb rational}{\v\f2 RATIONAL} -{\f2 , have an -effect on the denominator of an expression. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DF} - -${\footnote \pard\plain \sl240 \fs20 $ DF} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0784} - - K{\footnote \pard\plain \sl240 \fs20 K partial derivative;derivative;DF operator;operator} - -}{\b\f2 DF}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 df} {\f2 operator finds partial derivatives with respect to one or -more variables. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 df} {\f4 (}{\f3 ,} {\f4 - [}{\f3 ,} {\f4 ] - \{}{\f3 ,} {\f4 [ }{\f3 ,} {\f4 ] \}) -\par -\par -}{\f2 \par - can be any valid REDUCE algebraic expression. -must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 , and is the differentiation variable. - must be a non-negative integer. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -df(x**2,x); \par - \par - 2*X \par - \par - \par - \par -df(x**2*y + sin(y),y); \par - \par - 2 \par - COS(Y) + X \par - \par - \par - \par -df((x+y)**10,z); \par - \par - 0 \par - \par - \par - \par - \par -df(1/x**2,x,2); \par - \par - 6 \par - --- \par - 4 \par - X \par - \par - \par - \par -df(x**4*y + sin(y),y,x,3); \par - \par - 24*X \par - \par - \par - \par -for all x let df(tan(x),x) = sec(x)**2; \par - \par - \par -df(tan(3*x),x); \par - \par - 2 \par - 3*SEC(3*X) \par - \par -\pard \sl240 }{\f2 An error message results if a non-kernel is entered as a differentiation -operator. If the optional number is omitted, it is assumed to be 1. -See the declaration } -{\f2\uldb depend}{\v\f2 DEPEND} -{\f2 to establish dependencies for implicit -differentiation. -\par -\par -You can define your own differentiation rules, expanding REDUCE's -capabilities, using the } -{\f2\uldb let}{\v\f2 LET} -{\f2 command as shown in the last example -above. Note that once you add your own rule for differentiating a -function, it supersedes REDUCE's normal handling of that function for the -duration of the REDUCE session. If you clear the rule -(} -{\f2\uldb clearrules}{\v\f2 CLEARRULES} -{\f2 ), you don't get back -to the previous rule. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EXPAND\_CASES} - -${\footnote \pard\plain \sl240 \fs20 $ EXPAND_CASES} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0785} - - K{\footnote \pard\plain \sl240 \fs20 K solve;EXPAND_CASES operator;operator} - -}{\b\f2 EXPAND\_CASES}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -When a } -{\f2\uldb root_of}{\v\f2 ROOT\_OF} -{\f2 form in a result of } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 -has been converted to a } -{\f2\uldb one_of}{\v\f2 ONE\_OF} -{\f2 form, }{\f3 expand_cases} {\f2 -can be used to convert this into form corresponding to the -normal explicit results of } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 . See } -{\f2\uldb root_of}{\v\f2 ROOT\_OF} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EXPREAD} - -${\footnote \pard\plain \sl240 \fs20 $ EXPREAD} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0786} - - K{\footnote \pard\plain \sl240 \fs20 K input;EXPREAD operator;operator} - -}{\b\f2 EXPREAD}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 expread} {\f4 () -\par -\par -}{\f2 \par -}{\f3 expread} {\f2 reads one well-formed expression from the current input -buffer and returns its value. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -expread(); a+b; \par - \par - A + B \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FACTORIZE} - -${\footnote \pard\plain \sl240 \fs20 $ FACTORIZE} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0787} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;factorize;FACTORIZE operator;operator} - -}{\b\f2 FACTORIZE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 factorize} {\f2 operator factors a given expression. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 factorize} {\f4 () -\par -\par -}{\f2 \par - should be a polynomial, otherwise an error will result. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -fff := factorize(x^3 - y^3); \par - \par - 2 2 \par - \{X - Y,X + X*Y + Y \} \par - \par - \par -fac1 := first fff; \par - \par - FAC1 := X - Y \par - \par - \par -factorize(x^15 - 1); \par - \par - \{X - 1, \par - 2 \par - X + X + 1, \par - 4 3 2 \par - X + X + X + X + 1, \par - 8 7 6 5 4 \par - X - X + X - X + X - X + 1\} \par - \par - \par -lastone := part(ws,length ws); \par - \par - 8 7 6 5 4 \par - LASTONE := X - X + X - X + X - X + 1 \par - \par - \par -setmod 2; \par - \par - 1 \par - \par - \par -on modular; \par - \par -factorize(x^15 - 1); \par - \par - \{X + 1, \par - 2 \par - X + X + 1, \par - 4 \par - X + X + 1, \par - 4 3 \par - X + X + 1, \par - 4 3 2 \par - X + X + X + X + 1\} \par - \par -\pard \sl240 }{\f2 The }{\f3 factorize} {\f2 command returns the factors it finds as a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . -You can therefore use the usual list access methods (} -{\f2\uldb first}{\v\f2 FIRST} -{\f2 , -} -{\f2\uldb second}{\v\f2 SECOND} -{\f2 , } -{\f2\uldb third}{\v\f2 THIRD} -{\f2 , } -{\f2\uldb rest}{\v\f2 REST} -{\f2 , } -{\f2\uldb length}{\v\f2 LENGTH} -{\f2 and -} -{\f2\uldb part}{\v\f2 PART} -{\f2 ) to extract the factors. -\par -\par -If the given to }{\f3 factorize} {\f2 is an integer, it will be -factored into its prime components. To factor any integer factor of a -non-numerical expression, the switch } -{\f2\uldb ifactor}{\v\f2 IFACTOR} -{\f2 should be turned on. -Its default is off. } -{\f2\uldb ifactor}{\v\f2 IFACTOR} -{\f2 has effect only when factoring is -explicitly done by }{\f3 factorize} {\f2 , not when factoring is automatically -done with the } -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 switch. If full factorization is not -needed the switch } -{\f2\uldb limitedfactors}{\v\f2 LIMITEDFACTORS} -{\f2 allows you to reduce the -computing time of calls to }{\f3 factorize} {\f2 . -\par -\par -Factoring can be done in a modular domain by calling }{\f3 factorize} {\f2 when -} -{\f2\uldb modular}{\v\f2 MODULAR} -{\f2 is on. You can set the modulus with the } -{\f2\uldb setmod}{\v\f2 SETMOD} -{\f2 -command. The last example above shows factoring modulo 2. -\par -\par -For general comments on factoring, see comments under the switch -} -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # HYPOT} - -${\footnote \pard\plain \sl240 \fs20 $ HYPOT} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0788} - - K{\footnote \pard\plain \sl240 \fs20 K HYPOT operator;operator} - -}{\b\f2 HYPOT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 hypot(,) -\par -\par -}{\f2 \par -If }{\f3 rounded} {\f2 is on, and the two arguments evaluate to numbers, this -operator returns the square root of the sums of the squares of the -arguments in a manner that avoids intermediate overflow. In other cases, -an expression in the original operator is returned. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -hypot(3,4); \par - \par - HYPOT(3,4) \par - \par - \par -on rounded; \par - \par -ws; \par - \par - 5.0 \par - \par - \par -hypot(a,b); \par - \par - HYPOT(A,B) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # IMPART} - -${\footnote \pard\plain \sl240 \fs20 $ IMPART} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0789} - - K{\footnote \pard\plain \sl240 \fs20 K complex;imaginary part;IMPART operator;operator} - -}{\b\f2 IMPART}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 impart} {\f4 () or }{\f3 impart} {\f4 -\par -\par -}{\f2 \par -This operator returns the imaginary part of an expression, if that -argument has an numerical value. A non-numerical argument is returned as -an expression in the operators } -{\f2\uldb repart}{\v\f2 REPART} -{\f2 and }{\f3 impart} {\f2 . - \par -examples: \par -\pard \tx3420 }{\f4 \par -impart(1+i); \par - \par - 1 \par - \par - \par -impart(a+i*b); \par - \par - REPART(B) + IMPART(A) \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INT} - -${\footnote \pard\plain \sl240 \fs20 $ INT} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0790} - - K{\footnote \pard\plain \sl240 \fs20 K integration;INT operator;operator} - -}{\b\f2 INT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 int} {\f2 operator performs analytic integration on a variety of -functions. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 int} {\f4 (,) -\par -\par -}{\f2 \par - can be any scalar expression. involving polynomials, log -functions, exponential functions, or tangent or arctangent expressions. -}{\f3 int} {\f2 attempts expressions involving error functions, dilogarithms -and other trigonometric expressions. Integrals involving algebraic -extensions (such as square roots) may not succeed. must be a -REDUCE } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -int(x**3 + 3,x); \par - \par - 3 \par - X*(X + 12) \par - ----------- \par - 4 \par - \par - \par - \par -int(sin(x)*exp(2*x),x); \par - \par - \par - 2*X \par - E *(COS(X) - 2*SIN(X)) \par - - ------------------------ \par - 5 \par - \par - \par -int(1/(x^2-2),x); \par - \par - \par - SQRT(2)*(LOG( - SQRT(2) + X) - LOG(SQRT(2) + X)) \par - ------------------------------------------------ \par - 4 \par - \par - \par -int(sin(x)/(4 + cos(x)**2),x); \par - \par - \par - COS(X) \par - ATAN(------) \par - 2 \par - - ------------ \par - 2 \par - \par - \par - \par -int(1/sqrt(x^2-x),x); \par - \par - SQRT(X)*SQRT(X - 1) \par - INT(-------------------,X) \par - 2 \par - X -X \par - \par -\pard \sl240 }{\f2 Note that REDUCE couldn't handle the last integral with its default -integrator, since the integrand involves a square root. However, -the integral can be found using the } -{\f2\uldb algint}{\v\f2 ALGINT} -{\f2 package. -Alternatively, you could add a rule using the } -{\f2\uldb let}{\v\f2 LET} -{\f2 statement -to evaluate this integral. -\par -\par -The arbitrary constant of integration is not shown. Definite integrals can -be found by evaluating the result at the limits of integration (use -} -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 ) and subtracting the lower from the higher. Evaluation can -be easily done by the } -{\f2\uldb sub}{\v\f2 SUB} -{\f2 operator. -\par -\par -When }{\f3 int} {\f2 cannot find an integral it returns an expression -involving formal }{\f3 int} {\f2 expressions unless the switch -} -{\f2\uldb failhard}{\v\f2 FAILHARD} -{\f2 has been set. If not all of the expression -can be integrated, the switch } -{\f2\uldb nolnr}{\v\f2 NOLNR} -{\f2 controls whether a partially -integrated result should be returned or not. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INTERPOL} - -${\footnote \pard\plain \sl240 \fs20 $ INTERPOL} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0791} - - K{\footnote \pard\plain \sl240 \fs20 K approximation;polynomial;interpolation;INTERPOL operator;operator} - -}{\b\f2 INTERPOL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -}{\f3 interpol} {\f2 generates an interpolation polynomial. - \par -syntax: \par -}{\f4 \par -\par -interpol(,,) -\par -\par -}{\f2 \par - and are } -{\f2\uldb list}{\v\f2 LIST} -{\f2 s of equal length and - is an algebraic expression (preferably a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 ). -The interpolation polynomial is generated in the given variable of degree -length()-1. The unique polynomial }{\f3 f} {\f2 is defined by the -property that for corresponding elements }{\f3 v} {\f2 of and -}{\f3 p} {\f2 of the relation }{\f3 f(p)=v} {\f2 holds. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -f := for i:=1:4 collect(i**3-1); \par - \par - F := 0,7,26,63 \par - \par - \par -p := \{1,2,3,4\}; \par - \par - P := 1,2,3,4 \par - \par - \par -interpol(f,x,p); \par - \par - 3 \par - X - 1 \par - \par -\pard \sl240 }{\f2 The Aitken-Neville interpolation algorithm is used which guarantees a -stable result even with rounded numbers and an ill-conditioned problem. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LCOF} - -${\footnote \pard\plain \sl240 \fs20 $ LCOF} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0792} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;coefficient;LCOF operator;operator} - -}{\b\f2 LCOF}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 lcof} {\f2 operator returns the leading coefficient of a given expression -with respect to a given variable. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 lcof} {\f4 (,) -\par -\par -}{\f2 \par - is ordinarily a polynomial. If } -{\f2\uldb ratarg}{\v\f2 RATARG} -{\f2 is on, -a rational expression may also be used, otherwise an error results. - must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -lcof((x+2*y)**5,y); \par - \par - 32 \par - \par - \par -lcof((x + y*sin(x))**2 + cos(x)*sin(x)**2,sin(x)); \par - \par - \par - \par - 2 \par - COS(X) + Y \par - \par - \par -lcof(x**2 + 3*x + 17,y); \par - \par - 2 \par - X + 3*X + 17 \par - \par -\pard \sl240 }{\f2 If the kernel does not appear in the expression, }{\f3 lcof} {\f2 returns the -expression. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LENGTH} - -${\footnote \pard\plain \sl240 \fs20 $ LENGTH} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0793} - - K{\footnote \pard\plain \sl240 \fs20 K list;LENGTH operator;operator} - -}{\b\f2 LENGTH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 length} {\f2 operator returns the number of items in a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 , the -number of -terms in an expression, or the dimensions of an array or matrix. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 length} {\f4 () or }{\f3 length} {\f4 -\par -\par -}{\f2 \par - can be a list structure, an array, a matrix, or a scalar expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -alist := \{a,b,\{ww,xx,yy,zz\}\}; \par - \par - ALIST := \{A,B,\{WW,XX,YY,ZZ\}\} \par - \par - \par -length alist; \par - \par - 3 \par - \par - \par -length third alist; \par - \par - 4 \par - \par - \par -dlist := \{d\}; \par - \par - DLIST := \{D\} \par - \par - \par -length rest dlist; \par - \par - 0 \par - \par - \par -matrix mmm(4,5); \par - \par -length mmm; \par - \par - \{4,5\} \par - \par - \par -array aaa(5,3,2); \par - \par -length aaa; \par - \par - \{6,4,3\} \par - \par - \par -eex := (x+3)**2/(x-y); \par - \par - 2 \par - X + 6*X + 9 \par - EEX := ------------ \par - X - Y \par - \par - \par -length eex; \par - \par - 5 \par - \par -\pard \sl240 }{\f2 An item in a list that is itself a list only counts as one item. An error -message will be printed if }{\f3 length} {\f2 is called on a matrix which has -not had its dimensions set. The }{\f3 length} {\f2 of an array includes the -zeroth element of each dimension, showing the full number of elements -allocated. (Declaring an array A with n elements -allocates A(0),A(1),...,A(n).) The -}{\f3 length} {\f2 of an expression is the total number of additive terms -appearing in the numerator and denominator of the expression. Note that -subtraction of a term is represented internally as addition of a negative -term. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LHS} - -${\footnote \pard\plain \sl240 \fs20 $ LHS} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0794} - - K{\footnote \pard\plain \sl240 \fs20 K equation;left-hand side;LHS operator;operator} - -}{\b\f2 LHS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 lhs} {\f2 operator returns the left-hand side of an } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 , -such as those -returned in a list by } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 lhs} {\f4 () or }{\f3 lhs} {\f4 -\par -\par -\par -\par -}{\f2 must be an equation of the form -\par -\par -}{\f3 left-hand side} {\f3 =} {\f3 right-hand side} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -polly := (x+3)*(x^4+2x+1); \par - \par - 5 4 2 \par - POLLY := X + 3*X + 2*X + 7*X + 3 \par - \par - \par -pollyroots := solve(polly,x); \par - \par - POLLYROOTS := \{X=ROOT F(X3 - X2 + X + 1,X , \par - O ) \par - X=-1, \par - X=-3\} \par - \par - \par -variable := lhs first pollyroots; \par - \par - VARIABLE := X \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LIMIT} - -${\footnote \pard\plain \sl240 \fs20 $ LIMIT} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0795} - - K{\footnote \pard\plain \sl240 \fs20 K l'Hopital's rule;limit;LIMIT operator;operator} - -}{\b\f2 LIMIT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -LIMITS is a fast limit package for REDUCE for functions which are -continuous except for computable poles and singularities, based on -some earlier work by Ian Cohen and John P. Fitch. The Truncated -Power Series package is used for non-critical points, at which -the value of the function is the constant term in the expansion -around that point. l'Hopital's rule is used in critical cases, -with preprocessing of 1-1 forms and reformatting of product forms -in order to apply l'Hopital's rule. A limited amount of bounded -arithmetic is also employed where applicable. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 limit} {\f4 (,,) or -\par -\par -}{\f3 limit!+} {\f4 (,,) or -\par -\par -}{\f3 limit!-} {\f4 (,,) -\par -\par -}{\f2 \par -where is an expression depending of the variable -(a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 ) and is the limit point. -If the limit depends upon the direction of approach to the , -the operators }{\f3 limit!+} {\f2 and }{\f3 limit!-} {\f2 may be used. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -limit(x*cot(x),x,0); \par - \par - 0 \par - \par - \par -limit((2x+5)/(3x-2),x,infinity); \par - \par - 2 \par - -- \par - 3 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LPOWER} - -${\footnote \pard\plain \sl240 \fs20 $ LPOWER} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0796} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;leading power;LPOWER operator;operator} - -}{\b\f2 LPOWER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 lpower} {\f2 operator returns the leading power of an expression with -respect to a kernel. 1 is returned if the expression does not depend on -the kernel. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 lpower} {\f4 (,) -\par -\par -}{\f2 \par - is ordinarily a polynomial. If } -{\f2\uldb ratarg}{\v\f2 RATARG} -{\f2 is on, -a rational expression may also be used, otherwise an error results. - must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -lpower((x+2*y)**6,y); \par - \par - 6 \par - Y \par - \par - \par -lpower((x + cos(x))**8 + df(x**2,x),cos(x)); \par - \par - \par - \par - 8 \par - COS(X) \par - \par - \par -lpower(x**3 + 3*x,y); \par - \par - 1 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LTERM} - -${\footnote \pard\plain \sl240 \fs20 $ LTERM} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0797} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;leading term;LTERM operator;operator} - -}{\b\f2 LTERM}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 lterm} {\f2 operator returns the leading term of an expression with -respect to a kernel. The expression is returned if it does not depend on -the kernel. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 lterm} {\f4 (,) -\par -\par -}{\f2 \par - is ordinarily a polynomial. If } -{\f2\uldb ratarg}{\v\f2 RATARG} -{\f2 is on, -a rational expression may also be used, otherwise an error results. - must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -lterm((x+2*y)**6,y); \par - \par - 6 \par - 64*Y \par - \par - \par -lterm((x + cos(x))**8 + df(x**2,x),cos(x)); \par - \par - \par - \par - 8 \par - COS(X) \par - \par - \par -lterm(x**3 + 3*x,y); \par - \par - 3 \par - X + 3X \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MAINVAR} - -${\footnote \pard\plain \sl240 \fs20 $ MAINVAR} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0798} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;main variable;MAINVAR operator;operator} - -}{\b\f2 MAINVAR}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 mainvar} {\f2 operator returns the main variable (in the system's -internal representation) of its argument. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 mainvar} {\f4 () -\par -\par -\par -\par -}{\f2 is usually a polynomial, but may be any valid REDUCE -scalar expression. In the case of a rational function, the main variable -of the numerator is returned. The main variable returned is a -} -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -test := (a + b + c)**2; \par - \par - 2 2 2 \par - TEST := A + 2*A*B + 2*A*C + B + 2*B*C + C \par - \par - \par -mainvar(test); \par - \par - A \par - \par - \par -korder c,b,a; \par - \par -mainvar(test); \par - \par - C \par - \par - \par -mainvar(2*cos(x)**2); \par - \par - COS(X) \par - \par - \par -mainvar(17); \par - \par - 0 \par - \par -\pard \sl240 }{\f2 The main variable is the first variable in the canonical ordering of -kernels. Generally, alphabetically ordered functions come first, then -alphabetically ordered identifiers (variables). Numbers come last, and as -far as }{\f3 mainvar} {\f2 is concerned belong in the family }{\f3 0} {\f2 . The -canonical ordering can be changed by the declaration } -{\f2\uldb korder}{\v\f2 KORDER} -{\f2 , as -shown above. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MAP} - -${\footnote \pard\plain \sl240 \fs20 $ MAP} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0799} - - K{\footnote \pard\plain \sl240 \fs20 K composite structure;map;MAP operator;operator} - -}{\b\f2 MAP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 map} {\f2 operator applies a uniform evaluation pattern -to all members of a composite structure: a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 , -a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 or the arguments of an } -{\f2\uldb operator}{\v\f2 OPERATOR} -{\f2 expression. -The evaluation pattern can be a -unary procedure, an operator, or an algebraic expression with -one free variable. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 map} {\f4 (,) -\par -\par -}{\f2 \par - is a list, a matrix or an operator expression. -\par -\par - is -the name of an operator for a single argument: the operator - is evaluated once with each element of as its single argument, -\par -\par -or an algebraic expression with exactly one } -{\f2\uldb free variable}{\v\f2 Free_Variable} -{\f2 , that is -a variable preceded by the tilde symbol: the expression - is evaluated for each element of where the element is - substituted for the free variable, -\par -\par -or a replacement } -{\f2\uldb rule}{\v\f2 RULE} -{\f2 of the form - \par -syntax: \par -}{\f4 \par -\par -}{\f3 var} {\f4 => }{\f3 rep} {\f4 -\par -\par -}{\f2 \par -where is a variable (a without subscript) - and is an expression which contains . - Here }{\f3 rep} {\f2 is evaluated for each element of where - the element is substituted for }{\f3 var} {\f2 . }{\f3 var} {\f2 may be - optionally preceded by a tilde. -\par -\par -The rule form for is needed when more than -one free variable occurs. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -map(abs,\{1,-2,a,-a\}); \par - \par - 1,2,abs(a),abs(a) \par - \par - \par -map(int(~w,x), mat((x^2,x^5),(x^4,x^5))); \par - \par - \par - [ 3 6 ] \par - [ x x ] \par - [---- ----] \par - [ 3 6 ] \par - [ ] \par - [ 5 6 ] \par - [ x x ] \par - [---- ----] \par - [ 5 6 ] \par - \par - \par -map(~w*6, x^2/3 = y^3/2 -1); \par - \par - 2 3 \par - 2*x =3*(y -2) \par - \par -\pard \sl240 }{\f2 You can use }{\f3 map} {\f2 in nested expressions. It is not allowed to -apply }{\f3 map} {\f2 for a non-composed object, e.g. an identifier or a number. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MKID} - -${\footnote \pard\plain \sl240 \fs20 $ MKID} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0800} - - K{\footnote \pard\plain \sl240 \fs20 K identifier;MKID command;command} - -}{\b\f2 MKID}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 mkid} {\f2 command constructs an identifier, given a stem and an identifier -or an integer. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 mkid} {\f4 (,) -\par -\par -}{\f2 \par - can be any valid REDUCE identifier that does not include escaped -special characters. may be an integer, including one given by a -local variable in a } -{\f2\uldb for}{\v\f2 FOR} -{\f2 loop, or any other legal group of -characters. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -mkid(x,3); \par - \par - X3 \par - \par - \par -factorize(x^15 - 1); \par - \par - \{X - 1, \par - 2 \par - X + X + 1, \par - 4 3 2 \par - X + X + X + X + 1, \par - 8 7 5 4 3 \par - X - X + X - X + X - X + 1\} \par - \par - \par - \par -for i := 1:length ws do write set(mkid(f,i),part(ws,i)); \par - \par - \par - \par - 8 7 5 4 3 \par - X - X + X - X + X - X + 1 \par - 4 3 2 \par - X + X + X + X + 1 \par - 2 \par - X + X + 1 \par - X - 1 \par - \par -\pard \sl240 }{\f2 You can use }{\f3 mkid} {\f2 to construct identifiers from inside procedures. This -allows you to handle an unknown number of factors, or deal with variable -amounts of data. It is particularly helpful to attach identifiers to the -answers returned by }{\f3 factorize} {\f2 and }{\f3 solve} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NPRIMITIVE} - -${\footnote \pard\plain \sl240 \fs20 $ NPRIMITIVE} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0801} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;primitive part;NPRIMITIVE operator;operator} - -}{\b\f2 NPRIMITIVE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 nprimitive} {\f4 () or }{\f3 nprimitive} {\f4 - -\par -\par -}{\f2 \par -This operator returns the numerically-primitive part of any scalar -expression. In other words, any overall integer factors in the expression -are removed. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -nprimitive((2x+2y)^2); \par - \par - 2 2 \par - X + 2*X*Y + Y \par - \par - \par -nprimitive(3*a*b*c); \par - \par - 3*A*B*C \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NUM} - -${\footnote \pard\plain \sl240 \fs20 $ NUM} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0802} - - K{\footnote \pard\plain \sl240 \fs20 K rational expression;numerator;NUM operator;operator} - -}{\b\f2 NUM}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 num} {\f2 operator returns the numerator of its argument. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 num} {\f4 () or }{\f3 num} {\f4 -\par -\par -}{\f2 \par - can be any valid REDUCE scalar expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -num(100/6); \par - \par - 50 \par - \par - \par -num(a/5 + b/6); \par - \par - 6*A + 5*B \par - \par - \par -num(sin(x)); \par - \par - SIN(X) \par - \par -\pard \sl240 }{\f2 }{\f3 num} {\f2 returns the numerator of the expression after it has been simplified -by REDUCE. As seen in the examples, this includes putting sums of rational -expressions over a common denominator, and reducing common factors where -possible. If the expression is not a rational expression, it is returned -unchanged. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ODESOLVE} - -${\footnote \pard\plain \sl240 \fs20 $ ODESOLVE} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0803} - - K{\footnote \pard\plain \sl240 \fs20 K solve;differential equation;ODESOLVE operator;operator} - -}{\b\f2 ODESOLVE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 odesolve} {\f2 package is a solver for ordinary differential -equations. At the present time it has still limited capabilities: -\par -\par -1. it can handle only a single scalar equation presented as an - algebraic expression or equation, and -\par -\par -2. it can solve only first-order equations of simple types, linear - equations with constant coefficients and Euler equations. -\par -\par -These solvable types are exactly those for which Lie symmetry -techniques give no useful information. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 odesolve} {\f4 (,,) -\par -\par -\par -\par -}{\f2 is a single scalar expression such that =0 -is the ordinary differential equation (ODE for short) to be solved, or -is an equivalent } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 . -\par -\par - is the name of the dependent variable, - is the name of the independent variable. -\par -\par -A differential in is expressed using the } -{\f2\uldb df}{\v\f2 DF} -{\f2 -operator. Note that in most cases you must declare explicitly - to depend of using a } -{\f2\uldb depend}{\v\f2 DEPEND} -{\f2 -declaration -- otherwise the derivative might be evaluated to -zero on input to }{\f3 odesolve} {\f2 . -\par -\par -The returned value is a list containing the equation giving the general -solution of the ODE (for simultaneous equations this will be a -list of equations eventually). It will contain occurrences of -the operator }{\f3 arbconst} {\f2 for the arbitrary constants in the general -solution. The arguments of }{\f3 arbconst} {\f2 should be new. -A counter }{\f3 !!arbconst} {\f2 is used to arrange this. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -depend y,x; \par - \par -\% A first-order linear equation, with an initial condition \par - \par -ode:=df(y,x) + y * sin x/cos x - 1/cos x$ \par - \par -odesolve(ode,y,x); \par - \par - \{y=arbconst(1)*cos(x) + sin(x)\} \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ONE\_OF} - -${\footnote \pard\plain \sl240 \fs20 $ ONE_OF} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0804} - - K{\footnote \pard\plain \sl240 \fs20 K ONE_OF type;type} - -}{\b\f2 ONE\_OF}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par -\par - -The operator }{\f3 one_of} {\f2 is used to represent an indefinite choice -of one element from a finite set of objects. - \par -examples: \par -\pard \tx3420 }{\f4 \par -x=one_of\{1,2,5\} \par -\pard \sl240 }{\f2 this equation encodes that x can take one of the values -1,2 or 5}{\f4 \pard \tx3420 \par -\pard \sl240 }{\f2 -REDUCE generates a }{\f3 one_of} {\f2 form in cases when an implicit -}{\f3 root_of} {\f2 expression could be converted to an explicit solution set. -A }{\f3 one_of} {\f2 form can be converted to a }{\f3 solve} {\f2 solution using -} -{\f2\uldb expand_cases}{\v\f2 EXPAND\_CASES} -{\f2 . See } -{\f2\uldb root_of}{\v\f2 ROOT\_OF} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PART} - -${\footnote \pard\plain \sl240 \fs20 $ PART} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0805} - - K{\footnote \pard\plain \sl240 \fs20 K decomposition;PART operator;operator} - -}{\b\f2 PART}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 part} {\f2 permits the extraction of various parts or -operators of expressions and } -{\f2\uldb list}{\v\f2 LIST} -{\f3 s} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 part} {\f4 (\{,\}*) -\par -\par -}{\f2 \par - can be any valid REDUCE expression or a list, -integer may be an expression that evaluates to a positive or negative -integer or 0. A positive integer picks up the n th term, -counting from the first term toward the end. A negative integer n -picks up the n th term, counting from the back toward the front. The -integer 0 picks up the operator (which is }{\f3 LIST} {\f2 when the expression -is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 ). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -part((x + y)**5,4); \par - \par - 2 3 \par - 10*X *Y \par - \par - \par -part((x + y)**5,4,2); \par - \par - 2 \par - X \par - \par - \par -part((x + y)**5,4,2,1); \par - \par - X \par - \par - \par -part((x + y)**5,0); \par - \par - PLUS \par - \par - \par -part((x + y)**5,-5); \par - \par - 4 \par - 5*X *Y \par - \par - \par -part((x + y)**5,4) := sin(x); \par - \par - 5 4 3 2 4 5 \par - X + 5*X *Y + 10*X *Y + SIN(X) + 5*X*Y + Y \par - \par - \par -alist := \{x,y,\{aa,bb,cc\},x**2*sqrt(y)\}; \par - \par - \par - 2 \par - ALIST := \{X,Y,\{AA,BB,CC\},SQRT(Y)*X \} \par - \par - \par -part(alist,3,2); \par - \par - BB \par - \par - \par -part(alist,4,0); \par - \par - TIMES \par - \par -\pard \sl240 }{\f2 Additional integer arguments after the first one examine the -terms recursively, as shown above. In the third line, the fourth term -is picked from the original polynomial, 10x^2y^3, -then the second term from that, x^2, and finally the first -component, x. If an integer's absolute value is too large for -the appropriate expression, a message is given. -\par -\par -}{\f3 part} {\f2 works on the form of the expression as printed, or as it would -have been printed at that point of the calculation, bearing in mind the -current switch settings. It is important to realize that the switch settings -change the operation of }{\f3 part} {\f2 . } -{\f2\uldb pri}{\v\f2 PRI} -{\f2 must be on when -}{\f3 part} {\f2 is used. -\par -\par -When }{\f3 part} {\f2 is used on a polynomial expression that has minus signs, the -}{\f3 +} {\f2 is always returned as the top-level operator. The minus is found -as a unary operator attached to the negative term. -\par -\par -}{\f3 part} {\f2 can also be used to change the relevant part of the expression or -list as shown in the sixth example line. The }{\f3 part} {\f2 operator returns the -changed expression, though original expression is not changed. You can -also use }{\f3 part} {\f2 to change the operator. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PF} - -${\footnote \pard\plain \sl240 \fs20 $ PF} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0806} - - K{\footnote \pard\plain \sl240 \fs20 K rational expression;partial fraction;PF operator;operator} - -}{\b\f2 PF}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 pf(,) -\par -\par -}{\f2 \par -}{\f3 pf} {\f2 transforms into a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of partial fraction -s -with respect to the main variable, . }{\f3 pf} {\f2 does a -complete partial fraction decomposition, and as the algorithms used are -fairly unsophisticated (factorization and the extended Euclidean -algorithm), the code may be unacceptably slow in complicated cases. - \par -examples: \par -\pard \tx3420 }{\f4 \par -pf(2/((x+1)^2*(x+2)),x); \par - \par - 2 -2 2 \par - \{-----,-----,------------\} \par - X + 2 X + 1 2 \par - X + 2*X + 1 \par - \par - \par -off exp; \par - \par -pf(2/((x+1)^2*(x+2)),x); \par - \par - \par - 2 - 2 2 \par - \{-----,-----,--------\} \par - X + 2 X + 1 2 \par - (X + 1) \par - \par - \par -for each j in ws sum j; \par - \par - 2 \par - ---------------- \par - 2 \par - ( + 2)*(X + 1) \par - \par -\pard \sl240 }{\f2 \par -\par -If you want the denominators in factored form, turn } -{\f2\uldb exp}{\v\f2 EXP} -{\f2 off, as -shown in the second example above. As shown in the final example, the -} -{\f2\uldb for}{\v\f2 FOR} -{\f2 }{\f3 each} {\f2 construct can be used to recombine the terms. -Alternatively, one can use the operations on lists to extract any desired -term. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PROD} - -${\footnote \pard\plain \sl240 \fs20 $ PROD} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0807} - - K{\footnote \pard\plain \sl240 \fs20 K product;Gosper algorithm;PROD operator;operator} - -}{\b\f2 PROD}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 prod} {\f2 returns -the indefinite or definite product of a given expression. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 prod} {\f4 (,[, [, ]]) -\par -\par -\par -\par -}{\f2 where is the expression to be multiplied, is the -control variable (a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 ), and and -uplim are the optional lower and upper limits. If is -not supplied the upper limit is taken as . The -Gosper algorithm is used. If there is no closed form solution, -the operator returns the input unchanged. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -prod(k/(k-2),k); \par - \par - k*( - k + 1) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # REDUCT} - -${\footnote \pard\plain \sl240 \fs20 $ REDUCT} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0808} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;reductum;REDUCT operator;operator} - -}{\b\f2 REDUCT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 reduct} {\f2 operator returns the remainder of its expression after the -leading term with respect to the kernel in the second argument is removed. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 reduct} {\f4 (,) -\par -\par -}{\f2 \par - is ordinarily a polynomial. If } -{\f2\uldb ratarg}{\v\f2 RATARG} -{\f2 is on, -a rational expression may also be used, otherwise an error results. - must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -reduct((x+y)**3,x); \par - \par - 2 2 \par - Y*(3*X + 3*X*Y + Y ) \par - \par - \par -reduct(x + sin(x)**3,sin(x)); \par - \par - X \par - \par - \par -reduct(x + sin(x)**3,y); \par - \par - 0 \par - \par -\pard \sl240 }{\f2 If the expression does not contain the kernel, }{\f3 reduct} {\f2 returns 0. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # REPART} - -${\footnote \pard\plain \sl240 \fs20 $ REPART} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0809} - - K{\footnote \pard\plain \sl240 \fs20 K complex;real part;REPART operator;operator} - -}{\b\f2 REPART}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 repart} {\f4 () or }{\f3 repart} {\f4 -\par -\par -}{\f2 \par -This operator returns the real part of an expression, if that argument has an -numerical value. A non-numerical argument is returned as an expression in -the operators }{\f3 repart} {\f2 and } -{\f2\uldb impart}{\v\f2 IMPART} -{\f2 . - \par -examples: \par -\pard \tx3420 }{\f4 \par -repart(1+i); \par - \par - 1 \par - \par - \par -repart(a+i*b); \par - \par - REPART(A) - IMPART(B) \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RESULTANT} - -${\footnote \pard\plain \sl240 \fs20 $ RESULTANT} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0810} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;RESULTANT operator;operator} - -}{\b\f2 RESULTANT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 resultant} {\f2 operator computes the resultant of two polynomials with -respect to a given variable. If the resultant is 0, the polynomials have -a root in common. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 resultant} {\f4 (,,) -\par -\par -}{\f2 \par - must be a polynomial containing ; - must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -resultant(x**2 + 2*x + 1,x+1,x); \par - \par - 0 \par - \par - \par -resultant(x**2 + 2*x + 1,x-3,x); \par - \par - 16 \par - \par - \par -resultant(z**3 + z**2 + 5*z + 5, \par - z**4 - 6*z**3 + 16*z**2 - 30*z + 55, \par - z); \par -\pard \sl240 \par - \par - 0 \par - \par - \par -resultant(x**3*y + 4*x*y + 10,y**2 + 6*y + 4,y); \par - \par - \par - 6 5 4 3 2 \par - Y + 18*Y + 120*Y + 360*Y + 480*Y + 288*Y + 64 \par - \par -\pard \sl240 }{\f2 The resultant is the determinant of the Sylvester matrix, formed from the -coefficients of the two polynomials in the following way: -\par -\par -Given two polynomials: -\par -\par -\pard \tx3420 }{\f4 \par - n n-1 \par - a x + a1 x + ... + an \par - \par -\pard \sl240 }{\f2 and -\par -\par -\pard \tx3420 }{\f4 \par - m m-1 \par - b x + b1 x + ... + bm \par - \par -\pard \sl240 }{\f2 form the (m+n)x(m+n-1) Sylvester matrix by the following means: -\par -\par -\pard \tx3420 }{\f4 \par - 0.......0 a a1 .......... an \par - 0....0 a a1 .......... an 0 \par - . . . . \par - a0 a1 .......... an 0.......0 \par - 0.......0 b b1 .......... bm \par - 0....0 b b1 .......... bm 0 \par - . . . . \par - b b1 .......... bm 0.......0 \par - \par -\pard \sl240 }{\f2 If the determinant of this matrix is 0, the two polynomials have a common -root. Finding the resultant of large expressions is time-consuming, due -to the time needed to find a large determinant. -\par -\par -The sign conventions }{\f3 resultant} {\f2 uses are those given in the article, -``Computing in Algebraic Extensions,'' by R. Loos, appearing in -, 2nd ed., -edited by B. Buchberger, G.E. Collins and R. Loos, and published by -Springer-Verlag, 1983. -These are: -\par -\par -\pard \tx3420 }{\f4 \par - resultant(p(x),q(x),x) = (-1)^\{deg p(x)*deg q(x)\} * resultant(q(x),p(x),x), \par - resultant(a,p(x),x) = a^\{deg p(x)\}, \par - resultant(a,b,x) = 1 \par -\pard \sl240 }{\f2 where p(x) and q(x) are polynomials which have x as a variable, and -a and b are free of x. -\par -\par -Error messages are given if }{\f3 resultant} {\f2 is given a non-polynomial -expression, or a non-kernel variable. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RHS} - -${\footnote \pard\plain \sl240 \fs20 $ RHS} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0811} - - K{\footnote \pard\plain \sl240 \fs20 K equation;right-hand side;RHS operator;operator} - -}{\b\f2 RHS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 rhs} {\f2 operator returns the right-hand side of an } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 , -such as those returned in a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 by } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 rhs} {\f4 () or }{\f3 rhs} {\f4 <\{equation> -\par -\par -}{\f2 \par - must be an equation of the form left-hand side = right-hand -side. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -roots := solve(x**2 + 6*x*y + 5x + 3y**2,x); \par - \par - \par - 2 \par - SQRT(24*Y + 60*Y + 25) + 6*Y + 5 \par - ROOTS := \{X= - ---------------------------------, \par - 2 \par - 2 \par - SQRT(24*Y + 60*Y + 25) - 6*Y - 5 \par - X= ---------------------------------\} \par - 2 \par - \par - \par -root1 := rhs first roots; \par - \par - 2 \par - SQRT(24*Y + 60*Y + 25) + 6*Y + 5 \par - ROOT1 := - --------------------------------- \par - 2 \par - \par - \par -root2 := rhs second roots; \par - \par - 2 \par - SQRT(24*Y + 60*Y + 25) - 6*Y - 5 \par - ROOT2 := ---------------------------------- \par - 2 \par - \par -\pard \sl240 }{\f2 An error message is given if }{\f3 rhs} {\f2 is applied to something other than an -equation. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROOT\_OF} - -${\footnote \pard\plain \sl240 \fs20 $ ROOT_OF} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0812} - - K{\footnote \pard\plain \sl240 \fs20 K solve;roots;ROOT_OF operator;operator} - -}{\b\f2 ROOT\_OF}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -When the operator } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 is unable to find an explicit solution -or if that solution would be too complicated, the result is presented -as formal root expression using the internal operator }{\f3 root_of} {\f2 -and a new local variable. An expression with a top level }{\f3 root_of} {\f2 -is implicitly a list with an unknown number of elements since we -can't always know how many solutions an equation has. If a -substitution is made into such an expression, closed form solutions -can emerge. If this occurs, the }{\f3 root_of} {\f2 construct is -replaced by an operator } -{\f2\uldb one_of}{\v\f2 ONE\_OF} -{\f2 . At this point it is -of course possible to transform the result if the original }{\f3 solve} {\f2 -operator expression into a standard }{\f3 solve} {\f2 solution. To -effect this, the operator } -{\f2\uldb expand_cases}{\v\f2 EXPAND\_CASES} -{\f2 can be used. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -solve(a*x^7-x^2+1,x); \par - \par - 7 2 \par - \{x=root_of(a*x_ - x_ + 1,x_)\} \par - \par - \par -sub(a=0,ws); \par - \par - \{x=one_of(1,-1)\} \par - \par - \par -expand_cases ws; \par - \par - x=1,x=-1 \par - \par -\pard \sl240 }{\f2 The components of }{\f3 root_of} {\f2 and }{\f3 one_of} {\f2 expressions can be -processed as usual with operators } -{\f2\uldb arglength}{\v\f2 ARGLENGTH} -{\f2 and } -{\f2\uldb part}{\v\f2 PART} -{\f2 . -A higher power of a }{\f3 root_of} {\f2 expression with a polynomial -as first argument is simplified by using the polynomial as a side relation. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SELECT} - -${\footnote \pard\plain \sl240 \fs20 $ SELECT} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0813} - - K{\footnote \pard\plain \sl240 \fs20 K list;map;SELECT operator;operator} - -}{\b\f2 SELECT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 select} {\f2 operator extracts from a list -or from the arguments of an n--ary operator elements corresponding -to a boolean predicate. The predicate pattern can be a -unary procedure, an operator or an algebraic expression with -one } -{\f2\uldb free variable}{\v\f2 Free_Variable} -{\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 select} {\f4 (,) -\par -\par -}{\f2 \par - is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . -\par -\par - is -the name of an operator for a single argument: the operator - is evaluated once with each element of as its single argument, -\par -\par -or an algebraic expression with exactly one } -{\f2\uldb free variable}{\v\f2 Free_Variable} -{\f2 , that is -a variable preceded by the tilde symbol: the expression - is evaluated for each element of where the element is - substituted for the free variable, -\par -\par -or a replacement } -{\f2\uldb rule}{\v\f2 RULE} -{\f2 of the form - \par -syntax: \par -}{\f4 \par -\par -}{\f3 var} {\f4 => }{\f3 rep} {\f4 -\par -\par -}{\f2 \par -where is a variable (a without subscript) - and is an expression which contains . - Here }{\f3 rep} {\f2 is evaluated for each element of where - the element is substituted for }{\f3 var} {\f2 . }{\f3 var} {\f2 may be - optionally preceded by a tilde. -\par -\par -The rule form for is needed when more than -one free variable occurs. The evaluation result of is -interpreted as } -{\f2\uldb boolean value}{\v\f2 boolean_value} -{\f2 corresponding to the conventions of -REDUCE. The result value is built with the leading operator of the -input expression. - \par -examples: \par -\pard \tx3420 }{\f4 \par - select( ~w>0 , \{1,-1,2,-3,3\}) \par - \par - \{1,2,3\} \par - \par - \par - q:=(part((x+y)^5,0):=list) \par - \par - select(evenp deg(~w,y),q); \par - \par - 5 3 2 4 \par - \{x ,10*x *y ,5*x*y \} \par - \par - \par - select(evenp deg(~w,x),2x^2+3x^3+4x^4); \par - \par - \par - 2 4 \par - 2x +4x \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SHOWRULES} - -${\footnote \pard\plain \sl240 \fs20 $ SHOWRULES} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0814} - - K{\footnote \pard\plain \sl240 \fs20 K output;rule;SHOWRULES operator;operator} - -}{\b\f2 SHOWRULES}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 showrules} {\f4 () or - }{\f3 showrules} {\f4 -\par -\par -}{\f2 \par -}{\f3 showrules} {\f2 returns in } -{\f2\uldb rule}{\v\f2 RULE} -{\f3 -list} {\f2 form any -} -{\f2\uldb operator}{\v\f2 OPERATOR} -{\f2 rules associated with its argument. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -showrules log; \par - \par - \{LOG(E) => 1, \par - LOG(1) => 0, \par - ~X \par - LOG(E ) => ~X, \par - 1 \par - DF(LOG(~X),~X) => --\} \par - ~X \par - \par -\pard \sl240 }{\f2 Such rules can then be manipulated further as with any } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . For -example -}{\f3 rhs first ws;} {\f2 has the value 1. -\par -\par -An operator may have properties that cannot be displayed in such a form, -such as the fact it is an } -{\f2\uldb odd}{\v\f2 ODD} -{\f2 function, or has a definition defined -as a procedure. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SOLVE} - -${\footnote \pard\plain \sl240 \fs20 $ SOLVE} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0815} - - K{\footnote \pard\plain \sl240 \fs20 K solve;root;equation system;equation solving;equation;SOLVE operator;operator} - -}{\b\f2 SOLVE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 solve} {\f2 operator solves a single algebraic } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 or a -system of simultaneous equations. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 solve} {\f4 ( [ , ]) or -\par -\par -}{\f3 solve} {\f4 (\{,...\} [ ,\{ ,...\}] ) -\par -\par -}{\f2 \par -\par -If the number of equations equals the number of distinct kernels, the -optional kernel argument(s) may be omitted. is either a -scalar expression or an } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 . -When more than one expression is given, -the } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of expressions is surrounded by curly braces. -The optional list -of } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 s follows, also in curly braces. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -sss := solve(x^2 + 7); \par - \par - Unknown: X \par - SSS := \{X= - SQRT(7)*I, \par - X=SQRT(7)*I\} \par - \par - \par -rhs first sss; \par - \par - - SQRT(7)*I \par - \par - \par -solve(sin(x^2*y),y); \par - \par - 2*ARBINT(1)*PI \par - \{Y=--------------- \par - 2 \par - X \par - PI*(2*ARBINT(1) + 1) \par - Y=--------------------\} \par - 2 \par - X \par - \par - \par -off allbranch; \par - \par -solve(sin(x**2*y),y); \par - \par - \{Y=0\} \par - \par - \par -solve(\{3x + 5y = -4,2*x + y = -10\},\{x,y\}); \par - \par - \par - \par - 22 46 \par - \{\{X= - --,Y=--\}\} \par - 7 7 \par - \par - \par -solve(\{x + a*y + z,2x + 5\},\{x,y\}); \par - \par - \par - \par - 5 2*Z - 5 \par - \{\{X= - -,Y= - -------\}\} \par - 2 2*A \par - \par - \par -ab := (x+2)^2*(x^6 + 17x + 1); \par - \par - \par - 8 7 6 3 2 \par - AB := X + 4*X + 4*X + 17*X + 69*X + 72*X + 4 \par - \par - \par -www := solve(ab,x); \par - \par - \{X=ROOT F(X6 + 17*X + 1),X=-2\} \par - O \par - \par - \par -root_multiplicities; \par - \par - \{1,2\} \par - \par -\pard \sl240 }{\f2 Results of the }{\f3 solve} {\f2 operator are returned as } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f3 s} {\f2 -in a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . -You can use the usual list access methods (} -{\f2\uldb first}{\v\f2 FIRST} -{\f2 , -} -{\f2\uldb second}{\v\f2 SECOND} -{\f2 , } -{\f2\uldb third}{\v\f2 THIRD} -{\f2 , } -{\f2\uldb rest}{\v\f2 REST} -{\f2 and } -{\f2\uldb part}{\v\f2 PART} -{\f2 ) to -extract the desired equation, and then use the operators } -{\f2\uldb rhs}{\v\f2 RHS} -{\f2 and -} -{\f2\uldb lhs}{\v\f2 LHS} -{\f2 to access the right-hand or left-hand expression of the -equation. When }{\f3 solve} {\f2 is unable to solve an equation, it returns the -unsolved part as the argument of }{\f3 root_of} {\f2 , with the variable renamed -to avoid confusion, as shown in the last example above. -\par -\par -For one equation, }{\f3 solve} {\f2 uses square-free factorization, roots of -unity, and the known inverses of the } -{\f2\uldb log}{\v\f2 LOG} -{\f2 , } -{\f2\uldb sin}{\v\f2 SIN} -{\f2 , -} -{\f2\uldb cos}{\v\f2 COS} -{\f2 , } -{\f2\uldb acos}{\v\f2 ACOS} -{\f2 , } -{\f2\uldb asin}{\v\f2 ASIN} -{\f2 , and -exponentiation operators. The quadratic, cubic and quartic formulas are -used if necessary, but these are applied only when the switch -} -{\f2\uldb fullroots}{\v\f2 FULLROOTS} -{\f2 is set on; otherwise or when no closed form is available -the result is returned as -} -{\f2\uldb root_of}{\v\f2 ROOT\_OF} -{\f2 expression. The switch } -{\f2\uldb trigform}{\v\f2 TRIGFORM} -{\f2 -determines which type of cubic and quartic formula is used. -The multiplicity of each solution is given in a list as -the system variable } -{\f2\uldb root_multiplicities}{\v\f2 ROOT\_MULTIPLICITIES} -{\f2 . For systems of -simultaneous linear equations, matrix inversion is used. For nonlinear -systems, the Groebner basis method is used. -\par -\par -Linear equation system solving is influenced by the switch } -{\f2\uldb cramer}{\v\f2 CRAMER} -{\f2 . -\par -\par -Singular systems can be solved when the switch } -{\f2\uldb solvesingular}{\v\f2 SOLVESINGULAR} -{\f2 is -on, which is the default setting. An empty list is returned the system of -equations is inconsistent. For a linear inconsistent system with parameters -the variable } -{\f2\uldb requirements}{\v\f2 requirements} -{\f2 constraints -conditions for the system to become consistent. -\par -\par -For a solvable linear and polynomial system with parameters -the variable } -{\f2\uldb assumptions}{\v\f2 assumptions} -{\f2 -contains a list side relations for the parameters: the solution is -valid only as long as none of these expressions is zero. -\par -\par -If the switch } -{\f2\uldb varopt}{\v\f2 VAROPT} -{\f2 is on (default), the system rearranges the -variable sequence for minimal computation time. Without }{\f3 varopt} {\f2 -the user supplied variable sequence is maintained. -\par -\par -If the solution has free variables (dimension of the solution is greater -than zero), these are represented by } -{\f2\uldb arbcomplex}{\v\f2 ARBCOMPLEX} -{\f2 expressions -as long as the switch } -{\f2\uldb arbvars}{\v\f2 ARBVARS} -{\f2 is on (default). Without -}{\f3 arbvars} {\f2 no explicit equations are generated for free variables. -\par -\par -\par - \par -related: \par -\par -\tab } -{\f2\uldb allbranch}{\v\f2 ALLBRANCH} -{\f2 switch -\par -\tab } -{\f2\uldb arbvars}{\v\f2 ARBVARS} -{\f2 switch -\par -\tab } -{\f2\uldb assumptions}{\v\f2 assumptions} -{\f2 variable -\par -\tab } -{\f2\uldb fullroots}{\v\f2 FULLROOTS} -{\f2 switch -\par -\tab } -{\f2\uldb requirements}{\v\f2 requirements} -{\f2 variable -\par -\tab } -{\f2\uldb roots}{\v\f2 ROOTS} -{\f2 operator -\par -\tab } -{\f2\uldb root_of}{\v\f2 ROOT\_OF} -{\f2 operator -\par -\tab } -{\f2\uldb trigform}{\v\f2 TRIGFORM} -{\f2 switch -\par -\tab } -{\f2\uldb varopt}{\v\f2 VAROPT} -{\f2 switch -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SORT} - -${\footnote \pard\plain \sl240 \fs20 $ SORT} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0816} - - K{\footnote \pard\plain \sl240 \fs20 K sorting;SORT operator;operator} - -}{\b\f2 SORT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 sort} {\f2 operator sorts the elements of a list according to -an arbitrary comparison operator. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 sort} {\f4 (,) -\par -\par -}{\f2 \par - is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of algebraic expressions. - is a comparison operator which defines a partial -ordering among the members of . may be -one of the builtin comparison operators like -}{\f3 <} {\f2 (} -{\f2\uldb lessp}{\v\f2 LESSP} -{\f2 ), }{\f3 <=} {\f2 (} -{\f2\uldb leq}{\v\f2 LEQ} -{\f2 ) -etc., or may be the name of a comparison procedure. -Such a procedure has two arguments, and it returns -} -{\f2\uldb true}{\v\f2 TRUE} -{\f2 if the first argument -ranges before the second one, and 0 or } -{\f2\uldb nil}{\v\f2 NIL} -{\f2 otherwise. -The result of }{\f3 sort} {\f2 is a new list which contains the -elements of in a sequence corresponding to . - \par -examples: \par -\pard \tx3420 }{\f4 \par - procedure ce(a,b); \par - \par - if evenp a and not evenp b then 1 else 0; \par - \par -for i:=1:10 collect random(50)$ \par - \par -sort(ws,>=); \par - \par - \{41,38,33,30,28,25,20,17,8,5\} \par - \par - \par -sort(ws,<); \par - \par - \{5,8,17,20,25,28,30,33,38,41\} \par - \par - \par -sort(ws,ce); \par - \par - \{8,20,28,30,38,5,17,25,33,41\} \par - \par - \par - procedure cd(a,b); \par - \par - if deg(a,x)>deg(b,x) then 1 else \par - \par - if deg(a,x)deg(b,y) then 1 else 0; \par - \par -sort(\{x^2,y^2,x*y\},cd); \par - \par - 2 2 \par - \{x ,x*y,y \} \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # STRUCTR} - -${\footnote \pard\plain \sl240 \fs20 $ STRUCTR} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0817} - - K{\footnote \pard\plain \sl240 \fs20 K decomposition;STRUCTR operator;operator} - -}{\b\f2 STRUCTR}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 structr} {\f2 operator breaks its argument expression into named -subexpressions. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 structr} {\f4 ( [,[, ...]]) -\par -\par -}{\f2 \par - may be any valid REDUCE scalar expression. - may be any valid REDUCE }{\f3 identifier} {\f2 . The first -identifier -is the stem for subexpression names, the second is the name to be assigned -to the structured expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -structr(sqrt(x**2 + 2*x) + sin(x**2*z)); \par - \par - \par - ANS1 + ANS2 \par - where \par - 2 \par - ANS2 := SIN(X *Z) \par - 1/2 \par - ANS1 := ((X + 2)*X) \par - \par - \par -ans3; \par - \par - ANS3 \par - \par - \par -on fort; \par - \par -structr((x+1)**5 + tan(x*y*z),var,aa); \par - \par - \par - VAR1=TAN(X*Y*Z) \par - AA=VAR1+X**5+5.*X**4+10.*X**3+10.X**2+5.*X+1 \par - \par -\pard \sl240 }{\f2 The second argument to }{\f3 structr} {\f2 is optional. If it is not given, the -default stem }{\f3 ANS} {\f2 is used by REDUCE to construct names for the -subexpression. The names are only for display purposes: REDUCE does not -store the names and their values unless the switch } -{\f2\uldb savestructr}{\v\f2 SAVESTRUCTR} -{\f2 is -on. -\par -\par -If a third argument is given, the structured expression as a whole is named by -this argument, when } -{\f2\uldb fort}{\v\f2 FORT} -{\f2 is on. The expression is not stored -under this -name. You can send these structured Fortran expressions to a file with the -}{\f3 out} {\f2 command. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SUB} - -${\footnote \pard\plain \sl240 \fs20 $ SUB} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0818} - - K{\footnote \pard\plain \sl240 \fs20 K substitution;SUB operator;operator} - -}{\b\f2 SUB}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 sub} {\f2 operator substitutes a new expression for a kernel in an -expression. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 sub} {\f4 (}{\f3 =} {\f4 - \{,}{\f3 =} {\f4 \}*, - ) or -\par -\par -}{\f3 sub} {\f4 (\{}{\f3 =} {\f4 *, - }{\f3 =} {\f3 expression} {\f4 \},) -\par -\par -}{\f2 \par - must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 , can be any REDUCE -scalar expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -sub(x=3,y=4,(x+y)**3); \par - \par - 343 \par - \par - \par -x; \par - \par - X \par - \par - \par -sub(\{cos=sin,sin=cos\},cos a+sin b\} \par - \par - \par - COS(B) + SIN(A) \par - \par -\pard \sl240 }{\f2 Note in the second example that operators can be replaced using the -}{\f3 sub} {\f2 operator. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SUM} - -${\footnote \pard\plain \sl240 \fs20 $ SUM} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0819} - - K{\footnote \pard\plain \sl240 \fs20 K summation;Gosper algorithm;SUM operator;operator} - -}{\b\f2 SUM}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 sum} {\f2 returns -the indefinite or definite summation of a given expression. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 sum} {\f4 (,[, [, ]]) -\par -\par -\par -\par -}{\f2 where is the expression to be added, is the -control variable (a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 ), and and -are the optional lower and upper limits. If is -not supplied the upper limit is taken as . The Gosper -algorithm is used. If there is no closed form solution, the operator -returns the input unchanged. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -sum(4n**3,n); \par - \par - 2 2 \par - n *(n + 2*n + 1) \par - \par - \par -sum(2a+2k*r,k,0,n-1); \par - \par - n*(2*a + n*r - r) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # WS} - -${\footnote \pard\plain \sl240 \fs20 $ WS} - -+{\footnote \pard\plain \sl240 \fs20 + g8:0820} - - K{\footnote \pard\plain \sl240 \fs20 K interactive;work space;WS operator;operator} - -}{\b\f2 WS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 ws} {\f2 operator alone returns the last result; }{\f3 ws} {\f2 with a -number argument returns the results of the REDUCE statement executed after -that numbered prompt. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 ws} {\f4 or }{\f3 ws} {\f4 () -\par -\par -}{\f2 \par - must be an integer between 1 and the current REDUCE prompt number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 (In the following examples, unlike most others, the numbered -prompt is shown.)}{\f4 \pard \tx3420 \par - \par -1: df(sin y,y); \par - \par - COS(Y) \par - \par - \par -2: ws^2; \par - \par - 2 \par - COS(Y) \par - \par - \par -3: df(ws 1,y); \par - \par - -SIN(Y) \par - \par -\pard \sl240 }{\f2 -\par -\par -}{\f3 ws} {\f2 and }{\f3 ws} {\f3 (} {\f2 }{\f3 )} {\f2 can be used anywhere the -expression they stand for can be used. Calling a number for which no -result was produced, such as a switch setting, will give an error message. -\par -\par -The current workspace always contains the results of the last REDUCE -command that produced an expression, even if several input statements -that do not produce expressions have intervened. For example, if you do -a differentiation, producing a result expression, then change several -switches, the operator }{\f3 ws;} {\f2 returns the results of the differentiation. -The current workspace (}{\f3 ws} {\f2 ) can also be used inside files, though the -numbered workspace contains only the }{\f3 in} {\f2 command that input the file. -\par -\par -There are three history lists kept in your REDUCE session. The first -stores raw input, suitable for the statement editor. The second stores -parsed input, ready to execute and accessible by } -{\f2\uldb input}{\v\f2 INPUT} -{\f2 . The -third stores results, when they are produced by statements, which are -accessible by the }{\f3 ws} {\f2 < n> operator. If your session is very -long, storage space begins to fill up with these expressions, so it is a -good idea to end the session once in a while, saving needed expressions to -files with the } -{\f2\uldb saveas}{\v\f2 SAVEAS} -{\f2 and } -{\f2\uldb out}{\v\f2 OUT} -{\f2 commands. -\par -\par -An error message is given if a reference number has not yet been used. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g8} - -${\footnote \pard\plain \sl240 \fs20 $ Algebraic Operators} - -+{\footnote \pard\plain \sl240 \fs20 + index:0008} -}{\b\f2 Algebraic Operators}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb APPEND operator} -{\v\f2 APPEND}{\f2 \par -}{\f2 \tab}{\f2\uldb ARBINT operator} -{\v\f2 ARBINT}{\f2 \par -}{\f2 \tab}{\f2\uldb ARBCOMPLEX operator} -{\v\f2 ARBCOMPLEX}{\f2 \par -}{\f2 \tab}{\f2\uldb ARGLENGTH operator} -{\v\f2 ARGLENGTH}{\f2 \par -}{\f2 \tab}{\f2\uldb COEFF operator} -{\v\f2 COEFF}{\f2 \par -}{\f2 \tab}{\f2\uldb COEFFN operator} -{\v\f2 COEFFN}{\f2 \par -}{\f2 \tab}{\f2\uldb CONJ operator} -{\v\f2 CONJ}{\f2 \par -}{\f2 \tab}{\f2\uldb CONTINUED_FRACTION operator} -{\v\f2 CONTINUED_FRACTION}{\f2 \par -}{\f2 \tab}{\f2\uldb DECOMPOSE operator} -{\v\f2 DECOMPOSE}{\f2 \par -}{\f2 \tab}{\f2\uldb DEG operator} -{\v\f2 DEG}{\f2 \par -}{\f2 \tab}{\f2\uldb DEN operator} -{\v\f2 DEN}{\f2 \par -}{\f2 \tab}{\f2\uldb DF operator} -{\v\f2 DF}{\f2 \par -}{\f2 \tab}{\f2\uldb EXPAND\_CASES operator} -{\v\f2 EXPAND\_CASES}{\f2 \par -}{\f2 \tab}{\f2\uldb EXPREAD operator} -{\v\f2 EXPREAD}{\f2 \par -}{\f2 \tab}{\f2\uldb FACTORIZE operator} -{\v\f2 FACTORIZE}{\f2 \par -}{\f2 \tab}{\f2\uldb HYPOT operator} -{\v\f2 HYPOT}{\f2 \par -}{\f2 \tab}{\f2\uldb IMPART operator} -{\v\f2 IMPART}{\f2 \par -}{\f2 \tab}{\f2\uldb INT operator} -{\v\f2 INT}{\f2 \par -}{\f2 \tab}{\f2\uldb INTERPOL operator} -{\v\f2 INTERPOL}{\f2 \par -}{\f2 \tab}{\f2\uldb LCOF operator} -{\v\f2 LCOF}{\f2 \par -}{\f2 \tab}{\f2\uldb LENGTH operator} -{\v\f2 LENGTH}{\f2 \par -}{\f2 \tab}{\f2\uldb LHS operator} -{\v\f2 LHS}{\f2 \par -}{\f2 \tab}{\f2\uldb LIMIT operator} -{\v\f2 LIMIT}{\f2 \par -}{\f2 \tab}{\f2\uldb LPOWER operator} -{\v\f2 LPOWER}{\f2 \par -}{\f2 \tab}{\f2\uldb LTERM operator} -{\v\f2 LTERM}{\f2 \par -}{\f2 \tab}{\f2\uldb MAINVAR operator} -{\v\f2 MAINVAR}{\f2 \par -}{\f2 \tab}{\f2\uldb MAP operator} -{\v\f2 MAP}{\f2 \par -}{\f2 \tab}{\f2\uldb MKID command} -{\v\f2 MKID}{\f2 \par -}{\f2 \tab}{\f2\uldb NPRIMITIVE operator} -{\v\f2 NPRIMITIVE}{\f2 \par -}{\f2 \tab}{\f2\uldb NUM operator} -{\v\f2 NUM}{\f2 \par -}{\f2 \tab}{\f2\uldb ODESOLVE operator} -{\v\f2 ODESOLVE}{\f2 \par -}{\f2 \tab}{\f2\uldb ONE\_OF type} -{\v\f2 ONE\_OF}{\f2 \par -}{\f2 \tab}{\f2\uldb PART operator} -{\v\f2 PART}{\f2 \par -}{\f2 \tab}{\f2\uldb PF operator} -{\v\f2 PF}{\f2 \par -}{\f2 \tab}{\f2\uldb PROD operator} -{\v\f2 PROD}{\f2 \par -}{\f2 \tab}{\f2\uldb REDUCT operator} -{\v\f2 REDUCT}{\f2 \par -}{\f2 \tab}{\f2\uldb REPART operator} -{\v\f2 REPART}{\f2 \par -}{\f2 \tab}{\f2\uldb RESULTANT operator} -{\v\f2 RESULTANT}{\f2 \par -}{\f2 \tab}{\f2\uldb RHS operator} -{\v\f2 RHS}{\f2 \par -}{\f2 \tab}{\f2\uldb ROOT\_OF operator} -{\v\f2 ROOT\_OF}{\f2 \par -}{\f2 \tab}{\f2\uldb SELECT operator} -{\v\f2 SELECT}{\f2 \par -}{\f2 \tab}{\f2\uldb SHOWRULES operator} -{\v\f2 SHOWRULES}{\f2 \par -}{\f2 \tab}{\f2\uldb SOLVE operator} -{\v\f2 SOLVE}{\f2 \par -}{\f2 \tab}{\f2\uldb SORT operator} -{\v\f2 SORT}{\f2 \par -}{\f2 \tab}{\f2\uldb STRUCTR operator} -{\v\f2 STRUCTR}{\f2 \par -}{\f2 \tab}{\f2\uldb SUB operator} -{\v\f2 SUB}{\f2 \par -}{\f2 \tab}{\f2\uldb SUM operator} -{\v\f2 SUM}{\f2 \par -}{\f2 \tab}{\f2\uldb WS operator} -{\v\f2 WS}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ALGEBRAIC} - -${\footnote \pard\plain \sl240 \fs20 $ ALGEBRAIC} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0821} - - K{\footnote \pard\plain \sl240 \fs20 K evaluation;ALGEBRAIC command;command} - -}{\b\f2 ALGEBRAIC}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 algebraic} {\f2 command changes REDUCE's mode of operation to -algebraic. When }{\f3 algebraic} {\f2 is used as an operator (with an -argument inside parentheses) that argument is evaluated in algebraic -mode, but REDUCE's mode is not changed. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -algebraic; \par - \par -symbolic; \par - \par - NIL \par - \par - \par -algebraic(x**2); \par - \par - 2 \par - X \par - \par - \par -x**2; \par - \par - ***** The symbol X has no value. \par - \par -\pard \sl240 }{\f2 REDUCE's symbolic mode does not know about most algebraic commands. -Error messages in this mode may also depend on the particular Lisp -used for the REDUCE implementation. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ANTISYMMETRIC} - -${\footnote \pard\plain \sl240 \fs20 $ ANTISYMMETRIC} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0822} - - K{\footnote \pard\plain \sl240 \fs20 K ANTISYMMETRIC declaration;declaration} - -}{\b\f2 ANTISYMMETRIC}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -When an operator is declared }{\f3 antisymmetric} {\f2 , its arguments are -reordered to conform to the internal ordering of the system. If an odd -number of argument interchanges are required to do this ordering, -the sign of the expression is changed. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 antisymmetric} {\f4 \{}{\f3 ,} {\f4 \}* -\par -\par -}{\f2 \par - is an identifier that has been declared as an operator. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator m,n; \par - \par -antisymmetric m,n; \par - \par -m(x,n(1,2)); \par - \par - - M( - N(2,1),X) \par - \par - \par -operator p; \par - \par -antisymmetric p; \par - \par -p(a,b,c); \par - \par - P(A,B,C) \par - \par - \par -p(b,a,c); \par - \par - - P(A,B,C) \par - \par -\pard \sl240 }{\f2 If has not been declared an operator, the flag -}{\f3 antisymmetric} {\f2 is still attached to it. When is -subsequently used as an operator, the message }{\f3 Declare} {\f2 - }{\f3 operator? (Y or N)} {\f2 is printed. If the user replies }{\f3 y} {\f2 , the -antisymmetric property of the operator is used. -\par -\par -Note in the first example, identifiers are customarily ordered -alphabetically, while numbers are ordered from largest to smallest. -The operators may have any desired number of arguments (less than 128). -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ARRAY} - -${\footnote \pard\plain \sl240 \fs20 $ ARRAY} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0823} - - K{\footnote \pard\plain \sl240 \fs20 K ARRAY declaration;declaration} - -}{\b\f2 ARRAY}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The }{\f3 array} {\f2 declaration declares a list of identifiers to be of type -}{\f3 array} {\f2 , and sets all their entries to 0. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 array} {\f4 () - \{}{\f3 ,} {\f4 ()\}* -\par -\par -}{\f2 \par - may be any valid REDUCE identifier. If the identifier -was already an array, a warning message is given that the array has been -redefined. are of form - \{,\}*. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -array a(2,5),b(3,3,3),c(200); \par - \par -array a(3,5); \par - \par - *** ARRAY A REDEFINED \par - \par - \par -a(3,4); \par - \par - 0 \par - \par - \par -length a; \par - \par - 4,6 \par - \par -\pard \sl240 }{\f2 Arrays are always global, even if defined inside a procedure or block -statement. Their status as an array remains until the variable is -reset by } -{\f2\uldb clear}{\v\f2 CLEAR} -{\f2 . Arrays may not have the same names as operators, -procedures or scalar variables. -\par -\par -Array elements are referred to by the usual notation: }{\f3 a(i,j)} {\f2 -returns the jth element of the ith row. The } -{\f2\uldb assign}{\v\f2 assign} -{\f2 ment operator -}{\f3 :=} {\f2 is used to put values into the array. Arrays as a whole -cannot be subject to assignment by } -{\f2\uldb let}{\v\f2 LET} -{\f2 or }{\f3 :=} {\f2 ; the -assignment operator }{\f3 :=} {\f2 is only valid for individual elements. -\par -\par -When you use } -{\f2\uldb let}{\v\f2 LET} -{\f2 on an array element, the contents of that -element become the argument to }{\f3 let} {\f2 . Thus, if the element -contains a number or some other expression that is not a valid argument -for this command, you get an error message. If the element contains an -identifier, the identifier has the substitution rule attached to it -globally. The same behavior occurs with } -{\f2\uldb clear}{\v\f2 CLEAR} -{\f2 . If the array -element contains an identifier or simple_expression, it is cleared. Do - use }{\f3 clear} {\f2 to try to set an array element to 0. Because -of the side effects of either }{\f3 let} {\f2 or }{\f3 clear} {\f2 , it is unwise -to apply either of these to array elements. -\par -\par -Array indices always start with 0, so that the declaration }{\f3 array a(5)} {\f2 -sets aside 6 units of space, indexed from 0 through 5, and initializes -them to 0. The } -{\f2\uldb length}{\v\f2 LENGTH} -{\f2 command returns a list of the true number of -elements in each dimension. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CLEAR} - -${\footnote \pard\plain \sl240 \fs20 $ CLEAR} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0824} - - K{\footnote \pard\plain \sl240 \fs20 K CLEAR command;command} - -}{\b\f2 CLEAR}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 clear} {\f2 command is used to remove assignments or remove substitution -rules from any expression. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 clear} {\f4 \{,\}+ or -\par -\par - }{\f3 clear} {\f4 -\par -\par -}{\f2 \par - can be any }{\f3 scalar} {\f2 , } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 , -or } -{\f2\uldb array}{\v\f2 ARRAY} -{\f2 variable or -} -{\f2\uldb procedure}{\v\f2 PROCEDURE} -{\f2 name. can be any general -or specific } -{\f2\uldb let}{\v\f2 LET} -{\f2 statement (see below in Comments). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -array a(2,3); \par - \par -a(2,2) := 15; \par - \par - A(2,2) := 15 \par - \par - \par -clear a; \par - \par -a(2,2); \par - \par - Declare A operator? (Y or N) \par - \par - \par -let x = y + z; \par - \par -sin(x); \par - \par - SIN(Y + Z) \par - \par - \par -clear x; \par - \par -sin(x); \par - \par - SIN(X) \par - \par - \par -let x**5 = 7; \par - \par -clear x; \par - \par -x**5; \par - \par - 7 \par - \par - \par -clear x**5; \par - \par -x**5; \par - \par - 5 \par - X \par - \par -\pard \sl240 }{\f2 Although it is not a good idea, operators of the same name but taking -different numbers of arguments can be defined. Using a }{\f3 clear} {\f2 statement -on any of these operators clears every one with the same name, even if the -number of arguments is different. -\par -\par -The }{\f3 clear} {\f2 command is used to ``forget" matrices, arrays, operators -and scalar variables, returning their identifiers to the pristine state -to be used for other purposes. When }{\f3 clear} {\f2 is applied to array -elements, the contents of the array element becomes the argument for -}{\f3 clear} {\f2 . Thus, you get an error message if the element contains a -number, or some other expression that is not a legal argument to -}{\f3 clear} {\f2 . If the element contains an identifier, it is cleared. -When clear is applied to matrix elements, an error message is returned -if the element evaluates to a number, otherwise there is no effect. Do - not try to use }{\f3 clear} {\f2 to set array or matrix elements to 0. -You will not be pleased with the results. -\par -\par -If you are trying to clear power or product substitution rules made with -either } -{\f2\uldb let}{\v\f2 LET} -{\f2 or } -{\f2\uldb forall}{\v\f2 FORALL} -{\f2 ...}{\f3 let} {\f2 , you must -reproduce the rule, exactly as you typed it with the same arguments, up to -but not including the equal sign, using the word }{\f3 clear} {\f2 instead of -the word }{\f3 let} {\f2 . This is shown in the last example. Any other type of -}{\f3 let} {\f2 or }{\f3 forall} {\f2 ...}{\f3 let} {\f2 substitution can be cleared -with just the variable or operator name. } -{\f2\uldb match}{\v\f2 MATCH} -{\f2 behaves the same as -} -{\f2\uldb let}{\v\f2 LET} -{\f2 in this situation. There is a more complicated example under -} -{\f2\uldb forall}{\v\f2 FORALL} -{\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CLEARRULES} - -${\footnote \pard\plain \sl240 \fs20 $ CLEARRULES} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0825} - - K{\footnote \pard\plain \sl240 \fs20 K rule;CLEARRULES command;command} - -}{\b\f2 CLEARRULES}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 clearrules} {\f4 \{,\}+ -\par -\par -}{\f2 \par -The operator }{\f3 clearrules} {\f2 is used to remove previously defined -} -{\f2\uldb rule}{\v\f2 RULE} -{\f2 lists from the system. can be an explicit rule -list, or evaluate to a rule list. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -trig1 := \{cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2, \par - cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, \par - sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, \par - cos(~x)^2 => (1+cos(2*x))/2, \par - sin(~x)^2 => (1-cos(2*x))/2\}$ \par - \par -let trig1; \par -cos(a)*cos(b); \par - \par - COS(A - B) + COS(A + B) \par - ----------------------- \par - 2 \par - \par - \par -clearrules trig1; \par -cos(a)*cos(b); \par - \par - COS(A)*COS(B) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DEFINE} - -${\footnote \pard\plain \sl240 \fs20 $ DEFINE} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0826} - - K{\footnote \pard\plain \sl240 \fs20 K DEFINE command;command} - -}{\b\f2 DEFINE}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The command }{\f3 define} {\f2 allows you to supply a new name for an identifier -or replace it by any valid REDUCE expression. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 define} {\f4 }{\f3 =} {\f4 - \{}{\f3 ,} {\f4 }{\f3 =} {\f4 \}* -\par -\par -}{\f2 \par - is any valid REDUCE identifier, can be a -number, an identifier, an operator, a reserved word, or an expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -define is= :=, xx=y+z; \par - \par - \par -a is 10; \par - \par - A := 10 \par - \par - \par - \par -xx**2; \par - \par - 2 2 \par - Y + 2*Y*Z + Z \par - \par - \par - \par -xx := 10; \par - \par - Y + Z := 10 \par - \par -\pard \sl240 }{\f2 The renaming is done at the input level, and therefore takes precedence -over any other replacement or substitution declared for the same identifier. -It remains in effect until the end of the REDUCE session. Be careful with -it, since you cannot easily undo it without ending the session. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DEPEND} - -${\footnote \pard\plain \sl240 \fs20 $ DEPEND} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0827} - - K{\footnote \pard\plain \sl240 \fs20 K dependency;DEPEND declaration;declaration} - -}{\b\f2 DEPEND}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -}{\f3 depend} {\f2 declares that its first argument depends on the rest of its -arguments. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 depend} {\f4 \{}{\f3 ,} {\f4 \}+ -\par -\par -}{\f2 \par - must be a legal variable name or a prefix operator (see -} -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 ). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -depend y,x; \par - \par - \par -df(y**2,x); \par - \par - 2*DF(Y,X)*Y \par - \par - \par - \par -depend z,cos(x),y; \par - \par - \par -df(sin(z),cos(x)); \par - \par - COS(Z)*DF(Z,COS(X)) \par - \par - \par - \par -df(z**2,x); \par - \par - 2*DF(Z,X)*Z \par - \par - \par - \par -nodepend z,y; \par - \par - \par -df(z**2,x); \par - \par - 2*DF(Z,X)*Z \par - \par - \par - \par -cc := df(y**2,x); \par - \par - CC := 2*DF(Y,X)*Y \par - \par - \par - \par -y := tan x; \par - \par - Y := TAN(X); \par - \par - \par - \par -cc; \par - \par - 2 \par - 2*TAN(X)*(TAN(X) + 1) \par - \par -\pard \sl240 }{\f2 Dependencies can be removed by using the declaration } -{\f2\uldb nodepend}{\v\f2 NODEPEND} -{\f2 . -The differentiation operator uses this information, as shown in the -examples above. Linear operators also use knowledge of dependencies -(see } -{\f2\uldb linear}{\v\f2 LINEAR} -{\f2 ). Note that dependencies can be nested: Having -declared y to depend on x, and z -to depend on y, we -see that the chain rule was applied to the derivative of a function of -z with respect to x. If the explicit function of the -dependency is later entered into the system, terms with }{\f3 DF(Y,X)} {\f2 , -for example, are expanded when they are displayed again, as shown in the -last example. The boolean operator } -{\f2\uldb freeof}{\v\f2 FREEOF} -{\f2 allows you to -check the dependency between two algebraic objects. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EVEN} - -${\footnote \pard\plain \sl240 \fs20 $ EVEN} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0828} - - K{\footnote \pard\plain \sl240 \fs20 K EVEN declaration;declaration} - -}{\b\f2 EVEN}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -syntax: \par -}{\f4 \par -\par -}{\f3 even} {\f4 \{,\}* -\par -\par -}{\f2 \par -This declaration is used to declare an operator even in its first -argument. Expressions involving an operator declared in this manner are -transformed if the first argument contains a minus sign. Any other -arguments are not affected. - \par -examples: \par -\pard \tx3420 }{\f4 \par - even f; \par - \par - f(-a) \par - \par - F(A) \par - \par - \par - f(-a,-b) \par - \par - F(A,-B) \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FACTOR_declaration} - -${\footnote \pard\plain \sl240 \fs20 $ FACTOR_declaration} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0829} - - K{\footnote \pard\plain \sl240 \fs20 K output;FACTOR declaration;declaration} - -}{\b\f2 FACTOR}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -When a kernel is declared by }{\f3 factor} {\f2 , all terms involving fixed -powers of that kernel are printed as a product of the fixed powers and -the rest of the terms. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 factor} {\f4 \{}{\f3 ,} {\f4 \}* -\par -\par -}{\f2 \par - must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 or a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of -}{\f3 kernel} {\f2 s. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := (x + y + z)**2; \par - \par - 2 2 2 \par - A := X + 2*X*Y + 2*X*Z + Y + 2*Y*Z + Z \par - \par - \par -factor y; \par - \par -a; \par - \par - 2 2 2 \par - Y + 2*Y*(X + Z) + X + 2*X*Z + Z \par - \par - \par -factor sin(x); \par - \par -c := df(sin(x)**4*x**2*z,x); \par - \par - 4 3 2 \par - C := 2*SIN(X) *X*Z + 4*SIN(X) *COS(X)*X *Z \par - \par - \par -remfac sin(x); \par - \par -c; \par - \par - 3 \par - 2*SIN(X) *X*Z*(2*COS(X)*X + SIN(X)) \par - \par -\pard \sl240 }{\f2 Use the }{\f3 factor} {\f2 declaration to display variables of interest so that -you can see their powers more clearly, as shown in the example. Remove -this special treatment with the declaration } -{\f2\uldb remfac}{\v\f2 REMFAC} -{\f2 . The -}{\f3 factor} {\f2 declaration is only effective when the switch } -{\f2\uldb pri}{\v\f2 PRI} -{\f2 -is on. -\par -\par -The }{\f3 factor} {\f2 declaration is not a factoring command; to factor -expressions use the } -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 switch or the } -{\f2\uldb factorize}{\v\f2 FACTORIZE} -{\f2 command. -\par -\par -The }{\f3 factor} {\f2 declaration is helpful in such cases as Taylor polynomials -where the explicit powers of the variable are expected at the top level, not -buried in various factored forms. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FORALL} - -${\footnote \pard\plain \sl240 \fs20 $ FORALL} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0830} - - K{\footnote \pard\plain \sl240 \fs20 K substitution;FORALL command;command} - -}{\b\f2 FORALL}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 forall} {\f2 or (preferably) }{\f3 for all} {\f2 command is used as a -modifier for } -{\f2\uldb let}{\v\f2 LET} -{\f2 statements, indicating the universal applicability -of the rule, with possible qualifications. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 for all} {\f4 \{,\}* }{\f3 let} {\f4 - -\par -\par -or -\par -\par -}{\f3 for all} {\f4 \{,\}* - }{\f3 such that} {\f4 }{\f3 let} {\f4 -\par -\par -}{\f2 \par - may be any valid REDUCE identifier, -can be an operator, a product or power, or a group or block statement. - must be a logical or comparison operator returning true or -false. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -for all x let f(x) = sin(x**2); \par - \par - \par - \par - Declare F operator ? (Y or N) \par - \par - \par -y \par - \par -f(a); \par - \par - 2 \par - SIN(A ) \par - \par - \par -operator pos; \par - \par -for all x such that x>=0 let pos(x) = sqrt(x + 1); \par - \par -pos(5); \par - \par - SQRT(6) \par - \par - \par -pos(-5); \par - \par - POS(-5) \par - \par - \par -clear pos; \par - \par -pos(5); \par - \par - Declare POS operator ? (Y or N) \par - \par - \par -for all a such that numberp a let x**a = 1; \par - \par -x**4; \par - \par - 1 \par - \par - \par -clear x**a; \par - \par - *** X**A not found \par - \par - \par -for all a clear x**a; \par - \par -x**4; \par - \par - 1 \par - \par - \par -for all a such that numberp a clear x**a; \par - \par -x**4; \par - \par - 4 \par - X \par - \par -\pard \sl240 }{\f2 Substitution rules defined by }{\f3 for all} {\f2 or }{\f3 for -all} {\f2 ...}{\f3 such that} {\f2 commands that involve products or powers are -cleared by reproducing the command, with exactly the same variable names -used, up to but not including the equal sign, with } -{\f2\uldb clear}{\v\f2 CLEAR} -{\f2 -replacing }{\f3 let} {\f2 , as shown in the last example. Other substitutions -involving variables or operator names can be cleared with just the name, -like any other variable. -\par -\par -The } -{\f2\uldb match}{\v\f2 MATCH} -{\f2 command can also be used in product and power substitutions. -The syntax of its use and clearing is exactly like }{\f3 let} {\f2 . A }{\f3 match} {\f2 -substitution only replaces the term if it is exactly like the pattern, for -example }{\f3 match x**5 = 1} {\f2 replaces only terms of }{\f3 x**5} {\f2 and not -terms of higher powers. -\par -\par -It is easier to declare your potential operator before defining the -}{\f3 for all} {\f2 rule, since the system will ask you to declare it an -operator anyway. Names of declared arrays or matrices or scalar -variables are invalid as operator names, to avoid ambiguity. Either -}{\f3 for all} {\f2 ...}{\f3 let} {\f2 statements or procedures are often used to define -operators. One difference is that procedures implement ``call by value" -meaning that assignments involving their formal parameters do not change -the calling variables that replace them. If you use assignment statements -on the formal parameters in a }{\f3 for all} {\f2 ...}{\f3 let} {\f2 statement, the -effects are seen in the calling variables. Be careful not to redefine a -system operator unless you mean it: the statement }{\f3 for all x let -sin(x)=0;} {\f2 has exactly that effect, and the usual definition for sin(x) has -been lost for the remainder of the REDUCE session. \par -\par - -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INFIX} - -${\footnote \pard\plain \sl240 \fs20 $ INFIX} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0831} - - K{\footnote \pard\plain \sl240 \fs20 K operator;INFIX declaration;declaration} - -}{\b\f2 INFIX}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -}{\f3 infix} {\f2 declares identifiers to be infix operators. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 infix} {\f4 \{,\}* -\par -\par -}{\f2 \par - can be any valid REDUCE identifier, which has not already -been declared an operator, array or matrix, and is not reserved by the -system. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -infix aa; \par - \par -for all x,y let aa(x,y) = cos(x)*cos(y) - sin(x)*sin(y); \par - \par -x aa y; \par - \par - COS(X)*COS(Y) - SIN(X)*SIN(Y) \par - \par - \par -pi/3 aa pi/2; \par - \par - SQRT(3) \par - - ------- \par - 2 \par - \par - \par -aa(pi,pi); \par - \par - 1 \par - \par -\pard \sl240 }{\f2 A } -{\f2\uldb let}{\v\f2 LET} -{\f2 statement must be used to attach functionality to -the operator. Note that the operator is defined in prefix form in -the }{\f3 let} {\f2 statement. -After its definition, the operator may be used in either prefix or infix -mode. The above operator aa finds the cosine of the sum -of two angles by the formula -\par -\par -cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y). -\par -\par -Precedence may be attached to infix operators with the -} -{\f2\uldb precedence}{\v\f2 PRECEDENCE} -{\f2 declaration. -\par -\par -User-defined infix operators may be used in prefix form. If they are used -in infix form, a space must be left on each side of the operator to avoid -ambiguity. Infix operators are always binary. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INTEGER} - -${\footnote \pard\plain \sl240 \fs20 $ INTEGER} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0832} - - K{\footnote \pard\plain \sl240 \fs20 K INTEGER declaration;declaration} - -}{\b\f2 INTEGER}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The }{\f3 integer} {\f2 declaration must be made immediately after a -} -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 (or other variable declaration such as } -{\f2\uldb real}{\v\f2 REAL} -{\f2 -and } -{\f2\uldb scalar}{\v\f2 SCALAR} -{\f2 ) and declares local integer variables. They are -initialized to 0. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 integer} {\f4 \{,\}* -\par -\par -}{\f2 \par - may be any valid REDUCE identifier, except -}{\f3 t} {\f2 or }{\f3 nil} {\f2 . -\par -\par -Integer variables remain local, and do not share values with variables of -the same name outside the } -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 ...}{\f3 end} {\f2 block. When the -block is finished, the variables are removed. You may use the words -} -{\f2\uldb real}{\v\f2 REAL} -{\f2 or } -{\f2\uldb scalar}{\v\f2 SCALAR} -{\f2 in the place of }{\f3 integer} {\f2 . -}{\f3 integer} {\f2 does not indicate typechecking by the -current REDUCE; it is only for your own information. Declaration -statements must immediately follow the }{\f3 begin} {\f2 , without a semicolon -between }{\f3 begin} {\f2 and the first variable declaration. -\par -\par -Any variables used inside }{\f3 begin} {\f2 ...}{\f3 end} {\f2 blocks that were not -declared }{\f3 scalar} {\f2 , }{\f3 real} {\f2 or }{\f3 integer} {\f2 are global, and any -change made to them inside the block affects their global value. Any -} -{\f2\uldb array}{\v\f2 ARRAY} -{\f2 or } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 declared inside a block is always global. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # KORDER} - -${\footnote \pard\plain \sl240 \fs20 $ KORDER} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0833} - - K{\footnote \pard\plain \sl240 \fs20 K order;variable order;kernel order;KORDER declaration;declaration} - -}{\b\f2 KORDER}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -The }{\f3 korder} {\f2 declaration changes the internal canonical ordering of -kernels. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 korder} {\f4 \{}{\f3 ,} {\f4 \}* -\par -\par -}{\f2 \par - must be a REDUCE } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 or a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of -}{\f3 kernel} {\f2 s. -\par -\par -The declaration }{\f3 korder} {\f2 changes the internal ordering, but not the print -ordering, so the effects cannot be seen on output. However, in some -calculations, the order of the variables can have significant effects on the -time and space demands of a calculation. If you are doing a demanding -calculation with several kernels, you can experiment with changing the -canonical ordering to improve behavior. -\par -\par -The first kernel in the argument list is given the highest priority, the -second gets the next highest, and so on. Kernels not named in a -}{\f3 korder} {\f2 ordering otherwise. A new }{\f3 korder} {\f2 declaration replaces -the previous one. To return to canonical ordering, use the command -}{\f3 korder nil} {\f2 . -\par -\par -To change the print ordering, use the declaration } -{\f2\uldb order}{\v\f2 ORDER} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LET} - -${\footnote \pard\plain \sl240 \fs20 $ LET} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0834} - - K{\footnote \pard\plain \sl240 \fs20 K rule;substitution;LET command;command} - -}{\b\f2 LET}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 let} {\f2 command defines general or specific substitution rules. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 let} {\f4 }{\f3 =} {\f4 \{, -}{\f3 =} {\f4 \}* -\par -\par -}{\f2 \par - can be any valid REDUCE identifier except an array, and in -some cases can be an expression; can be any valid REDUCE -expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -let a = sin(x); \par - \par -b := a; \par - \par - B := SIN X; \par - \par - \par -let c = a; \par - \par -exp(a); \par - \par - SIN(X) \par - E \par - \par - \par -a := x**2; \par - \par - 2 \par - A := X \par - \par - \par -exp(a); \par - \par - 2 \par - X \par - E \par - \par - \par -exp(b); \par - \par - SIN(X) \par - E \par - \par - \par -exp(c); \par - \par - 2 \par - X \par - E \par - \par - \par -let m + n = p; \par - \par -(m + n)**5; \par - \par - 5 \par - P \par - \par - \par -operator h; \par - \par -let h(u,v) = u - v; \par - \par -h(u,v); \par - \par - U - V \par - \par - \par -h(x,y); \par - \par - H(X,Y) \par - \par - \par -array q(10); \par - \par -let q(1) = 15; \par - \par - ***** Substitution for 0 not allowed \par - \par -\pard \sl240 }{\f2 The }{\f3 let} {\f2 command is also used to activate a }{\f3 rule sets} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 let} {\f4 \{,\}+ -\par -\par -}{\f2 \par - can be an explicit } -{\f2\uldb rule}{\v\f2 RULE} -{\f2 }{\f3 list} {\f2 , or evaluate -to a rule list. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -trig1 := \{cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2, \par - cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, \par - sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, \par - cos(~x)^2 => (1+cos(2*x))/2, \par - sin(~x)^2 => (1-cos(2*x))/2\}$ \par - \par -let trig1; \par -cos(a)*cos(b); \par - \par - COS(A - B) + COS(A + B) \par - ------------------------ \par - 2 \par - \par -\pard \sl240 }{\f2 A }{\f3 let} {\f2 command returns no value, though the substitution rule is -entered. Assignment rules made by } -{\f2\uldb assign}{\v\f2 assign} -{\f2 and }{\f3 let} {\f2 -rules are at the -same level, and cancel each other. There is a difference in their -operation, however, as shown in the first example: a }{\f3 let} {\f2 assignment -tracks the changes in what it is assigned to, while a }{\f3 :=} {\f2 assignment -is fixed at the value it originally had. -\par -\par -The use of expressions as left-hand sides of }{\f3 let} {\f2 statements is a -little complicated. The rules of operation are: -\par -\par -\tab (i) -Expressions of the form A*B = C do not change A, B or C, but set A*B to C. -\par -\par -\tab (ii) -Expressions of the form A+B = C substitute C - B for A, but do not change -B or C. -\par -\par -\tab (iii) -Expressions of the form A-B = C substitute B + C for A, but do not change -B or C. -\par -\par -\tab (iv) -Expressions of the form A/B = C substitute B*C for A, but do not change B or -C. -\par -\par -\tab (v) -Expressions of the form A**N = C substitute C for A**N in every expression of -a power of A to N or greater. An asymptotic command such as A**N = 0 sets -all terms involving A to powers greater than or equal to N to 0. Finite -fields may be generated by requiring modular arithmetic (the } -{\f2\uldb modular}{\v\f2 MODULAR} -{\f2 -switch) and defining the primitive polynomial via a }{\f3 let} {\f2 statement. -\par -\par -\par -}{\f3 let} {\f2 substitutions involving expressions are cleared by using -the } -{\f2\uldb clear}{\v\f2 CLEAR} -{\f2 command with exactly the same expression. -\par -\par -Note when a simple }{\f3 let} {\f2 statement is used to assign functionality to an -operator, it is valid only for the exact identifiers used. For the use of the -}{\f3 let} {\f2 command to attach more general functionality to an operator, -see } -{\f2\uldb forall}{\v\f2 FORALL} -{\f2 . -\par -\par -Arrays as a whole cannot be arguments to }{\f3 let} {\f2 statements, but -matrices as a whole can be legal arguments, provided both arguments are -matrices. However, it is important to note that the two matrices are then -linked. Any change to an element of one matrix changes the corresponding -value in the other. Unless you want this behavior, you should not use -}{\f3 let} {\f2 for matrices. The assignment operator } -{\f2\uldb assign}{\v\f2 assign} -{\f2 can be used -for non-tracking assignments, avoiding the side effects. Matrices are -redimensioned as needed in }{\f3 let} {\f2 statements. -\par -\par -When array or matrix elements are used as the left-hand side of }{\f3 let} {\f2 -statements, the contents of that element is used as the argument. When the -contents is a number or some other expression that is not a valid left-hand -side for }{\f3 let} {\f2 , you get an error message. If the contents is an -identifier or simple expression, the }{\f3 let} {\f2 rule is globally attached -to that identifier, and is in effect not only inside the array or matrix, -but everywhere. Because of such unwanted side effects, you should not -use }{\f3 let} {\f2 with array or matrix elements. The assignment operator -}{\f3 :=} {\f2 can be used to put values into array or matrix elements without -the side effects. -\par -\par -Local variables declared inside }{\f3 begin} {\f2 ...}{\f3 end} {\f2 blocks cannot -be used as the left-hand side of }{\f3 let} {\f2 statements. However, -} -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 ...}{\f3 end} {\f2 blocks themselves can be used as the -right-hand side of }{\f3 let} {\f2 statements. The construction: - \par -syntax: \par -}{\f4 \par -\par -}{\f3 for all} {\f4 - }{\f3 let} {\f4 ()}{\f3 =} {\f4 -\par -\par -}{\f2 \par -is an alternative to the - \par -syntax: \par -}{\f4 \par -\par -}{\f3 procedure} {\f4 ()}{\f3 ;} {\f4 -\par -\par -}{\f2 \par -construction. One important difference between the two constructions is that -the as formal parameters to a procedure have their global values -protected against change by the procedure, while the of a -}{\f3 let} {\f2 statement are changed globally by its actions. -\par -\par -Be careful in using a construction such as }{\f3 let x = x + 1} {\f2 except inside -a controlled loop statement. The process of resubstitution continues until -a stack overflow message is given. -\par -\par -The }{\f3 let} {\f2 statement may be used to make global changes to variables from -inside procedures. If }{\f3 x} {\f2 is a formal parameter to a procedure, the -command }{\f3 let x = } {\f2 ... makes the change to the calling variable. -For example, if a procedure was defined by -\pard \tx3420 }{\f4 \par - procedure f(x,y); \par - let x = 15; \par -\pard \sl240 }{\f2 \par -\par -and the procedure was called as -\pard \tx3420 }{\f4 \par - f(a,b); \par -\pard \sl240 }{\f2 \par -\par -}{\f3 a} {\f2 would have its value changed to 15. Be careful when using }{\f3 let} {\f2 -statements inside procedures to avoid unwanted side effects. -\par -\par -It is also important to be careful when replacing }{\f3 let} {\f2 statements with -other }{\f3 let} {\f2 statements. The overlapping of these substitutions can be -unpredictable. Ordinarily the latest-entered rule is the first to be applied. -Sometimes the previous rule is superseded completely; other times it stays -around as a special case. The order of entering a set of related }{\f3 let} {\f2 -expressions is very important to their eventual behavior. The best -approach is to assume that the rules will be applied in an arbitrary order. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LINEAR} - -${\footnote \pard\plain \sl240 \fs20 $ LINEAR} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0835} - - K{\footnote \pard\plain \sl240 \fs20 K operator;LINEAR declaration;declaration} - -}{\b\f2 LINEAR}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -An operator can be declared linear in its first argument over powers of -its second argument by the declaration }{\f3 linear.} {\f2 - \par -syntax: \par -}{\f4 \par -\par -}{\f3 linear} {\f4 \{}{\f3 ,} {\f4 \}* -\par -\par -}{\f2 \par - must have been declared to be an operator. Be careful not -to use a system operator name, because this command may change its definition. -The operator being declared must have at least two arguments, and the -second one must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator f; \par - \par -linear f; \par - \par -f(0,x); \par - \par - 0 \par - \par - \par -f(-y,x); \par - \par - - F(1,X)*Y \par - \par - \par -f(y+z,x); \par - \par - F(1,X)*(Y + Z) \par - \par - \par -f(y*z,x); \par - \par - F(1,X)*Y*Z \par - \par - \par -depend z,x; \par - \par -f(y*z,x); \par - \par - F(Z,X)*Y \par - \par - \par -f(y/z,x); \par - \par - 1 \par - F(-,X)*Y \par - Z \par - \par - \par -depend y,x; \par - \par -f(y/z,x); \par - \par - Y \par - F(-,X) \par - Z \par - \par - \par -nodepend z,x; \par - \par -f(y/z,x); \par - \par - F(Y,X) \par - ------ \par - Z \par - \par - \par -f(2*e**sin(x),x); \par - \par - SIN(X) \par - 2*F(E ,X) \par - \par -\pard \sl240 }{\f2 Even when the operator has not had its functionality attached, it exhibits -linear properties as shown in the examples. Notice the difference when -dependencies are added. Dependencies are also in effect when the operator's -first argument contains its second, as in the last line above. -\par -\par -For a fully-developed example of the use of linear operators, refer to the -article in the , Vol. 14 (1974), pp. -301-317, ``Analytic Computation of Some Integrals in Fourth Order Quantum -Electrodynamics," by J.A. Fox and A.C. Hearn. The article includes the -complete listing of REDUCE procedures used for this work. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LINELENGTH} - -${\footnote \pard\plain \sl240 \fs20 $ LINELENGTH} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0836} - - K{\footnote \pard\plain \sl240 \fs20 K output;LINELENGTH declaration;declaration} - -}{\b\f2 LINELENGTH}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -The }{\f3 linelength} {\f2 declaration sets the length of the output line. Default -is 80. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 linelength} {\f4 -\par -\par -}{\f2 \par -To change the linelength, - must evaluate to a positive integer less than 128 -(although this varies from system to system), and should not be less than -20 or so for proper operation. -\par -\par -}{\f3 linelength} {\f2 returns the previous linelength. If you want the current -linelength value, but not change it, say }{\f3 linelength nil} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LISP} - -${\footnote \pard\plain \sl240 \fs20 $ LISP} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0837} - - K{\footnote \pard\plain \sl240 \fs20 K LISP command;command} - -}{\b\f2 LISP}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 lisp} {\f2 command changes REDUCE's mode of operation to symbolic. When -}{\f3 lisp} {\f2 is followed by an expression, that expression is evaluated in -symbolic mode, but REDUCE's mode is not changed. This command is -equivalent to } -{\f2\uldb symbolic}{\v\f2 SYMBOLIC} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -lisp; \par - \par - NIL \par - \par - \par -car '(a b c d e); \par - \par - A \par - \par - \par -algebraic; \par - \par -c := (lisp car '(first second))**2; \par - \par - \par - \par - 2 \par - C := FIRST \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LISTARGP} - -${\footnote \pard\plain \sl240 \fs20 $ LISTARGP} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0838} - - K{\footnote \pard\plain \sl240 \fs20 K argument;list;LISTARGP declaration;declaration} - -}{\b\f2 LISTARGP}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 listargp} {\f4 \{}{\f3 ,} {\f4 \}* -\par -\par -}{\f2 \par -If an operator other than those specifically defined for lists is given a -single argument that is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 , then the result of this -operation will be a list in which that operator is applied to each element -of the list. -This process can be inhibited for a specific operator, or list of operators, -by using the declaration }{\f3 listargp} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -log \{a,b,c\}; \par - \par - LOG(A),LOG(B),LOG(C) \par - \par - \par -listargp log; \par - \par -log \{a,b,c\}; \par - \par - LOG(A,B,C) \par - \par -\pard \sl240 }{\f2 It is possible to inhibit such distribution globally by turning on the -switch } -{\f2\uldb listargs}{\v\f2 LISTARGS} -{\f2 . In addition, if an operator has more than one -argument, no such distribution occurs, so }{\f3 listargp} {\f2 has no effect. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NODEPEND} - -${\footnote \pard\plain \sl240 \fs20 $ NODEPEND} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0839} - - K{\footnote \pard\plain \sl240 \fs20 K depend;NODEPEND declaration;declaration} - -}{\b\f2 NODEPEND}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -The }{\f3 nodepend} {\f2 declaration removes the dependency declared with -} -{\f2\uldb depend}{\v\f2 DEPEND} -{\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 nodepend} {\f4 \{,\}+ -\par -\par -\par -\par -}{\f2 must be a kernel that has had a dependency declared upon the -one or more other kernels that are its other arguments. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -depend y,x,z; \par - \par -df(sin y,x); \par - \par - COS(Y)*DF(Y,X) \par - \par - \par -df(sin y,x,z); \par - \par - COS(Y)*DF(Y,X,Z) - DF(Y,X)*DF(Y,Z)*SIN(Y) \par - \par - \par -nodepend y,z; \par - \par -df(sin y,x); \par - \par - COS(Y)*DF(Y,X) \par - \par - \par -df(sin y,x,z); \par - \par - 0 \par - \par -\pard \sl240 }{\f2 A warning message is printed if the dependency had not been declared by -}{\f3 depend} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MATCH} - -${\footnote \pard\plain \sl240 \fs20 $ MATCH} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0840} - - K{\footnote \pard\plain \sl240 \fs20 K substitution;MATCH command;command} - -}{\b\f2 MATCH}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 match} {\f2 command is similar to the } -{\f2\uldb let}{\v\f2 LET} -{\f2 command, except -that it matches only explicit powers in substitution. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 match} {\f4 }{\f3 =} {\f4 \{, - }{\f3 =} {\f4 \}* -\par -\par -}{\f2 \par - is generally a term involving powers, and is limited by -the rules for the } -{\f2\uldb let}{\v\f2 LET} -{\f2 command. may be -any valid REDUCE scalar expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -match c**2*a**2 = d; \par -(a+c)**4; \par - \par - 4 3 3 4 \par - A + 4*A *C + 4*A*C + C + 6*D \par - \par - \par -match a+b = c; \par - \par -a + 2*b; \par - \par - B + C \par - \par - \par -(a + b + c)**2; \par - \par - 2 2 2 \par - A - B + 2*B*C + 3*C \par - \par - \par -clear a+b; \par - \par -(a + b + c)**2; \par - \par - 2 2 2 \par - A + 2*A*B + 2*A*C + B + 2*B*C + C \par - \par - \par -let p*r = s; \par - \par -match p*q = ss; \par - \par -(a + p*r)**2; \par - \par - 2 2 \par - A + 2*A*S + S \par - \par - \par -(a + p*q)**2; \par - \par - 2 2 2 \par - A + 2*A*SS + P *Q \par - \par -\pard \sl240 }{\f2 Note in the last example that }{\f3 a + b} {\f2 has been explicitly matched -after the squaring was done, replacing each single power of }{\f3 a} {\f2 by -}{\f3 c - b} {\f2 . This kind of substitution, although following the rules, is -confusing and could lead to unrecognizable results. It is better to use -}{\f3 match} {\f2 with explicit powers or products only. }{\f3 match} {\f2 should -not be used inside procedures for the same reasons that }{\f3 let} {\f2 should -not be. -\par -\par -Unlike } -{\f2\uldb let}{\v\f2 LET} -{\f2 substitutions, }{\f3 match} {\f2 substitutions are executed -after all other operations are complete. The last example shows the -difference. }{\f3 match} {\f2 commands can be cleared by using } -{\f2\uldb clear}{\v\f2 CLEAR} -{\f2 , -with exactly the expression that the original }{\f3 match} {\f2 took. -}{\f3 match} {\f2 commands can also be done more generally with }{\f3 for all} {\f2 -or } -{\f2\uldb forall}{\v\f2 FORALL} -{\f2 ...}{\f3 such that} {\f2 commands. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NONCOM} - -${\footnote \pard\plain \sl240 \fs20 $ NONCOM} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0841} - - K{\footnote \pard\plain \sl240 \fs20 K operator;non commutative;commutative;NONCOM declaration;declaration} - -}{\b\f2 NONCOM}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -}{\f3 noncom} {\f2 declares that already-declared operators are noncommutative -under multiplication. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 noncom} {\f4 \{,\}* -\par -\par -}{\f2 \par - must have been declared an } -{\f2\uldb operator}{\v\f2 OPERATOR} -{\f2 , or a warning -message is given. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator f,h; \par - \par -noncom f; \par - \par -f(a)*f(b) - f(b)*f(a); \par - \par - F(A)*F(B) - F(B)*F(A) \par - \par - \par -h(a)*h(b) - h(b)*h(a); \par - \par - 0 \par - \par - \par -operator comm; \par - \par -for all x,y such that x neq y and ordp(x,y) \par - let f(x)*f(y) = f(y)*f(x) + comm(x,y); \par -\pard \sl240 \par - \par -f(1)*f(2); \par - \par - F(1)*F(2) \par - \par - \par -f(2)*f(1); \par - \par - COMM(2,1) + F(1)*F(2) \par - \par -\pard \sl240 }{\f2 The last example introduces the commutator of }{\f4 f(x)}{\f2 and }{\f4 f(y)}{\f2 -for all x and y. The equality check is to prevent an infinite loop. The -operator f can have other functionality attached to it if desired, or it -can remain an indeterminate operator. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NONZERO} - -${\footnote \pard\plain \sl240 \fs20 $ NONZERO} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0842} - - K{\footnote \pard\plain \sl240 \fs20 K operator;NONZERO declaration;declaration} - -}{\b\f2 NONZERO}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 nonzero} {\f4 \{,\}* -\par -\par -}{\f2 \par -If an } -{\f2\uldb operator}{\v\f2 OPERATOR} -{\f2 }{\f3 f} {\f2 is declared } -{\f2\uldb odd}{\v\f2 ODD} -{\f2 , then }{\f3 f(0)} {\f2 -is replaced by zero unless }{\f3 f} {\f2 is also declared non zero by the -declaration }{\f3 nonzero} {\f2 . - \par -examples: \par -\pard \tx3420 }{\f4 \par - odd f; \par - \par - f(0) \par - \par - 0 \par - \par - \par - nonzero f; \par - \par - f(0) \par - \par - F(0) \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ODD} - -${\footnote \pard\plain \sl240 \fs20 $ ODD} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0843} - - K{\footnote \pard\plain \sl240 \fs20 K operator;ODD declaration;declaration} - -}{\b\f2 ODD}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 odd} {\f4 \{,\}* -\par -\par -}{\f2 \par -This declaration is used to declare an operator odd in its first -argument. Expressions involving an operator declared in this manner are -transformed if the first argument contains a minus sign. Any other -arguments are not affected. - \par -examples: \par -\pard \tx3420 }{\f4 \par - odd f; \par - \par - f(-a) \par - \par - -F(A) \par - \par - \par - f(-a,-b) \par - \par - -F(A,-B) \par - \par - \par - f(a,-b) \par - \par - F(A,-B) \par - \par -\pard \sl240 }{\f2 \par -\par -If say }{\f3 f} {\f2 is declared odd, then }{\f3 f(0)} {\f2 is replaced by zero -unless }{\f3 f} {\f2 is also declared non zero by the declaration -} -{\f2\uldb nonzero}{\v\f2 NONZERO} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # OFF} - -${\footnote \pard\plain \sl240 \fs20 $ OFF} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0844} - - K{\footnote \pard\plain \sl240 \fs20 K switch;OFF command;command} - -}{\b\f2 OFF}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 off} {\f2 command is used to turn switches off. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 off} {\f4 \{,\}* -\par -\par -}{\f2 \par - can be any }{\f3 switch} {\f2 name. There is no problem if the -switch is already off. If the switch name is mistyped, an error message is -given. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ON} - -${\footnote \pard\plain \sl240 \fs20 $ ON} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0845} - - K{\footnote \pard\plain \sl240 \fs20 K switch;ON command;command} - -}{\b\f2 ON}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 on} {\f2 command is used to turn switches on. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 on} {\f4 \{,\}* -\par -\par -}{\f2 \par - can be any }{\f3 switch} {\f2 name. There is no problem if the -switch is already on. If the switch name is mistyped, an error message is -given. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # OPERATOR} - -${\footnote \pard\plain \sl240 \fs20 $ OPERATOR} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0846} - - K{\footnote \pard\plain \sl240 \fs20 K OPERATOR declaration;declaration} - -}{\b\f2 OPERATOR}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -Use the }{\f3 operator} {\f2 declaration to declare your own operators. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 operator} {\f4 \{,\}* -\par -\par -}{\f2 \par - can be any valid REDUCE identifier, which is not the name -of a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 , } -{\f2\uldb array}{\v\f2 ARRAY} -{\f2 , scalar variable or previously-defined -operator. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator dis,fac; \par - \par -let dis(~x,~y) = sqrt(x^2 + y^2); \par - \par -dis(1,2); \par - \par - SQRT(5) \par - \par - \par -dis(a,10); \par - \par - 2 \par - SQRT(A + 100) \par - \par - \par -on rounded; \par - \par -dis(1.5,7.2); \par - \par - 7.35459040329 \par - \par - \par -let fac(~n) = if n=0 then 1 \par - else if not(fixp n and n>0) \par - then rederr "choose non-negative integer" \par - else for i := 1:n product i; \par -\pard \sl240 \par - \par -fac(5); \par - \par - 120 \par - \par - \par -fac(-2); \par - \par - ***** choose non-negative integer \par - \par -\pard \sl240 }{\f2 The first operator is the Euclidean distance metric, the distance of point -}{\f4 (x,y)}{\f2 from the origin. The second operator is the factorial. -\par -\par -Operators can have various properties assigned to them; they can be -declared } -{\f2\uldb infix}{\v\f2 INFIX} -{\f2 , } -{\f2\uldb linear}{\v\f2 LINEAR} -{\f2 , } -{\f2\uldb symmetric}{\v\f2 SYMMETRIC} -{\f2 , -} -{\f2\uldb antisymmetric}{\v\f2 ANTISYMMETRIC} -{\f2 , or } -{\f2\uldb noncom}{\v\f2 NONCOM} -{\f3 mutative} {\f2 . -The default operator is prefix, nonlinear, and commutative. -Precedence can also be assigned to operators using the declaration -} -{\f2\uldb precedence}{\v\f2 PRECEDENCE} -{\f2 . -\par -\par -Functionality is assigned to an operator by a } -{\f2\uldb let}{\v\f2 LET} -{\f2 statement or -a } -{\f2\uldb forall}{\v\f2 FORALL} -{\f2 ...}{\f3 let} {\f2 statement, -(or possibly by a procedure with the name -of the operator). Be careful not to redefine a system operator by -accident. REDUCE permits you to redefine system operators, giving you a -warning message that the operator was already defined. This flexibility -allows you to add mathematical rules that do what you want them to do, but -can produce odd or erroneous behavior if you are not careful. -\par -\par -You can declare operators from inside } -{\f2\uldb procedure}{\v\f2 PROCEDURE} -{\f2 s, as long as they -are not local variables. Operators defined inside procedures are global. -A formal parameter may be declared as an operator, and has the effect of -declaring the calling variable as the operator. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ORDER} - -${\footnote \pard\plain \sl240 \fs20 $ ORDER} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0847} - - K{\footnote \pard\plain \sl240 \fs20 K output;variable order;order;ORDER declaration;declaration} - -}{\b\f2 ORDER}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -The }{\f3 order} {\f2 declaration changes the order of precedence of kernels for -display purposes only. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 order} {\f4 \{,\}* -\par -\par -}{\f2 \par - must be a valid } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 or } -{\f2\uldb operator}{\v\f2 OPERATOR} -{\f2 name -complete with argument or a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of such objects. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x + y + z + cos(a); \par - \par - COS(A) + X + Y + Z \par - \par - \par -order z,y,x,cos(a); \par - \par -x + y + z + cos(a); \par - \par - Z + Y + X + COS(A) \par - \par - \par -(x + y)**2; \par - \par - 2 2 \par - Y + 2*Y*X + X \par - \par - \par -order nil; \par - \par -(z + cos(z))**2; \par - \par - 2 2 \par - COS(Z) + 2*COS(Z)*Z + Z \par - \par -\pard \sl240 }{\f2 }{\f3 order} {\f2 affects the printing order of the identifiers only; internal -order is unchanged. Change internal order of evaluation with the -declaration } -{\f2\uldb korder}{\v\f2 KORDER} -{\f2 . You can use }{\f3 order} {\f2 to feature variables -or functions you are particularly interested in. -\par -\par -Declarations made with }{\f3 order} {\f2 are cumulative: kernels in new order -declarations are ordered behind those in previous declarations, and -previous declarations retain their relative order. Of course, specific -kernels named in new declarations are removed from previous ones and given -the new priority. Return to the standard canonical printing order with the -statement }{\f3 order nil} {\f2 . -\par -\par -The print order specified by }{\f3 order} {\f2 commands is not in effect if the -switch } -{\f2\uldb pri}{\v\f2 PRI} -{\f2 is off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PRECEDENCE} - -${\footnote \pard\plain \sl240 \fs20 $ PRECEDENCE} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0848} - - K{\footnote \pard\plain \sl240 \fs20 K operator;PRECEDENCE declaration;declaration} - -}{\b\f2 PRECEDENCE}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -The }{\f3 precedence} {\f2 declaration attaches a precedence to an infix operator. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 precedence} {\f4 , -\par -\par -}{\f2 \par - should have been declared an operator but may be a REDUCE -identifier that is not already an operator, array, or matrix. - must be a system infix operator or have had its -precedence already declared. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator f,h; \par - \par -precedence f,+; \par - \par -precedence h,*; \par - \par -a + f(1,2)*c; \par - \par - (1 F 2)*C + A \par - \par - \par -a + h(1,2)*c; \par - \par - 1 H 2*C + A \par - \par - \par -a*1 f 2*c; \par - \par - A F 2*C \par - \par - \par -a*1 h 2*c; \par - \par - 1 H 2*A*C \par - \par -\pard \sl240 }{\f2 The operator whose precedence is being declared is inserted into the infix -operator precedence list at the next higher place than . -\par -\par -Attaching a precedence to an operator has the side effect of declaring the -operator to be infix. If the identifier argument for }{\f3 precedence} {\f2 has -not been declared to be an operator, an attempt to use it causes an error -message. After declaring it to be an operator, it becomes an infix operator -with the precedence previously given. Infix operators may be used in prefix -form; if they are used in infix form, a space must be left on each side of -the operator to avoid ambiguity. Declared infix operators are always binary. -\par -\par -To see the infix operator precedence list, enter symbolic mode and type -}{\f3 preclis!*;} {\f2 . The lowest precedence operator is listed first. -\par -\par -All prefix operators have precedence higher than infix operators. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PRECISION} - -${\footnote \pard\plain \sl240 \fs20 $ PRECISION} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0849} - - K{\footnote \pard\plain \sl240 \fs20 K floating point;rounded;PRECISION declaration;declaration} - -}{\b\f2 PRECISION}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -The }{\f3 precision} {\f2 declaration sets the number of decimal places used when -} -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on. Default is system dependent, and normally about 12. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 precision} {\f4 () or }{\f3 precision} {\f4 -\par -\par -}{\f2 \par - must be a positive integer. When is 0, the -current precision is displayed, but not changed. There is no upper limit, -but precision of greater than several hundred causes unpleasantly slow -operation on numeric calculations. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -7/9; \par - \par - 0.777777777778 \par - \par - \par -precision 20; \par - \par - 20 \par - \par - \par -7/9; \par - \par - 0.77777777777777777778 \par - \par - \par -sin(pi/4); \par - \par - 0.7071067811865475244 \par - \par -\pard \sl240 }{\f2 Trailing zeroes are dropped, so sometimes fewer than 20 decimal places are -printed as in the last example. Turn on the switch } -{\f2\uldb fullprec}{\v\f2 FULLPREC} -{\f2 if -you want to print all significant digits. The } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 mode -carries calculations to two more places than given by }{\f3 precision} {\f2 , and -rounds off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PRINT\_PRECISION} - -${\footnote \pard\plain \sl240 \fs20 $ PRINT_PRECISION} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0850} - - K{\footnote \pard\plain \sl240 \fs20 K rounded;floating point;output;PRINT_PRECISION declaration;declaration} - -}{\b\f2 PRINT\_PRECISION}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 print_precision} {\f4 () - or }{\f3 print_precision} {\f4 -\par -\par -}{\f2 \par -In } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 mode, numbers are normally printed to the specified -precision. If the user wishes to print such numbers with less precision, -the printing precision can be set by the declaration }{\f3 print_precision} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -1/3; \par - \par - 0.333333333333 \par - \par - \par -print_precision 5; \par - \par -1/3 \par - \par - 0.33333 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # REAL} - -${\footnote \pard\plain \sl240 \fs20 $ REAL} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0851} - - K{\footnote \pard\plain \sl240 \fs20 K REAL declaration;declaration} - -}{\b\f2 REAL}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The }{\f3 real} {\f2 declaration must be made immediately after a -} -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 (or other variable declaration such as } -{\f2\uldb integer}{\v\f2 INTEGER} -{\f2 -and } -{\f2\uldb scalar}{\v\f2 SCALAR} -{\f2 ) and declares local integer variables. They are -initialized to zero. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 real} {\f4 \{,\}* -\par -\par -}{\f2 \par - may be any valid REDUCE identifier, except -}{\f3 t} {\f2 or }{\f3 nil} {\f2 . -\par -\par -Real variables remain local, and do not share values with variables of the -same name outside the } -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 ...}{\f3 end} {\f2 block. When the -block is finished, the variables are removed. You may use the words -} -{\f2\uldb integer}{\v\f2 INTEGER} -{\f2 or } -{\f2\uldb scalar}{\v\f2 SCALAR} -{\f2 in the place of }{\f3 real} {\f2 . -}{\f3 real} {\f2 does not indicate typechecking by the current REDUCE; it is -only for your own information. Declaration statements must immediately -follow the }{\f3 begin} {\f2 , without a semicolon between }{\f3 begin} {\f2 and the -first variable declaration. -\par -\par -Any variables used inside a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 -that were not declared }{\f3 scalar} {\f2 , }{\f3 real} {\f2 or }{\f3 integer} {\f2 are -global, and any change made to them inside the block affects their global -value. Any } -{\f2\uldb array}{\v\f2 ARRAY} -{\f2 or } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 declared inside a block is always -global. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # REMFAC} - -${\footnote \pard\plain \sl240 \fs20 $ REMFAC} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0852} - - K{\footnote \pard\plain \sl240 \fs20 K output;factor;REMFAC declaration;declaration} - -}{\b\f2 REMFAC}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -The }{\f3 remfac} {\f2 declaration removes the special factoring treatment of its -arguments that was declared with } -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 remfac} {\f4 \{,\}+ -\par -\par -}{\f2 \par - must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 or } -{\f2\uldb operator}{\v\f2 OPERATOR} -{\f2 name that -was declared as special with the } -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 declaration. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SCALAR} - -${\footnote \pard\plain \sl240 \fs20 $ SCALAR} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0853} - - K{\footnote \pard\plain \sl240 \fs20 K SCALAR declaration;declaration} - -}{\b\f2 SCALAR}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The }{\f3 scalar} {\f2 declaration must be made immediately after a -} -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 (or other variable declaration such as } -{\f2\uldb integer}{\v\f2 INTEGER} -{\f2 -and } -{\f2\uldb real}{\v\f2 REAL} -{\f2 ) and declares local scalar variables. They are -initialized to 0. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 scalar} {\f4 \{,\}* -\par -\par -}{\f2 \par - may be any valid REDUCE identifier, except }{\f3 t} {\f2 or -}{\f3 nil} {\f2 . -\par -\par -Scalar variables remain local, and do not share values with variables of -the same name outside the } -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 . -When the block is finished, the variables are removed. You may use the -words } -{\f2\uldb real}{\v\f2 REAL} -{\f2 or } -{\f2\uldb integer}{\v\f2 INTEGER} -{\f2 in the place of }{\f3 scalar} {\f2 . -}{\f3 real} {\f2 and }{\f3 integer} {\f2 do not indicate typechecking by the current -REDUCE; they are only for your own information. Declaration statements -must immediately follow the }{\f3 begin} {\f2 , without a semicolon between -}{\f3 begin} {\f2 and the first variable declaration. -\par -\par -Any variables used inside }{\f3 begin} {\f2 ...}{\f3 end} {\f2 blocks that were not -declared }{\f3 scalar} {\f2 , }{\f3 real} {\f2 or }{\f3 integer} {\f2 are global, and any -change made to them inside the block affects their global value. Arrays -declared inside a block are always global. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SCIENTIFIC\_NOTATION} - -${\footnote \pard\plain \sl240 \fs20 $ SCIENTIFIC_NOTATION} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0854} - - K{\footnote \pard\plain \sl240 \fs20 K rounded;floating point;output;SCIENTIFIC_NOTATION declaration;declaration} - -}{\b\f2 SCIENTIFIC\_NOTATION}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 scientific_notation} {\f4 () or -}{\f3 scientific_notation} {\f4 (\{,\}) -\par -\par -}{\f2 \par - and are positive integers. -}{\f3 scientific_notation} {\f2 controls the output format of floating point -numbers. At the default settings, any number with five or less digits -before the decimal point is printed in a fixed-point notation, e.g., -12345.6. Numbers with more than five digits are printed in scientific -notation, e.g., 1.234567E+5. Similarly, by default, any number with -eleven or more zeros after the decimal point is printed in scientific -notation. -\par -\par -When }{\f3 scientific_notation} {\f2 is called with the numerical argument - m a number with more than m digits before the decimal point, -or m or more zeros after the decimal point, is printed in scientific -notation. When }{\f3 scientific_notation} {\f2 is called with a list -\{,\}, a number with more than m digits before the -decimal point, or n or more zeros after the decimal point is -printed in scientific notation. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -on rounded; \par - \par - \par -12345.6; \par - \par - 12345.6 \par - \par - \par - \par -123456.5; \par - \par - 1.234565e+5 \par - \par - \par - \par -0.00000000000000012; \par - \par - 1.2e-16 \par - \par - \par - \par -scientific_notation 20; \par - \par - 5,11 \par - \par - \par - \par -5: 123456.7; \par - \par - 123456.7 \par - \par - \par - \par -0.00000000000000012; \par - \par - 0.00000000000000012 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SHARE} - -${\footnote \pard\plain \sl240 \fs20 $ SHARE} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0855} - - K{\footnote \pard\plain \sl240 \fs20 K SHARE declaration;declaration} - -}{\b\f2 SHARE}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The }{\f3 share} {\f2 declaration allows access to its arguments by both -algebraic and symbolic modes. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 share} {\f4 \{,\}* -\par -\par -}{\f2 \par - can be any valid REDUCE identifier. -\par -\par -Programming in } -{\f2\uldb symbolic}{\v\f2 SYMBOLIC} -{\f2 as well as algebraic mode allows -you a wider range -of techniques than just algebraic mode alone. Expressions do not cross the -boundary since they have different representations, unless the }{\f3 share} {\f2 -declaration is used. For more information on using symbolic mode, see -the , and the . -\par -\par -You should be aware that a previously-declared array is destroyed by the -}{\f3 share} {\f2 declaration. Scalar variables retain their values. You can -share a declared } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 that has not yet -been dimensioned so that it can be -used by both modes. Values that are later put into the matrix are -accessible from symbolic mode too, but not by the usual matrix reference -mechanism. In symbolic mode, a matrix is stored as a list whose first -element is } -{\f2\uldb MAT}{\v\f2 MAT} -{\f2 , and whose next elements are the rows of the matrix -stored as lists of the individual elements. Access in symbolic mode is by -the operators } -{\f2\uldb first}{\v\f2 FIRST} -{\f2 , } -{\f2\uldb second}{\v\f2 SECOND} -{\f2 , } -{\f2\uldb third}{\v\f2 THIRD} -{\f2 and -} -{\f2\uldb rest}{\v\f2 REST} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SYMBOLIC} - -${\footnote \pard\plain \sl240 \fs20 $ SYMBOLIC} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0856} - - K{\footnote \pard\plain \sl240 \fs20 K SYMBOLIC command;command} - -}{\b\f2 SYMBOLIC}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 symbolic} {\f2 command changes REDUCE's mode of operation to symbolic. -When }{\f3 symbolic} {\f2 is followed by an expression, that expression is -evaluated in symbolic mode, but REDUCE's mode is not changed. It is -equivalent to the } -{\f2\uldb lisp}{\v\f2 LISP} -{\f2 command. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -symbolic; \par - \par - NIL \par - \par - \par -cdr '(a b c); \par - \par - (B C) \par - \par - \par -algebraic; \par - \par -x + symbolic car '(y z); \par - \par - X + Y \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SYMMETRIC} - -${\footnote \pard\plain \sl240 \fs20 $ SYMMETRIC} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0857} - - K{\footnote \pard\plain \sl240 \fs20 K operator;SYMMETRIC declaration;declaration} - -}{\b\f2 SYMMETRIC}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -When an operator is declared }{\f3 symmetric} {\f2 , its arguments are reordered -to conform to the internal ordering of the system. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 symmetric} {\f4 \{,\}* -\par -\par -}{\f2 \par - is an identifier that has been declared an operator. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator m,n; \par - \par -symmetric m,n; \par - \par -m(y,a,sin(x)); \par - \par - M(SIN(X),A,Y) \par - \par - \par -n(z,m(b,a,q)); \par - \par - N(M(A,B,Q),Z) \par - \par -\pard \sl240 }{\f2 If has not been declared to be an operator, the flag -}{\f3 symmetric} {\f2 is still attached to it. When is -subsequently used as an operator, the message }{\f3 Declare} {\f2 - }{\f3 operator ? (Y or N)} {\f2 is printed. If the user replies }{\f3 y} {\f2 , the -symmetric property of the operator is used. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TR} - -${\footnote \pard\plain \sl240 \fs20 $ TR} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0858} - - K{\footnote \pard\plain \sl240 \fs20 K trace;TR declaration;declaration} - -}{\b\f2 TR}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -The }{\f3 tr} {\f2 declaration is used to trace system or user-written procedures. -It is only useful to those with a good knowledge of both Lisp and the -internal formats used by REDUCE. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 tr} {\f4 \{,\}* -\par -\par -}{\f2 \par - is the name of a REDUCE system procedure or one of your own -procedures. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 The system procedure }{\f3 prepsq} {\f2 is traced, - which prepares REDUCE standard -forms for printing by converting them to a Lisp prefix form.}{\f4 \pard \tx3420 \par - \par -tr prepsq; \par - \par - (PREPSQ) \par - \par - \par -x**2 + y; \par - \par - PREPSQ entry: \par - Arg 1: (((((X . 2) . 1) ((Y . 1) . 1)) . 1) \par - PREPSQ return value = (PLUS (EXPT X 2) Y) \par - PREPSQ entry: \par - Arg 1: (1 . 1) \par - PREPSQ return value = 1 \par - 2 \par - X + Y \par - \par - \par -untr prepsq; \par - \par - (PREPSQ) \par - \par -\pard \sl240 }{\f2 -\par -\par -This example is for a PSL-based system; the above format will vary if -other Lisp systems are used. -\par -\par -When a procedure is traced, the first lines show entry to the procedure and -the arguments it is given. The value returned by the procedure is printed -upon exit. If you are tracing several procedures, with a call to one of -them inside the other, the inner trace will be indented showing procedure -nesting. There are no trace options. However, the format of the trace -depends on the underlying Lisp system used. The trace can be removed with -the command } -{\f2\uldb untr}{\v\f2 UNTR} -{\f2 . Note that }{\f3 trace} {\f2 , below, is a matrix -operator, while }{\f3 tr} {\f2 does procedure tracing. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # UNTR} - -${\footnote \pard\plain \sl240 \fs20 $ UNTR} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0859} - - K{\footnote \pard\plain \sl240 \fs20 K trace;UNTR declaration;declaration} - -}{\b\f2 UNTR}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - - \par -\par -The }{\f3 untr} {\f2 declaration is used to remove a trace from system or -user-written procedures declared with } -{\f2\uldb tr}{\v\f2 TR} -{\f2 . It is only useful to -those with a good knowledge of both Lisp and the internal formats used by -REDUCE. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 untr} {\f4 \{,\}* -\par -\par -}{\f2 \par - is the name of a REDUCE system procedure or one of your own -procedures that has previously been the argument of a }{\f3 tr} {\f2 -declaration. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # VARNAME} - -${\footnote \pard\plain \sl240 \fs20 $ VARNAME} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0860} - - K{\footnote \pard\plain \sl240 \fs20 K VARNAME declaration;declaration} - -}{\b\f2 VARNAME}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The declaration }{\f3 varname} {\f2 instructs REDUCE to use its argument as the -default Fortran (when } -{\f2\uldb fort}{\v\f2 FORT} -{\f2 is on) or } -{\f2\uldb structr}{\v\f2 STRUCTR} -{\f2 identifier -and identifier stem, rather than using }{\f3 ANS} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 varname} {\f4 -\par -\par -}{\f2 \par - can be any combination of one or more alphanumeric -characters. Try to avoid REDUCE reserved words. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -varname ident; \par - \par - IDENT \par - \par - \par -on fort; \par - \par -x**2 + 1; \par - \par - IDENT=X**2+1. \par - \par - \par -off fort,exp; \par - \par -structr(((x+y)**2 + z)**3); \par - \par - 3 \par - IDENT2 \par - where \par - 2 \par - IDENT2 := IDENT1 + Z \par - IDENT1 := X + Y \par - \par -\pard \sl240 }{\f2 } -{\f2\uldb exp}{\v\f2 EXP} -{\f2 was turned off so that } -{\f2\uldb structr}{\v\f2 STRUCTR} -{\f2 could show the -structure. If }{\f3 exp} {\f2 had been on, the expression would have been -expanded into a polynomial. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # WEIGHT} - -${\footnote \pard\plain \sl240 \fs20 $ WEIGHT} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0861} - - K{\footnote \pard\plain \sl240 \fs20 K WEIGHT command;command} - -}{\b\f2 WEIGHT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 weight} {\f2 command is used to attach weights to kernels for asymptotic -constraints. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 weight} {\f4 }{\f3 =} {\f4 -\par -\par -}{\f2 \par - must be a REDUCE } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 , must be a -positive integer, not 0. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := (x+y)**4; \par - \par - 4 3 2 2 3 4 \par - A := X + 4*X *Y + 6*X *Y + 4*X*Y + Y \par - \par - \par -weight x=2,y=3; \par - \par -wtlevel 8; \par - \par -a; \par - \par - 4 \par - X \par - \par - \par -wtlevel 10; \par - \par -a; \par - \par - 2 2 2 \par - X *(6*Y + 4*X*Y + X ) \par - \par - \par -int(x**2,x); \par - \par - ***** X invalid as KERNEL \par - \par -\pard \sl240 }{\f2 Weights and } -{\f2\uldb wtlevel}{\v\f2 WTLEVEL} -{\f2 are used for asymptotic constraints, where -higher-order terms are considered insignificant. -\par -\par -Weights are originally equivalent to 0 until set by a }{\f3 weight} {\f2 -command. To remove a weight from a kernel, use the } -{\f2\uldb clear}{\v\f2 CLEAR} -{\f2 -command. Weights once assigned cannot be changed without clearing the -identifier. Once a weight is assigned to a kernel, it is no longer a -kernel and cannot be used in any REDUCE commands or operators that require -kernels, until the weight is cleared. Note that terms are ordered by -greatest weight. -\par -\par -The weight level of the system is set by } -{\f2\uldb wtlevel}{\v\f2 WTLEVEL} -{\f2 , initially at -2. Since no kernels have weights, no effect from }{\f3 wtlevel} {\f2 can be -seen. Once you assign weights to kernels, you must set }{\f3 wtlevel} {\f2 -correctly for the desired operation. When weighted variables appear in a -term, their weights are summed for the total weight of the term (powers of -variables multiply their weights). When a term exceeds the weight level -of the system, it is discarded from the result expression. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # WHERE} - -${\footnote \pard\plain \sl240 \fs20 $ WHERE} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0862} - - K{\footnote \pard\plain \sl240 \fs20 K substitution;WHERE operator;operator} - -}{\b\f2 WHERE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 where} {\f2 operator provides an infix notation for one-time -substitutions for kernels in expressions. - \par -syntax: \par -}{\f4 \par -\par - }{\f3 where} {\f4 - }{\f3 =} {\f4 - \{, }{\f3 =} {\f4 \}* -\par -\par -}{\f2 \par - can be any REDUCE scalar expression, must -be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 . Alternatively a } -{\f2\uldb rule}{\v\f2 RULE} -{\f2 or a }{\f3 rule list} {\f2 -can be a member of the right-hand part of a }{\f3 where} {\f2 expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x**2 + 17*x*y + 4*y**2 where x=1,y=2; \par - \par - \par - 51 \par - \par - \par -for i := 1:5 collect x**i*q where q= for j := 1:i product j; \par - \par - \par - \par - 2 3 4 5 \par - \{X,2*X ,6*X ,24*X ,120*X \} \par - \par - \par -x**2 + y + z where z=y**3,y=3; \par - \par - 2 3 \par - X + Y + 3 \par - \par -\pard \sl240 }{\f2 Substitution inside a }{\f3 where} {\f2 expression has no effect upon the values -of the kernels outside the expression. The }{\f3 where} {\f2 operator has the -lowest precedence of all the infix operators, which are lower than prefix -operators, so that the substitutions apply to the entire expression -preceding the }{\f3 where} {\f2 operator. However, }{\f3 where} {\f2 is applied -before command keywords such as }{\f3 then} {\f2 , }{\f3 repeat} {\f2 , or }{\f3 do} {\f2 . -\par -\par -A } -{\f2\uldb rule}{\v\f2 RULE} -{\f2 or a }{\f3 rule set} {\f2 in the right-hand part of the -}{\f3 where} {\f2 expression act as if the rules were activated by } -{\f2\uldb let}{\v\f2 LET} -{\f2 -immediately before the evaluation of the expression and deactivated -by } -{\f2\uldb clearrules}{\v\f2 CLEARRULES} -{\f2 immediately afterwards. -\par -\par -}{\f3 where} {\f2 gives you a natural notation for auxiliary variables in -expressions. As the second example shows, the substitute expression can be -a command to be evaluated. The substitute assignments are made in -parallel, rather than sequentially, as the last example shows. The -expression resulting from the first round of substitutions is not -reexamined to see if any further such substitutions can be made. -}{\f3 where} {\f2 can also be used to define auxiliary variables in -} -{\f2\uldb procedure}{\v\f2 PROCEDURE} -{\f2 definitions. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # WHILE} - -${\footnote \pard\plain \sl240 \fs20 $ WHILE} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0863} - - K{\footnote \pard\plain \sl240 \fs20 K loop;WHILE command;command} - -}{\b\f2 WHILE}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 while} {\f2 command causes a statement to be repeatedly executed until a -given condition is true. If the condition is initially false, the statement -is not executed at all. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 while} {\f4 }{\f3 do} {\f4 -\par -\par -}{\f2 \par - is given by a logical operator, must be a -single REDUCE statement, or a } -{\f2\uldb group}{\v\f2 group} -{\f2 (}{\f3 <<} {\f2 ...}{\f3 >>} {\f2 ) or -} -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 10; \par - \par - A := 10 \par - \par - \par -while a <= 12 do <>; \par - \par - \par - \par - 10 \par - \par - \par - 11 \par - \par - 12 \par - \par -while a < 5 do <>; \par - \par - \par - \par - nothing is printed \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # WTLEVEL} - -${\footnote \pard\plain \sl240 \fs20 $ WTLEVEL} - -+{\footnote \pard\plain \sl240 \fs20 + g9:0864} - - K{\footnote \pard\plain \sl240 \fs20 K WTLEVEL command;command} - -}{\b\f2 WTLEVEL}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -In conjunction with } -{\f2\uldb weight}{\v\f2 WEIGHT} -{\f2 , }{\f3 wtlevel} {\f2 is used to implement -asymptotic constraints. Its default value is 2. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 wtlevel} {\f4 -\par -\par -}{\f2 \par -To change the weight level, must evaluate to a positive -integer that is the greatest weight term to be retained in expressions -involving kernels with weight assignments. }{\f3 wtlevel} {\f2 returns the -new weight level. If you want the current weight level, but not -change it, say }{\f3 wtlevel nil} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -(x+y)**4; \par - \par - \par - 4 3 2 2 3 4 \par - X + 4*X *Y + 6*X *Y + 4*X*Y + Y \par - \par - \par -weight x=2,y=3; \par - \par -wtlevel 8; \par - \par -(x+y)**4; \par - \par - 4 \par - X \par - \par - \par -wtlevel 10; \par - \par -(x+y)**4; \par - \par - 2 2 2 \par - X *(6*Y + 4*X*Y + X ) \par - \par - \par -int(x**2,x); \par - \par - ***** X invalid as KERNEL \par - \par -\pard \sl240 }{\f2 }{\f3 wtlevel} {\f2 is used in conjunction with the command } -{\f2\uldb weight}{\v\f2 WEIGHT} -{\f2 to -enable asymptotic constraints. Weight of a term is computed by multiplying -the weights of each variable in it by the power to which it has been -raised, and adding the resulting weights for each variable. If the weight -of the term is greater than }{\f3 wtlevel} {\f2 , the term is dropped from the -expression, and not used in any further computation involving the -expression. -\par -\par -Once a weight has been attached to a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 , it is no longer -recognized by the system as a kernel, though still a variable. It cannot -be used in REDUCE commands and operators that need kernels. The weight -attachment can be undone with a } -{\f2\uldb clear}{\v\f2 CLEAR} -{\f2 command. }{\f3 wtlevel} {\f2 can -be changed as desired. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g9} - -${\footnote \pard\plain \sl240 \fs20 $ Declarations} - -+{\footnote \pard\plain \sl240 \fs20 + index:0009} -}{\b\f2 Declarations}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb ALGEBRAIC command} -{\v\f2 ALGEBRAIC}{\f2 \par -}{\f2 \tab}{\f2\uldb ANTISYMMETRIC declaration} -{\v\f2 ANTISYMMETRIC}{\f2 \par -}{\f2 \tab}{\f2\uldb ARRAY declaration} -{\v\f2 ARRAY}{\f2 \par -}{\f2 \tab}{\f2\uldb CLEAR command} -{\v\f2 CLEAR}{\f2 \par -}{\f2 \tab}{\f2\uldb CLEARRULES command} -{\v\f2 CLEARRULES}{\f2 \par -}{\f2 \tab}{\f2\uldb DEFINE command} -{\v\f2 DEFINE}{\f2 \par -}{\f2 \tab}{\f2\uldb DEPEND declaration} -{\v\f2 DEPEND}{\f2 \par -}{\f2 \tab}{\f2\uldb EVEN declaration} -{\v\f2 EVEN}{\f2 \par -}{\f2 \tab}{\f2\uldb FACTOR declaration} -{\v\f2 FACTOR_declaration}{\f2 \par -}{\f2 \tab}{\f2\uldb FORALL command} -{\v\f2 FORALL}{\f2 \par -}{\f2 \tab}{\f2\uldb INFIX declaration} -{\v\f2 INFIX}{\f2 \par -}{\f2 \tab}{\f2\uldb INTEGER declaration} -{\v\f2 INTEGER}{\f2 \par -}{\f2 \tab}{\f2\uldb KORDER declaration} -{\v\f2 KORDER}{\f2 \par -}{\f2 \tab}{\f2\uldb LET command} -{\v\f2 LET}{\f2 \par -}{\f2 \tab}{\f2\uldb LINEAR declaration} -{\v\f2 LINEAR}{\f2 \par -}{\f2 \tab}{\f2\uldb LINELENGTH declaration} -{\v\f2 LINELENGTH}{\f2 \par -}{\f2 \tab}{\f2\uldb LISP command} -{\v\f2 LISP}{\f2 \par -}{\f2 \tab}{\f2\uldb LISTARGP declaration} -{\v\f2 LISTARGP}{\f2 \par -}{\f2 \tab}{\f2\uldb NODEPEND declaration} -{\v\f2 NODEPEND}{\f2 \par -}{\f2 \tab}{\f2\uldb MATCH command} -{\v\f2 MATCH}{\f2 \par -}{\f2 \tab}{\f2\uldb NONCOM declaration} -{\v\f2 NONCOM}{\f2 \par -}{\f2 \tab}{\f2\uldb NONZERO declaration} -{\v\f2 NONZERO}{\f2 \par -}{\f2 \tab}{\f2\uldb ODD declaration} -{\v\f2 ODD}{\f2 \par -}{\f2 \tab}{\f2\uldb OFF command} -{\v\f2 OFF}{\f2 \par -}{\f2 \tab}{\f2\uldb ON command} -{\v\f2 ON}{\f2 \par -}{\f2 \tab}{\f2\uldb OPERATOR declaration} -{\v\f2 OPERATOR}{\f2 \par -}{\f2 \tab}{\f2\uldb ORDER declaration} -{\v\f2 ORDER}{\f2 \par -}{\f2 \tab}{\f2\uldb PRECEDENCE declaration} -{\v\f2 PRECEDENCE}{\f2 \par -}{\f2 \tab}{\f2\uldb PRECISION declaration} -{\v\f2 PRECISION}{\f2 \par -}{\f2 \tab}{\f2\uldb PRINT\_PRECISION declaration} -{\v\f2 PRINT\_PRECISION}{\f2 \par -}{\f2 \tab}{\f2\uldb REAL declaration} -{\v\f2 REAL}{\f2 \par -}{\f2 \tab}{\f2\uldb REMFAC declaration} -{\v\f2 REMFAC}{\f2 \par -}{\f2 \tab}{\f2\uldb SCALAR declaration} -{\v\f2 SCALAR}{\f2 \par -}{\f2 \tab}{\f2\uldb SCIENTIFIC\_NOTATION declaration} -{\v\f2 SCIENTIFIC\_NOTATION}{\f2 \par -}{\f2 \tab}{\f2\uldb SHARE declaration} -{\v\f2 SHARE}{\f2 \par -}{\f2 \tab}{\f2\uldb SYMBOLIC command} -{\v\f2 SYMBOLIC}{\f2 \par -}{\f2 \tab}{\f2\uldb SYMMETRIC declaration} -{\v\f2 SYMMETRIC}{\f2 \par -}{\f2 \tab}{\f2\uldb TR declaration} -{\v\f2 TR}{\f2 \par -}{\f2 \tab}{\f2\uldb UNTR declaration} -{\v\f2 UNTR}{\f2 \par -}{\f2 \tab}{\f2\uldb VARNAME declaration} -{\v\f2 VARNAME}{\f2 \par -}{\f2 \tab}{\f2\uldb WEIGHT command} -{\v\f2 WEIGHT}{\f2 \par -}{\f2 \tab}{\f2\uldb WHERE operator} -{\v\f2 WHERE}{\f2 \par -}{\f2 \tab}{\f2\uldb WHILE command} -{\v\f2 WHILE}{\f2 \par -}{\f2 \tab}{\f2\uldb WTLEVEL command} -{\v\f2 WTLEVEL}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # IN} - -${\footnote \pard\plain \sl240 \fs20 $ IN} - -+{\footnote \pard\plain \sl240 \fs20 + g10:0865} - - K{\footnote \pard\plain \sl240 \fs20 K input;IN command;command} - -}{\b\f2 IN}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 in} {\f2 command takes a list of file names and inputs each file into -the system. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 in} {\f4 \{,\}* -\par -\par -}{\f2 \par - must be in the current directory, or be a valid pathname. -If the file name is not an identifier, double quote marks (}{\f3 "} {\f2 ) are -needed around the file name. -\par -\par -A message is given if the file cannot be found, or has a mistake -in it. -\par -\par -Ending the command with a semicolon causes the file to be echoed to the -screen; ending it with a dollar sign does not echo the file. If you want -some but not all of a file echoed, turn the switch } -{\f2\uldb echo}{\v\f2 ECHO} -{\f2 on or off -in the file. -\par -\par -An efficient way to develop procedures in REDUCE is to write them into a file -using a system editor of your choice, and then input the -files into an active REDUCE session. REDUCE reparses the procedure as -it takes information from the file, overwriting the previous procedure -definition. When it accepts the procedure, it echoes its name to the screen. -Data can also be input to the system from files. -\par -\par -Files to be read in should always end in } -{\f2\uldb end}{\v\f2 END} -{\f3 ;} {\f2 to avoid -end-of-file problems. Note that this is an additional }{\f3 end;} {\f2 to any -ending procedures in the file. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INPUT} - -${\footnote \pard\plain \sl240 \fs20 $ INPUT} - -+{\footnote \pard\plain \sl240 \fs20 + g10:0866} - - K{\footnote \pard\plain \sl240 \fs20 K interactive;INPUT command;command} - -}{\b\f2 INPUT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 input} {\f2 command returns the input expression to the REDUCE numbered -prompt that is its argument. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 input} {\f4 () or }{\f3 input} {\f4 -\par -\par -\par -\par -}{\f2 must be between 1 and the current REDUCE prompt number. -\par -\par -An expression brought back by }{\f3 input} {\f2 can be reexecuted with new -values or switch settings, or used as an argument in another expression. -The command } -{\f2\uldb ws}{\v\f2 WS} -{\f2 brings back the results of a numbered REDUCE -statement. Two lists contain every input and every output statement since -the beginning of the session. If your session is very long, storage space -begins to fill up with these expressions, so it is a good idea to end the -session once in a while, saving needed expressions to files with the -} -{\f2\uldb saveas}{\v\f2 SAVEAS} -{\f2 and } -{\f2\uldb out}{\v\f2 OUT} -{\f2 commands. -\par -\par -Switch settings and } -{\f2\uldb let}{\v\f2 LET} -{\f2 statements can also be reexecuted by using -}{\f3 input} {\f2 . -\par -\par -An error message is given if a number is called for that has not yet been used. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # OUT} - -${\footnote \pard\plain \sl240 \fs20 $ OUT} - -+{\footnote \pard\plain \sl240 \fs20 + g10:0867} - - K{\footnote \pard\plain \sl240 \fs20 K open;output;OUT command;command} - -}{\b\f2 OUT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 out} {\f2 command directs output to the filename that is its argument, -until another }{\f3 out} {\f2 changes the output file, or } -{\f2\uldb shut}{\v\f2 SHUT} -{\f2 closes it. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 out} {\f4 or }{\f3 out "} {\f4 }{\f3 "} {\f4 or }{\f3 out t} {\f4 -\par -\par -}{\f2 \par - must be in the current directory, or be a valid complete -file description for your system. If the file name is not -in the current directory, quote marks are needed around the file name. -If the file already exists, a message is printed allowing you to decide -whether to supersede the contents of the file with new material. -\par -\par -To restore output to the terminal, type }{\f3 out t} {\f2 , or } -{\f2\uldb shut}{\v\f2 SHUT} -{\f2 the -file. When you use }{\f3 out t} {\f2 , the file remains available, and if you -open it again (with another }{\f3 out} {\f2 ), new material is appended rather -than overwriting. -\par -\par -To write a file using }{\f3 out} {\f2 that can be input at a later time, the -switch } -{\f2\uldb nat}{\v\f2 NAT} -{\f2 must be turned off, so that the standard linear form -is saved that can be read in by } -{\f2\uldb in}{\v\f2 IN} -{\f2 . If }{\f3 nat} {\f2 is on, exponents -are printed on the line above the expression, which causes trouble -when REDUCE tries to read the file. -\par -\par -There is a slight complication if you are using the }{\f3 out} {\f2 command from -inside a file to create another file. The } -{\f2\uldb echo}{\v\f2 ECHO} -{\f2 switch is normally -off at the top-level and on while reading files (so you can see what is -being read in). If you create a file using }{\f3 out} {\f2 at the top-level, -the result lines are printed into the file as you want them. But if you -create such a file from inside a file, the }{\f3 echo} {\f2 switch is on, and -every line is echoed, first as you typed it, then as REDUCE parsed it, and -then once more for the file. Therefore, when you create a file from -a file, you need to turn }{\f3 echo} {\f2 off explicitly before the }{\f3 out} {\f2 -command, and turn it back on when you }{\f3 shut} {\f2 the created file, so your -executing file echoes as it should. This behavior also means that as you -watch the file execute, you cannot see the lines that are being put into -the }{\f3 out} {\f2 file. As soon as you turn }{\f3 echo} {\f2 on, you can see -output again. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SHUT} - -${\footnote \pard\plain \sl240 \fs20 $ SHUT} - -+{\footnote \pard\plain \sl240 \fs20 + g10:0868} - - K{\footnote \pard\plain \sl240 \fs20 K close;output;SHUT command;command} - -}{\b\f2 SHUT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 shut} {\f2 command closes output files. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 shut} {\f4 \{,\}* -\par -\par -}{\f2 \par - must have been a file opened by } -{\f2\uldb out}{\v\f2 OUT} -{\f2 . -\par -\par -A file that has been opened by } -{\f2\uldb out}{\v\f2 OUT} -{\f2 must be }{\f3 shut} {\f2 before it is -brought in by } -{\f2\uldb in}{\v\f2 IN} -{\f2 . Files that have been opened by }{\f3 out} {\f2 should -always be }{\f3 shut} {\f2 before the end of the REDUCE session, to avoid either -loss of information or the printing of extraneous information into the file. -In most systems, terminating a session by } -{\f2\uldb bye}{\v\f2 BYE} -{\f2 closes all open -output files. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g10} - -${\footnote \pard\plain \sl240 \fs20 $ Input and Output} - -+{\footnote \pard\plain \sl240 \fs20 + index:0010} -}{\b\f2 Input and Output}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb IN command} -{\v\f2 IN}{\f2 \par -}{\f2 \tab}{\f2\uldb INPUT command} -{\v\f2 INPUT}{\f2 \par -}{\f2 \tab}{\f2\uldb OUT command} -{\v\f2 OUT}{\f2 \par -}{\f2 \tab}{\f2\uldb SHUT command} -{\v\f2 SHUT}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ACOS} - -${\footnote \pard\plain \sl240 \fs20 $ ACOS} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0869} - - K{\footnote \pard\plain \sl240 \fs20 K arccosine;ACOS operator;operator} - -}{\b\f2 ACOS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 acos} {\f2 operator returns the arccosine of its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 acos} {\f4 () or }{\f3 acos} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -acos(ab); \par - \par - ACOS(AB) \par - \par - \par -acos 15; \par - \par - ACOS(15) \par - \par - \par -df(acos(x*y),x); \par - \par - 2 2 \par - SQRT( - X *Y + 1)*Y \par - -------------------- \par - 2 2 \par - X *Y - 1 \par - \par - \par -on rounded; \par - \par -res := acos(sqrt(2)/2); \par - \par - RES := 0.785398163397 \par - \par - \par -res-pi/4; \par - \par - 0 \par - \par -\pard \sl240 }{\f2 An explicit numeric value is not given unless the switch } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is -on and the argument has an absolute numeric value less than or equal to 1. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ACOSH} - -${\footnote \pard\plain \sl240 \fs20 $ ACOSH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0870} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic arccosine;ACOSH operator;operator} - -}{\b\f2 ACOSH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -}{\f3 acosh} {\f2 represents the hyperbolic arccosine of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of -}{\f3 acosh} {\f2 is known to the system. Numerical values may also be found by -turning on the switch } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 acosh} {\f4 () or }{\f3 acosh} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -acosh a; \par - \par - ACOSH(A) \par - \par - \par -acosh(0); \par - \par - ACOSH(0) \par - \par - \par -df(acosh(a**2),a); \par - \par - 4 \par - 2*SQRT(A - 1)*A \par - ---------------- \par - 4 \par - A - 1 \par - \par - \par -int(acosh(x),x); \par - \par - INT(ACOSH(X),X) \par - \par -\pard \sl240 }{\f2 You may attach functionality by defining }{\f3 acosh} {\f2 to be the inverse of -}{\f3 cosh} {\f2 . This is done by the commands -\pard \tx3420 }{\f4 \par - put('cosh,'inverse,'acosh); \par - put('acosh,'inverse,'cosh); \par -\pard \sl240 }{\f2 \par -\par -You can write a procedure to attach integrals or other -functions to }{\f3 acosh} {\f2 . You may wish to add a check to see that its -argument is properly restricted. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ACOT} - -${\footnote \pard\plain \sl240 \fs20 $ ACOT} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0871} - - K{\footnote \pard\plain \sl240 \fs20 K arccotangent;ACOT operator;operator} - -}{\b\f2 ACOT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -}{\f3 acot} {\f2 represents the arccotangent of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of -}{\f3 acot} {\f2 is known to the system. Numerical values may also be found by -turning on the switch } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 acot} {\f4 () or }{\f3 acot} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -You can add functionality yourself with }{\f3 let} {\f2 and procedures. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ACOTH} - -${\footnote \pard\plain \sl240 \fs20 $ ACOTH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0872} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic cotangent;ACOTH operator;operator} - -}{\b\f2 ACOTH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -}{\f3 acoth} {\f2 represents the inverse hyperbolic cotangent of its argument. -It takes an arbitrary scalar expression as its argument. The derivative -of }{\f3 acoth} {\f2 is known to the system. Numerical values may also be found -by turning on the switch } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 acoth} {\f4 () or }{\f3 acoth} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, -matrix or vector expression. must be a single -identifier or begin with a prefix operator name. You can add -functionality yourself with }{\f3 let} {\f2 and procedures. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ACSC} - -${\footnote \pard\plain \sl240 \fs20 $ ACSC} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0873} - - K{\footnote \pard\plain \sl240 \fs20 K arccosecant;ACSC operator;operator} - -}{\b\f2 ACSC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 acsc} {\f2 operator returns the arccosecant of its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 acsc} {\f4 () or }{\f3 acsc} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -acsc(ab); \par - \par - ACSC(AB) \par - \par - \par -acsc 15; \par - \par - ACSC(15) \par - \par - \par -df(acsc(x*y),x); \par - \par - 2 2 \par - -SQRT(X *Y - 1) \par - ---------------- \par - 2 2 \par - X*(X *Y - 1) \par - \par - \par -on rounded; \par - \par -res := acsc(2/sqrt(3)); \par - \par - RES := 1.0471975512 \par - \par - \par -res-pi/3; \par - \par - 0 \par - \par -\pard \sl240 }{\f2 An explicit numeric value is not given unless the switch }{\f3 rounded} {\f2 is -on and the argument has an absolute numeric value less than or equal to 1. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ACSCH} - -${\footnote \pard\plain \sl240 \fs20 $ ACSCH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0874} - - K{\footnote \pard\plain \sl240 \fs20 K arccosecant;ACSCH operator;operator} - -}{\b\f2 ACSCH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 acsch} {\f2 operator returns the hyperbolic arccosecant of its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 acsch} {\f4 () or }{\f3 acsch} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -acsch(ab); \par - \par - ACSCH(AB) \par - \par - \par -acsch 15; \par - \par - ACSCH(15) \par - \par - \par -df(acsch(x*y),x); \par - \par - 2 2 \par - -SQRT(X *Y + 1) \par - ---------------- \par - 2 2 \par - X*(X *Y + 1) \par - \par - \par -on rounded; \par - \par -res := acsch(3); \par - \par - RES := 0.327450150237 \par - \par -\pard \sl240 }{\f2 An explicit numeric value is not given unless the switch }{\f3 rounded} {\f2 is -on and the argument has an absolute numeric value less than or equal to 1. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ASEC} - -${\footnote \pard\plain \sl240 \fs20 $ ASEC} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0875} - - K{\footnote \pard\plain \sl240 \fs20 K arccosecant;ASEC operator;operator} - -}{\b\f2 ASEC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 asec} {\f2 operator returns the arccosecant of its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 asec} {\f4 () or }{\f3 asec} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -asec(ab); \par - \par - ASEC(AB) \par - \par - \par -asec 15; \par - \par - ASEC(15) \par - \par - \par -df(asec(x*y),x); \par - \par - 2 2 \par - SQRT(X *Y - 1) \par - --------------- \par - 2 2 \par - X*(X *Y - 1) \par - \par - \par -on rounded; \par - \par -res := asec sqrt(2); \par - \par - RES := 0.785398163397 \par - \par - \par -res-pi/4; \par - \par - 0 \par - \par -\pard \sl240 }{\f2 An explicit numeric value is not given unless the switch }{\f3 rounded} {\f2 is -on and the argument has an absolute numeric value greater or equal to 1. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ASECH} - -${\footnote \pard\plain \sl240 \fs20 $ ASECH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0876} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic arccosecant;ASECH operator;operator} - -}{\b\f2 ASECH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -}{\f3 asech} {\f2 represents the hyperbolic arccosecant of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of -}{\f3 asech} {\f2 is known to the system. Numerical values may also be found by -turning on the switch } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 asech} {\f4 () or }{\f3 asech} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -asech a; \par - \par - ASECH(A) \par - \par - \par -asech(1); \par - \par - 0 \par - \par - \par -df(acosh(a**2),a); \par - \par - 4 \par - 2*SQRT(- A + 1) \par - ---------------- \par - 4 \par - A*(A - 1) \par - \par - \par -int(asech(x),x); \par - \par - INT(ASECH(X),X) \par - \par -\pard \sl240 }{\f2 You may attach functionality by defining }{\f3 asech} {\f2 to be the inverse of -}{\f3 sech} {\f2 . This is done by the commands -\pard \tx3420 }{\f4 \par - put('sech,'inverse,'asech); \par - put('asech,'inverse,'sech); \par -\pard \sl240 }{\f2 \par -\par -You can write a procedure to attach integrals or other -functions to }{\f3 asech} {\f2 . You may wish to add a check to see that its -argument is properly restricted. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ASIN} - -${\footnote \pard\plain \sl240 \fs20 $ ASIN} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0877} - - K{\footnote \pard\plain \sl240 \fs20 K arcsine;ASIN operator;operator} - -}{\b\f2 ASIN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 asin} {\f2 operator returns the arcsine of its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 asin} {\f4 () or }{\f3 asin} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -asin(givenangle); \par - \par - ASIN(GIVENANGLE) \par - \par - \par -asin(5); \par - \par - ASIN(5) \par - \par - \par -df(asin(2*x),x); \par - \par - 2 \par - 2*SQRT( - 4*X + 1)) \par - - -------------------- \par - 2 \par - 4*X - 1 \par - \par - \par -on rounded; \par - \par -asin .5; \par - \par - 0.523598775598 \par - \par - \par -asin(sqrt(3)); \par - \par - ASIN(1.73205080757) \par - \par - \par -asin(sqrt(3)/2); \par - \par - 1.04719755120 \par - \par -\pard \sl240 }{\f2 A numeric value is not returned by }{\f3 asin} {\f2 unless the switch -}{\f3 rounded} {\f2 is on and its argument has an absolute value less than or -equal to 1. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ASINH} - -${\footnote \pard\plain \sl240 \fs20 $ ASINH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0878} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic arcsine;ASINH operator;operator} - -}{\b\f2 ASINH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 asinh} {\f2 operator returns the hyperbolic arcsine of its argument. -The derivative of }{\f3 asinh} {\f2 and some simple transformations are known -to the system. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 asinh} {\f4 () or }{\f3 asinh} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -asinh d; \par - \par - ASINH(D) \par - \par - \par -asinh(1); \par - \par - ASINH(1) \par - \par - \par -df(asinh(2*x),x); \par - \par - 2 \par - 2*SQRT(4*X + 1)) \par - ----------------- \par - 2 \par - 4*X + 1 \par - \par -\pard \sl240 }{\f2 You may attach further functionality by defining }{\f3 asinh} {\f2 to be the -inverse of }{\f3 sinh} {\f2 . This is done by the commands -\pard \tx3420 }{\f4 \par - put('sinh,'inverse,'asinh); \par - put('asinh,'inverse,'sinh); \par -\pard \sl240 }{\f2 \par -\par -A numeric value is not returned by }{\f3 asinh} {\f2 unless the switch -}{\f3 rounded} {\f2 is on and its argument evaluates to a number. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ATAN} - -${\footnote \pard\plain \sl240 \fs20 $ ATAN} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0879} - - K{\footnote \pard\plain \sl240 \fs20 K arctangent;ATAN operator;operator} - -}{\b\f2 ATAN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 atan} {\f2 operator returns the arctangent of its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 atan} {\f4 () or }{\f3 atan} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -atan(middle); \par - \par - ATAN(MIDDLE) \par - \par - \par -on rounded; \par - \par -atan 45; \par - \par - 1.54857776147 \par - \par - \par -off rounded; \par - \par -int(atan(x),x); \par - \par - 2 \par - 2*ATAN(X)*X - LOG(X + 1) \par - ------------------------- \par - 2 \par - \par - \par -df(atan(y**2),y); \par - \par - 2*Y \par - ------- \par - 4 \par - Y + 1 \par - \par -\pard \sl240 }{\f2 A numeric value is not returned by }{\f3 atan} {\f2 unless the switch -} -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on and its argument evaluates to a number. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ATANH} - -${\footnote \pard\plain \sl240 \fs20 $ ATANH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0880} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic arctangent;ATANH operator;operator} - -}{\b\f2 ATANH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 atanh} {\f2 operator returns the hyperbolic arctangent of its argument. -The derivative of }{\f3 asinh} {\f2 and some simple transformations are known -to the system. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 atanh} {\f4 () or }{\f3 atanh} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -atanh aa; \par - \par - ATANH(AA) \par - \par - \par -atanh(1); \par - \par - ATANH(1) \par - \par - \par -df(atanh(x*y),y); \par - \par - - X \par - ---------- \par - 2 2 \par - X *Y - 1 \par - \par -\pard \sl240 }{\f2 A numeric value is not returned by }{\f3 asinh} {\f2 unless the switch -}{\f3 rounded} {\f2 is on and its argument evaluates to a number. -You may attach additional functionality by defining }{\f3 atanh} {\f2 to be the -inverse of }{\f3 tanh} {\f2 . This is done by the commands -\par -\par -\pard \tx3420 }{\f4 \par - put('tanh,'inverse,'atanh); \par - put('atanh,'inverse,'tanh); \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ATAN2} - -${\footnote \pard\plain \sl240 \fs20 $ ATAN2} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0881} - - K{\footnote \pard\plain \sl240 \fs20 K ATAN2 operator;operator} - -}{\b\f2 ATAN2}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 atan2} {\f4 (,) -\par -\par -}{\f2 \par - is any valid scalar REDUCE expression. In -} -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 mode, if a numerical value exists, }{\f3 atan2} {\f2 returns -the principal value of the arc tangent of the second argument divided by -the first in the range [-pi,+pi] radians, using the signs of both -arguments to determine the quadrant of the return value. An expression in -terms of }{\f3 atan2} {\f2 is returned in other cases. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -atan2(3,2); \par - \par - ATAN2(3,2); \par - \par - \par -on rounded; \par - \par -atan2(3,2); \par - \par - 0.982793723247 \par - \par - \par -atan2(a,b); \par - \par - ATAN2(A,B); \par - \par - \par -atan2(1,0); \par - \par - 1.57079632679 \par - \par -\pard \sl240 }{\f2 }{\f3 atan2} {\f2 returns a numeric value only if } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on. Then -}{\f3 atan2} {\f2 is calculated to the current degree of floating point precision. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COS} - -${\footnote \pard\plain \sl240 \fs20 $ COS} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0882} - - K{\footnote \pard\plain \sl240 \fs20 K COS operator;operator} - -}{\b\f2 COS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 cos} {\f2 operator returns the cosine of its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 cos} {\f4 () or }{\f3 cos} {\f4 -\par -\par -}{\f2 \par - is any valid scalar REDUCE expression, - is a single identifier or begins with a prefix -operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par - \par -cos abc; \par - \par - COS(ABC) \par - \par - \par - \par -cos(pi); \par - \par - -1 \par - \par - \par - \par -cos 4; \par - \par - COS(4) \par - \par - \par - \par -on rounded; \par - \par - \par -cos(4); \par - \par - - 0.653643620864 \par - \par - \par - \par -cos log 5; \par - \par - - 0.0386319699339 \par - \par -\pard \sl240 }{\f2 }{\f3 cos} {\f2 returns a numeric value only if } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on. Then the -cosine is calculated to the current degree of floating point precision. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COSH} - -${\footnote \pard\plain \sl240 \fs20 $ COSH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0883} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic cosine;COSH operator;operator} - -}{\b\f2 COSH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 cosh} {\f2 operator returns the hyperbolic cosine of its argument. -The derivative of }{\f3 cosh} {\f2 and some simple transformations are known -to the system. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 cosh} {\f4 () or }{\f3 cosh} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -cosh b; \par - \par - COSH(B) \par - \par - \par - \par -cosh(0); \par - \par - 1 \par - \par - \par - \par -df(cosh(x*y),x); \par - \par - SINH(X*Y)*Y \par - \par - \par - \par -int(cosh(x),x); \par - \par - SINH(X) \par - \par -\pard \sl240 }{\f2 You may attach further functionality by defining its inverse (see -} -{\f2\uldb acosh}{\v\f2 ACOSH} -{\f2 ). -A numeric value is not returned by }{\f3 cosh} {\f2 unless the switch -} -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on and its argument evaluates to a number. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COT} - -${\footnote \pard\plain \sl240 \fs20 $ COT} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0884} - - K{\footnote \pard\plain \sl240 \fs20 K COT operator;operator} - -}{\b\f2 COT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -}{\f3 cot} {\f2 represents the cotangent of its argument. It takes an arbitrary -scalar expression as its argument. The derivative of }{\f3 acot} {\f2 and some -simple properties are known to the system. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 cot} {\f4 () or }{\f3 cot} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression. -must be a single identifier or begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -cot(a)*tan(a); \par - \par - COT(A)*TAN(A)) \par - \par - \par -cot(1); \par - \par - COT(1) \par - \par - \par -df(cot(2*x),x); \par - \par - 2 \par - - 2*(COT(2*X) + 1) \par - \par -\pard \sl240 }{\f2 Numerical values of expressions involving }{\f3 cot} {\f2 may be found by -turning on the switch } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COTH} - -${\footnote \pard\plain \sl240 \fs20 $ COTH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0885} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic cotangent;COTH operator;operator} - -}{\b\f2 COTH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 coth} {\f2 operator returns the hyperbolic cotangent of its argument. -The derivative of }{\f3 coth} {\f2 and some simple transformations are known -to the system. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 coth} {\f4 () or }{\f3 coth} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression. -must be a single identifier or begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -df(coth(x*y),x); \par - \par - 2 \par - - Y*(COTH(X*Y) - 1) \par - \par - \par - \par -coth acoth z; \par - \par - Z \par - \par -\pard \sl240 }{\f2 You can write } -{\f2\uldb let}{\v\f2 LET} -{\f2 statements and procedures to add further -functionality to }{\f3 coth} {\f2 if you wish. Numerical values of expressions -involving }{\f3 coth} {\f2 may also be found by turning on the switch -} -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CSC} - -${\footnote \pard\plain \sl240 \fs20 $ CSC} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0886} - - K{\footnote \pard\plain \sl240 \fs20 K cosecant;CSC operator;operator} - -}{\b\f2 CSC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 csc} {\f2 operator returns the cosecant of its argument. -The derivative of }{\f3 csc} {\f2 and some simple transformations are known -to the system. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 csc} {\f4 () or }{\f3 csc} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression. -must be a single identifier or begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -csc(q)*sin(q); \par - \par - CSC(Q)*SIN(Q) \par - \par - \par - \par -df(csc(x*y),x); \par - \par - -COT(X*Y)*CSC(X*Y)*Y \par - \par -\pard \sl240 }{\f2 You can write } -{\f2\uldb let}{\v\f2 LET} -{\f2 statements and procedures to add further -functionality to }{\f3 csc} {\f2 if you wish. Numerical values of expressions -involving }{\f3 csc} {\f2 may also be found by turning on the switch -} -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CSCH} - -${\footnote \pard\plain \sl240 \fs20 $ CSCH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0887} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic cosecan;CSCH operator;operator} - -}{\b\f2 CSCH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 cosh} {\f2 operator returns the hyperbolic cosecant of its argument. -The derivative of }{\f3 csch} {\f2 and some simple transformations are known -to the system. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 csch} {\f4 () or }{\f3 csch} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -csch b; \par - \par - CSCH(B) \par - \par - \par - \par -csch(0); \par - \par - 0 \par - \par - \par - \par -df(csch(x*y),x); \par - \par - - COTH(X*Y)*CSCH(X*Y)*Y \par - \par - \par - \par -int(csch(x),x); \par - \par - INT(CSCH(X),X) \par - \par -\pard \sl240 }{\f2 A numeric value is not returned by }{\f3 csch} {\f2 unless the switch -} -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on and its argument evaluates to a number. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ERF} - -${\footnote \pard\plain \sl240 \fs20 $ ERF} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0888} - - K{\footnote \pard\plain \sl240 \fs20 K error function;ERF operator;operator} - -}{\b\f2 ERF}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 erf} {\f2 operator represents the error function, defined by -\par -\par -erf(x) = (2/sqrt(pi))*int(e^(-x^2),x) -\par -\par -A limited number of its properties are known to the system, including the -fact that it is an odd function. Its derivative is known, and from this, -some integrals may be computed. However, a complete integration procedure -for this operator is not currently included. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -erf(0); \par - \par - 0 \par - \par - \par -erf(-a); \par - \par - - ERF(A) \par - \par - \par -df(erf(x**2),x); \par - \par - 4*SQRT(PI)*X \par - ------------ \par - 4 \par - X \par - E *PI \par - \par - \par - \par -int(erf(x),x); \par - \par - 2 \par - X \par - E *ERF(X)*PI*X + SQRT(PI) \par - --------------------------- \par - 2 \par - X \par - E *PI \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EXP} - -${\footnote \pard\plain \sl240 \fs20 $ EXP} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0889} - - K{\footnote \pard\plain \sl240 \fs20 K exponential function;EXP operator;operator} - -}{\b\f2 EXP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 exp} {\f2 operator returns }{\f3 e} {\f2 raised to the power of its argument. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 exp} {\f4 () or }{\f3 exp} {\f4 -\par -\par -}{\f2 \par - can be any valid REDUCE scalar expression. - must be a single identifier or begin with a -prefix operator. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -exp(sin(x)); \par - \par - SIN X \par - E \par - \par - \par -exp(11); \par - \par - 11 \par - E \par - \par - \par -on rounded; \par - \par -exp sin(pi/3); \par - \par - 2.37744267524 \par - \par -\pard \sl240 }{\f2 Numeric values are returned only when }{\f3 rounded} {\f2 is on. -The single letter }{\f3 e} {\f2 with the exponential operator }{\f3 ^} {\f2 or -}{\f3 **} {\f2 may be substituted for }{\f3 exp} {\f2 without change of function. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SEC} - -${\footnote \pard\plain \sl240 \fs20 $ SEC} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0890} - - K{\footnote \pard\plain \sl240 \fs20 K SEC operator;operator} - -}{\b\f2 SEC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 sec} {\f2 operator returns the secant of its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 sec} {\f4 () or }{\f3 sec} {\f4 -\par -\par -}{\f2 \par - is any valid scalar REDUCE expression, - is a single identifier or begins with a prefix -operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par - \par -sec abc; \par - \par - SEC(ABC) \par - \par - \par - \par -sec(pi); \par - \par - -1 \par - \par - \par - \par -sec 4; \par - \par - SEC(4) \par - \par - \par - \par -on rounded; \par - \par - \par -sec(4); \par - \par - - 1.52988565647 \par - \par - \par - \par -sec log 5; \par - \par - - 25.8852966005 \par - \par -\pard \sl240 }{\f2 }{\f3 sec} {\f2 returns a numeric value only if } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on. Then the -secant is calculated to the current degree of floating point precision. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SECH} - -${\footnote \pard\plain \sl240 \fs20 $ SECH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0891} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic secant;SECH operator;operator} - -}{\b\f2 SECH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 sech} {\f2 operator returns the hyperbolic secant of its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 sech} {\f4 () or }{\f3 sech} {\f4 -\par -\par -}{\f2 \par - is any valid scalar REDUCE expression, - is a single identifier or begins with a prefix -operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -sech abc; \par - \par - SECH(ABC) \par - \par - \par - \par -sech(0); \par - \par - 1 \par - \par - \par - \par -sech 4; \par - \par - SECH(4) \par - \par - \par - \par -on rounded; \par - \par - \par -sech(4); \par - \par - 0.0366189934737 \par - \par - \par - \par -sech log 5; \par - \par - 0.384615384615 \par - \par -\pard \sl240 }{\f2 }{\f3 sech} {\f2 returns a numeric value only if } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on. Then the -expression is calculated to the current degree of floating point precision. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SIN} - -${\footnote \pard\plain \sl240 \fs20 $ SIN} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0892} - - K{\footnote \pard\plain \sl240 \fs20 K sine;SIN operator;operator} - -}{\b\f2 SIN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 sin} {\f2 operator returns the sine of its argument. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 sin} {\f4 () or }{\f3 sin} {\f4 -\par -\par -}{\f2 \par - is any valid scalar REDUCE expression, - is a single identifier or begins with a prefix -operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -sin aa; \par - \par - SIN(AA) \par - \par - \par -sin(pi/2); \par - \par - 1 \par - \par - \par -on rounded; \par - \par -sin 3; \par - \par - 0.14112000806 \par - \par - \par -sin(pi/2); \par - \par - 1.0 \par - \par -\pard \sl240 }{\f2 }{\f3 sin} {\f2 returns a numeric value only if }{\f3 rounded} {\f2 is on. -Then the sine is calculated to the current degree of floating point precision. -The argument in this case is assumed to be in radians. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SINH} - -${\footnote \pard\plain \sl240 \fs20 $ SINH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0893} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic sine;SINH operator;operator} - -}{\b\f2 SINH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 sinh} {\f2 operator returns the hyperbolic sine of its argument. -The derivative of }{\f3 sinh} {\f2 and some simple transformations are known -to the system. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 sinh} {\f4 () or }{\f3 sinh} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -sinh b; \par - \par - SINH(B) \par - \par - \par - \par -sinh(0); \par - \par - 0 \par - \par - \par -df(sinh(x**2),x); \par - \par - 2 \par - 2*COSH(X )*X \par - \par - \par -int(sinh(4*x),x); \par - \par - COSH(4*X) \par - --------- \par - 4 \par - \par - \par -on rounded; \par - \par -sinh 4; \par - \par - 27.2899171971 \par - \par -\pard \sl240 }{\f2 You may attach further functionality by defining its inverse (see -} -{\f2\uldb asinh}{\v\f2 ASINH} -{\f2 ). -A numeric value is not returned by }{\f3 sinh} {\f2 unless the switch -} -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on and its argument evaluates to a number. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TAN} - -${\footnote \pard\plain \sl240 \fs20 $ TAN} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0894} - - K{\footnote \pard\plain \sl240 \fs20 K TAN operator;operator} - -}{\b\f2 TAN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 tan} {\f2 operator returns the tangent of its argument. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 tan} {\f4 () or }{\f3 tan} {\f4 -\par -\par -\par -\par -}{\f2 is any valid scalar REDUCE expression, - is a single identifier or begins with a prefix -operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -tan a; \par - \par - TAN(A) \par - \par - \par -tan(pi/5); \par - \par - PI \par - TAN(--) \par - 5 \par - \par - \par -on rounded; \par -tan(pi/5); \par - \par - 0.726542528005 \par - \par -\pard \sl240 }{\f2 }{\f3 tan} {\f2 returns a numeric value only if }{\f3 rounded} {\f2 is on. Then the -tangent is calculated to the current degree of floating point accuracy. -\par -\par -When } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on, -no check is made to see if the argument of }{\f3 tan} {\f2 is a multiple of -}{\f4 pi/2}{\f2 , for which the tangent goes to positive or negative infinity. -(Of course, since REDUCE uses a fixed-point representation of }{\f4 pi/2}{\f2 , -it produces a large but not infinite number.) You need to make a check for -multiples of }{\f4 pi/2}{\f2 in any program you use that might possibly ask -for the tangent of such a quantity. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TANH} - -${\footnote \pard\plain \sl240 \fs20 $ TANH} - -+{\footnote \pard\plain \sl240 \fs20 + g11:0895} - - K{\footnote \pard\plain \sl240 \fs20 K hyperbolic tangent;TANH operator;operator} - -}{\b\f2 TANH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 tanh} {\f2 operator returns the hyperbolic tangent of its argument. -The derivative of }{\f3 tanh} {\f2 and some simple transformations are known -to the system. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 tanh} {\f4 () or }{\f3 tanh} {\f4 -\par -\par -}{\f2 \par - may be any scalar REDUCE expression, not an array, matrix or -vector expression. must be a single identifier or -begin with a prefix operator name. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -tanh b; \par - \par - TANH(B) \par - \par - \par -tanh(0); \par - \par - 0 \par - \par - \par -df(tanh(x*y),x); \par - \par - 2 \par - Y*( - TANH(X*Y) + 1) \par - \par - \par -int(tanh(x),x); \par - \par - 2*X \par - LOG(E + 1) - X \par - \par - \par -on rounded; tanh 2; \par - \par - 0.964027580076 \par - \par -\pard \sl240 }{\f2 You may attach further functionality by defining its inverse (see -} -{\f2\uldb atanh}{\v\f2 ATANH} -{\f2 ). -A numeric value is not returned by }{\f3 tanh} {\f2 unless the switch -} -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on and its argument evaluates to a number. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g11} - -${\footnote \pard\plain \sl240 \fs20 $ Elementary Functions} - -+{\footnote \pard\plain \sl240 \fs20 + index:0011} -}{\b\f2 Elementary Functions}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb ACOS operator} -{\v\f2 ACOS}{\f2 \par -}{\f2 \tab}{\f2\uldb ACOSH operator} -{\v\f2 ACOSH}{\f2 \par -}{\f2 \tab}{\f2\uldb ACOT operator} -{\v\f2 ACOT}{\f2 \par -}{\f2 \tab}{\f2\uldb ACOTH operator} -{\v\f2 ACOTH}{\f2 \par -}{\f2 \tab}{\f2\uldb ACSC operator} -{\v\f2 ACSC}{\f2 \par -}{\f2 \tab}{\f2\uldb ACSCH operator} -{\v\f2 ACSCH}{\f2 \par -}{\f2 \tab}{\f2\uldb ASEC operator} -{\v\f2 ASEC}{\f2 \par -}{\f2 \tab}{\f2\uldb ASECH operator} -{\v\f2 ASECH}{\f2 \par -}{\f2 \tab}{\f2\uldb ASIN operator} -{\v\f2 ASIN}{\f2 \par -}{\f2 \tab}{\f2\uldb ASINH operator} -{\v\f2 ASINH}{\f2 \par -}{\f2 \tab}{\f2\uldb ATAN operator} -{\v\f2 ATAN}{\f2 \par -}{\f2 \tab}{\f2\uldb ATANH operator} -{\v\f2 ATANH}{\f2 \par -}{\f2 \tab}{\f2\uldb ATAN2 operator} -{\v\f2 ATAN2}{\f2 \par -}{\f2 \tab}{\f2\uldb COS operator} -{\v\f2 COS}{\f2 \par -}{\f2 \tab}{\f2\uldb COSH operator} -{\v\f2 COSH}{\f2 \par -}{\f2 \tab}{\f2\uldb COT operator} -{\v\f2 COT}{\f2 \par -}{\f2 \tab}{\f2\uldb COTH operator} -{\v\f2 COTH}{\f2 \par -}{\f2 \tab}{\f2\uldb CSC operator} -{\v\f2 CSC}{\f2 \par -}{\f2 \tab}{\f2\uldb CSCH operator} -{\v\f2 CSCH}{\f2 \par -}{\f2 \tab}{\f2\uldb ERF operator} -{\v\f2 ERF}{\f2 \par -}{\f2 \tab}{\f2\uldb EXP operator} -{\v\f2 EXP}{\f2 \par -}{\f2 \tab}{\f2\uldb SEC operator} -{\v\f2 SEC}{\f2 \par -}{\f2 \tab}{\f2\uldb SECH operator} -{\v\f2 SECH}{\f2 \par -}{\f2 \tab}{\f2\uldb SIN operator} -{\v\f2 SIN}{\f2 \par -}{\f2 \tab}{\f2\uldb SINH operator} -{\v\f2 SINH}{\f2 \par -}{\f2 \tab}{\f2\uldb TAN operator} -{\v\f2 TAN}{\f2 \par -}{\f2 \tab}{\f2\uldb TANH operator} -{\v\f2 TANH}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SWITCHES} - -${\footnote \pard\plain \sl240 \fs20 $ SWITCHES} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0896} - - K{\footnote \pard\plain \sl240 \fs20 K SWITCHES introduction;introduction} - -}{\b\f2 SWITCHES}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par -\par - -Switches are set on or off using the commands } -{\f2\uldb on}{\v\f2 ON} -{\f2 or -} -{\f2\uldb off}{\v\f2 OFF} -{\f2 , respectively. -The default setting of the switches described in this section is -} -{\f2\uldb off}{\v\f2 OFF} -{\f2 unless stated otherwise. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ALGINT} - -${\footnote \pard\plain \sl240 \fs20 $ ALGINT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0897} - - K{\footnote \pard\plain \sl240 \fs20 K integration;ALGINT switch;switch} - -}{\b\f2 ALGINT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 algint} {\f2 switch is on, the algebraic integration module (which -must be loaded from the REDUCE library) is used for integration. -\par -\par -Loading }{\f3 algint} {\f2 from the library automatically turns on the -}{\f3 algint} {\f2 switch. An error message will be given if }{\f3 algint} {\f2 is -turned on when the }{\f3 algint} {\f2 has not been loaded from the library. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ALLBRANCH} - -${\footnote \pard\plain \sl240 \fs20 $ ALLBRANCH} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0898} - - K{\footnote \pard\plain \sl240 \fs20 K ALLBRANCH switch;switch} - -}{\b\f2 ALLBRANCH}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -When }{\f3 allbranch} {\f2 is on, the operator } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 selects all -branches of solutions. -When }{\f3 allbranch} {\f2 is off, it selects only the principal -branches. Default is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -solve(log(sin(x+3)),x); \par - \par - \{X=2*ARBINT(1)*PI - ASIN(1) - 3, \par - X=2*ARBINT(1)*PI + ASIN(1) + PI - 3\} \par - \par - \par -off allbranch; \par - \par -solve(log(sin(x+3)),x); \par - \par - X=ASIN(1) - 3 \par - \par -\pard \sl240 }{\f2 } -{\f2\uldb arbint}{\v\f2 ARBINT} -{\f2 (1) indicates an arbitrary integer, which is given a -unique identifier by REDUCE, showing that there are infinitely many -solutions of this type. When }{\f3 allbranch} {\f2 is off, the single -canonical solution is given. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ALLFAC} - -${\footnote \pard\plain \sl240 \fs20 $ ALLFAC} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0899} - - K{\footnote \pard\plain \sl240 \fs20 K output;ALLFAC switch;switch} - -}{\b\f2 ALLFAC}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -The }{\f3 allfac} {\f2 switch, when on, causes REDUCE to factor out automatically -common products in the output of expressions. Default is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x + x*y**3 + x**2*cos(z); \par - \par - 3 \par - X*(COS(Z)*X + Y + 1) \par - \par - \par -off allfac; \par - \par -x + x*y**3 + x**2*cos(z); \par - \par - 2 3 \par - COS(Z)*X + X*Y + X \par - \par -\pard \sl240 }{\f2 The }{\f3 allfac} {\f2 switch has no effect when }{\f3 pri} {\f2 is off. Although the -switch setting stays as it was, printing behavior is as if it were off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ARBVARS} - -${\footnote \pard\plain \sl240 \fs20 $ ARBVARS} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0900} - - K{\footnote \pard\plain \sl240 \fs20 K solve;ARBVARS switch;switch} - -}{\b\f2 ARBVARS}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 arbvars} {\f2 is on, the solutions of singular or underdetermined -systems of equations are presented in terms of arbitrary complex variables -(see } -{\f2\uldb arbcomplex}{\v\f2 ARBCOMPLEX} -{\f2 ). Otherwise, the solution is parametrized in -terms of some of the input variables. Default is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -solve(\{2x + y,4x + 2y\},\{x,y\}); \par - \par - arbcomplex(1) \par - \{\{x= - -------------,y=arbcomplex(1)\}\} \par - 2 \par - \par - \par -solve(\{sqrt(x)+ y**3-1\},\{x,y\}); \par - \par - \par - 6 3 \par - \{\{y=arbcomplex(2),x=y - 2*y + 1\}\} \par - \par - \par -off arbvars; \par - \par -solve(\{2x + y,4x + 2y\},\{x,y\}); \par - \par - y \par - \{\{x= - -\}\} \par - 2 \par - \par - \par -solve(\{sqrt(x)+ y**3-1\},\{x,y\}); \par - \par - \par - 6 3 \par - \{\{x=y - 2*y + 1\}\} \par - \par -\pard \sl240 }{\f2 With }{\f3 arbvars} {\f2 off, the return value }{\f3 \{\{\}\}} {\f2 means that the -equations given to } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 imply no relation among the input -variables. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BALANCED\_MOD} - -${\footnote \pard\plain \sl240 \fs20 $ BALANCED_MOD} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0901} - - K{\footnote \pard\plain \sl240 \fs20 K modular;BALANCED_MOD switch;switch} - -}{\b\f2 BALANCED\_MOD}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -} -{\f2\uldb modular}{\v\f2 MODULAR} -{\f2 numbers are normally produced in the range [0,...), -where - is the current modulus. With }{\f3 balanced_mod} {\f2 on, the range -[-/2,/2] is used instead. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -setmod 7; \par - \par - 1 \par - \par - \par -on modular; \par - \par -4; \par - \par - 4 \par - \par - \par -on balanced_mod; \par - \par -4; \par - \par - -3 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BFSPACE} - -${\footnote \pard\plain \sl240 \fs20 $ BFSPACE} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0902} - - K{\footnote \pard\plain \sl240 \fs20 K floating point;output;BFSPACE switch;switch} - -}{\b\f2 BFSPACE}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -Floating point numbers are normally printed in a compact notation (either -fixed point or in scientific notation if } -{\f2\uldb SCIENTIFIC_NOTATION}{\v\f2 SCIENTIFIC\_NOTATION} -{\f2 -has been used). In some (but not all) cases, it helps comprehensibility -if spaces are inserted in the number at regular intervals. The switch -}{\f3 bfspace} {\f2 , if on, will cause a blank to be inserted in the number after -every five characters. - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -1.2345678; \par - \par - 1.2345678 \par - \par - \par -on bfspace; \par - \par -1.2345678; \par - \par - 1.234 5678 \par - \par -\pard \sl240 }{\f2 \par -\par -}{\f3 bfspace} {\f2 is normally off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COMBINEEXPT} - -${\footnote \pard\plain \sl240 \fs20 $ COMBINEEXPT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0903} - - K{\footnote \pard\plain \sl240 \fs20 K exponent simplification;COMBINEEXPT switch;switch} - -}{\b\f2 COMBINEEXPT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -REDUCE is in general poor at surd simplification. However, when the -switch }{\f3 combineexpt} {\f2 is on, the system attempts to combine -exponentials whenever possible. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -3^(1/2)*3^(1/3)*3^(1/6); \par - \par - 1/3 1/6 \par - SQRT(3)*3 *3 \par - \par - \par -on combineexpt; \par - \par -ws; \par - \par - 1 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COMBINELOGS} - -${\footnote \pard\plain \sl240 \fs20 $ COMBINELOGS} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0904} - - K{\footnote \pard\plain \sl240 \fs20 K logarithm;COMBINELOGS switch;switch} - -}{\b\f2 COMBINELOGS}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -In many cases it is desirable to expand product arguments of logarithms, -or collect a sum of logarithms into a single logarithm. Since these are -inverse operations, it is not possible to provide rules for doing both at -the same time and preserve the REDUCE concept of idempotent evaluation. -As an alternative, REDUCE provides two switches } -{\f2\uldb expandlogs}{\v\f2 EXPANDLOGS} -{\f2 and -}{\f3 combinelogs} {\f2 to carry out these operations. - \par -examples: \par -\pard \tx3420 }{\f4 \par -on expandlogs; \par - \par -log(x*y); \par - \par - LOG(X) + LOG(Y) \par - \par - \par -on combinelogs; \par - \par -ws; \par - \par - LOG(X*Y) \par - \par -\pard \sl240 }{\f2 \par -\par -At the present time, it is possible to have both switches on at once, -which could lead to infinite recursion. However, an expression is -switched from one form to the other in this case. Users should not rely -on this behavior, since it may change in the next release. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COMP} - -${\footnote \pard\plain \sl240 \fs20 $ COMP} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0905} - - K{\footnote \pard\plain \sl240 \fs20 K compiler;COMP switch;switch} - -}{\b\f2 COMP}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 comp} {\f2 is on, any succeeding function definitions are compiled -into a faster-running form. Default is }{\f3 off} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 The following procedure finds Fibonacci numbers recursively. -Create a new file ``refib" in your current directory with the following -lines in it:}{\f4 \pard \tx3420 \par - \par -procedure refib(n); \par - if fixp n and n >= 0 then \par - if n <= 1 then 1 \par - else refib(n-1) + refib(n-2) \par - else rederr "nonnegative integer only"; \par - \par -end; \par -\pard \sl240 \par -\pard \sl240 }{\f2 Now load REDUCE and run the following:}{\f4 \pard \tx3420 \par - \par -on time; \par - \par - Time: 100 ms \par - \par - \par - \par -in "refib"$ \par - \par - Time: 0 ms \par - \par - \par - \par - \par - \par - REFIB \par - \par - \par - \par - \par - \par - Time: 260 ms \par - \par - \par - \par - \par - \par - Time: 20 ms \par - \par - \par - \par -refib(80); \par - \par - 37889062373143906 \par - \par - \par - \par - \par - \par - Time: 14840 ms \par - \par - \par - \par -on comp; \par - \par - Time: 80 ms \par - \par - \par - \par -in "refib"$ \par - \par - Time: 20 ms \par - \par - \par - \par - \par - \par - REFIB \par - \par - \par - \par - \par - \par - Time: 640 ms \par - \par - \par - \par -refib(80); \par - \par - 37889062373143906 \par - \par - \par - \par - \par - \par - Time: 10940 ms \par - \par -\pard \sl240 }{\f2 -\par -\par -Note that the compiled procedure runs faster. Your time messages will -differ depending upon which system you have. Compiled functions remain so -for the duration of the REDUCE session, and are then lost. They must be -recompiled if wanted in another session. With the switch } -{\f2\uldb time}{\v\f2 TIME} -{\f2 on -as shown above, the CPU time used in executing the command is returned in -milliseconds. Be careful not to leave }{\f3 comp} {\f2 on unless you want it, -as it makes the processing of procedures much slower. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COMPLEX} - -${\footnote \pard\plain \sl240 \fs20 $ COMPLEX} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0906} - - K{\footnote \pard\plain \sl240 \fs20 K complex;COMPLEX switch;switch} - -}{\b\f2 COMPLEX}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 complex} {\f2 switch is on, full complex arithmetic is used in -simplification, function evaluation, and factorization. Default is }{\f3 off} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -factorize(a**2 + b**2); \par - \par - 2 2 \par - \{A + B \} \par - \par - \par -on complex; \par - \par - \par -factorize(a**2 + b**2); \par - \par - \{A - I*B,A + I*B\} \par - \par - \par - \par -(x**2 + y**2)/(x + i*y); \par - \par - X - I*Y \par - \par - \par - \par -on rounded; \par - \par - *** Domain mode COMPLEX changed to COMPLEX_FLOAT \par - \par - \par - \par -sqrt(-17); \par - \par - 4.12310562562*I \par - \par - \par - \par -log(7*i); \par - \par - 1.94591014906 + 1.57079632679*I \par - \par -\pard \sl240 }{\f2 Complex floating-point can be done by turning on } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 in -addition to }{\f3 complex} {\f2 . With }{\f3 complex} {\f2 off however, REDUCE knows -that i is the square root of -1 but will not -carry out more complicated complex operations. If you want complex -denominators cleared by multiplication by their conjugates, turn on the -switch } -{\f2\uldb rationalize}{\v\f2 RATIONALIZE} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CREF} - -${\footnote \pard\plain \sl240 \fs20 $ CREF} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0907} - - K{\footnote \pard\plain \sl240 \fs20 K cross reference;CREF switch;switch} - -}{\b\f2 CREF}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -The switch }{\f3 cref} {\f2 invokes the CREF cross-reference program that -processes a set of procedure definitions to produce a summary of their -entry points, undefined procedures, non-local variables and so on. The -program will also check that procedures are called with a consistent -number of arguments, and print a diagnostic message otherwise. -\par -\par -The output is alphabetized on the first seven characters of each function -name. -\par -\par -To invoke the cross-reference program, }{\f3 cref} {\f2 is first turned on. -This causes the program to load and the cross-referencing process to -begin. After all the required definitions are loaded, turning }{\f3 cref} {\f2 -off will cause a cross-reference listing to be produced. -\par -\par -Algebraic procedures in REDUCE are treated as if they were symbolic, so -that algebraic constructs will actually appear as calls to symbolic -functions, such as }{\f3 aeval} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CRAMER} - -${\footnote \pard\plain \sl240 \fs20 $ CRAMER} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0908} - - K{\footnote \pard\plain \sl240 \fs20 K solve;linear system;matrix;CRAMER switch;switch} - -}{\b\f2 CRAMER}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 cramer} {\f2 switch is on, } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 inversion -and linear equation -solving (operator } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 ) is done by Cramer's rule, through exterior -multiplication. Default is }{\f3 off} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on time; \par - \par - Time: 80 ms \par - \par - \par -off output; \par - \par - Time: 100 ms \par - \par - \par -mm := mat((a,b,c,d,f),(a,a,c,f,b),(b,c,a,c,d), (c,c,a,b,f), \par - (d,a,d,e,f)); \par -\pard \sl240 \par - \par - Time: 300 ms \par - \par - \par -inverse := 1/mm; \par - \par - Time: 18460 ms \par - \par - \par -on cramer; \par - \par - Time: 80 ms \par - \par - \par -cramersinv := 1/mm; \par - \par - Time: 9260 ms \par - \par -\pard \sl240 }{\f2 Your time readings will vary depending on the REDUCE version you use. -After you invert the matrix, turn on } -{\f2\uldb output}{\v\f2 OUTPUT} -{\f2 and ask for one of -the elements of the inverse matrix, such as }{\f3 cramersinv(3,2)} {\f2 , so that -you can see the size of the expressions produced. -\par -\par -Inversion of matrices and the solution of linear equations with dense -symbolic entries in many variables is generally considerably faster with -}{\f3 cramer} {\f2 on. However, inversion of numeric-valued matrices is -slower. Consider the matrices you're inverting before deciding whether to -turn }{\f3 cramer} {\f2 on or off. A substantial portion of the time in matrix -inversion is given to formatting the results for printing. To save this -time, turn }{\f3 output} {\f2 off, as shown in this example or terminate the -expression with a dollar sign instead of a semicolon. The results are -still available to you in the workspace associated with your prompt -number, or you can assign them to an identifier for further use. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DEFN} - -${\footnote \pard\plain \sl240 \fs20 $ DEFN} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0909} - - K{\footnote \pard\plain \sl240 \fs20 K lisp;DEFN switch;switch} - -}{\b\f2 DEFN}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the switch }{\f3 defn} {\f2 is on, the Standard Lisp equivalent of the -input statement or procedure is printed, but not evaluated. Default is -}{\f3 off} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -on defn; \par - \par - \par -17/3; \par - \par - (AEVAL (LIST 'QUOTIENT 17 3)) \par - \par - \par - \par -df(sin(x),x,2); \par - \par - \par - (AEVAL (LIST 'DF (LIST 'SIN 'X) 'X 2)) \par - \par - \par -procedure coshval(a); \par - begin scalar g; \par - g := (exp(a) + exp(-a))/2; \par - return g \par - end; \par -\pard \sl240 \par - \par - (AEVAL \par - (PROGN \par - (FLAG '(COSHVAL) 'OPFN) \par - (DE COSHVAL (A) \par - (PROG (G) \par - (SETQ G \par - (AEVAL \par - (LIST \par - 'QUOTIENT \par - (LIST \par - 'PLUS \par - (LIST 'EXP A) \par - (LIST 'EXP (LIST 'MINUS A))) \par - 2))) \par - (RETURN G)))) ) \par - \par - \par - \par -coshval(1); \par - \par - (AEVAL (LIST 'COSHVAL 1)) \par - \par - \par - \par -off defn; \par - \par - \par -coshval(1); \par - \par - Declare COSHVAL operator? (Y or N) \par - \par - \par - \par -n \par - \par -procedure coshval(a); \par - begin scalar g; \par - g := (exp(a) + exp(-a))/2; \par - return g \par - end; \par -\pard \sl240 \par - \par - COSHVAL \par - \par - \par - \par -on rounded; \par - \par - \par -coshval(1); \par - \par - 1.54308063482 \par - \par -\pard \sl240 }{\f2 The above function }{\f3 coshval} {\f2 finds the hyperbolic cosine (cosh) of its -argument. When }{\f3 defn} {\f2 is on, you can see the Standard Lisp equivalent -of the function, but it is not entered into the system as shown by the -message }{\f3 Declare COSHVAL operator?} {\f2 . It must be reentered with -}{\f3 defn} {\f2 off to be recognized. This procedure is used as an example; a -more efficient procedure would eliminate the unnecessary local variable -with -\pard \tx3420 }{\f4 \par - procedure coshval(a); \par - (exp(a) + exp(-a))/2; \par -\pard \sl240 }{\f2 \par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DEMO} - -${\footnote \pard\plain \sl240 \fs20 $ DEMO} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0910} - - K{\footnote \pard\plain \sl240 \fs20 K output;interactive;DEMO switch;switch} - -}{\b\f2 DEMO}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -The }{\f3 demo} {\f2 switch is used for interactive files, causing the system -to pause after each command in the file until you type a }{\f3 Return} {\f2 . -Default is }{\f3 off} {\f2 . -\par -\par -The switch }{\f3 demo} {\f2 has no effect on top level interactive -statements. Use it when you want to slow down operations in a file so -you can see what is happening. -\par -\par -You can either include the }{\f3 on demo} {\f2 command in the file, or enter -it from the top level before bringing in any file. Unlike the -} -{\f2\uldb pause}{\v\f2 PAUSE} -{\f2 command, }{\f3 on demo} {\f2 does not permit you to interrupt -the file for questions of your own. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DFPRINT} - -${\footnote \pard\plain \sl240 \fs20 $ DFPRINT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0911} - - K{\footnote \pard\plain \sl240 \fs20 K derivative;output;DFPRINT switch;switch} - -}{\b\f2 DFPRINT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 dfprint} {\f2 is on, expressions in the differentiation operator -} -{\f2\uldb df}{\v\f2 DF} -{\f2 are printed in a more ``natural'' notation, with the -differentiation variables appearing as subscripts. In addition, if the -switch } -{\f2\uldb noarg}{\v\f2 NOARG} -{\f2 is on (the default), the arguments of the -differentiated operator are suppressed. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator f; \par - \par -df(f x,x); \par - \par - DF(F(X),X); \par - \par - \par -on dfprint; \par - \par -ws; \par - \par - F \par - X \par - \par - \par -df(f(x,y),x,y); \par - \par - F \par - Y \par - \par - \par -off noarg; \par - \par -ws; \par - \par - F(X,Y) \par - X \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DIV} - -${\footnote \pard\plain \sl240 \fs20 $ DIV} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0912} - - K{\footnote \pard\plain \sl240 \fs20 K output;DIV switch;switch} - -}{\b\f2 DIV}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 div} {\f2 is on, the system divides any simple factors found in -the denominator of an expression into the numerator. Default is }{\f3 off} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -on div; \par - \par - \par -a := x**2/y**2; \par - \par - 2 -2 \par - A := X *Y \par - \par - \par - \par -b := a/(3*z); \par - \par - 1 2 -2 -1 \par - B := -*X *Y *Z \par - 3 \par - \par - \par - \par -off div; \par - \par - \par -a; \par - \par - 2 \par - X \par - --- \par - 2 \par - Y \par - \par - \par - \par -b; \par - \par - 2 \par - X \par - ------- \par - 2 \par - 3*Y *Z \par - \par -\pard \sl240 }{\f2 The }{\f3 div} {\f2 switch only has effect when the } -{\f2\uldb pri}{\v\f2 PRI} -{\f2 switch is on. -When }{\f3 pri} {\f2 is off, regardless of the setting of }{\f3 div} {\f2 , the -printing behavior is as if }{\f3 div} {\f2 were off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ECHO} - -${\footnote \pard\plain \sl240 \fs20 $ ECHO} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0913} - - K{\footnote \pard\plain \sl240 \fs20 K output;ECHO switch;switch} - -}{\b\f2 ECHO}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -The }{\f3 echo} {\f2 switch is normally off for top-level entry, and on when files -are brought in. If }{\f3 echo} {\f2 is turned on at the top level, your input -statements are echoed to the screen (thus appearing twice). Default -}{\f3 off} {\f2 (but note default }{\f3 on} {\f2 for files). -\par -\par -If you want to display certain portions of a file and not others, use the -commands }{\f3 off echo} {\f2 and }{\f3 on echo} {\f2 inside the file. If you want -no display of the file, use the input command -\par -\par - }{\f3 in} {\f2 filename}{\f3 $} {\f2 -\par -\par -rather than using the semicolon delimiter. -\par -\par -Be careful when you use commands within a file to generate another file. -Since }{\f3 echo} {\f2 is on for files, the output file echoes input statements -(unlike its behavior from the top level). You should explicitly turn off -}{\f3 echo} {\f2 when writing output, and turn it back on when you're done. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ERRCONT} - -${\footnote \pard\plain \sl240 \fs20 $ ERRCONT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0914} - - K{\footnote \pard\plain \sl240 \fs20 K error handling;ERRCONT switch;switch} - -}{\b\f2 ERRCONT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 errcont} {\f2 switch is on, error conditions do not stop file -execution. Error messages will be printed whether }{\f3 errcont} {\f2 is on or -off. -\par -\par -Default is }{\f3 off} {\f2 . -\par -\par -The following describes what happens when an error occurs in a file under -each setting of }{\f3 errcont} {\f2 and }{\f3 int} {\f2 : -\par -\par -Both off: Message is printed and parsing continues, but no further -statements are executed; no commands from keyboard accepted except bye or -end; -\par -\par -}{\f3 errcont} {\f2 off, }{\f3 int} {\f2 on: Message is printed, and you are asked -if you wish to continue. (This is the default behavior); -\par -\par -}{\f3 errcont} {\f2 on, }{\f3 int} {\f2 off: Message is printed, and file continues -to execute without pause; -\par -\par -Both on: Message is printed, and file continues to execute without pause. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EVALLHSEQP} - -${\footnote \pard\plain \sl240 \fs20 $ EVALLHSEQP} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0915} - - K{\footnote \pard\plain \sl240 \fs20 K equation;EVALLHSEQP switch;switch} - -}{\b\f2 EVALLHSEQP}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -Under normal circumstances, the right-hand-side of an } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 -is evaluated but not the left-hand-side. This also applies to any -substitutions made by the } -{\f2\uldb sub}{\v\f2 SUB} -{\f2 operator. If both sides are to be -evaluated, the switch }{\f3 evallhseqp} {\f2 should be turned on. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EXP_switch} - -${\footnote \pard\plain \sl240 \fs20 $ EXP_switch} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0916} - - K{\footnote \pard\plain \sl240 \fs20 K simplification;EXP switch;switch} - -}{\b\f2 EXP}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 exp} {\f2 switch is on, powers and products of expressions are -expanded. Default is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -(x+1)**3; \par - \par - 3 2 \par - X + 3*X + 3*X + 1 \par - \par - \par -(a + b*i)*(c + d*i); \par - \par - A*C + A*D*I + B*C*I - B*D \par - \par - \par -off exp; \par - \par -(x+1)**3; \par - \par - 3 \par - (X + 1) \par - \par - \par -(a + b*i)*(c + d*i); \par - \par - (A + B*I)*(C + D*I) \par - \par - \par -length((x+1)**2/(y+1)); \par - \par - 2 \par - \par -\pard \sl240 }{\f2 Note that REDUCE knows that i^2 = -1. -When }{\f3 exp} {\f2 is off, equivalent expressions may not simplify to the same -form, although zero expressions still simplify to zero. Several operators -that expect a polynomial argument behave differently when }{\f3 exp} {\f2 is -off, such as } -{\f2\uldb length}{\v\f2 LENGTH} -{\f2 . Be cautious about leaving }{\f3 exp} {\f2 off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EXPANDLOGS} - -${\footnote \pard\plain \sl240 \fs20 $ EXPANDLOGS} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0917} - - K{\footnote \pard\plain \sl240 \fs20 K logarithm;EXPANDLOGS switch;switch} - -}{\b\f2 EXPANDLOGS}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -In many cases it is desirable to expand product arguments of logarithms, -or collect a sum of logarithms into a single logarithm. Since these are -inverse operations, it is not possible to provide rules for doing both at -the same time and preserve the REDUCE concept of idempotent evaluation. -As an alternative, REDUCE provides two switches }{\f3 expandlogs} {\f2 and -} -{\f2\uldb combinelogs}{\v\f2 COMBINELOGS} -{\f2 to carry out these operations. Both are off by default. - \par -examples: \par -\pard \tx3420 }{\f4 \par -on expandlogs; \par - \par -log(x*y); \par - \par - LOG(X) + LOG(Y) \par - \par - \par -on combinelogs; \par - \par -ws; \par - \par - LOG(X*Y) \par - \par -\pard \sl240 }{\f2 \par -\par -At the present time, it is possible to have both switches on at once, -which could lead to infinite recursion. However, an expression is -switched from one form to the other in this case. Users should not rely -on this behavior, since it may change in the next release. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EZGCD} - -${\footnote \pard\plain \sl240 \fs20 $ EZGCD} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0918} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;greatest common divisor;EZGCD switch;switch} - -}{\b\f2 EZGCD}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 ezgcd} {\f2 and } -{\f2\uldb gcd}{\v\f2 GCD} -{\f2 are on, greatest common divisors are -computed using the EZ GCD algorithm that uses modular arithmetic (and is -usually faster). Default is }{\f3 off} {\f2 . -\par -\par -As a side effect of the gcd calculation, the expressions involved are -factored, though not the heavy-duty factoring of } -{\f2\uldb factorize}{\v\f2 FACTORIZE} -{\f2 . The -EZ GCD algorithm was introduced in a paper by J. Moses and D.Y.Y. Yun in -, 1973, pp. 159-166. -\par -\par -Note that the } -{\f2\uldb gcd}{\v\f2 GCD} -{\f2 switch must also be on for }{\f3 ezgcd} {\f2 to have -effect. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FACTOR} - -${\footnote \pard\plain \sl240 \fs20 $ FACTOR} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0919} - - K{\footnote \pard\plain \sl240 \fs20 K output;FACTOR switch;switch} - -}{\b\f2 FACTOR}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 factor} {\f2 switch is on, input expressions and results are -automatically factored. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -on factor; \par - \par - \par -aa := 3*x**3*a + 6*x**2*y*a + 3*x**3*b + 6*x**2*y*b \par - \par -+ x*y*a + 2*y**2*a + x*y*b + 2*y**2*b; \par - \par - \par - \par - 2 \par - AA := (A + B)*(3*X + Y)*(X + 2*Y) \par - \par - \par -off factor; \par - \par -aa; \par - \par - 3 2 2 3 2 \par - 3*A*X + 6*A*X *Y + A*X*Y + 2*A*Y + 3*B*X + 6*B*X *Y \par - \par - \par -+ B*X*Y + 2*B*Y^\{2\} \par - \par -on factor; \par - \par -ab := x**2 - 2; \par - \par - 2 \par - AB := X - 2 \par - \par -\pard \sl240 }{\f2 REDUCE factors univariate and multivariate polynomials with -integer coefficients, finding any factors that also have integer coefficients. -The factoring is done by reducing multivariate problems to univariate -ones with symbolic coefficients, and then solving the univariate ones modulo -small primes. The results of these calculations are merged to -determine the factors of the original polynomial. The factorizer normally -selects evaluation points and primes using a random number generator. -Thus, the detailed factoring behavior may be different each time any -particular problem is tackled. -\par -\par -When the }{\f3 factor} {\f2 switch is turned on, the } -{\f2\uldb exp}{\v\f2 EXP} -{\f2 switch is -turned off, and when the }{\f3 factor} {\f2 switch is turned off, the -} -{\f2\uldb exp}{\v\f2 EXP} -{\f2 switch is turned on, whether it was on previously or not. -\par -\par -When the switch } -{\f2\uldb trfac}{\v\f2 TRFAC} -{\f2 is on, informative messages are generated at -each call to the factorizer. The } -{\f2\uldb trallfac}{\v\f2 TRALLFAC} -{\f2 switch causes the -production of a more verbose trace message. It takes precedence over -}{\f3 trfac} {\f2 if they are both on. -\par -\par -To factor a polynomial explicitly and store the results, use the operator -} -{\f2\uldb factorize}{\v\f2 FACTORIZE} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FAILHARD} - -${\footnote \pard\plain \sl240 \fs20 $ FAILHARD} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0920} - - K{\footnote \pard\plain \sl240 \fs20 K integration;FAILHARD switch;switch} - -}{\b\f2 FAILHARD}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 failhard} {\f2 switch is on, the integration operator } -{\f2\uldb int}{\v\f2 INT} -{\f2 -terminates with an error message if the integral cannot be done in closed -terms. -Default is off. -\par -\par -Use the }{\f3 failhard} {\f2 switch when you are dealing with complicated integrals -and want to know immediately if REDUCE was unable to handle them. The -integration operator sometimes returns a formal integration form that is -more complicated than the original expression, when it is unable to -complete the integration. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FORT} - -${\footnote \pard\plain \sl240 \fs20 $ FORT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0921} - - K{\footnote \pard\plain \sl240 \fs20 K FORTRAN;FORT switch;switch} - -}{\b\f2 FORT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 fort} {\f2 is on, output is given Fortran-compatible syntax. Default -is }{\f3 off} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on fort; \par - \par -df(sin(7*x + y),x); \par - \par - ANS=7.*COS(7*X+Y) \par - \par - \par -on rounded; \par - \par -b := log(sin(pi/5 + n*pi)); \par - \par - B=LOG(SIN(3.14159265359*N+0.628318530718)) \par - \par -\pard \sl240 }{\f2 REDUCE results can be written to a file (using } -{\f2\uldb out}{\v\f2 OUT} -{\f2 ) and used as data -by Fortran programs when }{\f3 fort} {\f2 is in effect. }{\f3 fort} {\f2 knows about -correct statement length, continuation characters, defining a symbol when -it is first used, and other Fortran details. -\par -\par -The } -{\f2\uldb GENTRAN}{\v\f2 GENTRAN} -{\f2 package offers many more possibilities than the -}{\f3 fort} {\f2 switch. It produces Fortran (or C or Ratfor) code from REDUCE -procedures or structured specifications, including facilities for producing -double precision output. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FORTUPPER} - -${\footnote \pard\plain \sl240 \fs20 $ FORTUPPER} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0922} - - K{\footnote \pard\plain \sl240 \fs20 K FORTRAN;FORTUPPER switch;switch} - -}{\b\f2 FORTUPPER}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 fortupper} {\f2 is on, any Fortran-style output appears in upper case. -Default is }{\f3 off} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on fort; \par - \par -df(sin(7*x + y),x); \par - \par - ans=7.*cos(7*x+y) \par - \par - \par -on fortupper; \par - \par -df(sin(7*x + y),x); \par - \par - ANS=7.*COS(7*X+Y) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FULLPREC} - -${\footnote \pard\plain \sl240 \fs20 $ FULLPREC} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0923} - - K{\footnote \pard\plain \sl240 \fs20 K rounded;precision;FULLPREC switch;switch} - -}{\b\f2 FULLPREC}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -Trailing zeroes of rounded numbers to the full system precision are -normally not printed. If this information is needed, for example to get a -more understandable indication of the accuracy of certain data, the switch -}{\f3 fullprec} {\f2 can be turned on. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -1/2; \par - \par - 0.5 \par - \par - \par -on fullprec; \par - \par -ws; \par - \par - 0.500000000000 \par - \par -\pard \sl240 }{\f2 This is just an output options which neither influences -the accuracy of the computation nor does it give additional -information about the precision of the results. -See also } -{\f2\uldb scientific_notation}{\v\f2 SCIENTIFIC\_NOTATION} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FULLROOTS} - -${\footnote \pard\plain \sl240 \fs20 $ FULLROOTS} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0924} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;solve;FULLROOTS switch;switch} - -}{\b\f2 FULLROOTS}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -Since roots of cubic and quartic polynomials can often be very -messy, a switch }{\f3 fullroots} {\f2 controls the production -of results in closed form. } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 will apply the -formulas for explicit forms for degrees 3 and 4 only if -}{\f3 fullroots} {\f2 is }{\f3 on} {\f2 . Otherwise the result forms -are built using } -{\f2\uldb root_of}{\v\f2 ROOT\_OF} -{\f2 . Default is }{\f3 off} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GC} - -${\footnote \pard\plain \sl240 \fs20 $ GC} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0925} - - K{\footnote \pard\plain \sl240 \fs20 K memory;GC switch;switch} - -}{\b\f2 GC}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -With the }{\f3 gc} {\f2 switch, you can turn the garbage collection messages on -or off. The form of the message depends on the particular Lisp used for -the REDUCE implementation. -\par -\par -See } -{\f2\uldb reclaim}{\v\f2 RECLAIM} -{\f2 for an explanation of garbage collection. REDUCE does -garbage collection when needed even if you have turned the notices off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GCD_switch} - -${\footnote \pard\plain \sl240 \fs20 $ GCD_switch} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0926} - - K{\footnote \pard\plain \sl240 \fs20 K rational expression;greatest common divisor;GCD switch;switch} - -}{\b\f2 GCD}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 gcd} {\f2 is on, common factors in numerators and denominators of -expressions are canceled. Default is }{\f3 off} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -(2*(f*h)**2 - f**2*g*h - (f*g)**2 - f*h**3 + f*h*g**2 \par - - h**4 + g*h**3)/(f**2*h - f**2*g - f*h**2 + 2*f*g*h \par - - f*g**2 - g*h**2 + g**2*h); \par -\pard \sl240 \par - \par - 2 2 2 2 2 2 3 3 4 \par - F *G + F *G*H - 2*F *H - F*G *H + F*H - G*H + H \par - ---------------------------------------------------- \par - 2 2 2 2 2 2 \par - F *G - F *H + F*G - 2*F*G*H + F*H - G *H + G*H \par - \par - \par -on gcd; \par - \par -ws; \par - \par - 2 \par - F*G + 2*F*H + H \par - ---------------- \par - F + G \par - \par - \par -e2 := a*c + a*d + b*c + b*d; \par - \par - E2 := A*C + A*D + B*C + B*D \par - \par - \par -off exp; \par - \par -e2; \par - \par - (A + B)*(C + D) \par - \par -\pard \sl240 }{\f2 Even with }{\f3 gcd} {\f2 off, a check is automatically made for common variable -and numerical products in the numerators and denominators of expression, -and the appropriate cancellations made. Thus the example demonstrating the -use of }{\f3 gcd} {\f2 is somewhat complicated. Note when } -{\f2\uldb exp}{\v\f2 EXP} -{\f2 is off, -}{\f3 gcd} {\f2 has the side effect of factoring the expression. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # HORNER} - -${\footnote \pard\plain \sl240 \fs20 $ HORNER} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0927} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;output;HORNER switch;switch} - -}{\b\f2 HORNER}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 horner} {\f2 switch is on, polynomial expressions are printed -in Horner's form for faster and safer numerical evaluation. Default -is }{\f3 off} {\f2 . The leading variable of the expression is selected as -Horner variable. To select the Horner variable explicitly use the -} -{\f2\uldb korder}{\v\f2 KORDER} -{\f2 declaration. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on horner; \par - \par -(13p-4q)^3; \par - \par - 3 2 \par - ( - 64)*q + p*(624*q + p*(( - 2028)*q + p*2197)) \par - \par - \par -korder q; \par - \par -ws; \par - \par - 3 2 \par - 2197*p + q*(( - 2028)*p + q*(624*p + q*(-64))) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # IFACTOR} - -${\footnote \pard\plain \sl240 \fs20 $ IFACTOR} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0928} - - K{\footnote \pard\plain \sl240 \fs20 K factorize;integer;IFACTOR switch;switch} - -}{\b\f2 IFACTOR}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 ifactor} {\f2 switch is on, any integer terms appearing as a result -of the } -{\f2\uldb factorize}{\v\f2 FACTORIZE} -{\f2 command are factored themselves into primes. Default -is }{\f3 off} {\f2 . If the argument of }{\f3 factorize} {\f2 is an integer, -}{\f3 ifactor} {\f2 has no effect, since the integer is always factored. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -factorize(4*x**2 + 28*x + 48); \par - \par - \{4,X + 3,X + 4\} \par - \par - \par -factorize(22587); \par - \par - \{3,7529\} \par - \par - \par -on ifactor; \par - \par -factorize(4*x**2 + 28*x + 48); \par - \par - \{2,2,X + 4,X + 3\} \par - \par - \par -factorize(22587); \par - \par - \{3,7529\} \par - \par -\pard \sl240 }{\f2 Constant terms that appear within nonconstant -polynomial factors are not factored. -\par -\par -The }{\f3 ifactor} {\f2 switch affects only factoring done specifically -with } -{\f2\uldb factorize}{\v\f2 FACTORIZE} -{\f2 , not on factoring done automatically when the -} -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 switch is on. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INT_switch} - -${\footnote \pard\plain \sl240 \fs20 $ INT_switch} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0929} - - K{\footnote \pard\plain \sl240 \fs20 K interactive;INT switch;switch} - -}{\b\f2 INT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -The }{\f3 int} {\f2 switch specifies an interactive mode of operation. Default -}{\f3 on} {\f2 . -\par -\par -There is no reason to turn }{\f3 int} {\f2 off during interactive calculations, -since there are no benefits to be gained. If you do have }{\f3 int} {\f2 off -while inputting a file, and REDUCE finds an error, it prints the message -``Continuing with parsing only." In this state, REDUCE accepts only -} -{\f2\uldb end}{\v\f2 END} -{\f3 ;} {\f2 or } -{\f2\uldb bye}{\v\f2 BYE} -{\f3 ;} {\f2 from the keyboard; -everything else is ignored, even the command }{\f3 on int} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INTSTR} - -${\footnote \pard\plain \sl240 \fs20 $ INTSTR} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0930} - - K{\footnote \pard\plain \sl240 \fs20 K output;INTSTR switch;switch} - -}{\b\f2 INTSTR}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -If }{\f3 intstr} {\f2 (for ``internal structure'') is on, arguments of an -operator are printed in a more structured form. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator f; \par - \par -f(2x+2y); \par - \par - F(2*X + 2*Y) \par - \par - \par -on intstr; \par - \par -ws; \par - \par - F(2*(X + Y)) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LCM} - -${\footnote \pard\plain \sl240 \fs20 $ LCM} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0931} - - K{\footnote \pard\plain \sl240 \fs20 K rational expression;LCM switch;switch} - -}{\b\f2 LCM}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -The }{\f3 lcm} {\f2 switch instructs REDUCE to compute the least common multiple -of denominators whenever rational expressions occur. Default is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -off lcm; \par - \par -z := 1/(x**2 - y**2) + 1/(x-y)**2; \par - \par - \par - \par - 2*X*(X - Y) \par - Z := ------------------------- \par - 4 3 3 4 \par - X - 2*X *Y + 2*X*Y - Y \par - \par - \par -on lcm; \par - \par -z; \par - \par - 2*X*(X - Y) \par - ------------------------- \par - 4 3 3 4 \par - X - 2*X *Y + 2*X*Y - Y \par - \par - \par -zz := 1/(x**2 - y**2) + 1/(x-y)**2; \par - \par - \par - \par - 2*X \par - ZZ := --------------------- \par - 3 2 2 3 \par - X - X *Y - X*Y + Y \par - \par - \par -on gcd; \par - \par -z; \par - \par - 2*X \par - ---------------------- \par - 3 2 2 3 \par - X - X *Y - X*Y + Y \par - \par -\pard \sl240 }{\f2 Note that }{\f3 lcm} {\f2 has effect only when rational expressions are first -combined. It does not examine existing structures for simplifications on -display. That is shown above when z is entered with -}{\f3 lcm} {\f2 off. It remains unsimplified even after }{\f3 lcm} {\f2 is turned -back on. However, a new variable containing the same expression is -simplified on entry. The switch } -{\f2\uldb gcd}{\v\f2 GCD} -{\f2 does examine existing -structures, as shown in the last example line above. -\par -\par -Full greatest common divisor calculations become expensive if work with -large rational expressions is required. A considerable savings of time -can be had if a full gcd check is made only when denominators are combined, -and only a partial check for numerators. This is the effect of the }{\f3 lcm} {\f2 -switch. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LESSSPACE} - -${\footnote \pard\plain \sl240 \fs20 $ LESSSPACE} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0932} - - K{\footnote \pard\plain \sl240 \fs20 K output;LESSSPACE switch;switch} - -}{\b\f2 LESSSPACE}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -You can turn on the switch }{\f3 lessspace} {\f2 if you want fewer -blank lines in your output. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LIMITEDFACTORS} - -${\footnote \pard\plain \sl240 \fs20 $ LIMITEDFACTORS} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0933} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;factorize;LIMITEDFACTORS switch;switch} - -}{\b\f2 LIMITEDFACTORS}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -To get limited factorization in cases where it is too expensive to use -full multivariate polynomial factorization, the switch -}{\f3 limitedfactors} {\f2 can be turned on. In that case, only ``inexpensive'' -factoring operations, such as square-free factorization, will be used -when } -{\f2\uldb factorize}{\v\f2 FACTORIZE} -{\f2 is called. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := (y-x)^2*(y^3+2x*y+5)*(y^2-3x*y+7)$ \par - \par -factorize a; \par - \par - \{ - X + Y, \par - X - Y, \par - 3 \par - 2*X*Y + Y + 5, \par - 2 \par - 3*X*Y - Y - 7\} \par - \par - \par -on limitedfactors; \par - \par -factorize a; \par - \par - \{ - X + Y, \par - X - Y, \par - 2 2 4 3 5 3 2 \par - 6*X *Y + 3*X*Y - 2*X*Y + X*Y - Y - 7*Y - 5*Y - 35\} \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LIST_switch} - -${\footnote \pard\plain \sl240 \fs20 $ LIST_switch} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0934} - - K{\footnote \pard\plain \sl240 \fs20 K LIST switch;switch} - -}{\b\f2 LIST}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -The }{\f3 list} {\f2 switch causes REDUCE to print each term in any sum on -separate lines. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x**2*(y**2 + 2*y) + x*(y**2 + z)/(2*a); \par - \par - \par - \par - 2 2 \par - X*(2*A*X*Y + 4*A*X*Y + Y +Z) \par - ------------------------------ \par - 2*A \par - \par - \par -on list; \par - \par -ws; \par - \par - 2 \par - (X*(2*A*X*Y \par - + 4*A*X*Y \par - 2 \par - + Y \par - + Z))/(2*A) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LISTARGS} - -${\footnote \pard\plain \sl240 \fs20 $ LISTARGS} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0935} - - K{\footnote \pard\plain \sl240 \fs20 K operator;argument;list;LISTARGS switch;switch} - -}{\b\f2 LISTARGS}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -If an operator other than those specifically defined for lists is given a -single argument that is a list, then the result of this operation will be -a list in which that operator is applied to each element of the list. -This process can be inhibited globally by turning on the switch -}{\f3 listargs} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -log \{a,b,c\}; \par - \par - LOG(A),LOG(B),LOG(C) \par - \par - \par -on listargs; \par - \par -log \{a,b,c\}; \par - \par - LOG(A,B,C) \par - \par -\pard \sl240 }{\f2 It is possible to inhibit such distribution for a specific operator by -using the declaration } -{\f2\uldb listargp}{\v\f2 LISTARGP} -{\f2 . In addition, if an operator has -more than one argument, no such distribution occurs, so }{\f3 listargs} {\f2 -has no effect. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MCD} - -${\footnote \pard\plain \sl240 \fs20 $ MCD} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0936} - - K{\footnote \pard\plain \sl240 \fs20 K rational expression;MCD switch;switch} - -}{\b\f2 MCD}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 mcd} {\f2 is on, sums and differences of rational expressions are put -on a common denominator. Default is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a/(x+1) + b/5; \par - \par - 5*A + B*X + B \par - ------------- \par - 5*(X + 1) \par - \par - \par -off mcd; \par - \par -a/(x+1) + b/5; \par - \par - -1 \par - (X + 1) *A + 1/5*B \par - \par - \par -1/6 + 1/7; \par - \par - 13/42 \par - \par -\pard \sl240 }{\f2 Even with }{\f3 mcd} {\f2 off, rational expressions involving only numbers are still -put over a common denominator. -\par -\par -Turning }{\f3 mcd} {\f2 off is useful when explicit negative powers are needed, -or if no greatest common divisor calculations are desired, or when -differentiating complicated rational expressions. Results when }{\f3 mcd} {\f2 -is off are no longer in canonical form, and expressions equivalent to zero -may not simplify to 0. Some operations, such as factoring cannot be done -while }{\f3 mcd} {\f2 is off. This option should therefore be used with some -caution. Turning }{\f3 mcd} {\f2 off is most valuable in intermediate parts of -a complicated calculation, and should be turned back on for the last stage. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MODULAR} - -${\footnote \pard\plain \sl240 \fs20 $ MODULAR} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0937} - - K{\footnote \pard\plain \sl240 \fs20 K modular;MODULAR switch;switch} - -}{\b\f2 MODULAR}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 modular} {\f2 is on, polynomial coefficients are reduced by the -modulus set by } -{\f2\uldb setmod}{\v\f2 SETMOD} -{\f2 . If no modulus has been set, }{\f3 modular} {\f2 -has no effect. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -setmod 2; \par - \par - 1 \par - \par - \par -on modular; \par - \par -(x+y)**2; \par - \par - 2 2 \par - X + Y \par - \par - \par -145*x**2 + 20*x**3 + 17 + 15*x*y; \par - \par - \par - \par - 2 \par - X + X*Y + 1 \par - \par -\pard \sl240 }{\f2 Modular operations are only conducted on the coefficients, not the -exponents. The modulus is not restricted to being prime. When the modulus -is prime, division by a number not relatively prime to the modulus results -in a error message. When the modulus is a composite -number, division by a power of the modulus results in an error message, but -division by an integer which is a factor of the modulus does not. -The representation of modular number can be influenced by -} -{\f2\uldb balanced_mod}{\v\f2 BALANCED\_MOD} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MSG} - -${\footnote \pard\plain \sl240 \fs20 $ MSG} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0938} - - K{\footnote \pard\plain \sl240 \fs20 K output;MSG switch;switch} - -}{\b\f2 MSG}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 msg} {\f2 is off, the printing of warning messages is suppressed. Error -messages are still printed. -\par -\par -Warning messages include those about redimensioning an } -{\f2\uldb array}{\v\f2 ARRAY} -{\f2 -or declaring an } -{\f2\uldb operator}{\v\f2 OPERATOR} -{\f2 where one is expected. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MULTIPLICITIES} - -${\footnote \pard\plain \sl240 \fs20 $ MULTIPLICITIES} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0939} - - K{\footnote \pard\plain \sl240 \fs20 K solve;MULTIPLICITIES switch;switch} - -}{\b\f2 MULTIPLICITIES}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 is applied to a set of equations with multiple roots, -solution multiplicities are normally stored in the global variable -} -{\f2\uldb root_multiplicities}{\v\f2 ROOT\_MULTIPLICITIES} -{\f2 rather than the solution list. If you want -the multiplicities explicitly displayed, the switch }{\f3 multiplicities} {\f2 -should be turned on. In this case, }{\f3 root_multiplicities} {\f2 has no value. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -solve(x^2=2x-1,x); \par - \par - X=1 \par - \par - \par -root_multiplicities; \par - \par - 2 \par - \par - \par -on multiplicities; \par - \par -solve(x^2=2x-1,x); \par - \par - X=1,X=1 \par - \par - \par -root_multiplicities; \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NAT} - -${\footnote \pard\plain \sl240 \fs20 $ NAT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0940} - - K{\footnote \pard\plain \sl240 \fs20 K output;NAT switch;switch} - -}{\b\f2 NAT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 nat} {\f2 is on, output is printed to the screen in natural form, with -raised exponents. }{\f3 nat} {\f2 should be turned off when outputting expressions -to a file for future input. Default is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -(x + y)**3; \par - \par - 3 2 2 3 \par - X + 3*X *Y + 3*X*Y + Y \par - \par - \par -off nat; \par - \par -(x + y)**3; \par - \par - X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3$ \par - \par - \par -on fort; \par - \par -(x + y)**3; \par - \par - ANS=X**3+3.*X**2*Y+3.*X*Y**2+Y**3 \par - \par -\pard \sl240 }{\f2 With }{\f3 nat} {\f2 off, a dollar sign is printed at the end of each expression. -An output file written with }{\f3 nat} {\f2 off is ready to be read into REDUCE -using the command } -{\f2\uldb in}{\v\f2 IN} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NERO} - -${\footnote \pard\plain \sl240 \fs20 $ NERO} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0941} - - K{\footnote \pard\plain \sl240 \fs20 K output;NERO switch;switch} - -}{\b\f2 NERO}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 nero} {\f2 is on, zero assignments (such as matrix elements) are not -printed. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -matrix a; \par -a := mat((1,0),(0,1)); \par - \par - A(1,1) := 1 \par - A(1,2) := 0 \par - A(2,1) := 0 \par - A(2,2) := 1 \par - \par - \par -on nero; \par - \par -a; \par - \par - MAT(1,1) := 1 \par - MAT(2,2) := 1 \par - \par - \par -a(1,2); \pard \sl240 }{\f2 nothing is printed.}{\f4 \pard \tx3420 \par - \par - \par -b := 0; \pard \sl240 }{\f2 nothing is printed.}{\f4 \pard \tx3420 \par - \par - \par -off nero; \par - \par -b := 0; \par - \par - B := 0 \par - \par -\pard \sl240 }{\f2 -\par -\par -}{\f3 nero} {\f2 is often used when dealing with large sparse matrices, to avoid -being overloaded with zero assignments. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NOARG} - -${\footnote \pard\plain \sl240 \fs20 $ NOARG} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0942} - - K{\footnote \pard\plain \sl240 \fs20 K derivative;output;NOARG switch;switch} - -}{\b\f2 NOARG}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When } -{\f2\uldb dfprint}{\v\f2 DFPRINT} -{\f2 is on, expressions in the differentiation operator -} -{\f2\uldb df}{\v\f2 DF} -{\f2 are printed in a more ``natural'' notation, with the -differentiation variables appearing as subscripts. When }{\f3 noarg} {\f2 -is on (the default), the arguments of the differentiated operator are also -suppressed. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator f; \par - \par -df(f x,x); \par - \par - DF(F(X),X); \par - \par - \par -on dfprint; \par - \par -ws; \par - \par - F \par - X \par - \par - \par -off noarg; \par - \par -ws; \par - \par - F(X) \par - X \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NOLNR} - -${\footnote \pard\plain \sl240 \fs20 $ NOLNR} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0943} - - K{\footnote \pard\plain \sl240 \fs20 K integration;NOLNR switch;switch} - -}{\b\f2 NOLNR}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 nolnr} {\f2 is on, the linear properties of the integration operator -} -{\f2\uldb int}{\v\f2 INT} -{\f2 are suppressed if the integral cannot be found in closed terms. -\par -\par -REDUCE uses the linear properties of integration to attempt to break down -an integral into manageable pieces. If an integral cannot be found in -closed terms, these pieces are returned. When the }{\f3 nolnr} {\f2 switch is off, -as many of the pieces as possible are integrated. When it is on, if any piece -fails, the rest of them remain unevaluated. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NOSPLIT} - -${\footnote \pard\plain \sl240 \fs20 $ NOSPLIT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0944} - - K{\footnote \pard\plain \sl240 \fs20 K output;NOSPLIT switch;switch} - -}{\b\f2 NOSPLIT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -Under normal circumstances, the printing routines try to break an expression -across lines at a natural point. This is a fairly expensive process. If -you are not overly concerned about where the end-of-line breaks come, you -can speed up the printing of expressions by turning off the switch -}{\f3 nosplit} {\f2 . This switch is normally on. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NUMVAL} - -${\footnote \pard\plain \sl240 \fs20 $ NUMVAL} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0945} - - K{\footnote \pard\plain \sl240 \fs20 K rounded;NUMVAL switch;switch} - -}{\b\f2 NUMVAL}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -With } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 on, elementary functions with numerical arguments -will return a numerical answer where appropriate. If you wish to inhibit -this evaluation, }{\f3 numval} {\f2 should be turned off. It is normally on. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -cos 3.4; \par - \par - - 0.966798192579 \par - \par - \par -off numval; \par - \par -cos 3.4; \par - \par - COS(3.4) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # OUTPUT} - -${\footnote \pard\plain \sl240 \fs20 $ OUTPUT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0946} - - K{\footnote \pard\plain \sl240 \fs20 K output;OUTPUT switch;switch} - -}{\b\f2 OUTPUT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 output} {\f2 is off, no output is printed from any REDUCE calculation. -The calculations have their usual effects other than printing. Default is -}{\f3 on} {\f2 . -\par -\par -Turn output }{\f3 off} {\f2 if you do not wish to see output when executing -large files, or to save the time REDUCE spends formatting large expressions -for display. Results are still available with } -{\f2\uldb ws}{\v\f2 WS} -{\f2 , or in their -assigned variables. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # OVERVIEW} - -${\footnote \pard\plain \sl240 \fs20 $ OVERVIEW} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0947} - - K{\footnote \pard\plain \sl240 \fs20 K factorize;OVERVIEW switch;switch} - -}{\b\f2 OVERVIEW}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 overview} {\f2 is on, the amount of detail reported by the factorizer -switches } -{\f2\uldb trfac}{\v\f2 TRFAC} -{\f2 and } -{\f2\uldb trallfac}{\v\f2 TRALLFAC} -{\f2 is reduced. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PERIOD} - -${\footnote \pard\plain \sl240 \fs20 $ PERIOD} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0948} - - K{\footnote \pard\plain \sl240 \fs20 K integer;output;PERIOD switch;switch} - -}{\b\f2 PERIOD}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 period} {\f2 is on, periods are added after integers in -Fortran-compatible output (when } -{\f2\uldb fort}{\v\f2 FORT} -{\f2 is on). There is no effect -when }{\f3 fort} {\f2 is off. Default is }{\f3 on} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PRECISE} - -${\footnote \pard\plain \sl240 \fs20 $ PRECISE} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0949} - - K{\footnote \pard\plain \sl240 \fs20 K square root;simplification;PRECISE switch;switch} - -}{\b\f2 PRECISE}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 precise} {\f2 switch is on, simplification of roots of even -powers returns absolute values, a more precise answer mathematically. -Default is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -sqrt(x**2); \par - \par - X \par - \par - \par -(x**2)**(1/4); \par - \par - SQRT(X) \par - \par - \par -on precise; \par - \par -sqrt(x**2); \par - \par - ABS(X) \par - \par - \par -(x**2)**(1/4); \par - \par - SQRT(ABS(X)) \par - \par -\pard \sl240 }{\f2 In many types of mathematical work, simplification of powers and surds can -proceed by the fastest means of simplifying the exponents arithmetically. -When it is important to you that the positive root be returned, turn -}{\f3 precise} {\f2 on. One situation where this is important is when graphing -square-root expressions such as sqrt(x^2+y^2) to -avoid a spike caused by REDUCE simplifying -sqrt(y^2) to y when x is -zero. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PRET} - -${\footnote \pard\plain \sl240 \fs20 $ PRET} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0950} - - K{\footnote \pard\plain \sl240 \fs20 K output;PRET switch;switch} - -}{\b\f2 PRET}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 pret} {\f2 is on, input is printed in standard REDUCE format and then -evaluated. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on pret; \par - \par - (x+1)^3; \par - \par - (x + 1)**3; \par - 3 2 \par - X + 3*X + 3*X + 1 \par - \par - \par - \par -procedure fac(n); \par - if not (fixp(n) and n>=0) \par - then rederr "Choose nonneg. integer only" \par - else for i := 0:n-1 product i+1; \par -\pard \sl240 \par - \par - procedure fac n; \par - if not (fixp n and n>=0) \par - then rederr "Choose nonneg. integer only" \par - else for i := 0:n - 1 product i + 1; \par - FAC \par - \par - \par - \par -fac 5; \par - \par - fac 5; \par - 120 \par - \par -\pard \sl240 }{\f2 Note that all input is converted to lower case except strings (which keep -the same case) all operators with a single argument have had the -parentheses removed, and all infix operators have had a space added on each -side. In addition, syntactical constructs like -}{\f3 if} {\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 are printed in a standard format. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PRI} - -${\footnote \pard\plain \sl240 \fs20 $ PRI} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0951} - - K{\footnote \pard\plain \sl240 \fs20 K output;PRI switch;switch} - -}{\b\f2 PRI}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 pri} {\f2 is on, the declarations } -{\f2\uldb order}{\v\f2 ORDER} -{\f2 and } -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 can -be used, and the switches } -{\f2\uldb allfac}{\v\f2 ALLFAC} -{\f2 , } -{\f2\uldb div}{\v\f2 DIV} -{\f2 , } -{\f2\uldb rat}{\v\f2 RAT} -{\f2 , -and } -{\f2\uldb revpri}{\v\f2 REVPRI} -{\f2 take effect when they are on. Default is }{\f3 on} {\f2 . -\par -\par -Printing of expressions is faster with }{\f3 pri} {\f2 off. The expressions are -then returned in one standard form, without any of the display options that -can be used to feature or display various parts of the expression. You can -also gain insight into REDUCE's representation of expressions with -}{\f3 pri} {\f2 off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RAISE} - -${\footnote \pard\plain \sl240 \fs20 $ RAISE} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0952} - - K{\footnote \pard\plain \sl240 \fs20 K character;input;RAISE switch;switch} - -}{\b\f2 RAISE}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 raise} {\f2 is on, lower case letters are automatically converted to -upper case on input. }{\f3 raise} {\f2 is normally on. -\par -\par -This conversion affects the internal representation of the letter, and is -independent of the case with which a letter is printed, which is normally -lower case. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RAT} - -${\footnote \pard\plain \sl240 \fs20 $ RAT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0953} - - K{\footnote \pard\plain \sl240 \fs20 K output;RAT switch;switch} - -}{\b\f2 RAT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 rat} {\f2 switch is on, and kernels have been selected to display -with the } -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 declaration, the denominator is printed with each -term rather than one common denominator at the end of an expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -(x+1)/x + x**2/sin y; \par - \par - \par - 3 \par - SIN(Y)*X + SIN(Y) + X \par - ---------------------- factor x; \par - SIN(Y)*X \par - \par - \par -(x+1)/x + x**2/sin y; \par - \par - \par - 3 \par - X + X*SIN(Y) + SIN(Y) \par - ---------------------- on rat; \par - X*SIN(Y) \par - \par - \par -(x+1)/x + x**2/sin y; \par - \par - \par - 2 \par - X -1 \par - ------ + 1 + X \par - SIN(Y) \par - \par -\pard \sl240 }{\f2 The }{\f3 rat} {\f2 switch only has effect when the } -{\f2\uldb pri}{\v\f2 PRI} -{\f2 switch is on. -When }{\f3 pri} {\f2 is off, regardless of the setting of }{\f3 rat} {\f2 , the -printing behavior is as if }{\f3 rat} {\f2 were off. }{\f3 rat} {\f2 only has -effect upon the display of expressions, not their internal form. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RATARG} - -${\footnote \pard\plain \sl240 \fs20 $ RATARG} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0954} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;rational expression;RATARG switch;switch} - -}{\b\f2 RATARG}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 ratarg} {\f2 is on, rational expressions can be given to operators -such as } -{\f2\uldb coeff}{\v\f2 COEFF} -{\f2 and } -{\f2\uldb lterm}{\v\f2 LTERM} -{\f2 that normally require -polynomials in one of their arguments. When }{\f3 ratarg} {\f2 is off, rational -expressions cause an error message. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -aa := x/y**2 + 1/x + y/x**2; \par - \par - \par - 3 2 3 \par - X + X*Y + Y \par - AA := -------------- \par - 2 2 \par - X *Y \par - \par - \par -coeff(aa,x); \par - \par - 3 2 3 \par - X + X*Y + Y \par - ***** -------------- invalid as POLYNOMIAL \par - 2 2 \par - X *Y \par - \par - \par -on ratarg; \par - \par -coeff(aa,x); \par - \par - \par - Y 1 1 \par - \{--,--,0,-----\} \par - 2 2 2 2 \par - X X X *Y \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RATIONAL} - -${\footnote \pard\plain \sl240 \fs20 $ RATIONAL} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0955} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;rational expression;RATIONAL switch;switch} - -}{\b\f2 RATIONAL}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 rational} {\f2 is on, polynomial expressions with rational coefficients -are produced. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x/2 + 3*y/4; \par - \par - 2*X + 3*Y \par - --------- \par - 4 \par - \par - \par -(x**2 + 5*x + 17)/2; \par - \par - 2 \par - X + 5*X + 17 \par - ------------- \par - 2 \par - \par - \par -on rational; \par - \par -x/2 + 3y/4; \par - \par - 1 3 \par - -*(X + -*Y) \par - 2 2 \par - \par - \par -(x**2 + 5*x + 17)/2; \par - \par - 1 2 \par - -*(X + 5*X + 17) \par - 2 \par - \par -\pard \sl240 }{\f2 By using }{\f3 rational} {\f2 , polynomial expressions with rational -coefficients can be used in some commands that expect polynomials. With -}{\f3 rational} {\f2 off, such a polynomial becomes a rational expression, with -denominator the least common multiple of the denominators of the rational -number coefficients. \par -\par - -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RATIONALIZE} - -${\footnote \pard\plain \sl240 \fs20 $ RATIONALIZE} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0956} - - K{\footnote \pard\plain \sl240 \fs20 K complex;simplification;rational expression;RATIONALIZE switch;switch} - -}{\b\f2 RATIONALIZE}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 rationalize} {\f2 switch is on, denominators of rational expressions -that contain complex numbers or root expressions are simplified by -multiplication by their conjugates. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -qq := (1+sqrt(3))/(sqrt(3)-7); \par - \par - SQRT(3) + 1 \par - QQ := ----------- \par - SQRT(3) - 7 \par - \par - \par -on rationalize; \par - \par -qq; \par - \par - - 4*SQRT(3) - 5 \par - --------------- \par - 23 \par - \par - \par -2/(4 + 6**(1/3)); \par - \par - 2/3 1/3 \par - 6 - 4*6 + 16 \par - ------------------ \par - 35 \par - \par - \par -(i-1)/(i+3); \par - \par - 2*I - 1 \par - ------- \par - 5 \par - \par - \par -off rationalize; \par - \par -(i-1)/(i+3); \par - \par - I - 1 \par - ------ \par - I + 3 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RATPRI} - -${\footnote \pard\plain \sl240 \fs20 $ RATPRI} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0957} - - K{\footnote \pard\plain \sl240 \fs20 K rational expression;output;RATPRI switch;switch} - -}{\b\f2 RATPRI}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 ratpri} {\f2 switch is on, rational expressions and fractions are -printed as two lines separated by a fraction bar, rather than in a linear -style. Default is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -3/17; \par - \par - 3 \par - -- \par - 17 \par - \par - \par -2/b + 3/y; \par - \par - 3*B + 2*Y \par - --------- \par - B*Y \par - \par - \par -off ratpri; \par - \par -3/17; \par - \par - 3/17 \par - \par - \par -2/b + 3/y; \par - \par - (3*B + 2*Y)/(B*Y) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # REVPRI} - -${\footnote \pard\plain \sl240 \fs20 $ REVPRI} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0958} - - K{\footnote \pard\plain \sl240 \fs20 K output;REVPRI switch;switch} - -}{\b\f2 REVPRI}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When the }{\f3 revpri} {\f2 switch is on, terms are printed in reverse order from -the normal printing order. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x**5 + x**2 + 18 + sqrt(y); \par - \par - 5 2 \par - SQRT(Y) + X + X + 18 \par - \par - \par -a + b + c + w; \par - \par - A + B + C + W \par - \par - \par -on revpri; \par - \par -x**5 + x**2 + 18 + sqrt(y); \par - \par - 2 5 \par - 17 + X + X + SQRT(Y) \par - \par - \par -a + b + c + w; \par - \par - W + C + B + A \par - \par -\pard \sl240 }{\f2 Turn }{\f3 revpri} {\f2 on when you want to display a polynomial in ascending -rather than descending order. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RLISP88} - -${\footnote \pard\plain \sl240 \fs20 $ RLISP88} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0959} - - K{\footnote \pard\plain \sl240 \fs20 K lisp;RLISP88 switch;switch} - -}{\b\f2 RLISP88}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -Rlisp '88 is a superset of the Rlisp that has been traditionally used for -the support of REDUCE. It is fully documented in the book Marti, J.B., -``RLISP '88: An Evolutionary Approach to Program Design and Reuse'', -World Scientific, Singapore (1993). It supports different looping -constructs from the traditional Rlisp, and treats ``-'' as a letter unless -separated by spaces. Turning on the switch }{\f3 rlisp88} {\f2 converts to -Rlisp '88 parsing conventions in symbolic mode, and enables the use of -Rlisp '88 extensions. Turning off the switch reverts to the traditional -Rlisp and the previous mode ( (} -{\f2\uldb symbolic}{\v\f2 SYMBOLIC} -{\f2 or } -{\f2\uldb algebraic}{\v\f2 ALGEBRAIC} -{\f2 ) -in force before }{\f3 rlisp88} {\f2 was turned on. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROUNDALL} - -${\footnote \pard\plain \sl240 \fs20 $ ROUNDALL} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0960} - - K{\footnote \pard\plain \sl240 \fs20 K floating point;rational expression;rounded;ROUNDALL switch;switch} - -}{\b\f2 ROUNDALL}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -In } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 mode, rational numbers are normally converted to a -floating point representation. If }{\f3 roundall} {\f2 is off, this conversion -does not occur. }{\f3 roundall} {\f2 is normally }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -1/2; \par - \par - 0.5 \par - \par - \par -off roundall; \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROUNDBF} - -${\footnote \pard\plain \sl240 \fs20 $ ROUNDBF} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0961} - - K{\footnote \pard\plain \sl240 \fs20 K ROUNDBF switch;switch} - -}{\b\f2 ROUNDBF}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -When } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on, the normal defaults cause underflows to be -converted to zero. If you really want the small number that results in -such cases, }{\f3 roundbf} {\f2 can be turned on. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -exp(-100000.1^2); \par - \par - 0 \par - \par - \par -on roundbf; \par - \par -exp(-100000.1^2); \par - \par - 1.18441281937E-4342953505 \par - \par -\pard \sl240 }{\f2 If a polynomial is input in } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 mode at the default -precision into any } -{\f2\uldb roots}{\v\f2 ROOTS} -{\f2 function, and it is not possible to -represent any of the coefficients of the polynomial precisely in the -system floating point representation, the switch }{\f3 roundbf} {\f2 will be -automatically turned on. All rounded computation will use the internal -bigfloat representation until the user subsequently turns }{\f3 roundbf} {\f2 -off. (A message is output to indicate that this condition is in effect.) -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROUNDED} - -${\footnote \pard\plain \sl240 \fs20 $ ROUNDED} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0962} - - K{\footnote \pard\plain \sl240 \fs20 K floating point;ROUNDED switch;switch} - -}{\b\f2 ROUNDED}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 rounded} {\f2 is on, floating-point arithmetic is enabled, with -precision initially at a system default value, which is usually 12 digits. -The precise number can be found by the command } -{\f2\uldb precision}{\v\f2 PRECISION} -{\f2 (0). - \par -examples: \par -\pard \tx3420 }{\f4 \par -pi; \par - \par - PI \par - \par - \par -35/217; \par - \par - 5 \par - -- \par - 31 \par - \par - \par -on rounded; \par - \par -pi; \par - \par - 3.14159265359 \par - \par - \par -35/217; \par - \par - 0.161 \par - \par - \par -sqrt(3); \par - \par - 1.73205080756 \par - \par -\pard \sl240 }{\f2 \par -\par -If more than the default number of decimal places are required, use the -} -{\f2\uldb precision}{\v\f2 PRECISION} -{\f2 command to set the required number. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SAVESTRUCTR} - -${\footnote \pard\plain \sl240 \fs20 $ SAVESTRUCTR} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0963} - - K{\footnote \pard\plain \sl240 \fs20 K STRUCTR OPERATOR;SAVESTRUCTR switch;switch} - -}{\b\f2 SAVESTRUCTR}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 savestructr} {\f2 is on, results of the } -{\f2\uldb structr}{\v\f2 STRUCTR} -{\f2 command are -returned as a list whose first element is the representation for the -expression and the remaining elements are equations showing the -relationships of the generated variables. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -off exp; \par - \par -structr((x+y)^3 + sin(x)^2); \par - \par - ANS3 \par - where \par - 3 2 \par - ANS3 := ANS1 + ANS2 \par - ANS2 := SIN(X) \par - ANS1 := X + Y \par - \par - \par -ans3; \par - \par - ANS3 \par - \par - \par -on savestructr; \par - \par -structr((x+y)^\{3\} + sin(x)^\{2\}); \par - \par - 3 2 \par - ANS3,ANS3=ANS1 + ANS2 ,ANS2=SIN(X),ANS1=X + Y \par - \par - \par -ans3 where rest ws; \par - \par - 3 2 \par - (X + Y) + SIN(X) \par - \par -\pard \sl240 }{\f2 In normal operation, } -{\f2\uldb structr}{\v\f2 STRUCTR} -{\f2 is only a display command. With -}{\f3 savestructr} {\f2 on, you can access the various parts of the expression -produced by }{\f3 structr} {\f2 . -\par -\par -The generic system names use the stem }{\f3 ANS} {\f2 . You can change this to your -own stem by the command } -{\f2\uldb varname}{\v\f2 VARNAME} -{\f2 . REDUCE adds integers to this stem -to make unique identifiers. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SOLVESINGULAR} - -${\footnote \pard\plain \sl240 \fs20 $ SOLVESINGULAR} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0964} - - K{\footnote \pard\plain \sl240 \fs20 K solve;SOLVESINGULAR switch;switch} - -}{\b\f2 SOLVESINGULAR}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 solvesingular} {\f2 is on, singular or underdetermined systems of -linear equations are solved, using arbitrary real, complex or integer -variables in the answer. Default is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -solve(\{2x + y,4x + 2y\},\{x,y\}); \par - \par - ARBCOMPLEX(1) \par - \{\{X= - -------------,Y=ARBCOMPLEX(1)\}\} \par - 2 \par - \par - \par -solve(\{7x + 15y - z,x - y - z\},\{x,y,z\}); \par - \par - \par - 8*ARBCOMPLEX(3) \par - \{\{X=---------------- \par - 11 \par - 3*ARBCOMPLEX(3) \par - Y= - ---------------- \par - 11 \par - Z=ARBCOMPLEX(3)\}\} \par - \par - \par -off solvesingular; \par - \par -solve(\{2x + y,4x + 2y\},\{x,y\}); \par - \par - ***** SOLVE given singular equations \par - \par - \par -solve(\{7x + 15y - z,x - y - z\},\{x,y,z\}); \par - \par - \par - ***** SOLVE given singular equations \par - \par -\pard \sl240 }{\f2 The integer following the identifier } -{\f2\uldb arbcomplex}{\v\f2 ARBCOMPLEX} -{\f2 above is assigned by -the system, and serves to identify the variable uniquely. It has no other -significance. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TIME} - -${\footnote \pard\plain \sl240 \fs20 $ TIME} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0965} - - K{\footnote \pard\plain \sl240 \fs20 K time;TIME switch;switch} - -}{\b\f2 TIME}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 time} {\f2 is on, the system time used in executing each REDUCE -statement is printed after the answer is printed. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on time; \par - \par - Time: 4940 ms \par - \par - \par -df(sin(x**2 + y),y); \par - \par - 2 \par - COS(X + Y ) \par - Time: 180 ms \par - \par - \par -solve(x**2 - 6*y,x); \par - \par - \{X= - SQRT(Y)*SQRT(6), \par - X=SQRT(Y)*SQRT(6)\} \par - Time: 320 ms \par - \par -\pard \sl240 }{\f2 When }{\f3 time} {\f2 is first turned on, the time since the beginning of the -REDUCE session is printed. After that, the time used in computation, -(usually in milliseconds, though this is system dependent) is printed after -the results of each command. Idle time or time spent typing in commands is -not counted. If }{\f3 time} {\f2 is turned off, the first reading after it is -turned on again gives the time elapsed since it was turned off. The time -printed is CPU or wall clock time, depending on the system. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TRALLFAC} - -${\footnote \pard\plain \sl240 \fs20 $ TRALLFAC} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0966} - - K{\footnote \pard\plain \sl240 \fs20 K factorize;TRALLFAC switch;switch} - -}{\b\f2 TRALLFAC}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 trallfac} {\f2 is on, a more detailed trace of factorizer calls is -generated. -\par -\par -The }{\f3 trallfac} {\f2 switch takes precedence over } -{\f2\uldb trfac}{\v\f2 TRFAC} -{\f2 if they are -both on. }{\f3 trfac} {\f2 gives a factorization trace with less detail in it. -When the } -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 switch is on also, all input polynomials are sent to -the factorizer automatically and trace information is generated. The -} -{\f2\uldb out}{\v\f2 OUT} -{\f2 command saves the results of the factoring, but not the trace. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TRFAC} - -${\footnote \pard\plain \sl240 \fs20 $ TRFAC} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0967} - - K{\footnote \pard\plain \sl240 \fs20 K factorize;TRFAC switch;switch} - -}{\b\f2 TRFAC}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 trfac} {\f2 is on, a narrative trace of any calls to the factorizer is -generated. Default is }{\f3 off} {\f2 . -\par -\par -When the switch } -{\f2\uldb factor}{\v\f2 FACTOR} -{\f2 is on, and }{\f3 trfac} {\f2 is on, every input -polynomial is sent to the factorizer, and a trace generated. With -}{\f3 factor} {\f2 off, only polynomials that are explicitly factored with the -command } -{\f2\uldb factorize}{\v\f2 FACTORIZE} -{\f2 generate trace information. -\par -\par -The } -{\f2\uldb out}{\v\f2 OUT} -{\f2 command saves the results of the factoring, but not -the trace. The } -{\f2\uldb trallfac}{\v\f2 TRALLFAC} -{\f2 switch gives trace information to a -greater level of detail. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TRIGFORM} - -${\footnote \pard\plain \sl240 \fs20 $ TRIGFORM} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0968} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;solve;TRIGFORM switch;switch} - -}{\b\f2 TRIGFORM}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When } -{\f2\uldb fullroots}{\v\f2 FULLROOTS} -{\f2 is on, } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 will compute the -roots of a cubic or quartic polynomial is closed form. When -}{\f3 trigform} {\f2 is on, the roots will be expressed by trigonometric -forms. Otherwise nested surds are used. Default is }{\f3 on} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TRINT} - -${\footnote \pard\plain \sl240 \fs20 $ TRINT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0969} - - K{\footnote \pard\plain \sl240 \fs20 K integration;TRINT switch;switch} - -}{\b\f2 TRINT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 trint} {\f2 is on, a narrative tracing various steps in the -integration process is produced. -\par -\par -The } -{\f2\uldb out}{\v\f2 OUT} -{\f2 command saves the results of the integration, but not the -trace. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TRNONLNR} - -${\footnote \pard\plain \sl240 \fs20 $ TRNONLNR} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0970} - - K{\footnote \pard\plain \sl240 \fs20 K solve;TRNONLNR switch;switch} - -}{\b\f2 TRNONLNR}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 trnonlnr} {\f2 is on, a narrative tracing various steps in -the process for solving non-linear equations is produced. -\par -\par -}{\f3 trnonlnr} {\f2 can only be used after the solve package has been loaded -(e.g., by an explicit call of } -{\f2\uldb load_package}{\v\f2 LOAD\_PACKAGE} -{\f2 ). The } -{\f2\uldb out}{\v\f2 OUT} -{\f2 -command saves the results of the equation solving, but not the trace. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # VAROPT} - -${\footnote \pard\plain \sl240 \fs20 $ VAROPT} - -+{\footnote \pard\plain \sl240 \fs20 + g12:0971} - - K{\footnote \pard\plain \sl240 \fs20 K solve;VAROPT switch;switch} - -}{\b\f2 VAROPT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - \par -\par -When }{\f3 varopt} {\f2 is on, the sequence of variables is optimized by -} -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 with respect to execution speed. Otherwise, the sequence -given in the call to } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 is preserved. Default is }{\f3 on} {\f2 . -\par -\par -In combination with the switch } -{\f2\uldb arbvars}{\v\f2 ARBVARS} -{\f2 , }{\f3 varopt} {\f2 can be used -to control variable elimination. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -off arbvars; \par - \par -solve(\{x+2z,x-3y\},\{x,y,z\}); \par - \par - x x \par - \{\{y=-,z= - -\}\} \par - 3 2 \par - \par - \par -solve(\{x*y=1,z=x\},\{x,y,z\}); \par - \par - 1 \par - \{\{z=x,y=-\}\} \par - x \par - \par - \par -off varopt; \par - \par -solve(\{x+2z,x-3y\},\{x,y,z\}); \par - \par - 2*z \par - \{\{x= - 2*z,y= - ---\}\} \par - 3 \par - \par - \par -solve(\{x*y=1,z=x\},\{x,y,z\}); \par - \par - 1 \par - \{\{y=-,x=z\}\} \par - z \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g12} - -${\footnote \pard\plain \sl240 \fs20 $ General Switches} - -+{\footnote \pard\plain \sl240 \fs20 + index:0012} -}{\b\f2 General Switches}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb SWITCHES introduction} -{\v\f2 SWITCHES}{\f2 \par -}{\f2 \tab}{\f2\uldb ALGINT switch} -{\v\f2 ALGINT}{\f2 \par -}{\f2 \tab}{\f2\uldb ALLBRANCH switch} -{\v\f2 ALLBRANCH}{\f2 \par -}{\f2 \tab}{\f2\uldb ALLFAC switch} -{\v\f2 ALLFAC}{\f2 \par -}{\f2 \tab}{\f2\uldb ARBVARS switch} -{\v\f2 ARBVARS}{\f2 \par -}{\f2 \tab}{\f2\uldb BALANCED\_MOD switch} -{\v\f2 BALANCED\_MOD}{\f2 \par -}{\f2 \tab}{\f2\uldb BFSPACE switch} -{\v\f2 BFSPACE}{\f2 \par -}{\f2 \tab}{\f2\uldb COMBINEEXPT switch} -{\v\f2 COMBINEEXPT}{\f2 \par -}{\f2 \tab}{\f2\uldb COMBINELOGS switch} -{\v\f2 COMBINELOGS}{\f2 \par -}{\f2 \tab}{\f2\uldb COMP switch} -{\v\f2 COMP}{\f2 \par -}{\f2 \tab}{\f2\uldb COMPLEX switch} -{\v\f2 COMPLEX}{\f2 \par -}{\f2 \tab}{\f2\uldb CREF switch} -{\v\f2 CREF}{\f2 \par -}{\f2 \tab}{\f2\uldb CRAMER switch} -{\v\f2 CRAMER}{\f2 \par -}{\f2 \tab}{\f2\uldb DEFN switch} -{\v\f2 DEFN}{\f2 \par -}{\f2 \tab}{\f2\uldb DEMO switch} -{\v\f2 DEMO}{\f2 \par -}{\f2 \tab}{\f2\uldb DFPRINT switch} -{\v\f2 DFPRINT}{\f2 \par -}{\f2 \tab}{\f2\uldb DIV switch} -{\v\f2 DIV}{\f2 \par -}{\f2 \tab}{\f2\uldb ECHO switch} -{\v\f2 ECHO}{\f2 \par -}{\f2 \tab}{\f2\uldb ERRCONT switch} -{\v\f2 ERRCONT}{\f2 \par -}{\f2 \tab}{\f2\uldb EVALLHSEQP switch} -{\v\f2 EVALLHSEQP}{\f2 \par -}{\f2 \tab}{\f2\uldb EXP switch} -{\v\f2 EXP_switch}{\f2 \par -}{\f2 \tab}{\f2\uldb EXPANDLOGS switch} -{\v\f2 EXPANDLOGS}{\f2 \par -}{\f2 \tab}{\f2\uldb EZGCD switch} -{\v\f2 EZGCD}{\f2 \par -}{\f2 \tab}{\f2\uldb FACTOR switch} -{\v\f2 FACTOR}{\f2 \par -}{\f2 \tab}{\f2\uldb FAILHARD switch} -{\v\f2 FAILHARD}{\f2 \par -}{\f2 \tab}{\f2\uldb FORT switch} -{\v\f2 FORT}{\f2 \par -}{\f2 \tab}{\f2\uldb FORTUPPER switch} -{\v\f2 FORTUPPER}{\f2 \par -}{\f2 \tab}{\f2\uldb FULLPREC switch} -{\v\f2 FULLPREC}{\f2 \par -}{\f2 \tab}{\f2\uldb FULLROOTS switch} -{\v\f2 FULLROOTS}{\f2 \par -}{\f2 \tab}{\f2\uldb GC switch} -{\v\f2 GC}{\f2 \par -}{\f2 \tab}{\f2\uldb GCD switch} -{\v\f2 GCD_switch}{\f2 \par -}{\f2 \tab}{\f2\uldb HORNER switch} -{\v\f2 HORNER}{\f2 \par -}{\f2 \tab}{\f2\uldb IFACTOR switch} -{\v\f2 IFACTOR}{\f2 \par -}{\f2 \tab}{\f2\uldb INT switch} -{\v\f2 INT_switch}{\f2 \par -}{\f2 \tab}{\f2\uldb INTSTR switch} -{\v\f2 INTSTR}{\f2 \par -}{\f2 \tab}{\f2\uldb LCM switch} -{\v\f2 LCM}{\f2 \par -}{\f2 \tab}{\f2\uldb LESSSPACE switch} -{\v\f2 LESSSPACE}{\f2 \par -}{\f2 \tab}{\f2\uldb LIMITEDFACTORS switch} -{\v\f2 LIMITEDFACTORS}{\f2 \par -}{\f2 \tab}{\f2\uldb LIST switch} -{\v\f2 LIST_switch}{\f2 \par -}{\f2 \tab}{\f2\uldb LISTARGS switch} -{\v\f2 LISTARGS}{\f2 \par -}{\f2 \tab}{\f2\uldb MCD switch} -{\v\f2 MCD}{\f2 \par -}{\f2 \tab}{\f2\uldb MODULAR switch} -{\v\f2 MODULAR}{\f2 \par -}{\f2 \tab}{\f2\uldb MSG switch} -{\v\f2 MSG}{\f2 \par -}{\f2 \tab}{\f2\uldb MULTIPLICITIES switch} -{\v\f2 MULTIPLICITIES}{\f2 \par -}{\f2 \tab}{\f2\uldb NAT switch} -{\v\f2 NAT}{\f2 \par -}{\f2 \tab}{\f2\uldb NERO switch} -{\v\f2 NERO}{\f2 \par -}{\f2 \tab}{\f2\uldb NOARG switch} -{\v\f2 NOARG}{\f2 \par -}{\f2 \tab}{\f2\uldb NOLNR switch} -{\v\f2 NOLNR}{\f2 \par -}{\f2 \tab}{\f2\uldb NOSPLIT switch} -{\v\f2 NOSPLIT}{\f2 \par -}{\f2 \tab}{\f2\uldb NUMVAL switch} -{\v\f2 NUMVAL}{\f2 \par -}{\f2 \tab}{\f2\uldb OUTPUT switch} -{\v\f2 OUTPUT}{\f2 \par -}{\f2 \tab}{\f2\uldb OVERVIEW switch} -{\v\f2 OVERVIEW}{\f2 \par -}{\f2 \tab}{\f2\uldb PERIOD switch} -{\v\f2 PERIOD}{\f2 \par -}{\f2 \tab}{\f2\uldb PRECISE switch} -{\v\f2 PRECISE}{\f2 \par -}{\f2 \tab}{\f2\uldb PRET switch} -{\v\f2 PRET}{\f2 \par -}{\f2 \tab}{\f2\uldb PRI switch} -{\v\f2 PRI}{\f2 \par -}{\f2 \tab}{\f2\uldb RAISE switch} -{\v\f2 RAISE}{\f2 \par -}{\f2 \tab}{\f2\uldb RAT switch} -{\v\f2 RAT}{\f2 \par -}{\f2 \tab}{\f2\uldb RATARG switch} -{\v\f2 RATARG}{\f2 \par -}{\f2 \tab}{\f2\uldb RATIONAL switch} -{\v\f2 RATIONAL}{\f2 \par -}{\f2 \tab}{\f2\uldb RATIONALIZE switch} -{\v\f2 RATIONALIZE}{\f2 \par -}{\f2 \tab}{\f2\uldb RATPRI switch} -{\v\f2 RATPRI}{\f2 \par -}{\f2 \tab}{\f2\uldb REVPRI switch} -{\v\f2 REVPRI}{\f2 \par -}{\f2 \tab}{\f2\uldb RLISP88 switch} -{\v\f2 RLISP88}{\f2 \par -}{\f2 \tab}{\f2\uldb ROUNDALL switch} -{\v\f2 ROUNDALL}{\f2 \par -}{\f2 \tab}{\f2\uldb ROUNDBF switch} -{\v\f2 ROUNDBF}{\f2 \par -}{\f2 \tab}{\f2\uldb ROUNDED switch} -{\v\f2 ROUNDED}{\f2 \par -}{\f2 \tab}{\f2\uldb SAVESTRUCTR switch} -{\v\f2 SAVESTRUCTR}{\f2 \par -}{\f2 \tab}{\f2\uldb SOLVESINGULAR switch} -{\v\f2 SOLVESINGULAR}{\f2 \par -}{\f2 \tab}{\f2\uldb TIME switch} -{\v\f2 TIME}{\f2 \par -}{\f2 \tab}{\f2\uldb TRALLFAC switch} -{\v\f2 TRALLFAC}{\f2 \par -}{\f2 \tab}{\f2\uldb TRFAC switch} -{\v\f2 TRFAC}{\f2 \par -}{\f2 \tab}{\f2\uldb TRIGFORM switch} -{\v\f2 TRIGFORM}{\f2 \par -}{\f2 \tab}{\f2\uldb TRINT switch} -{\v\f2 TRINT}{\f2 \par -}{\f2 \tab}{\f2\uldb TRNONLNR switch} -{\v\f2 TRNONLNR}{\f2 \par -}{\f2 \tab}{\f2\uldb VAROPT switch} -{\v\f2 VAROPT}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COFACTOR} - -${\footnote \pard\plain \sl240 \fs20 $ COFACTOR} - -+{\footnote \pard\plain \sl240 \fs20 + g13:0972} - - K{\footnote \pard\plain \sl240 \fs20 K matrix;COFACTOR operator;operator} - -}{\b\f2 COFACTOR}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 cofactor} {\f2 returns the cofactor of the element in row - and column of a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . Errors occur -if or do not evaluate to integer expressions or if -the matrix is not square. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 cofactor} {\f4 (,,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -cofactor(mat((a,b,c),(d,e,f),(p,q,r)),2,2); \par - \par - \par - A*R - C*P \par - \par - \par -cofactor(mat((a,b,c),(d,e,f)),1,1); \par - \par - \par - ***** non-square matrix \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DET} - -${\footnote \pard\plain \sl240 \fs20 $ DET} - -+{\footnote \pard\plain \sl240 \fs20 + g13:0973} - - K{\footnote \pard\plain \sl240 \fs20 K determinant;matrix;DET operator;operator} - -}{\b\f2 DET}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 det} {\f2 operator returns the determinant of its -(square } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 ) argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 det} {\f4 () or }{\f3 det} {\f4 -\par -\par -}{\f2 \par - must evaluate to a square matrix. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -matrix m,n; \par - \par - \par -m := mat((a,b),(c,d)); \par - \par - M(1,1) := A \par - M(1,2) := B \par - M(2,1) := C \par - M(2,2) := D \par - \par - \par - \par -det m; \par - \par - A*D - B*C \par - \par - \par -n := mat((1,2),(1,2)); \par - \par - N(1,1) := 1 \par - N(1,2) := 2 \par - N(2,1) := 1 \par - N(2,2) := 2 \par - \par - \par - \par - \par -det(n); \par - \par - 0 \par - \par - \par - \par -det(5); \par - \par - 5 \par - \par -\pard \sl240 }{\f2 Given a numerical argument, }{\f3 det} {\f2 returns the number. However, given a -variable name that has not been declared of type matrix, or a non-square -matrix, }{\f3 det} {\f2 returns an error message. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MAT} - -${\footnote \pard\plain \sl240 \fs20 $ MAT} - -+{\footnote \pard\plain \sl240 \fs20 + g13:0974} - - K{\footnote \pard\plain \sl240 \fs20 K matrix;MAT operator;operator} - -}{\b\f2 MAT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 mat} {\f2 operator is used to represent a two-dimensional -} -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 mat} {\f4 ((\{,\}*) \{(\{}{\f3 ,} {\f4 \}*)\}*) -\par -\par -}{\f2 \par - may be any valid REDUCE scalar expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -mat((1,2),(3,4)); \par - \par - MAT(1,1) := 1 \par - MAT(2,3) := 2 \par - MAT(2,1) := 3 \par - MAT(2,2) := 4 \par - \par - \par -mat(2,1); \par - \par - ***** Matrix mismatch \par - Cont? (Y or N) \par - \par - \par -matrix qt; \par - \par -qt := ws; \par - \par - QT(1,1) := 1 \par - QT(1,2) := 2 \par - QT(2,1) := 3 \par - QT(2,2) := 4 \par - \par - \par -matrix a,b; \par - \par -a := mat((x),(y),(z)); \par - \par - A(1,1) := X \par - A(2,1) := Y \par - A(3,1) := Z \par - \par - \par -b := mat((sin x,cos x,1)); \par - \par - B(1,1) := SIN(X) \par - B(1,2) := COS(X) \par - B(1,3) := 1 \par - \par -\pard \sl240 }{\f2 Matrices need not have a size declared (unlike arrays). }{\f3 mat} {\f2 -redimensions a matrix variable as needed. It is necessary, of course, -that all rows be the same length. An anonymous matrix, as shown in the -first example, must be named before it can be referenced (note error -message). When using }{\f3 mat} {\f2 to fill a 1 x n -matrix, the row of values must be inside a second set of parentheses, to -eliminate ambiguity. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MATEIGEN} - -${\footnote \pard\plain \sl240 \fs20 $ MATEIGEN} - -+{\footnote \pard\plain \sl240 \fs20 + g13:0975} - - K{\footnote \pard\plain \sl240 \fs20 K eigenvalue;matrix;MATEIGEN operator;operator} - -}{\b\f2 MATEIGEN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 mateigen} {\f2 operator calculates the eigenvalue equation and the -corresponding eigenvectors of a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 mateigen} {\f4 (,) -\par -\par -}{\f2 \par - must be a declared matrix of values, and must be -a legal REDUCE identifier. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -aa := mat((2,5),(1,0))\$ \par - \par -mateigen(aa,alpha); \par - \par - 2 \par - \{\{ALPHA - 2*ALPHA - 5, \par - 1, \par - 5*ARBCOMPLEX(1) \par - MAT(1,1) := ---------------, \par - ALPHA - 2 \par - \\ \par - MAT(2,1) := ARBCOMPLEX(1) \par - \}\} \par - \par -charpoly := first first ws; \par - \par - 2 \par - CHARPOLY := ALPHA - 2*ALPHA - 5 \par - \par - \par -bb := mat((1,0,1),(1,1,0),(0,0,1))\$ \par - \par -mateigen(bb,lamb); \par - \par - \{\{LAMB - 1,3, \par - [ 0 ] \par - [ARBCOMPLEX(2)] \par - [ 0 ] \par - \}\} \par - \par -\pard \sl240 }{\f2 The }{\f3 mateigen} {\f2 operator returns a list of lists of three -elements. The first element is a square free factor of the characteristic -polynomial; the second element is its multiplicity; and the third element -is the corresponding eigenvector. If the characteristic polynomial can be -completely factored, the product of the first elements of all the sublists -will produce the minimal polynomial. You can access the various parts of -the answer with the usual list access operators. -\par -\par -If the matrix is degenerate, more than one eigenvector can be produced for -the same eigenvalue, as shown by more than one arbitrary variable in the -eigenvector. The identification numbers of the arbitrary complex variables -shown in the examples above may not be the same as yours. Note that since -}{\f3 lambda} {\f2 is a reserved word in REDUCE, you cannot use it as a -tag-id for this operator. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MATRIX} - -${\footnote \pard\plain \sl240 \fs20 $ MATRIX} - -+{\footnote \pard\plain \sl240 \fs20 + g13:0976} - - K{\footnote \pard\plain \sl240 \fs20 K MATRIX declaration;declaration} - -}{\b\f2 MATRIX}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -Identifiers are declared to be of type }{\f3 matrix} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 matrix} {\f4 \tab option (,) -\par -\par -\{, \tab option - (,)\}* -\par -\par -}{\f2 \par - must not be an already-defined operator or array or -the name of a scalar variable. Dimensions are optional, and if used appear -inside parentheses. must be a positive integer. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -matrix a,b(1,4),c(4,4); \par - \par -b(1,1); \par - \par - 0 \par - \par - \par -a(1,1); \par - \par - ***** Matrix A not set \par - \par - \par -a := mat((x0,y0),(x1,y1)); \par - \par - A(1,1) := X0 \par - A(1,2) := Y0 \par - A(2,1) := X0 \par - A(2,2) := X1 \par - \par - \par -length a; \par - \par - \{2,2\} \par - \par - \par -b := a**2; \par - \par - 2 \par - B(1,1) := X0 + X1*Y0 \par - B(1,2) := Y0*(X0 + Y1) \par - B(2,1) := X1*(X0 + Y1) \par - 2 \par - B(2,2) := X1*Y0 + Y1 \par - \par -\pard \sl240 }{\f2 When a matrix variable has not been dimensioned, matrix elements cannot be -referenced until the matrix is set by the } -{\f2\uldb mat}{\v\f2 MAT} -{\f2 operator. When a -matrix is dimensioned in its declaration, matrix elements are set to 0. -Matrix elements cannot stand for themselves. When you use } -{\f2\uldb let}{\v\f2 LET} -{\f2 on -a matrix element, there is no effect unless the element contains a -constant, in which case an error message is returned. The same behavior -occurs with } -{\f2\uldb clear}{\v\f2 CLEAR} -{\f2 . Do use } -{\f2\uldb clear}{\v\f2 CLEAR} -{\f2 to try to -set a matrix element to 0. } -{\f2\uldb let}{\v\f2 LET} -{\f2 statements can be applied to -matrices as a whole, if the right-hand side of the expression is a matrix -expression, and the left-hand side identifier has been declared to be a matrix. -\par -\par -Arithmetical operators apply to matrices of the correct dimensions. The -operators }{\f3 +} {\f2 and }{\f3 -} {\f2 can be used with matrices of the same -dimensions. The operator }{\f3 *} {\f2 can be used to multiply -m x n matrices by n x p -matrices. Matrix multiplication is non-commutative. Scalars can also be -multiplied with matrices, with the result that each element of the matrix -is multiplied by the scalar. The operator }{\f3 /} {\f2 applied to two -matrices computes the first matrix multiplied by the inverse of the -second, if the inverse exists, and produces an error message otherwise. -Matrices can be divided by scalars, which results in dividing each element -of the matrix. Scalars can also be divided by matrices when the matrices -are invertible, and the result is the multiplication of the scalar by the -inverse of the matrix. Matrix inverses can by found by }{\f3 1/A} {\f2 or -}{\f3 /A} {\f2 , where }{\f3 A} {\f2 is a matrix. Square matrices can be raised to -positive integer powers, and also to negative integer powers if they are -nonsingular. -\par -\par -When a matrix variable is assigned to the results of a calculation, the -matrix is redimensioned if necessary. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NULLSPACE} - -${\footnote \pard\plain \sl240 \fs20 $ NULLSPACE} - -+{\footnote \pard\plain \sl240 \fs20 + g13:0977} - - K{\footnote \pard\plain \sl240 \fs20 K matrix;NULLSPACE operator;operator} - -}{\b\f2 NULLSPACE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 nullspace} {\f4 () -\par -\par -}{\f2 \par - calculates for its } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 argument, -}{\f3 a} {\f2 , a list of -linear independent vectors (a basis) whose linear combinations satisfy the -equation }{\f4 a x = 0}{\f2 . The basis is provided in a form such that as many -upper components as possible are isolated. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -nullspace mat((1,2,3,4),(5,6,7,8)); \par - \par - \par - \{ \par - [ 1 ] \par - [ ] \par - [ 0 ] \par - [ ] \par - [ - 3] \par - [ ] \par - [ 2 ] \par - , \par - [ 0 ] \par - [ ] \par - [ 1 ] \par - [ ] \par - [ - 2] \par - [ ] \par - [ 1 ] \par - \} \par - \par -\pard \sl240 }{\f2 Note that with }{\f3 b := nullspace a} {\f2 , the expression }{\f3 length b} {\f2 is -the nullity/ of A, and that }{\f3 second length a - length b} {\f2 -calculates the rank/ of A. The rank of a matrix expression can -also be found more directly by the } -{\f2\uldb rank}{\v\f2 RANK} -{\f2 operator. -\par -\par -In addition to the REDUCE matrix form, }{\f3 nullspace} {\f2 accepts as input a -matrix given as a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of lists, that is interpreted as a row matrix. If -that form of input is chosen, the vectors in the result will be -represented by lists as well. This additional input syntax facilitates -the use of }{\f3 nullspace} {\f2 in applications different from classical linear -algebra. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RANK} - -${\footnote \pard\plain \sl240 \fs20 $ RANK} - -+{\footnote \pard\plain \sl240 \fs20 + g13:0978} - - K{\footnote \pard\plain \sl240 \fs20 K matrix;RANK operator;operator} - -}{\b\f2 RANK}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 rank} {\f4 () -\par -\par -}{\f2 \par -}{\f3 rank} {\f2 calculates the rank of its matrix argument. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -rank mat((a,b,c),(d,e,f)); \par - \par - 2 \par - \par -\pard \sl240 }{\f2 The argument to }{\f3 rank} {\f2 can also be a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of lists, interpreted -either as a row matrix or a set of equations. If that form of input is -chosen, the vectors in the result will be represented by lists as well. -This additional input syntax facilitates the use of }{\f3 rank} {\f2 in -applications different from classical linear algebra. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TP} - -${\footnote \pard\plain \sl240 \fs20 $ TP} - -+{\footnote \pard\plain \sl240 \fs20 + g13:0979} - - K{\footnote \pard\plain \sl240 \fs20 K matrix;transpose;TP operator;operator} - -}{\b\f2 TP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 tp} {\f2 operator returns the transpose of its } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 - argument. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 tp} {\f4 or }{\f3 tp} {\f4 () -\par -\par -}{\f2 \par - must be a matrix, which either has had its dimensions set -in its declaration, or has had values put into it by }{\f3 mat} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -matrix m,n; \par - \par -m := mat((1,2,3),(4,5,6))$ \par - \par -n := tp m; \par - \par - N(1,1) := 1 \par - N(1,2) := 4 \par - N(2,1) := 2 \par - N(2,2) := 5 \par - N(3,1) := 3 \par - N(3,2) := 6 \par - \par -\pard \sl240 }{\f2 In an assignment statement involving }{\f3 tp} {\f2 , the matrix identifier on the -left-hand side is redimensioned to the correct size for the transpose. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TRACE} - -${\footnote \pard\plain \sl240 \fs20 $ TRACE} - -+{\footnote \pard\plain \sl240 \fs20 + g13:0980} - - K{\footnote \pard\plain \sl240 \fs20 K matrix;TRACE operator;operator} - -}{\b\f2 TRACE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 trace} {\f2 operator finds the trace of its } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 argument. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 trace} {\f4 () or }{\f3 trace} {\f4 -\par -\par -}{\f2 \par - or must evaluate to a square -matrix. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -matrix a; \par - \par -a := mat((x1,y1),(x2,y2))\$ \par - \par -trace a; \par - \par - X1 + Y2 \par - \par -\pard \sl240 }{\f2 The trace is the sum of the entries along the diagonal of a square matrix. -Given a non-matrix expression, or a non-square matrix, }{\f3 trace} {\f2 returns -an error message. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g13} - -${\footnote \pard\plain \sl240 \fs20 $ Matrix Operations} - -+{\footnote \pard\plain \sl240 \fs20 + index:0013} -}{\b\f2 Matrix Operations}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb COFACTOR operator} -{\v\f2 COFACTOR}{\f2 \par -}{\f2 \tab}{\f2\uldb DET operator} -{\v\f2 DET}{\f2 \par -}{\f2 \tab}{\f2\uldb MAT operator} -{\v\f2 MAT}{\f2 \par -}{\f2 \tab}{\f2\uldb MATEIGEN operator} -{\v\f2 MATEIGEN}{\f2 \par -}{\f2 \tab}{\f2\uldb MATRIX declaration} -{\v\f2 MATRIX}{\f2 \par -}{\f2 \tab}{\f2\uldb NULLSPACE operator} -{\v\f2 NULLSPACE}{\f2 \par -}{\f2 \tab}{\f2\uldb RANK operator} -{\v\f2 RANK}{\f2 \par -}{\f2 \tab}{\f2\uldb TP operator} -{\v\f2 TP}{\f2 \par -}{\f2 \tab}{\f2\uldb TRACE operator} -{\v\f2 TRACE}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Groebner_bases} - -${\footnote \pard\plain \sl240 \fs20 $ Groebner_bases} - -+{\footnote \pard\plain \sl240 \fs20 + g14:0981} - - K{\footnote \pard\plain \sl240 \fs20 K Kredel-Weispfenning algorithm;Faugere-Gianni-Lazard-Mora algorithm;Hollmann algorithm;Buchberger algorithm;Groebner bases;Groebner bases introduction;introducti -on} - -}{\b\f2 GROEBNER BASES}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par -\par - -The GROEBNER package calculates }{\f3 Groebner bases} {\f2 using the - }{\f3 Buchberger algorithm} {\f2 and provides related algorithms -for arithmetic with ideal bases, such as ideal quotients, -Hilbert polynomials ( }{\f3 Hollmann algorithm} {\f2 ), -basis conversion ( - }{\f3 Faugere-Gianni-Lazard-Mora algorithm} {\f2 ), independent -variable set ( }{\f3 Kredel-Weispfenning algorithm} {\f2 ). -\par -\par -Some routines of the Groebner package are used by } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 - in -that context the package is loaded automatically. However, if you -want to use the package by explicit calls you must load it by -\pard \tx3420 }{\f4 \par - load_package groebner; \par -\pard \sl240 }{\f2 \par -\par -For the common parameter setting of most operators in this package -see } -{\f2\uldb ideal parameters}{\v\f2 Ideal_Parameters} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Ideal_Parameters} - -${\footnote \pard\plain \sl240 \fs20 $ Ideal_Parameters} - -+{\footnote \pard\plain \sl240 \fs20 + g14:0982} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;Ideal Parameters concept;concept} - -}{\b\f2 IDEAL PARAMETERS}{\f2 \par -\par - - \par -\par -Most operators of the }{\f3 Groebner} {\f2 package compute expressions in a -polynomial ring which given as [,,...] where - is the current REDUCE coefficient domain. All algebraically -exact domains of REDUCE are supported. The package can operate over rings -and fields. The operation mode is distinguished automatically. In -general the ring mode is a bit faster than the field mode. The factoring -variant can be applied only over domains which allow you factoring of -multivariate polynomials. -\par -\par -The variable sequence is either declared explicitly as argument -in form of a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 in } -{\f2\uldb torder}{\v\f2 TORDER} -{\f2 , or it is extracted -automatically from the expressions. In the second case the current REDUCE -system order is used (see } -{\f2\uldb korder}{\v\f2 KORDER} -{\f2 ) for arranging the variables. -If some kernels should play the role of formal parameters (the ground -domain then is the polynomial ring over these), the variable -sequences must be given explicitly. -\par -\par -All REDUCE } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 s can be used as variables. But please note, -that all variables are considered as independent. E.g. when using -}{\f3 sin(a)} {\f2 and }{\f3 cos(a)} {\f2 as variables, the basic relation -}{\f3 sin(a)^2+cos(a)^2-1=0} {\f2 must be explicitly added to an equation set -because the Groebner operators don't include such knowledge automatically. -\par -\par -The terms (monomials) in polynomials are arranged according to the current -} -{\f2\uldb term order}{\v\f2 Term_order} -{\f2 . Note that the algebraic properties of the computed -results only are valid as long as neither the ordering nor the variable -sequence changes. -\par -\par -The input expressions can be polynomials

, rational -functions / or equations = built from -polynomials or rational functions. Apart from the }{\f3 tracing} {\f2 -algorithms } -{\f2\uldb groebnert}{\v\f2 groebnert} -{\f2 and } -{\f2\uldb preducet}{\v\f2 preducet} -{\f2 , where the equations -have a specific meaning, equations are converted to simple expressions by -taking the difference of the left-hand and right-hand sides --=>

. Rational functions are converted to -polynomials by converting the expression to a common denominator form -first, and then using the numerator only =>

. So eventual -zeros of the denominators are ignored. -\par -\par -A basis on input or output of an algorithm is coded as } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of -expressions \{,,...\} . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Term_order} - -${\footnote \pard\plain \sl240 \fs20 $ Term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0983} - - K{\footnote \pard\plain \sl240 \fs20 K distributive polynomials;Term order introduction;introduction} - -}{\b\f2 TERM ORDER}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par -\par - - \par -\par -For all }{\f3 Groebner} {\f2 operations the polynomials are -represented in distributive form: a sum of terms (monomials). -The terms are ordered corresponding to the actual }{\f3 term order} {\f2 -which is set by the } -{\f2\uldb torder}{\v\f2 TORDER} -{\f2 operator, and to the -actual variable sequence which is either given as explicit -parameter or by the system } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 order. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TORDER} - -${\footnote \pard\plain \sl240 \fs20 $ TORDER} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0984} - - K{\footnote \pard\plain \sl240 \fs20 K TORDER operator;operator} - -}{\b\f2 TORDER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The operator }{\f3 torder} {\f2 sets the actual variable sequence and term order. -\par -\par -1. simple term order: - \par -syntax: \par -}{\f4 \par -\par -}{\f3 torder} {\f4 (, ) -\par -\par -}{\f2 \par -where is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 of variables (} -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 s) and - is the name of a simple } -{\f2\uldb term order}{\v\f2 Term_order} -{\f2 mode -} -{\f2\uldb lex term order}{\v\f2 lex_term_order} -{\f2 , } -{\f2\uldb gradlex term order}{\v\f2 gradlex_term_order} -{\f2 , -} -{\f2\uldb revgradlex term order}{\v\f2 revgradlex_term_order} -{\f2 or another implemented parameterless mode. -\par -\par -2. stepped term order: - \par -syntax: \par -}{\f4 \par -\par -}{\f3 torder} {\f4 (,,) -\par -\par -\par -\par -}{\f2 where is the name of a two step term order, one of -} -{\f2\uldb gradlexgradlex term order}{\v\f2 gradlexgradlex_term_order} -{\f2 , } -{\f2\uldb gradlexrevgradlex term order}{\v\f2 gradlexrevgradlex_term_order} -{\f2 , -} -{\f2\uldb lexgradlex term order}{\v\f2 lexgradlex_term_order} -{\f2 or } -{\f2\uldb lexrevgradlex term order}{\v\f2 lexrevgradlex_term_order} -{\f2 , and - is a positive integer. -\par -\par -3. weighted term order - \par -syntax: \par -}{\f4 \par -\par -}{\f3 torder} {\f4 (, }{\f3 weighted} {\f4 , ,,...); -\par -\par -}{\f2 \par -where the are positive integers, see } -{\f2\uldb weighted term order}{\v\f2 weighted_term_order} -{\f2 . -\par -\par -4. matrix term order - \par -syntax: \par -}{\f4 \par -\par -}{\f3 torder} {\f4 (, }{\f3 matrix} {\f4 , ); -\par -\par -}{\f2 \par -where is a matrix with integer elements, see -} -{\f2\uldb torder_compile}{\v\f2 torder_compile} -{\f2 . -\par -\par -5. compiled term order - \par -syntax: \par -}{\f4 \par -\par -}{\f3 torder} {\f4 (, }{\f3 co} {\f4 ); -\par -\par -}{\f2 \par -where is the name of a routine generated by -} -{\f2\uldb torder_compile}{\v\f2 torder_compile} -{\f2 . -\par -\par -}{\f3 torder} {\f2 sets the variable sequence and the term order mode. If the -an empty list is used as variable sequence, the automatic variable extraction -is activated. The defaults are the empty variable list an the -} -{\f2\uldb lex term order}{\v\f2 lex_term_order} -{\f2 . -The previous setting is returned as a list. -\par -\par -Alternatively to the above syntax the arguments of }{\f3 torder} {\f2 may be -collected in a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 and passed as one argument to -}{\f3 torder} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # torder_compile} - -${\footnote \pard\plain \sl240 \fs20 $ torder_compile} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0985} - - K{\footnote \pard\plain \sl240 \fs20 K term order;torder_compile operator;operator} - -}{\b\f2 TORDER_COMPILE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -A matrix can be converted into -a compilable LISP program for faster execution by using - \par -syntax: \par -}{\f4 \par -\par -}{\f3 torder_compile} {\f4 (,) -\par -\par -}{\f2 \par -where is an identifier for the new term order and -is an integer matrix to be used as } -{\f2\uldb matrix term order}{\v\f2 matrix_term_order} -{\f2 . Afterwards -the term order can be activated by using in a } -{\f2\uldb torder}{\v\f2 TORDER} -{\f2 -expression. The resulting program is compiled if the switch } -{\f2\uldb comp}{\v\f2 COMP} -{\f2 -is on, or if the }{\f3 torder_compile} {\f2 expression is part of a compiled -module. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # lex_term_order} - -${\footnote \pard\plain \sl240 \fs20 $ lex_term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0986} - - K{\footnote \pard\plain \sl240 \fs20 K variable elimination;term order;lex term order concept;concept} - -}{\b\f2 LEX TERM ORDER}{\f2 \par -\par - - \par -\par -The terms are ordered lexicographically: two terms t1 t2 -are compared for their degrees -along the fixed variable sequence: t1 is higher than t2 -if the first different degree is higher in t1. -This order has the }{\f3 elimination property} {\f2 -for }{\f3 groebner basis} {\f2 calculations. -If the ideal has a univariate polynomial in the last -variable the groebner basis will contain -such polynomial. }{\f3 Lex} {\f2 is best -suited for solving of polynomial equation systems. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gradlex_term_order} - -${\footnote \pard\plain \sl240 \fs20 $ gradlex_term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0987} - - K{\footnote \pard\plain \sl240 \fs20 K term order;gradlex term order concept;concept} - -}{\b\f2 GRADLEX TERM ORDER}{\f2 \par -\par - - \par -\par -The terms are ordered first with their total -degree, and if the total degree is identical -the comparison is } -{\f2\uldb lex term order}{\v\f2 lex_term_order} -{\f2 . -With }{\f3 groebner} {\f2 basis calculations this term order -produces polynomials of lowest degree. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # revgradlex_term_order} - -${\footnote \pard\plain \sl240 \fs20 $ revgradlex_term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0988} - - K{\footnote \pard\plain \sl240 \fs20 K term order;revgradlex term order concept;concept} - -}{\b\f2 REVGRADLEX TERM ORDER}{\f2 \par -\par - - \par -\par -The terms are ordered first with their total -degree (degree sum), and if the total degree is identical -the comparison is the inverse of } -{\f2\uldb lex term order}{\v\f2 lex_term_order} -{\f2 . -With } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 and } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 -calculations this term order -is similar to } -{\f2\uldb gradlex term order}{\v\f2 gradlex_term_order} -{\f2 ; it is known -as most efficient ordering with respect to computing time. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gradlexgradlex_term_order} - -${\footnote \pard\plain \sl240 \fs20 $ gradlexgradlex_term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0989} - - K{\footnote \pard\plain \sl240 \fs20 K term order;gradlexgradlex term order concept;concept} - -}{\b\f2 GRADLEXGRADLEX TERM ORDER}{\f2 \par -\par - - \par -\par -The terms are separated into two groups where the -second parameter of the } -{\f2\uldb torder}{\v\f2 TORDER} -{\f2 call determines -the length of the first group. For a comparison first -the total degrees of both variable groups are compared. -If both are equal -} -{\f2\uldb gradlex term order}{\v\f2 gradlex_term_order} -{\f2 comparison is applied to the first -group, and if that does not decide } -{\f2\uldb gradlex term order}{\v\f2 gradlex_term_order} -{\f2 -is applied for the second group. This order has the elimination -property for the variable groups. It can be used e.g. for -separating variables from parameters. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gradlexrevgradlex_term_order} - -${\footnote \pard\plain \sl240 \fs20 $ gradlexrevgradlex_term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0990} - - K{\footnote \pard\plain \sl240 \fs20 K term order;gradlexrevgradlex term order concept;concept} - -}{\b\f2 GRADLEXREVGRADLEX TERM ORDER}{\f2 \par -\par - - \par -\par -Similar to } -{\f2\uldb gradlexgradlex term order}{\v\f2 gradlexgradlex_term_order} -{\f2 , but using -} -{\f2\uldb revgradlex term order}{\v\f2 revgradlex_term_order} -{\f2 for the second group. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # lexgradlex_term_order} - -${\footnote \pard\plain \sl240 \fs20 $ lexgradlex_term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0991} - - K{\footnote \pard\plain \sl240 \fs20 K term order;lexgradlex term order concept;concept} - -}{\b\f2 LEXGRADLEX TERM ORDER}{\f2 \par -\par - - \par -\par -Similar to } -{\f2\uldb gradlexgradlex term order}{\v\f2 gradlexgradlex_term_order} -{\f2 , but using -} -{\f2\uldb lex term order}{\v\f2 lex_term_order} -{\f2 for the first group. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # lexrevgradlex_term_order} - -${\footnote \pard\plain \sl240 \fs20 $ lexrevgradlex_term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0992} - - K{\footnote \pard\plain \sl240 \fs20 K term order;lexrevgradlex term order concept;concept} - -}{\b\f2 LEXREVGRADLEX TERM ORDER}{\f2 \par -\par - - \par -\par -Similar to } -{\f2\uldb gradlexgradlex term order}{\v\f2 gradlexgradlex_term_order} -{\f2 , but using -} -{\f2\uldb lex term order}{\v\f2 lex_term_order} -{\f2 for the first group -} -{\f2\uldb revgradlex term order}{\v\f2 revgradlex_term_order} -{\f2 for the second group. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # weighted_term_order} - -${\footnote \pard\plain \sl240 \fs20 $ weighted_term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0993} - - K{\footnote \pard\plain \sl240 \fs20 K term order;weighted term order concept;concept} - -}{\b\f2 WEIGHTED TERM ORDER}{\f2 \par -\par - - \par -\par -establishes a graduated ordering -similar to } -{\f2\uldb gradlex term order}{\v\f2 gradlex_term_order} -{\f2 , where the exponents first are -multiplied by the given weights. If there are less weight values than -variables, the weight list is extended by ones. If the weighted degree -comparison is not decidable, the -} -{\f2\uldb lex term order}{\v\f2 lex_term_order} -{\f2 is used. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # graded_term_order} - -${\footnote \pard\plain \sl240 \fs20 $ graded_term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0994} - - K{\footnote \pard\plain \sl240 \fs20 K term order;graded term order concept;concept} - -}{\b\f2 GRADED TERM ORDER}{\f2 \par -\par - - \par -\par -establishes a cascaded term ordering: first a graduated ordering -similar to } -{\f2\uldb gradlex term order}{\v\f2 gradlex_term_order} -{\f2 is used, where the exponents first are -multiplied by the given weights. If there are less weight values than -variables, the weight list is extended by ones. If the weighted degree -comparison is not decidable, the term ordering described in the following -parameters of the } -{\f2\uldb torder}{\v\f2 TORDER} -{\f2 command is used. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # matrix_term_order} - -${\footnote \pard\plain \sl240 \fs20 $ matrix_term_order} - -+{\footnote \pard\plain \sl240 \fs20 + g15:0995} - - K{\footnote \pard\plain \sl240 \fs20 K term order;matrix term order concept;concept} - -}{\b\f2 MATRIX TERM ORDER}{\f2 \par -\par - - \par -\par -Any arbitrary term order mode can be installed by a matrix with -integer elements where the row length corresponds to the variable -number. The matrix must have at least as many rows as columns. -It must have full rank, and the top nonzero element of each column -must be positive. -\par -\par -The matrix }{\f3 term order mode} {\f2 -defines a term order where the exponent vectors of the monomials are -first multiplied by the matrix and the resulting vectors are compared -lexicographically. -\par -\par -If the switch } -{\f2\uldb comp}{\v\f2 COMP} -{\f2 is on, the matrix is converted into -a compiled LISP program for faster execution. A matrix can also be -compiled explicitly, see } -{\f2\uldb torder_compile}{\v\f2 torder_compile} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g15} - -${\footnote \pard\plain \sl240 \fs20 $ Term order} - -+{\footnote \pard\plain \sl240 \fs20 + index:0015} -}{\b\f2 Term order}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Term order introduction} -{\v\f2 Term_order}{\f2 \par -}{\f2 \tab}{\f2\uldb TORDER operator} -{\v\f2 TORDER}{\f2 \par -}{\f2 \tab}{\f2\uldb torder_compile operator} -{\v\f2 torder_compile}{\f2 \par -}{\f2 \tab}{\f2\uldb lex term order concept} -{\v\f2 lex_term_order}{\f2 \par -}{\f2 \tab}{\f2\uldb gradlex term order concept} -{\v\f2 gradlex_term_order}{\f2 \par -}{\f2 \tab}{\f2\uldb revgradlex term order concept} -{\v\f2 revgradlex_term_order}{\f2 \par -}{\f2 \tab}{\f2\uldb gradlexgradlex term order concept} -{\v\f2 gradlexgradlex_term_order}{\f2 \par -}{\f2 \tab}{\f2\uldb gradlexrevgradlex term order concept} -{\v\f2 gradlexrevgradlex_term_order}{\f2 \par -}{\f2 \tab}{\f2\uldb lexgradlex term order concept} -{\v\f2 lexgradlex_term_order}{\f2 \par -}{\f2 \tab}{\f2\uldb lexrevgradlex term order concept} -{\v\f2 lexrevgradlex_term_order}{\f2 \par -}{\f2 \tab}{\f2\uldb weighted term order concept} -{\v\f2 weighted_term_order}{\f2 \par -}{\f2 \tab}{\f2\uldb graded term order concept} -{\v\f2 graded_term_order}{\f2 \par -}{\f2 \tab}{\f2\uldb matrix term order concept} -{\v\f2 matrix_term_order}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GVARS} - -${\footnote \pard\plain \sl240 \fs20 $ GVARS} - -+{\footnote \pard\plain \sl240 \fs20 + g16:0996} - - K{\footnote \pard\plain \sl240 \fs20 K GVARS operator;operator} - -}{\b\f2 GVARS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -syntax: \par -}{\f4 \par -\par -}{\f3 gvars} {\f4 (\{,,... \}) -\par -\par -\par -\par -}{\f2 where are expressions or } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s. -\par -\par -}{\f3 gvars} {\f2 extracts from the expressions the } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f3 s} {\f2 -which can -play the role of variables for a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 or } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 -calculation. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GROEBNER} - -${\footnote \pard\plain \sl240 \fs20 $ GROEBNER} - -+{\footnote \pard\plain \sl240 \fs20 + g16:0997} - - K{\footnote \pard\plain \sl240 \fs20 K Buchberger algorithm;GROEBNER operator;operator} - -}{\b\f2 GROEBNER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 groebner} {\f4 (\{}{\f3 exp} {\f4 , ...\}) -\par -\par -\par -\par -}{\f2 where \{}{\f3 exp} {\f2 , ... \}is a list of -expressions or equations. -\par -\par -The operator }{\f3 groebner} {\f2 implements the Buchberger algorithm -for computing Groebner bases for a given set of -expressions with respect to the given set of variables in the order -given. As a side effect, the sequence of variables is stored as a REDUCE list -in the shared variable } -{\f2\uldb gvarslast}{\v\f2 gvarslast} -{\f2 - this is important in cases -where the algorithm rearranges the variable sequence because } -{\f2\uldb groebopt}{\v\f2 groebopt} -{\f2 -is }{\f3 on} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - groebner(\{x**2+y**2-1,x-y\}) \par - \par - \{X - Y,2*Y**2 -1\} \par - \par -\pard \sl240 }{\f2 \par -related: \par -\par -\tab } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 operator -\par -\tab } -{\f2\uldb gvarslast}{\v\f2 gvarslast} -{\f2 variable -\par -\tab } -{\f2\uldb groebopt}{\v\f2 groebopt} -{\f2 switch -\par -\tab } -{\f2\uldb groebprereduce}{\v\f2 groebprereduce} -{\f2 switch -\par -\tab } -{\f2\uldb groebfullreduction}{\v\f2 groebfullreduction} -{\f2 switch -\par -\tab } -{\f2\uldb gltbasis}{\v\f2 gltbasis} -{\f2 switch -\par -\tab } -{\f2\uldb gltb}{\v\f2 gltb} -{\f2 variable -\par -\tab } -{\f2\uldb glterms}{\v\f2 glterms} -{\f2 variable -\par -\tab } -{\f2\uldb groebstat}{\v\f2 groebstat} -{\f2 switch -\par -\tab } -{\f2\uldb trgroeb}{\v\f2 trgroeb} -{\f2 switch -\par -\tab } -{\f2\uldb trgroebs}{\v\f2 trgroebs} -{\f2 switch -\par -\tab } -{\f2\uldb groebprot}{\v\f2 groebprot} -{\f2 switch -\par -\tab } -{\f2\uldb groebprotfile}{\v\f2 groebprotfile} -{\f2 variable -\par -\tab } -{\f2\uldb groebnert}{\v\f2 groebnert} -{\f2 operator -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebopt} - -${\footnote \pard\plain \sl240 \fs20 $ groebopt} - -+{\footnote \pard\plain \sl240 \fs20 + g16:0998} - - K{\footnote \pard\plain \sl240 \fs20 K groebopt switch;switch} - -}{\b\f2 GROEBOPT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -If }{\f3 groebopt} {\f2 is set ON, the sequence of variables is optimized -with respect to execution speed of }{\f3 groebner} {\f2 calculations; -note that the final list of variables is available in } -{\f2\uldb gvarslast}{\v\f2 gvarslast} -{\f2 . -By default }{\f3 groebopt} {\f2 is off, conserving the original variable -sequence. -\par -\par -An explicitly declared dependency using the } -{\f2\uldb depend}{\v\f2 DEPEND} -{\f2 -declaration supersedes the variable optimization. - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 \par -\par -guarantees that a will be placed in front of x and y. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gvarslast} - -${\footnote \pard\plain \sl240 \fs20 $ gvarslast} - -+{\footnote \pard\plain \sl240 \fs20 + g16:0999} - - K{\footnote \pard\plain \sl240 \fs20 K gvarslast variable;variable} - -}{\b\f2 GVARSLAST}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - -After a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 or } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 calculation -the actual variable sequence is stored in the variable -}{\f3 gvarslast} {\f2 . If } -{\f2\uldb groebopt}{\v\f2 groebopt} -{\f2 is }{\f3 on} {\f2 -}{\f3 gvarslast} {\f2 shows the variable sequence after reordering. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebprereduce} - -${\footnote \pard\plain \sl240 \fs20 $ groebprereduce} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1000} - - K{\footnote \pard\plain \sl240 \fs20 K groebprereduce switch;switch} - -}{\b\f2 GROEBPREREDUCE}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -If }{\f3 groebprereduce} {\f2 set ON, } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 -and } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 try to simplify the -input expressions: if the head term of an input expression is a -multiple of the head term of another expression, it can be reduced; -these reductions are done cyclicly as long as possible in order to -shorten the main part of the algorithm. -\par -\par -By default }{\f3 groebprereduce} {\f2 is off. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebfullreduction} - -${\footnote \pard\plain \sl240 \fs20 $ groebfullreduction} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1001} - - K{\footnote \pard\plain \sl240 \fs20 K groebfullreduction switch;switch} - -}{\b\f2 GROEBFULLREDUCTION}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -If }{\f3 groebfullreduction} {\f2 set off, the polynomial reduction steps during -} -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 and } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 are limited to the pure head -term reduction; subsequent terms are reduced otherwise. -\par -\par -By default }{\f3 groebfullreduction} {\f2 is on. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gltbasis} - -${\footnote \pard\plain \sl240 \fs20 $ gltbasis} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1002} - - K{\footnote \pard\plain \sl240 \fs20 K gltbasis switch;switch} - -}{\b\f2 GLTBASIS}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -If }{\f3 gltbasis} {\f2 set on, the leading terms of the result basis -of a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 or } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 calculation are -extracted. They are collected as a basis of monomials, which is -available as value of the global variable } -{\f2\uldb gltb}{\v\f2 gltb} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gltb} - -${\footnote \pard\plain \sl240 \fs20 $ gltb} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1003} - - K{\footnote \pard\plain \sl240 \fs20 K gltb variable;variable} - -}{\b\f2 GLTB}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - -See } -{\f2\uldb gltbasis}{\v\f2 gltbasis} -{\f2 -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # glterms} - -${\footnote \pard\plain \sl240 \fs20 $ glterms} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1004} - - K{\footnote \pard\plain \sl240 \fs20 K glterms variable;variable} - -}{\b\f2 GLTERMS}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - -If the expressions in a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 or } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 -call contain parameters (symbols -which are not member of the variable list), the share variable -}{\f3 glterms} {\f2 is set to a list of expression which during the -calculation were assumed to be nonzero. The calculated bases -are valid only under the assumption that all these expressions do -not vanish. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebstat} - -${\footnote \pard\plain \sl240 \fs20 $ groebstat} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1005} - - K{\footnote \pard\plain \sl240 \fs20 K groebstat switch;switch} - -}{\b\f2 GROEBSTAT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -if }{\f3 groebstat} {\f2 is on, a summary of the -} -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 or } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 computation is printed -at the end -including the computing time, the number of intermediate -H polynomials and the counters for the criteria hits. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # trgroeb} - -${\footnote \pard\plain \sl240 \fs20 $ trgroeb} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1006} - - K{\footnote \pard\plain \sl240 \fs20 K trgroeb switch;switch} - -}{\b\f2 TRGROEB}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -if }{\f3 trgroeb} {\f2 is on, intermediate H polynomials are -printed during a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 -or } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 calculation. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # trgroebs} - -${\footnote \pard\plain \sl240 \fs20 $ trgroebs} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1007} - - K{\footnote \pard\plain \sl240 \fs20 K trgroebs switch;switch} - -}{\b\f2 TRGROEBS}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -if }{\f3 trgroebs} {\f2 is on, intermediate H and S polynomials are -printed during a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 or } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 calculation. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gzerodim_} - -${\footnote \pard\plain \sl240 \fs20 $ gzerodim_} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1008} - - K{\footnote \pard\plain \sl240 \fs20 K gzerodim? operator;operator} - -}{\b\f2 GZERODIM?}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -syntax: \par -}{\f4 \par -\par -}{\f3 gzerodim!?} {\f4 () -\par -\par -\par -\par -}{\f2 where is a Groebner basis in the current -} -{\f2\uldb term order}{\v\f2 Term_order} -{\f2 with the actual setting -(see } -{\f2\uldb ideal parameters}{\v\f2 Ideal_Parameters} -{\f2 ). -\par -\par -}{\f3 gzerodim!?} {\f2 tests whether the ideal spanned by the given basis -has dimension zero. If yes, the number of zeros is returned, -} -{\f2\uldb nil}{\v\f2 NIL} -{\f2 otherwise. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gdimension} - -${\footnote \pard\plain \sl240 \fs20 $ gdimension} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1009} - - K{\footnote \pard\plain \sl240 \fs20 K groebner;ideal dimension;gdimension operator;operator} - -}{\b\f2 GDIMENSION}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 gdimension} {\f4 () -\par -\par -\par -\par -}{\f2 where is a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 basis in the current -term order (see } -{\f2\uldb ideal parameters}{\v\f2 Ideal_Parameters} -{\f2 ). -}{\f3 gdimension} {\f2 computes the dimension of the ideal -spanned by the given basis and returns the dimension as an integer -number. The Kredel-Weispfenning algorithm is used: the dimension -is the length of the longest independent variable set, -see } -{\f2\uldb gindependent_sets}{\v\f2 gindependent\_sets} -{\f2 -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gindependent\_sets} - -${\footnote \pard\plain \sl240 \fs20 $ gindependent_sets} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1010} - - K{\footnote \pard\plain \sl240 \fs20 K Kredel-Weispfenning algorithm;groebner;ideal dimension;ideal variables;gindependent_sets operator;operator} - -}{\b\f2 GINDEPENDENT\_SETS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 gindependent_sets} {\f4 () -\par -\par -\par -\par -}{\f2 where is a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 basis in any }{\f3 term order} {\f2 -(which must be the current }{\f3 term order} {\f2 ) with the specified -variables (see } -{\f2\uldb ideal parameters}{\v\f2 Ideal_Parameters} -{\f2 ). -\par -\par -}{\f3 Gindependent_sets} {\f2 computes the maximal -left independent variable sets of the ideal, that are -the variable sets which play the role of free parameters in the -current ideal basis. Each set is a list which is a subset of the -variable list. The result is a list of these sets. For an -ideal with dimension zero the list is empty. -The Kredel-Weispfenning algorithm is used. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # dd_groebner} - -${\footnote \pard\plain \sl240 \fs20 $ dd_groebner} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1011} - - K{\footnote \pard\plain \sl240 \fs20 K dd_groebner operator;operator} - -}{\b\f2 DD_GROEBNER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -For a homogeneous system of polynomials under -} -{\f2\uldb graded term order}{\v\f2 graded_term_order} -{\f2 , } -{\f2\uldb gradlex term order}{\v\f2 gradlex_term_order} -{\f2 , -} -{\f2\uldb revgradlex term order}{\v\f2 revgradlex_term_order} -{\f2 \par -\par -or } -{\f2\uldb weighted term order}{\v\f2 weighted_term_order} -{\f2 -a Groebner Base can be computed with limiting the grade -of the intermediate S polynomials: - \par -syntax: \par -}{\f4 \par -\par -}{\f3 dd_groebner} {\f4 (,,) -\par -\par -}{\f2 \par -where is a non negative integer and is an integer -or ``infinity". A pair of polynomials is considered -only if the grade of the lcm of their head terms is between - and . -For the term orders }{\f3 graded} {\f2 or }{\f3 weighted} {\f2 the (first) weight -vector is used for the grade computation. Otherwise the total -degree of a term is used. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # glexconvert} - -${\footnote \pard\plain \sl240 \fs20 $ glexconvert} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1012} - - K{\footnote \pard\plain \sl240 \fs20 K univariate polynomial;term order;ideal variables;glexconvert operator;operator} - -}{\b\f2 GLEXCONVERT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 glexconvert} {\f4 ([,][,MAXDEG=] -[,NEWVARS=]) -\par -\par -\par -\par -}{\f2 where is a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 basis -in the current term order, (optional) is a positive -integer and (optional) is a list of variables -(see } -{\f2\uldb ideal parameters}{\v\f2 Ideal_Parameters} -{\f2 ). -\par -\par -The operator }{\f3 glexconvert} {\f2 converts the basis -of a zero-dimensional ideal (finite number -of isolated solutions) from arbitrary ordering into a basis under -} -{\f2\uldb lex term order}{\v\f2 lex_term_order} -{\f2 . -\par -\par -The parameter defines the new variable sequence. -If omitted, the -original variable sequence is used. If only a subset of variables is -specified here, the partial ideal basis is evaluated. -\par -\par -If is a list with one element, the minimal - }{\f3 univariate polynomial} {\f2 is computed. -\par -\par - is an upper limit for the degrees. The algorithm stops with -an error message, if this limit is reached. -\par -\par -A warning occurs, if the ideal is not zero dimensional. -\par -\par -During the call the }{\f3 term order} {\f2 of the input basis must -be active. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # greduce} - -${\footnote \pard\plain \sl240 \fs20 $ greduce} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1013} - - K{\footnote \pard\plain \sl240 \fs20 K greduce operator;operator} - -}{\b\f2 GREDUCE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -syntax: \par -}{\f4 \par -\par -}{\f3 greduce} {\f4 (exp, \{exp1, exp2, ... , expm\}) -\par -\par -\par -\par -}{\f2 where exp is an expression, and \{exp1, exp2, ... , expm\} is -a list of expressions or equations. -\par -\par -}{\f3 greduce} {\f2 is functionally equivalent with a call to -} -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 and then a call to } -{\f2\uldb preduce}{\v\f2 preduce} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # preduce} - -${\footnote \pard\plain \sl240 \fs20 $ preduce} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1014} - - K{\footnote \pard\plain \sl240 \fs20 K preduce operator;operator} - -}{\b\f2 PREDUCE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -syntax: \par -}{\f4 \par -\par -}{\f3 preduce} {\f4 (

, \{, ... \}) -\par -\par -\par -\par -}{\f2 where

is an expression, and \{, ... \}is -a list of expressions or equations. -\par -\par -}{\f3 preduce} {\f2 computes the remainder of }{\f3 exp} {\f2 -modulo the given set of polynomials resp. equations. -This result is unique (canonical) only if the given set -is a }{\f3 groebner} {\f2 basis under the current } -{\f2\uldb term order}{\v\f2 Term_order} -{\f2 -\par -\par -see also: } -{\f2\uldb preducet}{\v\f2 preducet} -{\f2 operator. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # idealquotient} - -${\footnote \pard\plain \sl240 \fs20 $ idealquotient} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1015} - - K{\footnote \pard\plain \sl240 \fs20 K idealquotient operator;operator} - -}{\b\f2 IDEALQUOTIENT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -syntax: \par -}{\f4 \par -\par -}{\f3 idealquotient} {\f4 (\{, ...\}, ) -\par -\par -\par -\par -}{\f2 where \{,...\} is a list of -expressions or equations, is a single expression or equation. -\par -\par -}{\f3 idealquotient} {\f2 computes the ideal quotient: -ideal spanned by the expressions \{,...\} -divided by the single polynomial/expression . The result -is the } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 basis of the quotient ideal. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # hilbertpolynomial} - -${\footnote \pard\plain \sl240 \fs20 $ hilbertpolynomial} - -+{\footnote \pard\plain \sl240 \fs20 + g16:1016} - - K{\footnote \pard\plain \sl240 \fs20 K Hollmann algorithm;hilbertpolynomial operator;operator} - -}{\b\f2 HILBERTPOLYNOMIAL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 hilbertpolynomial() -\par -\par -\par -\par -}{\f2 where is a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 basis in the -current } -{\f2\uldb term order}{\v\f2 Term_order} -{\f2 . -\par -\par -The degree of the }{\f3 Hilbert polynomial} {\f2 is the -dimension of the ideal spanned by the basis. For an -ideal of dimension zero the Hilbert polynomial is a -constant which is the number of common zeros of the -ideal (including eventual multiplicities). -The }{\f3 Hollmann algorithm} {\f2 is used. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g16} - -${\footnote \pard\plain \sl240 \fs20 $ Basic Groebner operators} - -+{\footnote \pard\plain \sl240 \fs20 + index:0016} -}{\b\f2 Basic Groebner operators}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb GVARS operator} -{\v\f2 GVARS}{\f2 \par -}{\f2 \tab}{\f2\uldb GROEBNER operator} -{\v\f2 GROEBNER}{\f2 \par -}{\f2 \tab}{\f2\uldb groebopt switch} -{\v\f2 groebopt}{\f2 \par -}{\f2 \tab}{\f2\uldb gvarslast variable} -{\v\f2 gvarslast}{\f2 \par -}{\f2 \tab}{\f2\uldb groebprereduce switch} -{\v\f2 groebprereduce}{\f2 \par -}{\f2 \tab}{\f2\uldb groebfullreduction switch} -{\v\f2 groebfullreduction}{\f2 \par -}{\f2 \tab}{\f2\uldb gltbasis switch} -{\v\f2 gltbasis}{\f2 \par -}{\f2 \tab}{\f2\uldb gltb variable} -{\v\f2 gltb}{\f2 \par -}{\f2 \tab}{\f2\uldb glterms variable} -{\v\f2 glterms}{\f2 \par -}{\f2 \tab}{\f2\uldb groebstat switch} -{\v\f2 groebstat}{\f2 \par -}{\f2 \tab}{\f2\uldb trgroeb switch} -{\v\f2 trgroeb}{\f2 \par -}{\f2 \tab}{\f2\uldb trgroebs switch} -{\v\f2 trgroebs}{\f2 \par -}{\f2 \tab}{\f2\uldb gzerodim? operator} -{\v\f2 gzerodim_}{\f2 \par -}{\f2 \tab}{\f2\uldb gdimension operator} -{\v\f2 gdimension}{\f2 \par -}{\f2 \tab}{\f2\uldb gindependent\_sets operator} -{\v\f2 gindependent\_sets}{\f2 \par -}{\f2 \tab}{\f2\uldb dd_groebner operator} -{\v\f2 dd_groebner}{\f2 \par -}{\f2 \tab}{\f2\uldb glexconvert operator} -{\v\f2 glexconvert}{\f2 \par -}{\f2 \tab}{\f2\uldb greduce operator} -{\v\f2 greduce}{\f2 \par -}{\f2 \tab}{\f2\uldb preduce operator} -{\v\f2 preduce}{\f2 \par -}{\f2 \tab}{\f2\uldb idealquotient operator} -{\v\f2 idealquotient}{\f2 \par -}{\f2 \tab}{\f2\uldb hilbertpolynomial operator} -{\v\f2 hilbertpolynomial}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebnerf} - -${\footnote \pard\plain \sl240 \fs20 $ groebnerf} - -+{\footnote \pard\plain \sl240 \fs20 + g17:1017} - - K{\footnote \pard\plain \sl240 \fs20 K groebnerf operator;operator} - -}{\b\f2 GROEBNERF}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -syntax: \par -}{\f4 \par -\par -}{\f3 groebnerf} {\f4 (\{, ...\}[,\{\},\{, ... \}]); -\par -\par -\par -\par -}{\f2 where \{, ... \}is a list of expressions or -equations, and \{,... \}is -an optional list of polynomials to be considered as non zero -for this calculation. An empty list must be passed as second argument -if the non-zero list is specified. -\par -\par -}{\f3 groebnerf} {\f2 tries to separate polynomials into individual factors and -to branch the computation in a recursive manner (factorization tree). -The result is a list of partial Groebner bases. -Multiplicities (one factor with a higher power, the same partial basis -twice) are deleted as early as possible in order to speed up the -calculation. -\par -\par -The third parameter of }{\f3 groebnerf} {\f2 declares some polynomials -nonzero. If any of these is found in a branch of the calculation -the branch is canceled. -\par -\par - \par -example: \par -\pard \tx3420 }{\f4 \par -groebnerf(\{ 3*x**2*y+2*x*y+y+9*x**2+5*x = 3, \par - 2*x**3*y-x*y-y+6*x**3-2*x**2-3*x = -3, \par - x**3*y+x**2*y+3*x**3+2*x**2 \}, \{y,x\}); \par - \par - \{\{Y - 3,X\}, \par - \par - 2 \par - \{2*Y + 2*X - 1,2*X - 5*X - 5\}\} \par -\pard \sl240 }{\f2 \par -related: \par -\par -\tab } -{\f2\uldb groebresmax}{\v\f2 groebresmax} -{\f2 variable -\par -\tab } -{\f2\uldb groebmonfac}{\v\f2 groebmonfac} -{\f2 variable -\par -\tab } -{\f2\uldb groebrestriction}{\v\f2 groebrestriction} -{\f2 variable -\par -\tab } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 operator -\par -\tab } -{\f2\uldb gvarslast}{\v\f2 gvarslast} -{\f2 variable -\par -\tab } -{\f2\uldb groebopt}{\v\f2 groebopt} -{\f2 switch -\par -\tab } -{\f2\uldb groebprereduce}{\v\f2 groebprereduce} -{\f2 switch -\par -\tab } -{\f2\uldb groebfullreduction}{\v\f2 groebfullreduction} -{\f2 switch -\par -\tab } -{\f2\uldb gltbasis}{\v\f2 gltbasis} -{\f2 switch -\par -\tab } -{\f2\uldb gltb}{\v\f2 gltb} -{\f2 variable -\par -\tab } -{\f2\uldb glterms}{\v\f2 glterms} -{\f2 variable -\par -\tab } -{\f2\uldb groebstat}{\v\f2 groebstat} -{\f2 switch -\par -\tab } -{\f2\uldb trgroeb}{\v\f2 trgroeb} -{\f2 switch -\par -\tab } -{\f2\uldb trgroebs}{\v\f2 trgroebs} -{\f2 switch -\par -\tab } -{\f2\uldb groebnert}{\v\f2 groebnert} -{\f2 operator -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebmonfac} - -${\footnote \pard\plain \sl240 \fs20 $ groebmonfac} - -+{\footnote \pard\plain \sl240 \fs20 + g17:1018} - - K{\footnote \pard\plain \sl240 \fs20 K groebmonfac variable;variable} - -}{\b\f2 GROEBMONFAC}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - -The variable }{\f3 groebmonfac} {\f2 is connected to -the handling of monomial factors. A monomial factor is a product -of variable powers as a factor, e.g. x**2*y in x**3*y - -2*x**2*y**2. A monomial factor represents a solution of the type - x = 0 or y = 0 with a certain multiplicity. With -} -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 the multiplicity of monomial factors is lowered -to the value of the shared variable }{\f3 groebmonfac} {\f2 -which by default is 1 (= monomial factors remain present, but their -multiplicity is brought down). With -}{\f3 groebmonfac} {\f2 := 0 -the monomial factors are suppressed completely. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebresmax} - -${\footnote \pard\plain \sl240 \fs20 $ groebresmax} - -+{\footnote \pard\plain \sl240 \fs20 + g17:1019} - - K{\footnote \pard\plain \sl240 \fs20 K groebresmax variable;variable} - -}{\b\f2 GROEBRESMAX}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - -The variable }{\f3 groebresmax} {\f2 -controls during } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 calculations -the number of partial results. Its default value is 300. If -more partial results are calculated, the calculation is -terminated. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebrestriction} - -${\footnote \pard\plain \sl240 \fs20 $ groebrestriction} - -+{\footnote \pard\plain \sl240 \fs20 + g17:1020} - - K{\footnote \pard\plain \sl240 \fs20 K groebrestriction variable;variable} - -}{\b\f2 GROEBRESTRICTION}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - -During } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 calculations -irrelevant branches can be excluded -by setting the variable }{\f3 groebrestriction} {\f2 . The -following restrictions are implemented: - \par -syntax: \par -}{\f4 \par -\par -}{\f3 groebrestriction} {\f4 := }{\f3 nonnegative} {\f4 -\par -\par -}{\f3 groebrestriction} {\f4 := }{\f3 positive} {\f4 -\par -\par -}{\f3 groebrestriction} {\f4 := }{\f3 zeropoint} {\f4 -\par -\par -}{\f2 \par -With }{\f3 nonnegative} {\f2 branches are excluded where one -polynomial has no nonnegative real zeros; with }{\f3 positive} {\f2 -the restriction is sharpened to positive zeros only. -The restriction }{\f3 zeropoint} {\f2 excludes all branches -which do not have the origin (0,0,...0) in their solution -set. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g17} - -${\footnote \pard\plain \sl240 \fs20 $ Factorizing Groebner bases} - -+{\footnote \pard\plain \sl240 \fs20 + index:0017} -}{\b\f2 Factorizing Groebner bases}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb groebnerf operator} -{\v\f2 groebnerf}{\f2 \par -}{\f2 \tab}{\f2\uldb groebmonfac variable} -{\v\f2 groebmonfac}{\f2 \par -}{\f2 \tab}{\f2\uldb groebresmax variable} -{\v\f2 groebresmax}{\f2 \par -}{\f2 \tab}{\f2\uldb groebrestriction variable} -{\v\f2 groebrestriction}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebprot} - -${\footnote \pard\plain \sl240 \fs20 $ groebprot} - -+{\footnote \pard\plain \sl240 \fs20 + g18:1021} - - K{\footnote \pard\plain \sl240 \fs20 K groebprot switch;switch} - -}{\b\f2 GROEBPROT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -If }{\f3 groebprot} {\f2 is }{\f3 ON} {\f2 the computation steps during -} -{\f2\uldb preduce}{\v\f2 preduce} -{\f2 , } -{\f2\uldb greduce}{\v\f2 greduce} -{\f2 and } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 -are collected in a list which is assigned to the variable -} -{\f2\uldb groebprotfile}{\v\f2 groebprotfile} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebprotfile} - -${\footnote \pard\plain \sl240 \fs20 $ groebprotfile} - -+{\footnote \pard\plain \sl240 \fs20 + g18:1022} - - K{\footnote \pard\plain \sl240 \fs20 K groebprotfile variable;variable} - -}{\b\f2 GROEBPROTFILE}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - -See } -{\f2\uldb groebprot}{\v\f2 groebprot} -{\f2 switch. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # groebnert} - -${\footnote \pard\plain \sl240 \fs20 $ groebnert} - -+{\footnote \pard\plain \sl240 \fs20 + g18:1023} - - K{\footnote \pard\plain \sl240 \fs20 K groebnert operator;operator} - -}{\b\f2 GROEBNERT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -syntax: \par -}{\f4 \par -\par -}{\f3 groebnert} {\f4 (\{=,...\}) -\par -\par -\par -\par -}{\f2 where are } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f3 s} {\f2 (simple or indexed variables), - are polynomials. -\par -\par -}{\f3 groebnert} {\f2 is functionally equivalent to a } -{\f2\uldb groebner}{\v\f2 GROEBNER} -{\f2 -call for \{,...\}, but the result is a set of -equations where the left-hand sides are the basis elements while -the right-hand sides are the same values expressed as combinations -of the input formulas, expressed in terms of the names - \par -example: \par -\pard \tx3420 }{\f4 \par - groebnert(\{p1=2*x**2+4*y**2-100,p2=2*x-y+1\}); \par - \par - GB1 := \{2*X - Y + 1=P2, \par - \par - 2 \par - 9*Y - 2*Y - 199= - 2*X*P2 - Y*P2 + 2*P1 + P2\} \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # preducet} - -${\footnote \pard\plain \sl240 \fs20 $ preducet} - -+{\footnote \pard\plain \sl240 \fs20 + g18:1024} - - K{\footnote \pard\plain \sl240 \fs20 K preducet operator;operator} - -}{\b\f2 PREDUCET}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -syntax: \par -}{\f4 \par -\par -}{\f3 preduce} {\f4 (

,\{=...\}) -\par -\par -}{\f2 \par -where

is an expression, are kernels -(simple or indexed variables), -}{\f3 exp} {\f2 are polynomials. -\par -\par -}{\f3 preducet} {\f2 computes the remainder of

modulo \{,...\} -similar to } -{\f2\uldb preduce}{\v\f2 preduce} -{\f2 , but the result is an equation -which expresses the remainder as combination of the polynomials. - \par -example: \par -\pard \tx3420 }{\f4 \par - \par - GB2 := \{G1=2*X - Y + 1,G2=9*Y**2 - 2*Y - 199\} \par - preducet(q=x**2,gb2); \par - \par - - 16*Y + 208= - 18*X*G1 - 9*Y*G1 + 36*Q + 9*G1 - G2 \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g18} - -${\footnote \pard\plain \sl240 \fs20 $ Tracing Groebner bases} - -+{\footnote \pard\plain \sl240 \fs20 + index:0018} -}{\b\f2 Tracing Groebner bases}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb groebprot switch} -{\v\f2 groebprot}{\f2 \par -}{\f2 \tab}{\f2\uldb groebprotfile variable} -{\v\f2 groebprotfile}{\f2 \par -}{\f2 \tab}{\f2\uldb groebnert operator} -{\v\f2 groebnert}{\f2 \par -}{\f2 \tab}{\f2\uldb preducet operator} -{\v\f2 preducet}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Module} - -${\footnote \pard\plain \sl240 \fs20 $ Module} - -+{\footnote \pard\plain \sl240 \fs20 + g19:1025} - - K{\footnote \pard\plain \sl240 \fs20 K Module concept;concept} - -}{\b\f2 MODULE}{\f2 \par -\par - -Given a polynomial ring, e.g. R=Z[x,y,...] and an integer n>1. -The vectors with n elements of R form a free MODULE under -elementwise addition and multiplication with elements of R. -\par -\par -For a submodule given by a finite basis a Groebner basis -can be computed, and the facilities of the GROEBNER package -are available except the operators } -{\f2\uldb groebnerf}{\v\f2 groebnerf} -{\f2 -and }{\f3 groesolve} {\f2 . The vectors are encoded using auxiliary -variables which represent the unit vectors in the module. -These are declared in the share variable } -{\f2\uldb gmodule}{\v\f2 gmodule} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gmodule} - -${\footnote \pard\plain \sl240 \fs20 $ gmodule} - -+{\footnote \pard\plain \sl240 \fs20 + g19:1026} - - K{\footnote \pard\plain \sl240 \fs20 K gmodule variable;variable} - -}{\b\f2 GMODULE}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - -The vectors of a free } -{\f2\uldb module}{\v\f2 Module} -{\f2 over a polynomial ring R -are encoded as linear combinations with unit vectors of -M which are represented by auxiliary variables. These -must be collected in the variable }{\f3 gmodule} {\f2 before -any call to an operator of the Groebner package. -\par -\par -\pard \tx3420 }{\f4 \par - torder(\{x,y,v1,v2,v3\})$ \par - gmodule := \{v1,v2,v3\}$ \par - g:=groebner(\{x^2*v1 + y*v2,x*y*v1 - v3,2y*v1 + y*v3\}); \par -\pard \sl240 }{\f2 compute the Groebner basis of the submodule -\par -\par -\pard \tx3420 }{\f4 \par - ([x^2,y,0],[xy,0,-1],[0,2y,y]) \par -\pard \sl240 }{\f2 The members of the list }{\f3 gmodule} {\f2 are automatically -appended to the end of the variable list, if they are not -yet members there. They take part in the actual term ordering. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g19} - -${\footnote \pard\plain \sl240 \fs20 $ Groebner Bases for Modules} - -+{\footnote \pard\plain \sl240 \fs20 + index:0019} -}{\b\f2 Groebner Bases for Modules}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Module concept} -{\v\f2 Module}{\f2 \par -}{\f2 \tab}{\f2\uldb gmodule variable} -{\v\f2 gmodule}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gsort} - -${\footnote \pard\plain \sl240 \fs20 $ gsort} - -+{\footnote \pard\plain \sl240 \fs20 + g20:1027} - - K{\footnote \pard\plain \sl240 \fs20 K distributive polynomials;gsort operator;operator} - -}{\b\f2 GSORT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 gsort} {\f4 (

) -\par -\par -}{\f2 \par -where

is a polynomial or a list of polynomials. -\par -\par -The polynomials are reordered and sorted corresponding to -the current } -{\f2\uldb term order}{\v\f2 Term_order} -{\f2 . - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par - torder lex; \par - \par - gsort(x**2+2x*y+y**2,\{y,x\}); \par - \par - y**2+2y*x+x**2 \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gsplit} - -${\footnote \pard\plain \sl240 \fs20 $ gsplit} - -+{\footnote \pard\plain \sl240 \fs20 + g20:1028} - - K{\footnote \pard\plain \sl240 \fs20 K distributive polynomials;gsplit operator;operator} - -}{\b\f2 GSPLIT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 gsplit} {\f4 (

[,]); -\par -\par -}{\f2 \par -where

is a polynomial or a list of polynomials. -\par -\par -The polynomial is reordered corresponding to the -the current } -{\f2\uldb term order}{\v\f2 Term_order} -{\f2 and then -separated into leading term and reductum. Result is -a list with the leading term as first and the reductum -as second element. - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par - torder lex; \par - \par - gsplit(x**2+2x*y+y**2,\{y,x\}); \par - \par - \{y**2,2y*x+x**2\} \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gspoly} - -${\footnote \pard\plain \sl240 \fs20 $ gspoly} - -+{\footnote \pard\plain \sl240 \fs20 + g20:1029} - - K{\footnote \pard\plain \sl240 \fs20 K distributive polynomials;gspoly operator;operator} - -}{\b\f2 GSPOLY}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par - \par -syntax: \par -}{\f4 }{\f3 gspoly} {\f4 (,); -\par -\par -\par -\par -}{\f2 where and are polynomials. -\par -\par -The }{\f3 subtraction} {\f2 polynomial of p1 and p2 is computed -corresponding to the method of the Buchberger algorithm for -computing }{\f3 groebner bases} {\f2 : p1 and p2 are multiplied -with terms such that when subtracting them the leading terms -cancel each other. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g20} - -${\footnote \pard\plain \sl240 \fs20 $ Computing with distributive polynomials} - -+{\footnote \pard\plain \sl240 \fs20 + index:0020} -}{\b\f2 Computing with distributive polynomials}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb gsort operator} -{\v\f2 gsort}{\f2 \par -}{\f2 \tab}{\f2\uldb gsplit operator} -{\v\f2 gsplit}{\f2 \par -}{\f2 \tab}{\f2\uldb gspoly operator} -{\v\f2 gspoly}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g14} - -${\footnote \pard\plain \sl240 \fs20 $ Groebner package} - -+{\footnote \pard\plain \sl240 \fs20 + index:0014} -}{\b\f2 Groebner package}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Groebner bases introduction} -{\v\f2 Groebner_bases}{\f2 \par -}{\f2 \tab}{\f2\uldb Ideal Parameters concept} -{\v\f2 Ideal_Parameters}{\f2 \par -}{\f2 \tab}{\f2\uldb Term order} -{\v\f2 g15}{\f2 \par -}{\f2 \tab}{\f2\uldb Basic Groebner operators} -{\v\f2 g16}{\f2 \par -}{\f2 \tab}{\f2\uldb Factorizing Groebner bases} -{\v\f2 g17}{\f2 \par -}{\f2 \tab}{\f2\uldb Tracing Groebner bases} -{\v\f2 g18}{\f2 \par -}{\f2 \tab}{\f2\uldb Groebner Bases for Modules} -{\v\f2 g19}{\f2 \par -}{\f2 \tab}{\f2\uldb Computing with distributive polynomials} -{\v\f2 g20}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # HEPHYS} - -${\footnote \pard\plain \sl240 \fs20 $ HEPHYS} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1030} - - K{\footnote \pard\plain \sl240 \fs20 K HEPHYS introduction;introduction} - -}{\b\f2 HEPHYS}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par -\par - -The High-energy Physics package is historic for REDUCE, since REDUCE -originated as a program to aid in computations with Dirac expressions. -The commutation algebra of the gamma matrices is independent of their -representation, and is a natural subject for symbolic mathematics. Dirac -theory is applied to beta decay and the computation of -cross-sections and scattering. The high-energy physics operators are -available in the REDUCE main program, rather than as a module which must -be loaded. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # HE_dot} - -${\footnote \pard\plain \sl240 \fs20 $ HE_dot} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1031} - - K{\footnote \pard\plain \sl240 \fs20 K HE-dot operator;operator} - -}{\b\f2 .}{\f2 \tab }{\b\f2 HE-DOT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The . operator is used to denote the scalar product of two Lorentz -four-vectors. - \par -syntax: \par -}{\f4 \par -\par - }{\f3 .} {\f4 -\par -\par -}{\f2 \par - must be an identifier declared to be of type }{\f3 vector} {\f2 to have -the scalar product definition. When applied to arguments that are not -vectors, the } -{\f2\uldb cons}{\v\f2 CONS} -{\f2 operator is used, -whose symbol is also ``dot.'' -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -vector aa,bb,cc; \par - \par -let aa.bb = 0; \par - \par -aa.bb; \par - \par - 0 \par - \par - \par -aa.cc; \par - \par - AA.CC \par - \par - \par -q := aa.cc; \par - \par - Q := AA.CC \par - \par - \par -q; \par - \par - AA.CC \par - \par -\pard \sl240 }{\f2 Since vectors are special high-energy physics entities that do not contain -values, the . product will not return a true scalar product. You can -assign a scalar identifier to the result of a . operation, or assign a . -operation to have the value of the scalar you supply, as shown above. Note -that the result of a . operation is a scalar, not a vector. -\par -\par -The metric tensor g(u,v) can be represented by }{\f3 u.v} {\f2 . If contraction -over the indices is required, }{\f3 u} {\f2 and }{\f3 v} {\f2 should be declared to -be of type } -{\f2\uldb index}{\v\f2 INDEX} -{\f2 . -\par -\par -The dot operator has the highest precedence of the infix operators, so -expressions involving . and other operators have the scalar product -evaluated first before other operations are done. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EPS} - -${\footnote \pard\plain \sl240 \fs20 $ EPS} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1032} - - K{\footnote \pard\plain \sl240 \fs20 K EPS operator;operator} - -}{\b\f2 EPS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 eps} {\f2 operator denotes the completely antisymmetric tensor of -order 4 and its contraction with Lorentz four-vectors, as used in -high-energy physics calculations. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 eps} {\f4 (,,, -) -\par -\par -}{\f2 \par - must be a valid vector expression, and may be an index. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -vector g0,g1,g2,g3; \par - \par -eps(g1,g0,g2,g3); \par - \par - - EPS(G0,G1,G2,G3); \par - \par - \par -eps(g1,g2,g0,g3); \par - \par - EPS(G0,G1,G2,G3); \par - \par - \par -eps(g1,g2,g3,g1); \par - \par - 0 \par - \par -\pard \sl240 }{\f2 Vector identifiers are ordered alphabetically by REDUCE. When an odd number -of transpositions is required to restore the canonical order to the four -arguments of }{\f3 eps} {\f2 , the term is ordered and carries a minus sign. When an -even number of transpositions is required, the term is returned ordered and -positive. When one of the arguments is repeated, the value 0 is returned. -A contraction of the form -eps(_i j mu nu p_mu q_nu) -is represented by }{\f3 eps(i,j,p,q)} {\f2 when }{\f3 i} {\f2 and }{\f3 j} {\f2 have been -declared to be of type } -{\f2\uldb index}{\v\f2 INDEX} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # G} - -${\footnote \pard\plain \sl240 \fs20 $ G} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1033} - - K{\footnote \pard\plain \sl240 \fs20 K G operator;operator} - -}{\b\f2 G}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -}{\f3 g} {\f2 is an n-ary operator used to denote a product of gamma matrices -contracted with Lorentz four-vectors, in high-energy physics. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 g} {\f4 (, -\{,\}*) -\par -\par -}{\f2 \par - is a scalar identifier representing a fermion line -identifier, can be any valid vector expression, -representing a vector or a gamma matrix. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -vector aa,bb,cc; \par - \par -vector a; \par - \par -g(line1,aa,bb); \par - \par - AA.BB \par - \par - \par -g(line2,aa,a); \par - \par - 0 \par - \par - \par -g(id,aa,bb,cc); \par - \par - 0 \par - \par - \par -g(li1,aa,bb) + k; \par - \par - AA.BB + K \par - \par - \par -let aa.bb = m*k; \par - \par -g(ln1,aa)*g(ln1,bb); \par - \par - K*M \par - \par - \par -g(ln1,aa)*g(ln2,bb); \par - \par - 0 \par - \par -\pard \sl240 }{\f2 The vector }{\f3 A} {\f2 is reserved in arguments of }{\f3 g} {\f2 to denote the -special gamma matrix gamma_5. It must be declared to -be a vector before you use it. -\par -\par -Gamma matrix expressions are associated with fermion lines in a Feynman -diagram. If more than one line occurs in an expression, the gamma -matrices involved are separate (operating in independent spin space), as -shown in the last two example lines above. A product of gamma matrices -associated with a single line can be entered either as a single }{\f3 g} {\f2 -command with several vector arguments, or as products of separate }{\f3 g} {\f2 -commands each with a single argument. -\par -\par -While the product of vectors is not defined, the product, sum and -difference of several gamma expressions are defined, as is the product of -a gamma expression with a scalar. If an expression involving gamma -matrices includes a scalar, the scalar is treated as if it were the -product of itself with a unit 4 x 4 matrix. -\par -\par -Dirac expressions are evaluated by computing the trace of the expression -using the commutation algebra of gamma matrices. The algorithms used are -described in articles by J. S. R. Chisholm in Vol. -30, p. 426, 1963, and J. Kahane, , -Vol. 9, p. 1732, 1968. The trace is then divided by 4 to distinguish -between the trace of a scalar and the trace of an expression that is the -product of a scalar with a unit 4 x 4 matrix. -\par -\par -Trace calculations may be prevented over any line identifier by declaring it -to be } -{\f2\uldb nospur}{\v\f2 NOSPUR} -{\f2 . If it is later desired to evaluate these traces, -the declaration can be undone with the } -{\f2\uldb spur}{\v\f2 SPUR} -{\f2 declaration. -\par -\par -The notation of Bjorken and Drell, -1964, is assumed in all operations involving gamma matrices. For an -example of the use of }{\f3 g} {\f2 in a calculation, see the . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INDEX} - -${\footnote \pard\plain \sl240 \fs20 $ INDEX} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1034} - - K{\footnote \pard\plain \sl240 \fs20 K INDEX declaration;declaration} - -}{\b\f2 INDEX}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The declaration }{\f3 index} {\f2 flags a four-vector as an index for subsequent -high-energy physics calculations. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 index} {\f4 \{,\}* -\par -\par -}{\f2 \par - must have been declared of type }{\f3 vector} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -vector aa,bb,cc; \par - \par -index uu; \par - \par -let aa.bb = 0; \par - \par -(aa.uu)*(bb.uu); \par - \par - 0 \par - \par - \par -(aa.uu)*(cc.uu); \par - \par - AA.CC \par - \par -\pard \sl240 }{\f2 Index variables are used to represent contraction over components of -vectors when scalar products are taken by the . operator, as well as -indicating contraction for the } -{\f2\uldb eps}{\v\f2 EPS} -{\f2 operator or metric tensor. -\par -\par -The special status of a vector as an index can be revoked with the -declaration } -{\f2\uldb remind}{\v\f2 REMIND} -{\f2 . The object remains a vector, however. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MASS} - -${\footnote \pard\plain \sl240 \fs20 $ MASS} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1035} - - K{\footnote \pard\plain \sl240 \fs20 K MASS command;command} - -}{\b\f2 MASS}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 mass} {\f2 command associates a scalar variable as a mass with -the corresponding vector variable, in high-energy physics calculations. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 mass} {\f4 }{\f3 =} {\f4 -\{,}{\f3 =} {\f4 \}* -\par -\par -}{\f2 \par - can be a declared vector variable; }{\f3 mass} {\f2 will declare -it to be of type }{\f3 vector} {\f2 if it is not. This may override an existing -matrix variable by that name. must be a scalar variable. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -vector bb,cc; \par - \par -mass cc=m; \par - \par -mshell cc; \par - \par -cc.cc; \par - \par - 2 \par - M \par - \par -\pard \sl240 }{\f2 Once a mass has been attached to a vector with a }{\f3 mass} {\f2 declaration, -the } -{\f2\uldb mshell}{\v\f2 MSHELL} -{\f2 declaration puts the associated particle ``on the mass -shell.'' Subsequent scalar (.) products of the vector with itself will be -replaced by the square of the mass expression. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MSHELL} - -${\footnote \pard\plain \sl240 \fs20 $ MSHELL} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1036} - - K{\footnote \pard\plain \sl240 \fs20 K MSHELL command;command} - -}{\b\f2 MSHELL}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 mshell} {\f2 command puts particles on the mass shell in high-energy -physics calculations. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 mshell} {\f4 \{,\}* -\par -\par -}{\f2 \par - must have had a mass attached to it by a } -{\f2\uldb mass}{\v\f2 MASS} -{\f2 -declaration. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -vector v1,v2; \par - \par -mass v1=m,v2=q; \par - \par -mshell v1; \par - \par -v1.v1; \par - \par - 2 \par - M \par - \par - \par -v2.v2; \par - \par - V2.V2 \par - \par - \par -mshell v2; \par - \par -v1.v1*v2.v2; \par - \par - 2 2 \par - M *Q \par - \par -\pard \sl240 }{\f2 Even though a mass is attached to a vector variable representing a -particle, the replacement does not take place until the }{\f3 mshell} {\f2 -declaration is given for that vector variable. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NOSPUR} - -${\footnote \pard\plain \sl240 \fs20 $ NOSPUR} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1037} - - K{\footnote \pard\plain \sl240 \fs20 K NOSPUR declaration;declaration} - -}{\b\f2 NOSPUR}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The }{\f3 nospur} {\f2 declaration prevents the trace calculation over the given -line identifiers in high-energy physics calculations. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 nospur} {\f4 \{,\}* -\par -\par -}{\f2 \par - is a scalar identifier that will be used as a line identifier. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -vector a1,b1,c1; \par - \par -g(line1,a1,b1)*g(line2,b1,c1); \par - \par - A1.B1*B1.C1 \par - \par - \par -nospur line2; \par - \par -g(line1,a1,b1)*g(line2,b1,c1); \par - \par - A1.B1*G(LINE2,B1,C1) \par - \par -\pard \sl240 }{\f2 Nospur declarations can be removed by making the declaration } -{\f2\uldb spur}{\v\f2 SPUR} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # REMIND} - -${\footnote \pard\plain \sl240 \fs20 $ REMIND} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1038} - - K{\footnote \pard\plain \sl240 \fs20 K REMIND declaration;declaration} - -}{\b\f2 REMIND}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The }{\f3 remind} {\f2 declaration removes the special status of its arguments -as indices, which was set in the } -{\f2\uldb index}{\v\f2 INDEX} -{\f2 declaration, in -high-energy physics calculations. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 remind} {\f4 \{,\}* -\par -\par -}{\f2 \par - must have been declared to be of type } -{\f2\uldb index}{\v\f2 INDEX} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SPUR} - -${\footnote \pard\plain \sl240 \fs20 $ SPUR} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1039} - - K{\footnote \pard\plain \sl240 \fs20 K SPUR declaration;declaration} - -}{\b\f2 SPUR}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The }{\f3 spur} {\f2 declaration removes the special exemption from trace -calculations that was declared by } -{\f2\uldb nospur}{\v\f2 NOSPUR} -{\f2 , in high-energy physics -calculations. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 spur} {\f4 \{,\}* -\par -\par -}{\f2 \par - must be a line-identifier that has previously been declared -}{\f3 nospur} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # VECDIM} - -${\footnote \pard\plain \sl240 \fs20 $ VECDIM} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1040} - - K{\footnote \pard\plain \sl240 \fs20 K VECDIM command;command} - -}{\b\f2 VECDIM}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The command }{\f3 vecdim} {\f2 changes the vector dimension from 4 to an arbitrary -integer or symbol. Used in high-energy physics calculations. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 vecdim} {\f4 -\par -\par -}{\f2 \par - must be either an integer or a valid scalar identifier that -does not have a floating-point value. -\par -\par -The } -{\f2\uldb eps}{\v\f2 EPS} -{\f2 operator and the gamma_5 -symbol (}{\f3 A} {\f2 ) are not properly defined in anything except four -dimensions and will print an error message if you use them that way. The -other high-energy physics operators should work without problem. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # VECTOR} - -${\footnote \pard\plain \sl240 \fs20 $ VECTOR} - -+{\footnote \pard\plain \sl240 \fs20 + g21:1041} - - K{\footnote \pard\plain \sl240 \fs20 K VECTOR declaration;declaration} - -}{\b\f2 VECTOR}{\f2 \tab \tab \tab \tab }{\b\f2 declaration}{\f2 \par -\par - -The }{\f3 vector} {\f2 declaration declares that its arguments are of type }{\f3 vector} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 vector} {\f4 \{,\}* -\par -\par -}{\f2 \par - must be a valid REDUCE identifier. It may have already been -used for a matrix, array, operator or scalar variable. After an identifier -has been declared to be a vector, it may not be used as a scalar variable. -\par -\par -Vectors are special entities for high-energy physics calculations. You -cannot put values into their coordinates; they do not have coordinates. -They are legal arguments for the high-energy physics operators -} -{\f2\uldb eps}{\v\f2 EPS} -{\f2 , } -{\f2\uldb g}{\v\f2 G} -{\f2 and }{\f3 .} {\f2 (dot). Vector variables are -used to represent gamma matrices and gamma matrices contracted with Lorentz -4-vectors, since there are no Dirac variables per se in the system. -Vectors do follow the usual vector rules for arithmetic operations: -}{\f3 +} {\f2 and }{\f3 -} {\f2 operate upon two or more vectors, producing a -vector; }{\f3 *} {\f2 and }{\f3 /} {\f2 cannot be used between vectors; the -scalar product is represented by the . operator; and the product of a -scalar and vector expression is well defined, and is a vector. -\par -\par -You can represent components of vectors by including representations of unit -vectors in your system. For instance, letting }{\f3 E0} {\f2 represent the unit -vector (1,0,0,0), the command -\par -\par -}{\f3 V1.E0 := 0;} {\f2 would set up the substitution of zero for the first component of the vector -}{\f3 V1} {\f2 . -\par -\par -Identifiers that are declared by the }{\f3 index} {\f2 and }{\f3 mass} {\f2 declarations are -automatically declared to be vectors. -\par -\par -The following errors can occur in calculations using the high energy -physics package: -\par -\par -}{\f3 A represents only gamma5 in vector expressions} {\f2 You have tried to use A in some way other than gamma5 in a -high-energy physics expression. -\par -\par -\par -}{\f3 Gamma5 not allowed unless vecdim is 4} {\f2 You have used gamma_5 in a high-energy physics -computation involving a vector dimension other than 4. -\par -\par -\par - }{\f3 has no mass} {\f2 -\par -\par -One of the arguments to } -{\f2\uldb mshell}{\v\f2 MSHELL} -{\f2 has had no mass assigned to it, in -high-energy physics calculations. -\par -\par -\par -}{\f3 Missing arguments for G operator} {\f2 A line symbol is missing in a gamma matrix expression in high-energy physics -calculations. -\par -\par -\par -}{\f3 Unmatched index} {\f2 -\par -\par -The parser has found unmatched indices during the evaluation of a -gamma matrix expression in high-energy physics calculations. -\par -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g21} - -${\footnote \pard\plain \sl240 \fs20 $ High Energy Physics} - -+{\footnote \pard\plain \sl240 \fs20 + index:0021} -}{\b\f2 High Energy Physics}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb HEPHYS introduction} -{\v\f2 HEPHYS}{\f2 \par -}{\f2 \tab}{\f2\uldb HE-dot operator} -{\v\f2 HE_dot}{\f2 \par -}{\f2 \tab}{\f2\uldb EPS operator} -{\v\f2 EPS}{\f2 \par -}{\f2 \tab}{\f2\uldb G operator} -{\v\f2 G}{\f2 \par -}{\f2 \tab}{\f2\uldb INDEX declaration} -{\v\f2 INDEX}{\f2 \par -}{\f2 \tab}{\f2\uldb MASS command} -{\v\f2 MASS}{\f2 \par -}{\f2 \tab}{\f2\uldb MSHELL command} -{\v\f2 MSHELL}{\f2 \par -}{\f2 \tab}{\f2\uldb NOSPUR declaration} -{\v\f2 NOSPUR}{\f2 \par -}{\f2 \tab}{\f2\uldb REMIND declaration} -{\v\f2 REMIND}{\f2 \par -}{\f2 \tab}{\f2\uldb SPUR declaration} -{\v\f2 SPUR}{\f2 \par -}{\f2 \tab}{\f2\uldb VECDIM command} -{\v\f2 VECDIM}{\f2 \par -}{\f2 \tab}{\f2\uldb VECTOR declaration} -{\v\f2 VECTOR}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Numeric_Package} - -${\footnote \pard\plain \sl240 \fs20 $ Numeric_Package} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1042} - - K{\footnote \pard\plain \sl240 \fs20 K Numeric Package introduction;introduction} - -}{\b\f2 NUMERIC PACKAGE}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par -\par - -The numeric package supplies algorithms based on approximation -techniques of numerical mathematics. The algorithms use -the } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 mode arithmetic of REDUCE, including -the variable precision feature which is exploited in some -algorithms in an adaptive manner in order to reach the -desired accuracy. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Interval} - -${\footnote \pard\plain \sl240 \fs20 $ Interval} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1043} - - K{\footnote \pard\plain \sl240 \fs20 K Interval type;type} - -}{\b\f2 INTERVAL}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par -\par - -Intervals are generally coded as lower bound and -upper bound connected by the operator }{\f3 ..} {\f2 , usually -associated to a variable in an -equation. -\par -\par - \par -syntax: \par -}{\f4 = ( .. ) -\par -\par -}{\f2 \par -where is a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 and , are -numbers or expression which evaluate to numbers with <=. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 means that the variable x is taken in the range from 2.5 up to -3.5. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # numeric_accuracy} - -${\footnote \pard\plain \sl240 \fs20 $ numeric_accuracy} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1044} - - K{\footnote \pard\plain \sl240 \fs20 K numeric accuracy concept;concept} - -}{\b\f2 NUMERIC ACCURACY}{\f2 \par -\par - -The keyword parameters }{\f3 accuracy=a} {\f2 and }{\f3 iterations=i} {\f2 , -where }{\f3 a} {\f2 and }{\f3 i} {\f2 must be positive integer numbers, control the -iterative algorithms: the iteration is continued until -the local error is below 10**-a; if that is impossible -within }{\f3 i} {\f2 steps, the iteration is terminated with an -error message. The values reached so far are then returned -as the result. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TRNUMERIC} - -${\footnote \pard\plain \sl240 \fs20 $ TRNUMERIC} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1045} - - K{\footnote \pard\plain \sl240 \fs20 K TRNUMERIC switch;switch} - -}{\b\f2 TRNUMERIC}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - -Normally the algorithms produce only a minimum of printed -output during their operation. In cases of an unsuccessful -or unexpected long operation a }{\f3 trace of the iteration} {\f2 can be -printed by setting }{\f3 trnumeric} {\f2 }{\f3 on} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # num_min} - -${\footnote \pard\plain \sl240 \fs20 $ num_min} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1046} - - K{\footnote \pard\plain \sl240 \fs20 K Fletcher Reeves;steepest descent;minimum;num_min operator;operator} - -}{\b\f2 NUM_MIN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The Fletcher Reeves version of the }{\f3 steepest descent} {\f2 -algorithms is used to find the }{\f3 minimum} {\f2 of a -function of one or more variables. The -function must have continuous partial derivatives with respect to all -variables. The starting point of the search can be -specified; if not, random values are taken instead. -The steepest descent algorithms in general find only local -minima. - \par -\par - \par -syntax: \par -}{\f4 }{\f3 num_min} {\f4 (, - [=] [,[=] ... - [,accuracy=] [,iterations=]) -\par -\par -or -\par -\par -}{\f3 num_min} {\f4 (exp, \{ - [=] [,[=] ...] \} - [,accuracy=] [,iterations=]) -\par -\par -}{\f2 \par -where is a function expression, - are the variables in and - are the (optional) start values. -For and see } -{\f2\uldb numeric accuracy}{\v\f2 numeric_accuracy} -{\f2 . -\par -\par -}{\f3 Num_min} {\f2 tries to find the next local minimum along the descending -path starting at the given point. The result is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 -with the minimum function value as first element followed by a list -of } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f3 s} {\f2 , where the variables are equated to the coordinates -of the result point. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -num_min(sin(x)+x/5, x) \par - \par - \{4.9489585606,\{X=29.643767785\}\} \par - \par - \par -num_min(sin(x)+x/5, x=0) \par - \par - \{ - 1.3342267466,\{X= - 1.7721582671\}\} \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # num_solve} - -${\footnote \pard\plain \sl240 \fs20 $ num_solve} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1047} - - K{\footnote \pard\plain \sl240 \fs20 K Jacobian matrix;root;Newton iteration;equation system;equation solving;num_solve operator;operator} - -}{\b\f2 NUM_SOLVE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -An adaptively damped Newton iteration is used to find -an approximative root of a function (function vector) or the -solution of an } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 (equation system). The expressions -must have continuous derivatives for all variables. -A starting point for the iteration can be given. If not given -random values are taken instead. When the number of -forms is not equal to the number of variables, the -Newton method cannot be applied. Then the minimum -of the sum of absolute squares is located instead. -\par -\par -With } -{\f2\uldb complex}{\v\f2 COMPLEX} -{\f2 on, solutions with imaginary parts can be -found, if either the expression(s) or the starting point -contain a nonzero imaginary part. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 num_solve} {\f4 (, [=][,accuracy=][,iterations=]) -\par -\par -or -\par -\par -}{\f3 num_solve} {\f4 (\{,...,\}, [=],...,[=] - [,accuracy=][,iterations=]) -\par -\par -or -\par -\par -}{\f3 num_solve} {\f4 (\{,...,\}, \{[=],...,[=]\} - [,accuracy=][,iterations=]) - \par -\par -\par -\par -}{\f2 where are function expressions, - are the variables, - are optional start values. -For and see } -{\f2\uldb numeric accuracy}{\v\f2 numeric_accuracy} -{\f2 . - \par -\par -}{\f3 num_solve} {\f2 tries to find a zero/solution of the expression(s). -Result is a list of equations, where the variables are -equated to the coordinates of the result point. - \par -\par -The }{\f3 Jacobian matrix} {\f2 is stored as side effect the shared -variable }{\f3 jacobian} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -num_solve(\{sin x=cos y, x + y = 1\},\{x=1,y=2\}); \par - \par - \par - \{X= - 1.8561957251,Y=2.856195584\} \par - \par - \par -jacobian; \par - \par - [COS(X) SIN(Y)] \par - [ ] \par - [ 1 1 ] \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # num_int} - -${\footnote \pard\plain \sl240 \fs20 $ num_int} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1048} - - K{\footnote \pard\plain \sl240 \fs20 K integration;num_int operator;operator} - -}{\b\f2 NUM_INT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -For the numerical evaluation of univariate integrals -over a finite interval the following strategy is used: -If } -{\f2\uldb int}{\v\f2 INT} -{\f2 finds a formal antiderivative - which is bounded in the integration interval, this - is evaluated and the end points and the difference - is returned. -Otherwise a } -{\f2\uldb Chebyshev fit}{\v\f2 Chebyshev_fit} -{\f2 is computed, - starting with order 20, eventually up to order 80. - If that is recognized as sufficiently convergent - it is used for computing the integral by directly - integrating the coefficient sequence. -If none of these methods is successful, an - adaptive multilevel quadrature algorithm is used. -\par -\par -For multivariate integrals only the adaptive quadrature is used. -This algorithm tolerates isolated singularities. -The value }{\f3 iterations} {\f2 here limits the number of -local interval intersection levels. - is a measure for the relative total discretization -error (comparison of order 1 and order 2 approximations). -\par -\par - \par -syntax: \par -}{\f4 }{\f3 num_int} {\f4 (,=( .. ) - [,=( .. ),...] - [,accuracy=][,iterations=]) -\par -\par -}{\f2 \par -where is the function to be integrated, - are the integration variables, - are the lower bounds, - are the upper bounds. - \par -\par -Result is the value of the integral. - \par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -num_int(sin x,x=(0 .. 3.1415926)); \par - \par - 2.0000010334 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # num_odesolve} - -${\footnote \pard\plain \sl240 \fs20 $ num_odesolve} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1049} - - K{\footnote \pard\plain \sl240 \fs20 K ODE;initial value problem;Runge-Kutta;num_odesolve operator;operator} - -}{\b\f2 NUM_ODESOLVE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 Runge-Kutta} {\f2 method of order 3 finds an approximate graph for -the solution of real }{\f3 ODE initial value problem} {\f2 . - \par -\par - \par -syntax: \par -}{\f4 }{\f3 num_odesolve} {\f4 (,=, - =( .. ) - [,accuracy=][,iterations=]) -\par -\par -or -\par -\par -}{\f3 num_odesolve} {\f4 (\{,,...\}, - \{=,=,...\} - =( .. ) - [,accuracy=][,iterations=]) -\par -\par -\par -\par -}{\f2 where - and specify the dependent variable(s) -and the starting point value (vector), -, and specify the independent variable -and the integration interval (starting point and end point), - are equations or expressions which -contain the first derivative of the independent variable -with respect to the dependent variable. - \par -\par -The ODEs are converted to an explicit form, which then is -used for a Runge Kutta iteration over the given range. The -number of steps is controlled by the value of -(default: 20). If the steps are too coarse to reach the desired -accuracy in the neighborhood of the starting point, the number is -increased automatically. - \par -\par -Result is a list of pairs, each representing a point of the -approximate solution of the ODE problem. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -depend(y,x); \par - \par -num_odesolve(df(y,x)=y,y=1,x=(0 .. 1), iterations=5); \par - \par - \par - ,\{0.2,1.2214\},\{0.4,1.49181796\},\{0.6,1.8221064563\}, \par - \{0.8,2.2255208258\},\{1.0,2.7182511366\}\} \par - \par -\pard \sl240 }{\f2 In most cases you must declare the dependency relation -between the variables explicitly using } -{\f2\uldb depend}{\v\f2 DEPEND} -{\f2 ; -otherwise the formal derivative might be converted to zero. -\par -\par -The operator } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 is used to convert the form into -an explicit ODE. If that process fails or if it has no unique result, -the evaluation is stopped with an error message. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # bounds} - -${\footnote \pard\plain \sl240 \fs20 $ bounds} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1050} - - K{\footnote \pard\plain \sl240 \fs20 K bounds operator;operator} - -}{\b\f2 BOUNDS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par -Upper and lower bounds of a real valued function over an -} -{\f2\uldb interval}{\v\f2 Interval} -{\f2 or a rectangular multivariate domain are computed -by the operator }{\f3 bounds} {\f2 . The algorithmic basis is the computation -with inequalities: starting from the interval(s) of the -variables, the bounds are propagated in the expression -using the rules for inequality computation. Some knowledge -about the behavior of special functions like ABS, SIN, COS, EXP, LOG, -fractional exponentials etc. is integrated and can be evaluated -if the operator }{\f3 bounds} {\f2 is called with rounded mode on -(otherwise only algebraic evaluation rules are available). - \par -\par -If }{\f3 bounds} {\f2 finds a singularity within an interval, the evaluation -is stopped with an error message indicating the problem part -of the expression. - \par -\par - \par -syntax: \par -}{\f4 }{\f3 bounds} {\f4 (,=( .. ) - [,=( .. ) ...]) -\par -\par -or -\par -\par -}{\f3 bounds} {\f4 (,\{=( .. ) - [,=( .. ) ...]\}) -\par -\par -\par -\par -}{\f2 where is the function to be investigated, - are the variables of , - and specify the area as set of } -{\f2\uldb interval}{\v\f2 Interval} -{\f3 s} {\f2 . -\par -\par -}{\f3 bounds} {\f2 computes upper and lower bounds for the expression in the -given area. An } -{\f2\uldb interval}{\v\f2 Interval} -{\f2 is returned. - \par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -bounds(sin x,x=(1 .. 2)); \par - \par - -1 .. 1 \par - \par - \par -on rounded; \par - \par -bounds(sin x,x=(1 .. 2)); \par - \par - 0.84147098481 .. 1 \par - \par - \par -bounds(x**2+x,x=(-0.5 .. 0.5)); \par - \par - - 0.25 .. 0.75 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Chebyshev_fit} - -${\footnote \pard\plain \sl240 \fs20 $ Chebyshev_fit} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1051} - - K{\footnote \pard\plain \sl240 \fs20 K approximation;Chebyshev fit concept;concept} - -}{\b\f2 CHEBYSHEV FIT}{\f2 \par -\par - - \par -\par -The operator family }{\f3 Chebyshev_...} {\f2 implements approximation -and evaluation of functions by the Chebyshev method. -Let }{\f3 T(n,a,b,x)} {\f2 be the Chebyshev polynomial of order }{\f3 n} {\f2 -transformed to the interval }{\f3 (a,b)} {\f2 . -Then a function }{\f3 f(x)} {\f2 can be -approximated in }{\f3 (a,b)} {\f2 by a series -\par -\par -\pard \tx3420 }{\f4 \par - for i := 0:n sum c(i)*T(i,a,b,x) \par -\pard \sl240 }{\f2 The operator }{\f3 chebyshev_fit} {\f2 computes this approximation and -returns a list, which has as first element the sum expressed -as a polynomial and as second element the sequence -of Chebyshev coefficients. -}{\f3 Chebyshev_df} {\f2 and }{\f3 Chebyshev_int} {\f2 transform a Chebyshev -coefficient list into the coefficients of the corresponding -derivative or integral respectively. For evaluating a Chebyshev -approximation at a given point in the basic interval the -operator }{\f3 Chebyshev_eval} {\f2 can be used. -}{\f3 Chebyshev_eval} {\f2 is based on a recurrence relation which is -in general more stable than a direct evaluation of the -complete polynomial. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 chebyshev_fit} {\f4 (,=( .. ),) -\par -\par -}{\f3 chebyshev_eval} {\f4 (,=( .. ), - =) -\par -\par -}{\f3 chebyshev_df} {\f4 (,=( .. )) -\par -\par -}{\f3 chebyshev_int} {\f4 (,=( .. )) -\par -\par -}{\f2 \par -where is an algebraic expression (the target function), - is the variable of , - and are -numerical real values which describe an } -{\f2\uldb interval}{\v\f2 Interval} -{\f2 < , -the integer is the approximation order (set to 20 if missing), - is a number in the interval and is -a series of Chebyshev coefficients. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -on rounded; \par - \par - \par -w:=chebyshev_fit(sin x/x,x=(1 .. 3),5); \par - \par - \par - 3 2 \par - w := \{0.03824*x - 0.2398*x + 0.06514*x + 0.9778, \par - \{0.8991,-0.4066,-0.005198,0.009464,-0.00009511\}\} \par - \par - \par -chebyshev_eval(second w, x=(1 .. 3), x=2.1); \par - \par - \par - 0.4111 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # num_fit} - -${\footnote \pard\plain \sl240 \fs20 $ num_fit} - -+{\footnote \pard\plain \sl240 \fs20 + g22:1052} - - K{\footnote \pard\plain \sl240 \fs20 K least squares;approximation;num_fit operator;operator} - -}{\b\f2 NUM_FIT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 num_fit} {\f2 finds for a set of -points the linear combination of a given set of -functions (function basis) which approximates the -points best under the objective of the }{\f3 least squares} {\f2 -criterion (minimum of the sum of the squares of the deviation). -The solution is found as zero of the -gradient vector of the sum of squared errors. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 num_fit} {\f4 (,,=) -\par -\par -}{\f2 \par -where is a list of numeric values, - is a variable used for the approximation, - is a list of coordinate values which correspond to -, - is a set of functions varying in }{\f3 var} {\f2 which is used - for the approximation. - \par -\par -The result is a list containing as first element the -function which approximates the given values, and as -second element a list of coefficients which were used -to build this function from the basis. - \par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -pts:=for i:=1 step 1 until 5 collect i$ \par - \par -vals:=for i:=1 step 1 until 5 collect \par - \par - for j:=1:i product j$ \par - \par -num_fit(vals,\{1,x,x**2\},x=pts); \par - \par - 2 \par - \{14.571428571*X - 61.428571429*X + 54.6,\{54.6, \par - - 61.428571429,14.571428571\}\} \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g22} - -${\footnote \pard\plain \sl240 \fs20 $ Numeric Package} - -+{\footnote \pard\plain \sl240 \fs20 + index:0022} -}{\b\f2 Numeric Package}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Numeric Package introduction} -{\v\f2 Numeric_Package}{\f2 \par -}{\f2 \tab}{\f2\uldb Interval type} -{\v\f2 Interval}{\f2 \par -}{\f2 \tab}{\f2\uldb numeric accuracy concept} -{\v\f2 numeric_accuracy}{\f2 \par -}{\f2 \tab}{\f2\uldb TRNUMERIC switch} -{\v\f2 TRNUMERIC}{\f2 \par -}{\f2 \tab}{\f2\uldb num_min operator} -{\v\f2 num_min}{\f2 \par -}{\f2 \tab}{\f2\uldb num_solve operator} -{\v\f2 num_solve}{\f2 \par -}{\f2 \tab}{\f2\uldb num_int operator} -{\v\f2 num_int}{\f2 \par -}{\f2 \tab}{\f2\uldb num_odesolve operator} -{\v\f2 num_odesolve}{\f2 \par -}{\f2 \tab}{\f2\uldb bounds operator} -{\v\f2 bounds}{\f2 \par -}{\f2 \tab}{\f2\uldb Chebyshev fit concept} -{\v\f2 Chebyshev_fit}{\f2 \par -}{\f2 \tab}{\f2\uldb num_fit operator} -{\v\f2 num_fit}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Roots_Package} - -${\footnote \pard\plain \sl240 \fs20 $ Roots_Package} - -+{\footnote \pard\plain \sl240 \fs20 + g23:1053} - - K{\footnote \pard\plain \sl240 \fs20 K getroot;firstroot;rootval;rootsat-prec;rlrootno;isolater;polynomial;roots;Roots Package introduction;introduction} - -}{\b\f2 ROOTS PACKAGE}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par -\par - - \par -\par -The root finding package is designed so that it can -be used to find some or all of the roots of univariate -polynomials with real or complex coefficients, to the accuracy -specified by the user. -\par -\par -Not all operators of }{\f3 roots package} {\f2 are described here. For using -the operators -\par -\par -}{\f3 isolater} {\f2 (intervals isolating real roots) -\par -\par -}{\f3 rlrootno} {\f2 (number of real roots in an interval) -\par -\par -}{\f3 rootsat-prec} {\f2 (roots at system precision) -\par -\par -}{\f3 rootval} {\f2 (result in equation form) -\par -\par -}{\f3 firstroot} {\f2 (computing only one root) -\par -\par -}{\f3 getroot} {\f2 (selecting roots from a collection) -\par -\par -please consult the full documentation of the package. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MKPOLY} - -${\footnote \pard\plain \sl240 \fs20 $ MKPOLY} - -+{\footnote \pard\plain \sl240 \fs20 + g23:1054} - - K{\footnote \pard\plain \sl240 \fs20 K interpolation;roots;polynomial;MKPOLY operator;operator} - -}{\b\f2 MKPOLY}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -Given a roots list as returned by } -{\f2\uldb roots}{\v\f2 ROOTS} -{\f2 , -the operator }{\f3 mkpoly} {\f2 constructs a -polynomial which has these numbers as roots. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 mkpoly} {\f4 -\par -\par -}{\f2 \par -where is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 with equations, which -all have the same } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 on their left-hand sides -and numbers as right-hand sides. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -mkpoly\{x=1,x=-2,x=i,x=-i\}; \par - \par - x**4 + x**3 - x**2 + x - 2 \par - \par -\pard \sl240 }{\f2 Note that this polynomial is unique only up to a numeric -factor. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NEARESTROOT} - -${\footnote \pard\plain \sl240 \fs20 $ NEARESTROOT} - -+{\footnote \pard\plain \sl240 \fs20 + g23:1055} - - K{\footnote \pard\plain \sl240 \fs20 K solve;roots;NEARESTROOT operator;operator} - -}{\b\f2 NEARESTROOT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 nearestroot} {\f2 finds one root of a polynomial -with an iteration using a given starting point. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 nearestroot} {\f4 (

,) -\par -\par -}{\f2 \par -where

is a univariate polynomial -and is a number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -nearestroot(x^2+2,2); \par - \par - \{x=1.41421*i\} \par - \par -\pard \sl240 }{\f2 The minimal accuracy of the result values is controlled by -} -{\f2\uldb rootacc}{\v\f2 ROOTACC} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # REALROOTS} - -${\footnote \pard\plain \sl240 \fs20 $ REALROOTS} - -+{\footnote \pard\plain \sl240 \fs20 + g23:1056} - - K{\footnote \pard\plain \sl240 \fs20 K solve;roots;REALROOTS operator;operator} - -}{\b\f2 REALROOTS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 realroots} {\f2 finds that real roots of a polynomial -to an accuracy that is sufficient to separate them and which is -a minimum of 6 decimal places. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 realroots} {\f4 (

) or -\par -\par -}{\f3 realroots} {\f4 (

,,) -\par -\par -}{\f2 \par -where

is a univariate polynomial. -The optional parameters and classify -an interval: if given, exactly the real roots in this -interval will be returned. and -can also take the values }{\f3 infinity} {\f2 or }{\f3 -infinity} {\f2 . -If omitted all real roots will be returned. -Result is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 -of equations which represent the roots of the polynomial at the -given accuracy. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -realroots(x^5-2); \par - \par - \{x=1.1487\} \par - \par - \par -realroots(x^3-104*x^2+403*x-300,2,infinity); \par - \par - \par - \{x=3.0,x=100.0\} \par - \par - \par -realroots(x^3-104*x^2+403*x-300,-infinity,2); \par - \par - \par - \{x=1\} \par - \par -\pard \sl240 }{\f2 The minimal accuracy of the result values is controlled by -} -{\f2\uldb rootacc}{\v\f2 ROOTACC} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROOTACC} - -${\footnote \pard\plain \sl240 \fs20 $ ROOTACC} - -+{\footnote \pard\plain \sl240 \fs20 + g23:1057} - - K{\footnote \pard\plain \sl240 \fs20 K accuracy;roots;ROOTACC operator;operator} - -}{\b\f2 ROOTACC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 rootacc} {\f2 allows you to set the accuracy -up to which the roots package computes its results. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 rootacc} {\f4 () -\par -\par -}{\f2 \par -Here is an integer value. The internal accuracy of -the }{\f3 roots} {\f2 package is adjusted to a value of -}{\f3 max(6,n)} {\f2 . The default value is }{\f3 6} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROOTS} - -${\footnote \pard\plain \sl240 \fs20 $ ROOTS} - -+{\footnote \pard\plain \sl240 \fs20 + g23:1058} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;solve;roots;ROOTS operator;operator} - -}{\b\f2 ROOTS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 roots} {\f2 -is the main top level function of the roots package. -It will find all roots, real and complex, of the polynomial p -to an accuracy that is sufficient to separate them and which is -a minimum of 6 decimal places. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 roots} {\f4 (

) -\par -\par -}{\f2 \par -where

is a univariate polynomial. Result is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 -of equations which represent the roots of the polynomial at the -given accuracy. In addition, }{\f3 roots} {\f2 stores -separate lists of real roots and complex roots in the global -variables } -{\f2\uldb rootsreal}{\v\f2 ROOTSREAL} -{\f2 and } -{\f2\uldb rootscomplex}{\v\f2 ROOTSCOMPLEX} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -roots(x^5-2); \par - \par - \{x=-0.929316 + 0.675188*i, \par - x=-0.929316 - 0.675188*i, \par - x=0.354967 + 1.09248*i, \par - x=0.354967 - 1.09248*i, \par - x=1.1487\} \par - \par -\pard \sl240 }{\f2 The minimal accuracy of the result values is controlled by -} -{\f2\uldb rootacc}{\v\f2 ROOTACC} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROOT\_VAL} - -${\footnote \pard\plain \sl240 \fs20 $ ROOT_VAL} - -+{\footnote \pard\plain \sl240 \fs20 + g23:1059} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;solve;roots;ROOT_VAL operator;operator} - -}{\b\f2 ROOT\_VAL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The operator }{\f3 root_val} {\f2 computes the roots of a -univariate polynomial at system precision -(or greater if required for root separation) and presents -its result as a list of numbers. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 roots} {\f4 (

) -\par -\par -}{\f2 \par -where

is a univariate polynomial. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -root_val(x^5-2); \par - \par - \{-0.929316490603 + 0.6751879524*i, \par - -0.929316490603 - 0.6751879524*i, \par - 0.354967313105 + 1.09247705578*i, \par - 0.354967313105 - 1.09247705578*i, \par - 1.148698355\} \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROOTSCOMPLEX} - -${\footnote \pard\plain \sl240 \fs20 $ ROOTSCOMPLEX} - -+{\footnote \pard\plain \sl240 \fs20 + g23:1060} - - K{\footnote \pard\plain \sl240 \fs20 K complex;roots;ROOTSCOMPLEX variable;variable} - -}{\b\f2 ROOTSCOMPLEX}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -When the operator } -{\f2\uldb roots}{\v\f2 ROOTS} -{\f2 is called the complex -roots are collected in the global variable }{\f3 rootscomplex} {\f2 -as } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROOTSREAL} - -${\footnote \pard\plain \sl240 \fs20 $ ROOTSREAL} - -+{\footnote \pard\plain \sl240 \fs20 + g23:1061} - - K{\footnote \pard\plain \sl240 \fs20 K complex;roots;ROOTSREAL variable;variable} - -}{\b\f2 ROOTSREAL}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -When the operator } -{\f2\uldb roots}{\v\f2 ROOTS} -{\f2 is called the real -roots are collected in the global variable }{\f3 rootreal} {\f2 -as } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g23} - -${\footnote \pard\plain \sl240 \fs20 $ Roots Package} - -+{\footnote \pard\plain \sl240 \fs20 + index:0023} -}{\b\f2 Roots Package}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Roots Package introduction} -{\v\f2 Roots_Package}{\f2 \par -}{\f2 \tab}{\f2\uldb MKPOLY operator} -{\v\f2 MKPOLY}{\f2 \par -}{\f2 \tab}{\f2\uldb NEARESTROOT operator} -{\v\f2 NEARESTROOT}{\f2 \par -}{\f2 \tab}{\f2\uldb REALROOTS operator} -{\v\f2 REALROOTS}{\f2 \par -}{\f2 \tab}{\f2\uldb ROOTACC operator} -{\v\f2 ROOTACC}{\f2 \par -}{\f2 \tab}{\f2\uldb ROOTS operator} -{\v\f2 ROOTS}{\f2 \par -}{\f2 \tab}{\f2\uldb ROOT\_VAL operator} -{\v\f2 ROOT\_VAL}{\f2 \par -}{\f2 \tab}{\f2\uldb ROOTSCOMPLEX variable} -{\v\f2 ROOTSCOMPLEX}{\f2 \par -}{\f2 \tab}{\f2\uldb ROOTSREAL variable} -{\v\f2 ROOTSREAL}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Special_Function_Package} - -${\footnote \pard\plain \sl240 \fs20 $ Special_Function_Package} - -+{\footnote \pard\plain \sl240 \fs20 + g24:1062} - - K{\footnote \pard\plain \sl240 \fs20 K Special Function Package introduction;introduction} - -}{\b\f2 SPECIAL FUNCTION PACKAGE}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par -\par - -The REDUCE }{\f3 Special Function Package} {\f2 supplies extended -algebraic and numeric support for a wide class of objects. -This package was released together with REDUCE 3.5 (October 1993) -for the first time, a major update is released with REDUCE 3.6. -\par -\par -The functions included in this package are in most cases (unless otherwise -stated) defined and named like in the book by Abramowitz and Stegun: -Handbook of Mathematical Functions, Dover Publications. -\par -\par -The aim is to collect as much information on the special functions -and simplification capabilities as possible, -i.e. algebraic simplifications and numeric (rounded mode) code, limits -of the functions together -with the definitions of the functions, which are in most cases a power -series, a (definite) integral and/or a differential equation. -\par -\par -What can be found: Some famous constants, a variety of Bessel functions, -special polynomials, -the Gamma function, the (Riemann) Zeta function, Elliptic Functions, Elliptic -Integrals, 3J symbols (Clebsch-Gordan coefficients) and integral functions. -\par -\par -What is missing: Mathieu functions, LerchPhi, etc.. -The information about the special functions which solve certain -differential equation is very limited. -In several cases numerical approximation is restricted to real -arguments or is missing completely. -\par -\par -The implementation of this package uses REDUCE rule sets to a large extent, -which guarantees a high 'readability' of the functions definitions in the -source file directory. It makes extensions to the special -functions code easy in most cases too. To look at these rules -it may be convenient to use the showrules operator e.g. -\par -\par -} -{\f2\uldb showrules}{\v\f2 SHOWRULES} -{\f2 Besseli; -\par -\par -. -\par -\par -Some evaluations are improved if the special function package is loaded, -e.g. some (infinite) sums and products leading to expressions including -special functions are known in this case. -\par -\par -Note: The special function package has to be loaded explicitly by calling -\pard \tx3420 }{\f4 \par - load_package specfn; \par -\pard \sl240 }{\f2 \par -\par -The functions } -{\f2\uldb MeijerG}{\v\f2 MeijerG} -{\f2 and } -{\f2\uldb hypergeometric}{\v\f2 HYPERGEOMETRIC} -{\f2 require -additionally -\pard \tx3420 }{\f4 \par - load_package specfn2; \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Constants} - -${\footnote \pard\plain \sl240 \fs20 $ Constants} - -+{\footnote \pard\plain \sl240 \fs20 + g24:1063} - - K{\footnote \pard\plain \sl240 \fs20 K Golden_Ratio;Khinchin's constant;Catalan's constant;Euler's constant;Constants concept;concept} - -}{\b\f2 CONSTANTS}{\f2 \par -\par - - \par -\par -There are a few constants known to the special function package, namely -\par -\par -}{\f3 Euler's constant} {\f2 (which can be computed as -} -{\f2\uldb Psi}{\v\f2 PSI} -{\f2 (1)) and -\par -\par -}{\f3 Khinchin's constant} {\f2 (which is defined in Khinchin's book -\par -\par - ``Continued Fractions'') and -\par -\par -}{\f3 Golden_Ratio} {\f2 (which can be computed as (1 + sqrt 5)/2) and -\par -\par -}{\f3 Catalan's constant} {\f2 (which is known as an infinite sum of reciprocal -powers) -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par -Euler_Gamma; \par - \par - 0.577215664902 \par - \par - \par -Khinchin; \par - \par - 2.68545200107 \par - \par - \par -Catalan \par - \par - 0.915965594177 \par - \par - \par -Golden_Ratio \par - \par - 1.61803398875 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BERNOULLI} - -${\footnote \pard\plain \sl240 \fs20 $ BERNOULLI} - -+{\footnote \pard\plain \sl240 \fs20 + g25:1064} - - K{\footnote \pard\plain \sl240 \fs20 K BERNOULLI operator;operator} - -}{\b\f2 BERNOULLI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 bernoulli} {\f2 operator returns the nth Bernoulli number. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Bernoulli} {\f4 () -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par -bernoulli 20; \par - \par - - 174611 / 330 \par - \par - \par -bernoulli 17; \par - \par - 0 \par - \par -\pard \sl240 }{\f2 All Bernoulli numbers with odd indices except for 1 are zero. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BERNOULLIP} - -${\footnote \pard\plain \sl240 \fs20 $ BERNOULLIP} - -+{\footnote \pard\plain \sl240 \fs20 + g25:1065} - - K{\footnote \pard\plain \sl240 \fs20 K BERNOULLIP operator;operator} - -}{\b\f2 BERNOULLIP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 BernoulliP} {\f2 operator returns the nth Bernoulli Polynomial -evaluated at x. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 BernoulliP} {\f4 (,) -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par -BernoulliP(3,z); \par - \par - 2 \par - z*(2*z - 3*z + 1)/2 \par - \par - \par - \par -BernoulliP(10,3); \par - \par - 338585 / 66 \par - \par -\pard \sl240 }{\f2 The value of the nth Bernoulli Polynomial at 0 is the nth Bernoulli number. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EULER} - -${\footnote \pard\plain \sl240 \fs20 $ EULER} - -+{\footnote \pard\plain \sl240 \fs20 + g25:1066} - - K{\footnote \pard\plain \sl240 \fs20 K EULER operator;operator} - -}{\b\f2 EULER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 EULER} {\f2 operator returns the nth Euler number. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Euler} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Euler 20; \par - \par - 370371188237525 \par - \par - \par -Euler 0; \par - \par - 1 \par - \par -\pard \sl240 }{\f2 The }{\f3 Euler} {\f2 numbers are evaluated by a recursive algorithm which -makes it hard to compute Euler numbers above say 200. -\par -\par -Euler numbers appear in the coefficients of the power series -representation of 1/cos(z). -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EULERP} - -${\footnote \pard\plain \sl240 \fs20 $ EULERP} - -+{\footnote \pard\plain \sl240 \fs20 + g25:1067} - - K{\footnote \pard\plain \sl240 \fs20 K EULERP operator;operator} - -}{\b\f2 EULERP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 EulerP} {\f2 operator returns the nth Euler Polynomial. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 EulerP} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -EulerP(2,xx); \par - \par - xx*(xx - 1) \par - \par - \par -EulerP(10,3); \par - \par - 2046 \par - \par -\pard \sl240 }{\f2 The Euler numbers are the values of the Euler Polynomials at 1/2 -multiplied by 2**n. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ZETA} - -${\footnote \pard\plain \sl240 \fs20 $ ZETA} - -+{\footnote \pard\plain \sl240 \fs20 + g25:1068} - - K{\footnote \pard\plain \sl240 \fs20 K ZETA operator;operator} - -}{\b\f2 ZETA}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Zeta} {\f2 operator returns Riemann's Zeta function, -\par -\par -Zeta (z) := sum(1/(k**z),k,1,infinity) -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Zeta} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Zeta(2); \par - \par - 2 \par - pi / 6 \par - \par - \par -on rounded; \par - \par -Zeta 1.01; \par - \par - 100.577943338 \par - \par -\pard \sl240 }{\f2 Numerical computation for the Zeta function for arguments close to 1 are -tedious, because the series is converging very slowly. In this case a formula -(e.g. found in Bender/Orzag: Advanced Mathematical Methods for -Scientists and Engineers, McGraw-Hill) is used. -\par -\par -No numerical approximation for complex arguments is done. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g25} - -${\footnote \pard\plain \sl240 \fs20 $ Bernoulli Euler Zeta} - -+{\footnote \pard\plain \sl240 \fs20 + index:0025} -}{\b\f2 Bernoulli Euler Zeta}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb BERNOULLI operator} -{\v\f2 BERNOULLI}{\f2 \par -}{\f2 \tab}{\f2\uldb BERNOULLIP operator} -{\v\f2 BERNOULLIP}{\f2 \par -}{\f2 \tab}{\f2\uldb EULER operator} -{\v\f2 EULER}{\f2 \par -}{\f2 \tab}{\f2\uldb EULERP operator} -{\v\f2 EULERP}{\f2 \par -}{\f2 \tab}{\f2\uldb ZETA operator} -{\v\f2 ZETA}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BESSELJ} - -${\footnote \pard\plain \sl240 \fs20 $ BESSELJ} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1069} - - K{\footnote \pard\plain \sl240 \fs20 K BESSELJ operator;operator} - -}{\b\f2 BESSELJ}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 BesselJ} {\f2 operator returns the Bessel function of the first kind. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 BesselJ} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -BesselJ(1/2,pi); \par - \par - 0 \par - \par - \par -on rounded; \par - \par -BesselJ(0,1); \par - \par - 0.765197686558 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BESSELY} - -${\footnote \pard\plain \sl240 \fs20 $ BESSELY} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1070} - - K{\footnote \pard\plain \sl240 \fs20 K Weber's function;BESSELY operator;operator} - -}{\b\f2 BESSELY}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 BesselY} {\f2 operator returns the Bessel function of the second kind. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 BesselY} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -BesselY (1/2,pi); \par - \par - - sqrt(2) / pi \par - \par - \par -on rounded; \par - \par -BesselY (1,3); \par - \par - 0.324674424792 \par - \par -\pard \sl240 }{\f2 The operator }{\f3 BesselY} {\f2 is also called Weber's function. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # HANKEL1} - -${\footnote \pard\plain \sl240 \fs20 $ HANKEL1} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1071} - - K{\footnote \pard\plain \sl240 \fs20 K HANKEL1 operator;operator} - -}{\b\f2 HANKEL1}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Hankel1} {\f2 operator returns the Hankel function of the first kind. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Hankel1} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on complex; \par - \par -Hankel1 (1/2,pi); \par - \par - - i * sqrt(2) / pi \par - \par - \par -Hankel1 (1,pi); \par - \par - besselj(1,pi) + i*bessely(1,pi) \par - \par -\pard \sl240 }{\f2 The operator }{\f3 Hankel1} {\f2 is also called Bessel function of the third kind. -There is currently no numeric evaluation of Hankel functions. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # HANKEL2} - -${\footnote \pard\plain \sl240 \fs20 $ HANKEL2} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1072} - - K{\footnote \pard\plain \sl240 \fs20 K HANKEL2 operator;operator} - -}{\b\f2 HANKEL2}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Hankel2} {\f2 operator returns the Hankel function of the second kind. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Hankel2} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on complex; \par - \par -Hankel2 (1/2,pi); \par - \par - - i * sqrt(2) / pi \par - \par - \par -Hankel2 (1,pi); \par - \par - besselj(1,pi) - i*bessely(1,pi) \par - \par -\pard \sl240 }{\f2 The operator }{\f3 Hankel2} {\f2 is also called Bessel function of the third kind. -There is currently no numeric evaluation of Hankel functions. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BESSELI} - -${\footnote \pard\plain \sl240 \fs20 $ BESSELI} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1073} - - K{\footnote \pard\plain \sl240 \fs20 K BESSELI operator;operator} - -}{\b\f2 BESSELI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 BesselI} {\f2 operator returns the modified Bessel function I. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 BesselI} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -Besseli (1,1); \par - \par - 0.565159103992 \par - \par -\pard \sl240 }{\f2 The knowledge about the operator }{\f3 BesselI} {\f2 is currently fairly limited. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BESSELK} - -${\footnote \pard\plain \sl240 \fs20 $ BESSELK} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1074} - - K{\footnote \pard\plain \sl240 \fs20 K BESSELK operator;operator} - -}{\b\f2 BESSELK}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 BesselK} {\f2 operator returns the modified Bessel function K. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 BesselK} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -df(besselk(0,x),x); \par - \par - - besselk(1,x) \par - \par -\pard \sl240 }{\f2 There is currently no numeric support for the operator }{\f3 BesselK} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # StruveH} - -${\footnote \pard\plain \sl240 \fs20 $ StruveH} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1075} - - K{\footnote \pard\plain \sl240 \fs20 K StruveH operator;operator} - -}{\b\f2 STRUVEH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 StruveH} {\f2 operator returns Struve's H function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 StruveH} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -struveh(-3/2,x); \par - \par - - besselj(3/2,x) / i \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # StruveL} - -${\footnote \pard\plain \sl240 \fs20 $ StruveL} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1076} - - K{\footnote \pard\plain \sl240 \fs20 K StruveL operator;operator} - -}{\b\f2 STRUVEL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 StruveL} {\f2 operator returns the modified Struve L function . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 StruveL} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -struvel(-3/2,x); \par - \par - besseli(3/2,x) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # KummerM} - -${\footnote \pard\plain \sl240 \fs20 $ KummerM} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1077} - - K{\footnote \pard\plain \sl240 \fs20 K Confluent Hypergeometric function;KummerM operator;operator} - -}{\b\f2 KUMMERM}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 KummerM} {\f2 operator returns Kummer's M function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 KummerM} {\f4 (,,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -kummerm(1,1,x); \par - \par - x \par - e \par - \par - \par -on rounded; \par - \par -kummerm(1,3,1.3); \par - \par - 1.62046942914 \par - \par -\pard \sl240 }{\f2 Kummer's M function is one of the Confluent Hypergeometric functions. -For reference see the } -{\f2\uldb hypergeometric}{\v\f2 HYPERGEOMETRIC} -{\f2 operator. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # KummerU} - -${\footnote \pard\plain \sl240 \fs20 $ KummerU} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1078} - - K{\footnote \pard\plain \sl240 \fs20 K Confluent Hypergeometric function;KummerU operator;operator} - -}{\b\f2 KUMMERU}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 KummerU} {\f2 operator returns Kummer's U function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 KummerU} {\f4 (,,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -df(kummeru(1,1,x),x) \par - \par - - kummeru(2,2,x) \par - \par -\pard \sl240 }{\f2 Kummer's U function is one of the Confluent Hypergeometric functions. -For reference see the } -{\f2\uldb hypergeometric}{\v\f2 HYPERGEOMETRIC} -{\f2 operator. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # WhittakerW} - -${\footnote \pard\plain \sl240 \fs20 $ WhittakerW} - -+{\footnote \pard\plain \sl240 \fs20 + g26:1079} - - K{\footnote \pard\plain \sl240 \fs20 K Confluent Hypergeometric function;WhittakerW operator;operator} - -}{\b\f2 WHITTAKERW}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 WhittakerW} {\f2 operator returns Whittaker's W function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 WhittakerW} {\f4 (,,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -WhittakerW(2,2,2); \par - \par - 1 \par - 4*sqrt(2)*kummeru(-,5,2) \par - 2 \par - ------------------------- \par - e \par - \par -\pard \sl240 }{\f2 Whittaker's W function is one of the Confluent Hypergeometric functions. -For reference see the } -{\f2\uldb hypergeometric}{\v\f2 HYPERGEOMETRIC} -{\f2 operator. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g26} - -${\footnote \pard\plain \sl240 \fs20 $ Bessel Functions} - -+{\footnote \pard\plain \sl240 \fs20 + index:0026} -}{\b\f2 Bessel Functions}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb BESSELJ operator} -{\v\f2 BESSELJ}{\f2 \par -}{\f2 \tab}{\f2\uldb BESSELY operator} -{\v\f2 BESSELY}{\f2 \par -}{\f2 \tab}{\f2\uldb HANKEL1 operator} -{\v\f2 HANKEL1}{\f2 \par -}{\f2 \tab}{\f2\uldb HANKEL2 operator} -{\v\f2 HANKEL2}{\f2 \par -}{\f2 \tab}{\f2\uldb BESSELI operator} -{\v\f2 BESSELI}{\f2 \par -}{\f2 \tab}{\f2\uldb BESSELK operator} -{\v\f2 BESSELK}{\f2 \par -}{\f2 \tab}{\f2\uldb StruveH operator} -{\v\f2 StruveH}{\f2 \par -}{\f2 \tab}{\f2\uldb StruveL operator} -{\v\f2 StruveL}{\f2 \par -}{\f2 \tab}{\f2\uldb KummerM operator} -{\v\f2 KummerM}{\f2 \par -}{\f2 \tab}{\f2\uldb KummerU operator} -{\v\f2 KummerU}{\f2 \par -}{\f2 \tab}{\f2\uldb WhittakerW operator} -{\v\f2 WhittakerW}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Airy_Ai} - -${\footnote \pard\plain \sl240 \fs20 $ Airy_Ai} - -+{\footnote \pard\plain \sl240 \fs20 + g27:1080} - - K{\footnote \pard\plain \sl240 \fs20 K Airy_Ai operator;operator} - -}{\b\f2 AIRY_AI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Airy_Ai} {\f2 operator returns the Airy Ai function for a given argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Airy_Ai} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on complex; \par -on rounded; \par -Airy_Ai(0); \par - \par - \par - 0.355028053888 \par - \par - \par -Airy_Ai(3.45 + 17.97i); \par - \par - - 5.5561528511e+9 - 8.80397899932e+9*i \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Airy_Bi} - -${\footnote \pard\plain \sl240 \fs20 $ Airy_Bi} - -+{\footnote \pard\plain \sl240 \fs20 + g27:1081} - - K{\footnote \pard\plain \sl240 \fs20 K Airy_Bi operator;operator} - -}{\b\f2 AIRY_BI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Airy_Bi} {\f2 operator returns the Airy Bi function for a given -argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Airy_Bi} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Airy_Bi(0); \par - \par - 0.614926627446 \par - \par - \par -Airy_Bi(3.45 + 17.97i); \par - \par - 8.80397899932e+9 - 5.5561528511e+9*i \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Airy_Aiprime} - -${\footnote \pard\plain \sl240 \fs20 $ Airy_Aiprime} - -+{\footnote \pard\plain \sl240 \fs20 + g27:1082} - - K{\footnote \pard\plain \sl240 \fs20 K Airy_Aiprime operator;operator} - -}{\b\f2 AIRY_AIPRIME}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Airy_Aiprime} {\f2 operator returns the Airy Aiprime function for a -given argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Airy_Aiprime} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Airy_Aiprime(0); \par - \par - - 0.258819403793 \par - \par - \par -Airy_Aiprime(3.45+17.97i); \par - \par - - 3.83386421824e+19 + 2.16608828136e+19*i \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Airy_Biprime} - -${\footnote \pard\plain \sl240 \fs20 $ Airy_Biprime} - -+{\footnote \pard\plain \sl240 \fs20 + g27:1083} - - K{\footnote \pard\plain \sl240 \fs20 K Airy_Biprime operator;operator} - -}{\b\f2 AIRY_BIPRIME}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Airy_Biprime} {\f2 operator returns the Airy Biprime function for a -given argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Airy_Biprime} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Airy_Biprime(0); \par - \par - \par -Airy_Biprime(3.45 + 17.97i); \par - \par - 3.84251916792e+19 - 2.18006297399e+19*i \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g27} - -${\footnote \pard\plain \sl240 \fs20 $ Airy Functions} - -+{\footnote \pard\plain \sl240 \fs20 + index:0027} -}{\b\f2 Airy Functions}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Airy_Ai operator} -{\v\f2 Airy_Ai}{\f2 \par -}{\f2 \tab}{\f2\uldb Airy_Bi operator} -{\v\f2 Airy_Bi}{\f2 \par -}{\f2 \tab}{\f2\uldb Airy_Aiprime operator} -{\v\f2 Airy_Aiprime}{\f2 \par -}{\f2 \tab}{\f2\uldb Airy_Biprime operator} -{\v\f2 Airy_Biprime}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiSN} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiSN} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1084} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiSN operator;operator} - -}{\b\f2 JACOBISN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobisn} {\f2 operator returns the Jacobi Elliptic function sn. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobisn} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobisn(0.672, 0.36) \par - \par - 0.609519691792 \par - \par - \par -Jacobisn(1,0.9) \par - \par - 0.770085724907881 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiCN} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiCN} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1085} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiCN operator;operator} - -}{\b\f2 JACOBICN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobicn} {\f2 operator returns the Jacobi Elliptic function cn. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobicn} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobicn(7.2, 0.6) \par - \par - 0.837288298482018 \par - \par - \par -Jacobicn(0.11, 19) \par - \par - 0.994403862690043 - 1.6219006985556e-16*i \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiDN} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiDN} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1086} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiDN operator;operator} - -}{\b\f2 JACOBIDN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobidn} {\f2 operator returns the Jacobi Elliptic function dn. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobidn} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobidn(15, 0.683) \par - \par - 0.640574162024592 \par - \par - \par -Jacobidn(0,0) \par - \par - 1 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiCD} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiCD} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1087} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiCD operator;operator} - -}{\b\f2 JACOBICD}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobicd} {\f2 operator returns the Jacobi Elliptic function cd. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobicd} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobicd(1, 0.34) \par - \par - 0.657683337805273 \par - \par - \par -Jacobicd(0.8,0.8) \par - \par - 0.925587311582301 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiSD} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiSD} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1088} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiSD operator;operator} - -}{\b\f2 JACOBISD}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobisd} {\f2 operator returns the Jacobi Elliptic function sd. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobisd} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobisd(12, 0.4) \par - \par - 0.357189729437272 \par - \par - \par -Jacobisd(0.35,1) \par - \par - - 1.17713873203043 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiND} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiND} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1089} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiND operator;operator} - -}{\b\f2 JACOBIND}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobind} {\f2 operator returns the Jacobi Elliptic function nd. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobind} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobind(0.2, 17) \par - \par - 1.46553203037507 + 0.0000000000334032759313703*i \par - \par - \par -Jacobind(30, 0.001) \par - \par - 1.00048958438 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiDC} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiDC} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1090} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiDC operator;operator} - -}{\b\f2 JACOBIDC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobidc} {\f2 operator returns the Jacobi Elliptic function dc. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobidc} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobidc(0.003,1) \par - \par - 1 \par - \par - \par -Jacobidc(2, 0.75) \par - \par - 6.43472885111 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiNC} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiNC} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1091} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiNC operator;operator} - -}{\b\f2 JACOBINC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobinc} {\f2 operator returns the Jacobi Elliptic function nc. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobinc} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobinc(1,0) \par - \par - 1.85081571768093 \par - \par - \par -Jacobinc(56, 0.4387) \par - \par - 39.304842663512 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiSC} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiSC} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1092} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiSC operator;operator} - -}{\b\f2 JACOBISC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobisc} {\f2 operator returns the Jacobi Elliptic function sc. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobisc} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobisc(9, 0.88) \par - \par - - 1.16417697982095 \par - \par - \par -Jacobisc(0.34, 7) \par - \par - 0.305851938390775 - 9.8768100944891e-12*i \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiNS} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiNS} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1093} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiNS operator;operator} - -}{\b\f2 JACOBINS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobins} {\f2 operator returns the Jacobi Elliptic function ns. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobins} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobins(3, 0.9) \par - \par - 1.00945801599785 \par - \par - \par -Jacobins(0.887, 15) \par - \par - 0.683578280513975 - 0.85023411082469*i \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiDS} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiDS} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1094} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiDS operator;operator} - -}{\b\f2 JACOBIDS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobisn} {\f2 operator returns the Jacobi Elliptic function ds. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobids} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobids(98,0.223) \par - \par - - 1.061253961477 \par - \par - \par -Jacobids(0.36,0.6) \par - \par - 2.76693172243692 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiCS} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiCS} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1095} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiCS operator;operator} - -}{\b\f2 JACOBICS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Jacobics} {\f2 operator returns the Jacobi Elliptic function cs. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Jacobics} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Jacobics(0, 0.767) \par - \par - infinity \par - \par - \par -Jacobics(1.43, 0) \par - \par - 0.141734127352112 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiAMPLITUDE} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiAMPLITUDE} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1096} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiAMPLITUDE operator;operator} - -}{\b\f2 JACOBIAMPLITUDE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 JacobiAmplitude} {\f2 operator returns the amplitude of u. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 JacobiAmplitude} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -JacobiAmplitude(7.239, 0.427) \par - \par - 0.0520978301448978 \par - \par - \par -JacobiAmplitude(0,0.1) \par - \par - 0 \par - \par -\pard \sl240 }{\f2 Amplitude u = asin(}{\f3 Jacobisn(u,m)} {\f2 ) -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # AGM_FUNCTION} - -${\footnote \pard\plain \sl240 \fs20 $ AGM_FUNCTION} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1097} - - K{\footnote \pard\plain \sl240 \fs20 K AGM_FUNCTION operator;operator} - -}{\b\f2 AGM_FUNCTION}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 AGM_function} {\f2 operator returns a list of (N, AGM, - list of aNtoa0, list of bNtob0, list of cNtoc0) where a0, b0 and c0 -are the initial values; N is the index number of the last term -used to generate the AGM. AGM is the Arithmetic Geometric Mean. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 AGM_function} {\f4 (,,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -AGM_function(1,1,1) \par - \par - 1,1,1,1,1,1,0,1 \par - \par - \par -AGM_function(1, 0.1, 1.3) \par - \par - \{6, \par - 2.27985615996629, \par - \{2.27985615996629, 2.27985615996629, \par - 2.2798561599706, 2.2798624278857, \par - 2.28742283656583, 2.55, 1\}, \par - \{2.27985615996629, 2.27985615996629, \par - 2.27985615996198, 2.2798498920555, \par - 2.27230201920557, 2.02484567313166, 4.1\}, \par - \{0, 4.30803136219904e-12, 0.0000062679151007581, \par - 0.00756040868012758, 0.262577163434171, - 1.55, 5.9\}\} \par - \par -\pard \sl240 }{\f2 The other Jacobi functions use this function with initial values -a0=1, b0=sqrt(1-m), c0=sqrt(m). -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LANDENTRANS} - -${\footnote \pard\plain \sl240 \fs20 $ LANDENTRANS} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1098} - - K{\footnote \pard\plain \sl240 \fs20 K LANDENTRANS operator;operator} - -}{\b\f2 LANDENTRANS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 landentrans} {\f2 operator generates the descending landen -transformation of the given imput values, returning a list of these -values; initial to final in each case. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 landentrans} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -landentrans(0,0.1) \par - \par - \{\{0,0,0,0,0\},\{0.1,0.0025041751943776, \par - \par - \par - \par - \par - 0.00000156772498954046,6.1444078 9914461e-13,0\}\} \par - \par -\pard \sl240 }{\f2 The first list ascends in value, and the second descends in value. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EllipticF} - -${\footnote \pard\plain \sl240 \fs20 $ EllipticF} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1099} - - K{\footnote \pard\plain \sl240 \fs20 K EllipticF operator;operator} - -}{\b\f2 ELLIPTICF}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 EllipticF} {\f2 operator returns the Elliptic Integral of the -First Kind. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 EllitpicF} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -EllipticF(0.3, 8.222) \par - \par - 0.3 \par - \par - \par -EllipticF(7.396, 0.1) \par - \par - 7.58123216114307 \par - \par -\pard \sl240 }{\f2 The Complete Elliptic Integral of the First Kind can be found by -putting the first argument to pi/2 or by using }{\f3 EllipticK} {\f2 -and the second argument. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EllipticK} - -${\footnote \pard\plain \sl240 \fs20 $ EllipticK} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1100} - - K{\footnote \pard\plain \sl240 \fs20 K EllipticK operator;operator} - -}{\b\f2 ELLIPTICK}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 EllipticK} {\f2 operator returns the Elliptic value K. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 EllipticK} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -EllipticK(0.2) \par - \par - 1.65962359861053 \par - \par - \par -EllipticK(4.3) \par - \par - 0.808442364282734 - 1.05562492399206*i \par - \par - \par -EllipticK(0.000481) \par - \par - 1.57098526617635 \par - \par -\pard \sl240 }{\f2 The }{\f3 EllipticK} {\f2 function is the Complete Elliptic Integral of -the First Kind. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EllipticKprime} - -${\footnote \pard\plain \sl240 \fs20 $ EllipticKprime} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1101} - - K{\footnote \pard\plain \sl240 \fs20 K EllipticKprime operator;operator} - -}{\b\f2 ELLIPTICKPRIME}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 EllipticK'} {\f2 operator returns the Elliptic value K(m). -\par -\par - \par -syntax: \par -}{\f4 }{\f3 EllipticKprime} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -EllipticKprime(0.2) \par - \par - 2.25720532682085 \par - \par - \par -EllipticKprime(4.3) \par - \par - 1.05562492399206 \par - \par - \par -EllipticKprime(0.000481) \par - \par - 5.206621921966 \par - \par -\pard \sl240 }{\f2 The }{\f3 EllipticKprime} {\f2 function is the Complete Elliptic Integral of -the First Kind of (1-m). -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EllipticE} - -${\footnote \pard\plain \sl240 \fs20 $ EllipticE} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1102} - - K{\footnote \pard\plain \sl240 \fs20 K EllipticE operator;operator} - -}{\b\f2 ELLIPTICE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 EllipticE} {\f2 operator used with two arguments -returns the Elliptic Integral of the Second Kind. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 EllipticE} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -EllipticE(1.2,0.22) \par - \par - 1.15094019180949 \par - \par - \par -EllipticE(0,4.35) \par - \par - 0 \par - \par - \par -EllipticE(9,0.00719) \par - \par - 8.98312465929145 \par - \par -\pard \sl240 }{\f2 The Complete Elliptic Integral of the Second Kind can be obtained by -using just the second argument, or by using pi/2 as the first argument. -\par -\par -\par -The }{\f3 EllipticE} {\f2 operator used with one argument -returns the Elliptic value E. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 EllipticE} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -EllipticE(0.22) \par - \par - 1.48046637439519 \par - \par - \par -EllipticE(pi/2, 0.22) \par - \par - 1.48046637439519 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EllipticTHETA} - -${\footnote \pard\plain \sl240 \fs20 $ EllipticTHETA} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1103} - - K{\footnote \pard\plain \sl240 \fs20 K EllipticTHETA operator;operator} - -}{\b\f2 ELLIPTICTHETA}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 EllipticTheta} {\f2 operator returns one of the four Theta -functions. It cannot except any number other than 1,2,3 or 4 as -its first argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 EllipticTheta} {\f4 (,,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -EllipticTheta(1, 1.4, 0.72) \par - \par - 0.91634775373 \par - \par - \par -EllipticTheta(2, 3.9, 6.1 ) \par - \par - -48.0202736969 + 20.9881034377 i \par - \par - \par -EllipticTheta(3, 0.67, 0.2) \par - \par - 1.0083077448 \par - \par - \par -EllipticTheta(4, 8, 0.75) \par - \par - 0.894963369304 \par - \par - \par -EllipticTheta(5, 1, 0.1) \par - \par - ***** In EllipticTheta(a,u,m); a = 1,2,3 or 4. \par - \par -\pard \sl240 }{\f2 Theta functions are important because every one of the Jacobian -Elliptic functions can be expressed as the ratio of two theta functions. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiZETA} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiZETA} - -+{\footnote \pard\plain \sl240 \fs20 + g28:1104} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiZETA operator;operator} - -}{\b\f2 JACOBIZETA}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 JacobiZeta} {\f2 operator returns the Jacobian function Zeta. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 JacobiZeta} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -JacobiZeta(3.2, 0.8) \par - \par - - 0.254536403439 \par - \par - \par -JacobiZeta(0.2, 1.6) \par - \par - 0.171766095970451 - 0.0717028569800147*i \par - \par -\pard \sl240 }{\f2 The Jacobian function Zeta is related to the Jacobian function Theta. -But it is significantly different from Riemann's Zeta Function } -{\f2\uldb Zeta}{\v\f2 ZETA} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g28} - -${\footnote \pard\plain \sl240 \fs20 $ Jacobi's Elliptic Functions and Elliptic Integrals} - -+{\footnote \pard\plain \sl240 \fs20 + index:0028} -}{\b\f2 Jacobi's Elliptic Functions and Elliptic Integrals}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb JacobiSN operator} -{\v\f2 JacobiSN}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiCN operator} -{\v\f2 JacobiCN}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiDN operator} -{\v\f2 JacobiDN}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiCD operator} -{\v\f2 JacobiCD}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiSD operator} -{\v\f2 JacobiSD}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiND operator} -{\v\f2 JacobiND}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiDC operator} -{\v\f2 JacobiDC}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiNC operator} -{\v\f2 JacobiNC}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiSC operator} -{\v\f2 JacobiSC}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiNS operator} -{\v\f2 JacobiNS}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiDS operator} -{\v\f2 JacobiDS}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiCS operator} -{\v\f2 JacobiCS}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiAMPLITUDE operator} -{\v\f2 JacobiAMPLITUDE}{\f2 \par -}{\f2 \tab}{\f2\uldb AGM_FUNCTION operator} -{\v\f2 AGM_FUNCTION}{\f2 \par -}{\f2 \tab}{\f2\uldb LANDENTRANS operator} -{\v\f2 LANDENTRANS}{\f2 \par -}{\f2 \tab}{\f2\uldb EllipticF operator} -{\v\f2 EllipticF}{\f2 \par -}{\f2 \tab}{\f2\uldb EllipticK operator} -{\v\f2 EllipticK}{\f2 \par -}{\f2 \tab}{\f2\uldb EllipticKprime operator} -{\v\f2 EllipticKprime}{\f2 \par -}{\f2 \tab}{\f2\uldb EllipticE operator} -{\v\f2 EllipticE}{\f2 \par -}{\f2 \tab}{\f2\uldb EllipticTHETA operator} -{\v\f2 EllipticTHETA}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiZETA operator} -{\v\f2 JacobiZETA}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # POCHHAMMER} - -${\footnote \pard\plain \sl240 \fs20 $ POCHHAMMER} - -+{\footnote \pard\plain \sl240 \fs20 + g29:1105} - - K{\footnote \pard\plain \sl240 \fs20 K POCHHAMMER operator;operator} - -}{\b\f2 POCHHAMMER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par -The }{\f3 Pochhammer} {\f2 operator implements the Pochhammer notation -(shifted factorial). -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Pochhammer} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -pochhammer(17,4); \par - \par - 116280 \par - \par - \par - \par -pochhammer(1/2,z); \par - \par - factorial(2*z) \par - -------------------- \par - 2*z \par - (2 *factorial(z)) \par - \par -\pard \sl240 }{\f2 A number of complex rules for }{\f3 Pochhammer} {\f2 are inactive, because they -cause a huge system load in algebraic mode. If one wants to use more rules -for the simplification of Pochhammer's notation, one can do: -\par -\par -let special!*pochhammer!*rules; -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GAMMA} - -${\footnote \pard\plain \sl240 \fs20 $ GAMMA} - -+{\footnote \pard\plain \sl240 \fs20 + g29:1106} - - K{\footnote \pard\plain \sl240 \fs20 K GAMMA operator;operator} - -}{\b\f2 GAMMA}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Gamma} {\f2 operator returns the Gamma function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Gamma} {\f4 () -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par -gamma(10); \par - \par - 362880 \par - \par - \par -gamma(1/2); \par - \par - sqrt(pi) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BETA} - -${\footnote \pard\plain \sl240 \fs20 $ BETA} - -+{\footnote \pard\plain \sl240 \fs20 + g29:1107} - - K{\footnote \pard\plain \sl240 \fs20 K BETA operator;operator} - -}{\b\f2 BETA}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Beta} {\f2 operator returns the Beta function defined by -\par -\par -Beta (z,w) := defint(t**(z-1)* (1 - t)**(w-1),t,0,1) . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Beta} {\f4 (,) -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par -Beta(2,2); \par - \par - 1 / 6 \par - \par - \par -Beta(x,y); \par - \par - gamma(x)*gamma(y) / gamma(x + y) \par - \par -\pard \sl240 }{\f2 The operator }{\f3 Beta} {\f2 is simplified towards the } -{\f2\uldb GAMMA}{\v\f2 GAMMA} -{\f2 operator. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PSI} - -${\footnote \pard\plain \sl240 \fs20 $ PSI} - -+{\footnote \pard\plain \sl240 \fs20 + g29:1108} - - K{\footnote \pard\plain \sl240 \fs20 K Euler's constant;PSI operator;operator} - -}{\b\f2 PSI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 Psi} {\f2 operator returns the Psi (or DiGamma) function. -\par -\par -Psi(x) := df(Gamma(z),z)/ Gamma (z) -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Gamma} {\f4 () -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par -Psi(3); \par - \par - (2*log(2) + psi(1/2) + psi(1) + 3)/2 \par - \par - \par -on rounded; \par - \par -- Psi(1); \par - \par - 0.577215664902 \par - \par -\pard \sl240 }{\f2 Euler's constant can be found as - Psi(1). -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # POLYGAMMA} - -${\footnote \pard\plain \sl240 \fs20 $ POLYGAMMA} - -+{\footnote \pard\plain \sl240 \fs20 + g29:1109} - - K{\footnote \pard\plain \sl240 \fs20 K POLYGAMMA operator;operator} - -}{\b\f2 POLYGAMMA}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Polygamma} {\f2 operator returns the Polygamma function. -\par -\par -Polygamma(n,x) := df(Psi(z),z,n); -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Polygamma} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par - Polygamma(1,2); \par - \par - 2 \par - (pi - 6) / 6 \par - \par - \par -on rounded; \par - \par -Polygamma(1,2.35); \par - \par - 0.52849689109 \par - \par -\pard \sl240 }{\f2 The Polygamma function is used for simplification of the } -{\f2\uldb ZETA}{\v\f2 ZETA} -{\f2 -function for some arguments. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g29} - -${\footnote \pard\plain \sl240 \fs20 $ Gamma and Related Functions} - -+{\footnote \pard\plain \sl240 \fs20 + index:0029} -}{\b\f2 Gamma and Related Functions}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb POCHHAMMER operator} -{\v\f2 POCHHAMMER}{\f2 \par -}{\f2 \tab}{\f2\uldb GAMMA operator} -{\v\f2 GAMMA}{\f2 \par -}{\f2 \tab}{\f2\uldb BETA operator} -{\v\f2 BETA}{\f2 \par -}{\f2 \tab}{\f2\uldb PSI operator} -{\v\f2 PSI}{\f2 \par -}{\f2 \tab}{\f2\uldb POLYGAMMA operator} -{\v\f2 POLYGAMMA}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DILOG_extended} - -${\footnote \pard\plain \sl240 \fs20 $ DILOG_extended} - -+{\footnote \pard\plain \sl240 \fs20 + g30:1110} - - K{\footnote \pard\plain \sl240 \fs20 K dilogarithm function;Spence's Integral;DILOG extended operator;operator} - -}{\b\f2 DILOG EXTENDED}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The package }{\f3 specfn} {\f2 supplies an extended support for the -} -{\f2\uldb dilog}{\v\f2 DILOG} -{\f2 operator which implements the }{\f3 dilogarithm function} {\f2 . -\par -\par -dilog(x) := - defint(log(t)/(t - 1),t,1,x); -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Dilog} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -defint(log(t)/(t - 1),t,1,x); \par - \par - - dilog (x) \par - \par - \par -dilog 2; \par - \par - 2 \par - - pi /12 \par - \par - \par - \par -on rounded; \par - \par -Dilog 20; \par - \par - - 5.92783972438 \par - \par -\pard \sl240 }{\f2 The operator }{\f3 Dilog} {\f2 is sometimes called Spence's Integral for n = 2. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Lambert_W_function} - -${\footnote \pard\plain \sl240 \fs20 $ Lambert_W_function} - -+{\footnote \pard\plain \sl240 \fs20 + g30:1111} - - K{\footnote \pard\plain \sl240 \fs20 K Lambert_W function operator;operator} - -}{\b\f2 LAMBERT_W FUNCTION}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -Lambert's W function is the inverse of the function w * e^w. -It is used in the } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 package for equations containing -exponentials and logarithms. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Lambert_W} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Lambert_W(-1/e); \par - \par - -1 \par - \par - \par -solve(w + log(w),w); \par - \par - w=lambert_w(1) \par - \par - \par -on rounded; \par - \par -Lambert_W(-0.05); \par - \par - - 0.0527059835515 \par - \par -\pard \sl240 }{\f2 The current implementation will compute the principal branch in -rounded mode only. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g30} - -${\footnote \pard\plain \sl240 \fs20 $ Miscellaneous Functions} - -+{\footnote \pard\plain \sl240 \fs20 + index:0030} -}{\b\f2 Miscellaneous Functions}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb DILOG extended operator} -{\v\f2 DILOG_extended}{\f2 \par -}{\f2 \tab}{\f2\uldb Lambert_W function operator} -{\v\f2 Lambert_W_function}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ChebyshevT} - -${\footnote \pard\plain \sl240 \fs20 $ ChebyshevT} - -+{\footnote \pard\plain \sl240 \fs20 + g31:1112} - - K{\footnote \pard\plain \sl240 \fs20 K ChebyshevT operator;operator} - -}{\b\f2 CHEBYSHEVT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 ChebyshevT} {\f2 operator computes the nth Chebyshev T Polynomial (of the -first kind). -\par -\par - \par -syntax: \par -}{\f4 }{\f3 ChebyshevT} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -ChebyshevT(3,xx); \par - \par - 2 \par - xx*(4*xx - 3) \par - \par - \par - \par -ChebyshevT(3,4); \par - \par - 244 \par - \par -\pard \sl240 }{\f2 Chebyshev's T polynomials are computed using the recurrence relation: -\par -\par -ChebyshevT(n,x) := 2x*ChebyshevT(n-1,x) - ChebyshevT(n-2,x) with -\par -\par -ChebyshevT(0,x) := 0 and ChebyshevT(1,x) := x -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ChebyshevU} - -${\footnote \pard\plain \sl240 \fs20 $ ChebyshevU} - -+{\footnote \pard\plain \sl240 \fs20 + g31:1113} - - K{\footnote \pard\plain \sl240 \fs20 K ChebyshevU operator;operator} - -}{\b\f2 CHEBYSHEVU}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 ChebyshevU} {\f2 operator returns the nth Chebyshev U Polynomial (of the -second kind). -\par -\par - \par -syntax: \par -}{\f4 }{\f3 ChebyshevU} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -ChebyshevU(3,xx); \par - \par - 2 \par - 4*x*(2*x - 1) \par - \par - \par - \par -ChebyshevU(3,4); \par - \par - 496 \par - \par -\pard \sl240 }{\f2 Chebyshev's U polynomials are computed using the recurrence relation: -\par -\par -ChebyshevU(n,x) := 2x*ChebyshevU(n-1,x) - ChebyshevU(n-2,x) with -\par -\par -ChebyshevU(0,x) := 0 and ChebyshevU(1,x) := 2x -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # HermiteP} - -${\footnote \pard\plain \sl240 \fs20 $ HermiteP} - -+{\footnote \pard\plain \sl240 \fs20 + g31:1114} - - K{\footnote \pard\plain \sl240 \fs20 K HermiteP operator;operator} - -}{\b\f2 HERMITEP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 HermiteP} {\f2 operator returns the nth Hermite Polynomial. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 HermiteP} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -HermiteP(3,xx); \par - \par - 2 \par - 4*xx*(2*xx - 3) \par - \par - \par -HermiteP(3,4); \par - \par - 464 \par - \par -\pard \sl240 }{\f2 Hermite polynomials are computed using the recurrence relation: - \par -\par -HermiteP(n,x) := 2x*HermiteP(n-1,x) - 2*(n-1)*HermiteP(n-2,x) with -\par -\par -HermiteP(0,x) := 1 and HermiteP(1,x) := 2x -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LaguerreP} - -${\footnote \pard\plain \sl240 \fs20 $ LaguerreP} - -+{\footnote \pard\plain \sl240 \fs20 + g31:1115} - - K{\footnote \pard\plain \sl240 \fs20 K LaguerreP operator;operator} - -}{\b\f2 LAGUERREP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 LaguerreP} {\f2 operator computes the nth Laguerre Polynomial. -The two argument call of LaguerreP is a (common) abbreviation of -LaguerreP(n,0,x). -\par -\par - \par -syntax: \par -}{\f4 }{\f3 LaguerreP} {\f4 (,) or -\par -\par -}{\f3 LaguerreP} {\f4 (,,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -LaguerreP(3,xx); \par - \par - 3 2 \par - (- xx + 9*xx - 18*xx + 6)/6 \par - \par - \par - \par -LaguerreP(2,3,4); \par - \par - -2 \par - \par -\pard \sl240 }{\f2 Laguerre polynomials are computed using the recurrence relation: -\par -\par -LaguerreP(n,a,x) := (2n+a-1-x)/n*LaguerreP(n-1,a,x) - - (n+a-1) * LaguerreP(n-2,a,x) with -\par -\par -LaguerreP(0,a,x) := 1 and LaguerreP(2,a,x) := -x+1+a -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LegendreP} - -${\footnote \pard\plain \sl240 \fs20 $ LegendreP} - -+{\footnote \pard\plain \sl240 \fs20 + g31:1116} - - K{\footnote \pard\plain \sl240 \fs20 K LegendreP operator;operator} - -}{\b\f2 LEGENDREP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The binary }{\f3 LegendreP} {\f2 operator computes the nth Legendre -Polynomial which is -a special case of the nth Jacobi Polynomial with -\par -\par -LegendreP(n,x) := JacobiP(n,0,0,x) -\par -\par -The ternary form returns the associated Legendre Polynomial (see below). -\par -\par - \par -syntax: \par -}{\f4 }{\f3 LegendreP} {\f4 (,) or -\par -\par -}{\f3 LegendreP} {\f4 (,,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -LegendreP(3,xx); \par - \par - 2 \par - xx*(5*xx - 3) \par - ---------------- \par - 2 \par - \par - \par - \par -LegendreP(3,2,xx); \par - \par - 2 \par - 15*xx*( - xx + 1) \par - \par -\pard \sl240 }{\f2 The ternary form of the operator }{\f3 LegendreP} {\f2 is the associated -Legendre Polynomial defined as - \par -\par -P(n,m,x) = (-1)**m * (1-x**2)**(m/2) * df(LegendreP(n,x),x,m) -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # JacobiP} - -${\footnote \pard\plain \sl240 \fs20 $ JacobiP} - -+{\footnote \pard\plain \sl240 \fs20 + g31:1117} - - K{\footnote \pard\plain \sl240 \fs20 K JacobiP operator;operator} - -}{\b\f2 JACOBIP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 JacobiP} {\f2 operator computes the nth Jacobi Polynomial. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 JacobiP} {\f4 (,,, - ) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -JacobiP(3,4,5,xx); \par - \par - 3 2 \par - 7*(65*xx - 13*xx - 13*xx + 1) \par - ---------------------------------- \par - 8 \par - \par - \par - \par -JacobiP(3,4,5,6); \par - \par - 94465/8 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GegenbauerP} - -${\footnote \pard\plain \sl240 \fs20 $ GegenbauerP} - -+{\footnote \pard\plain \sl240 \fs20 + g31:1118} - - K{\footnote \pard\plain \sl240 \fs20 K ultraspherical polynomials;GegenbauerP operator;operator} - -}{\b\f2 GEGENBAUERP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 GegenbauerP} {\f2 operator computes Gegenbauer's (ultraspherical) -polynomials. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 GegenbauerP} {\f4 (,,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -GegenbauerP(3,2,xx); \par - \par - 2 \par - 4*xx*(8*xx - 3) \par - \par - \par - \par -GegenbauerP(3,2,4); \par - \par - 2000 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SolidHarmonicY} - -${\footnote \pard\plain \sl240 \fs20 $ SolidHarmonicY} - -+{\footnote \pard\plain \sl240 \fs20 + g31:1119} - - K{\footnote \pard\plain \sl240 \fs20 K Solid harmonic polynomials;SolidHarmonicY operator;operator} - -}{\b\f2 SOLIDHARMONICY}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 SolidHarmonicY} {\f2 operator computes Solid harmonic (Laplace) -polynomials. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 SolidHarmonicY} {\f4 (,, -,,,) -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -SolidHarmonicY(3,-2,x,y,z,r2); \par - \par - 2 2 \par - sqrt(105)*z*(-2*i*x*y + x - y ) \par - --------------------------------- \par - 4*sqrt(pi)*sqrt(2) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SphericalHarmonicY} - -${\footnote \pard\plain \sl240 \fs20 $ SphericalHarmonicY} - -+{\footnote \pard\plain \sl240 \fs20 + g31:1120} - - K{\footnote \pard\plain \sl240 \fs20 K Spherical harmonic polynomials;SphericalHarmonicY operator;operator} - -}{\b\f2 SPHERICALHARMONICY}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 SphericalHarmonicY} {\f2 operator computes Spherical harmonic (Laplace) -polynomials. These are special cases of the -solid harmonic polynomials, } -{\f2\uldb SolidHarmonicY}{\v\f2 SolidHarmonicY} -{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 SphericalHarmonicY} {\f4 (,, -,) -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par -SphericalHarmonicY(3,2,theta,phi); \par - \par - \par - 2 2 2 \par - sqrt(105)*cos(theta)*sin(theta) *(cos(phi) +2*cos(phi)*sin(phi)*i-sin(phi) ) \par - ----------------------------------------------------------------------------- \par - 4*sqrt(pi)*sqrt(2) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g31} - -${\footnote \pard\plain \sl240 \fs20 $ Orthogonal Polynomials} - -+{\footnote \pard\plain \sl240 \fs20 + index:0031} -}{\b\f2 Orthogonal Polynomials}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb ChebyshevT operator} -{\v\f2 ChebyshevT}{\f2 \par -}{\f2 \tab}{\f2\uldb ChebyshevU operator} -{\v\f2 ChebyshevU}{\f2 \par -}{\f2 \tab}{\f2\uldb HermiteP operator} -{\v\f2 HermiteP}{\f2 \par -}{\f2 \tab}{\f2\uldb LaguerreP operator} -{\v\f2 LaguerreP}{\f2 \par -}{\f2 \tab}{\f2\uldb LegendreP operator} -{\v\f2 LegendreP}{\f2 \par -}{\f2 \tab}{\f2\uldb JacobiP operator} -{\v\f2 JacobiP}{\f2 \par -}{\f2 \tab}{\f2\uldb GegenbauerP operator} -{\v\f2 GegenbauerP}{\f2 \par -}{\f2 \tab}{\f2\uldb SolidHarmonicY operator} -{\v\f2 SolidHarmonicY}{\f2 \par -}{\f2 \tab}{\f2\uldb SphericalHarmonicY operator} -{\v\f2 SphericalHarmonicY}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Si} - -${\footnote \pard\plain \sl240 \fs20 $ Si} - -+{\footnote \pard\plain \sl240 \fs20 + g32:1121} - - K{\footnote \pard\plain \sl240 \fs20 K integral function;Sine integral function;Si operator;operator} - -}{\b\f2 SI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 Si} {\f2 operator returns the Sine Integral function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Si} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -limit(Si(x),x,infinity); \par - \par - pi / 2 \par - \par - \par -on rounded; \par - \par -Si(0.35); \par - \par - 0.347626790989 \par - \par -\pard \sl240 }{\f2 The numeric values for the operator }{\f3 Si} {\f2 are computed via the -power series representation, which limits the argument range. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Shi} - -${\footnote \pard\plain \sl240 \fs20 $ Shi} - -+{\footnote \pard\plain \sl240 \fs20 + g32:1122} - - K{\footnote \pard\plain \sl240 \fs20 K integral function;hyperbolic sine integral function;Shi operator;operator} - -}{\b\f2 SHI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 Shi} {\f2 operator returns the hyperbolic Sine Integral function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Shi} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -df(shi(x),x); \par - \par - sinh(x) / x \par - \par - \par -on rounded; \par - \par -Shi(0.35); \par - \par - 0.352390716351 \par - \par -\pard \sl240 }{\f2 The numeric values for the operator }{\f3 Shi} {\f2 are computed via the -power series representation, which limits the argument range. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # s_i} - -${\footnote \pard\plain \sl240 \fs20 $ s_i} - -+{\footnote \pard\plain \sl240 \fs20 + g32:1123} - - K{\footnote \pard\plain \sl240 \fs20 K integral function;sine integral function;s_i operator;operator} - -}{\b\f2 S_I}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 s_i} {\f2 operator returns the Sine Integral function si. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 s_i} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -s_i(xx); \par - \par - (2*Si(xx) - pi) / 2 \par - \par - \par -df(s_i(x),x); \par - \par - sin(x) / x \par - \par -\pard \sl240 }{\f2 The operator name }{\f3 s_i} {\f2 is simplified towards } -{\f2\uldb SI}{\v\f2 Si} -{\f2 . -Since REDUCE is not case sensitive by default the name ``si'' can't be -used. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Ci} - -${\footnote \pard\plain \sl240 \fs20 $ Ci} - -+{\footnote \pard\plain \sl240 \fs20 + g32:1124} - - K{\footnote \pard\plain \sl240 \fs20 K cosine integral function;Ci operator;operator} - -}{\b\f2 CI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 Ci} {\f2 operator returns the Cosine Integral function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Ci} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -defint(cos(t)/t,t,x,infinity); \par - \par - - ci (x) \par - \par - \par -on rounded; \par - \par -Ci(0.35); \par - \par - - 0.50307556932 \par - \par -\pard \sl240 }{\f2 The numeric values for the operator }{\f3 Ci} {\f2 are computed via the -power series representation, which limits the argument range. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Chi} - -${\footnote \pard\plain \sl240 \fs20 $ Chi} - -+{\footnote \pard\plain \sl240 \fs20 + g32:1125} - - K{\footnote \pard\plain \sl240 \fs20 K integral function;hyperbolic cosine integral function;Chi operator;operator} - -}{\b\f2 CHI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 Chi} {\f2 operator returns the Hyperbolic Cosine Integral function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Chi} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -defint((cosh(t)-1)/t,t,0,x); \par - \par - - log(x) + psi(1) + chi(x) \par - \par - \par -on rounded; \par - \par -Chi(0.35); \par - \par - - 0.44182471827 \par - \par -\pard \sl240 }{\f2 The numeric values for the operator }{\f3 Chi} {\f2 are computed via the -power series representation, which limits the argument range. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ERF_extended} - -${\footnote \pard\plain \sl240 \fs20 $ ERF_extended} - -+{\footnote \pard\plain \sl240 \fs20 + g32:1126} - - K{\footnote \pard\plain \sl240 \fs20 K error function;ERF extended operator;operator} - -}{\b\f2 ERF EXTENDED}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The special function package supplies an extended support for the -} -{\f2\uldb erf}{\v\f2 ERF} -{\f2 operator which implements the }{\f3 error function} {\f2 -\par -\par -defint(e**(-x**2),x,0,infinity) * 2/sqrt(pi) -\par -\par -. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 erf} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -erf(-x); \par - \par - - erf(x) \par - \par - \par -on rounded; \par - \par -erf(0.35); \par - \par - 0.379382053562 \par - \par -\pard \sl240 }{\f2 The numeric values for the operator }{\f3 erf} {\f2 are computed via the -power series representation, which limits the argument range. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # erfc} - -${\footnote \pard\plain \sl240 \fs20 $ erfc} - -+{\footnote \pard\plain \sl240 \fs20 + g32:1127} - - K{\footnote \pard\plain \sl240 \fs20 K complementary error function;error function;erfc operator;operator} - -}{\b\f2 ERFC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 erfc} {\f2 operator returns the complementary Error function -\par -\par -1 - defint(e**(-x**2),x,0,infinity) * 2/sqrt(pi) -\par -\par -. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 erfc} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -erfc(xx); \par - \par - - erf(xx) + 1 \par - \par -\pard \sl240 }{\f2 The operator }{\f3 erfc} {\f2 is simplified towards the } -{\f2\uldb erf}{\v\f2 ERF} -{\f2 operator. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Ei} - -${\footnote \pard\plain \sl240 \fs20 $ Ei} - -+{\footnote \pard\plain \sl240 \fs20 + g32:1128} - - K{\footnote \pard\plain \sl240 \fs20 K exponential integral function;Ei operator;operator} - -}{\b\f2 EI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 Ei} {\f2 operator returns the Exponential Integral function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Ei} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -df(ei(x),x); \par - \par - x \par - e \par - --- \par - x \par - \par - \par -on rounded; \par - \par -Ei(0.35); \par - \par - - 0.0894340019184 \par - \par -\pard \sl240 }{\f2 The numeric values for the operator }{\f3 Ei} {\f2 are computed via the -power series representation, which limits the argument range. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Fresnel_C} - -${\footnote \pard\plain \sl240 \fs20 $ Fresnel_C} - -+{\footnote \pard\plain \sl240 \fs20 + g32:1129} - - K{\footnote \pard\plain \sl240 \fs20 K Fresnel_C operator;operator} - -}{\b\f2 FRESNEL_C}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Fresnel_C} {\f2 operator represents Fresnel's Cosine function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Fresnel_C} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -int(cos(t^2*pi/2),t,0,x); \par - \par - fresnel_c(x) \par - \par - \par -on rounded; \par - \par -fresnel_c(2.1); \par - \par - 0.581564135061 \par - \par -\pard \sl240 }{\f2 The operator }{\f3 Fresnel_C} {\f2 has a limited numeric evaluation of -large values of its argument. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Fresnel_S} - -${\footnote \pard\plain \sl240 \fs20 $ Fresnel_S} - -+{\footnote \pard\plain \sl240 \fs20 + g32:1130} - - K{\footnote \pard\plain \sl240 \fs20 K Fresnel_S operator;operator} - -}{\b\f2 FRESNEL_S}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Fresnel_S} {\f2 operator represents Fresnel's Sine Integral function. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Fresnel_S} {\f4 () -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -int(sin(t^2*pi/2),t,0,x); \par - \par - fresnel_s(x) \par - \par - \par -on rounded; \par - \par -fresnel_s(2.1); \par - \par - 0.374273359378 \par - \par -\pard \sl240 }{\f2 The operator }{\f3 Fresnel_S} {\f2 has a limited numeric evaluation of -large values of its argument. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g32} - -${\footnote \pard\plain \sl240 \fs20 $ Integral Functions} - -+{\footnote \pard\plain \sl240 \fs20 + index:0032} -}{\b\f2 Integral Functions}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Si operator} -{\v\f2 Si}{\f2 \par -}{\f2 \tab}{\f2\uldb Shi operator} -{\v\f2 Shi}{\f2 \par -}{\f2 \tab}{\f2\uldb s_i operator} -{\v\f2 s_i}{\f2 \par -}{\f2 \tab}{\f2\uldb Ci operator} -{\v\f2 Ci}{\f2 \par -}{\f2 \tab}{\f2\uldb Chi operator} -{\v\f2 Chi}{\f2 \par -}{\f2 \tab}{\f2\uldb ERF extended operator} -{\v\f2 ERF_extended}{\f2 \par -}{\f2 \tab}{\f2\uldb erfc operator} -{\v\f2 erfc}{\f2 \par -}{\f2 \tab}{\f2\uldb Ei operator} -{\v\f2 Ei}{\f2 \par -}{\f2 \tab}{\f2\uldb Fresnel_C operator} -{\v\f2 Fresnel_C}{\f2 \par -}{\f2 \tab}{\f2\uldb Fresnel_S operator} -{\v\f2 Fresnel_S}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BINOMIAL} - -${\footnote \pard\plain \sl240 \fs20 $ BINOMIAL} - -+{\footnote \pard\plain \sl240 \fs20 + g33:1131} - - K{\footnote \pard\plain \sl240 \fs20 K BINOMIAL operator;operator} - -}{\b\f2 BINOMIAL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Binomial} {\f2 operator returns the Binomial coefficient if both -parameter are integer and expressions involving the Gamma function otherwise. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Binomial} {\f4 (,) -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par -Binomial(49,6); \par - \par - 13983816 \par - \par - \par - \par -Binomial(n,3); \par - \par - gamma(n + 1) \par - --------------- \par - 6*gamma(n - 2) \par - \par -\pard \sl240 }{\f2 The operator }{\f3 Binomial} {\f2 evaluates the Binomial coefficients from -the explicit form and therefore it is not the best algorithm if you -want to compute many binomial coefficients with big indices in which -case a recursive algorithm is preferable. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # STIRLING1} - -${\footnote \pard\plain \sl240 \fs20 $ STIRLING1} - -+{\footnote \pard\plain \sl240 \fs20 + g33:1132} - - K{\footnote \pard\plain \sl240 \fs20 K STIRLING1 operator;operator} - -}{\b\f2 STIRLING1}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Stirling1} {\f2 operator returns the Stirling Numbers S(n,m) of the first -kind, i.e. the number of permutations of n symbols which have exactly m cycles -(divided by (-1)**(n-m)). -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Stirling1} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Stirling1 (17,4); \par - \par - -87077748875904 \par - \par - \par -Stirling1 (n,n-1); \par - \par - -gamma(n+1) \par - ------------- \par - 2*gamma(n-1) \par - \par -\pard \sl240 }{\f2 The operator }{\f3 Stirling1} {\f2 evaluates the Stirling numbers of the -first kind by rulesets for special cases or by a computing the closed -form, which is a series involving the operators } -{\f2\uldb BINOMIAL}{\v\f2 BINOMIAL} -{\f2 -and } -{\f2\uldb STIRLING2}{\v\f2 STIRLING2} -{\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # STIRLING2} - -${\footnote \pard\plain \sl240 \fs20 $ STIRLING2} - -+{\footnote \pard\plain \sl240 \fs20 + g33:1133} - - K{\footnote \pard\plain \sl240 \fs20 K STIRLING2 operator;operator} - -}{\b\f2 STIRLING2}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Stirling1} {\f2 operator returns the Stirling Numbers S(n,m) of the -second kind, i.e. the number of ways of partitioning a set of n elements -into m non-empty subsets. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Stirling2} {\f4 (,) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -Stirling2 (17,4); \par - \par - 694337290 \par - \par - \par -Stirling2 (n,n-1); \par - \par - gamma(n+1) \par - ------------- \par - 2*gamma(n-1) \par - \par -\pard \sl240 }{\f2 The operator }{\f3 Stirling2} {\f2 evaluates the Stirling numbers of the -second kind by rulesets for special cases or by a computing the closed -form. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g33} - -${\footnote \pard\plain \sl240 \fs20 $ Combinatorial Operators} - -+{\footnote \pard\plain \sl240 \fs20 + index:0033} -}{\b\f2 Combinatorial Operators}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb BINOMIAL operator} -{\v\f2 BINOMIAL}{\f2 \par -}{\f2 \tab}{\f2\uldb STIRLING1 operator} -{\v\f2 STIRLING1}{\f2 \par -}{\f2 \tab}{\f2\uldb STIRLING2 operator} -{\v\f2 STIRLING2}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ThreejSymbol} - -${\footnote \pard\plain \sl240 \fs20 $ ThreejSymbol} - -+{\footnote \pard\plain \sl240 \fs20 + g34:1134} - - K{\footnote \pard\plain \sl240 \fs20 K ThreejSymbol operator;operator} - -}{\b\f2 THREEJSYMBOL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 ThreejSymbol} {\f2 operator implements the 3j symbol. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 ThreejSymbol} {\f4 (,, -) -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -ThreejSymbol(\{j+1,m\},\{j+1,-m\},\{1,0\}); \par - \par - \par - j \par - ( - 1) *(abs(j - m + 1) - abs(j + m + 1)) \par - ------------------------------------------- \par - 3 2 m \par - 2*sqrt(2*j + 9*j + 13*j + 6)*( - 1) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Clebsch_Gordan} - -${\footnote \pard\plain \sl240 \fs20 $ Clebsch_Gordan} - -+{\footnote \pard\plain \sl240 \fs20 + g34:1135} - - K{\footnote \pard\plain \sl240 \fs20 K Clebsch_Gordan operator;operator} - -}{\b\f2 CLEBSCH_GORDAN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 Clebsch_Gordan} {\f2 operator implements the Clebsch_Gordan -coefficients. This is closely related to the } -{\f2\uldb Threejsymbol}{\v\f2 ThreejSymbol} -{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Clebsch_Gordan} {\f4 (,, -) -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par - Clebsch_Gordan(\{2,0\},\{2,0\},\{2,0\}); \par - \par - \par - -2 \par - --------- \par - sqrt(14) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SixjSymbol} - -${\footnote \pard\plain \sl240 \fs20 $ SixjSymbol} - -+{\footnote \pard\plain \sl240 \fs20 + g34:1136} - - K{\footnote \pard\plain \sl240 \fs20 K SixjSymbol operator;operator} - -}{\b\f2 SIXJSYMBOL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 SixjSymbol} {\f2 operator implements the 6j symbol. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 SixjSymbol} {\f4 (,) -\par -\par -\par -\par -}{\f2 \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -SixjSymbol(\{7,6,3\},\{2,4,6\}); \par - \par - 1 \par - ------------- \par - 14*sqrt(858) \par - \par -\pard \sl240 }{\f2 The operator }{\f3 SixjSymbol} {\f2 uses the } -{\f2\uldb ineq}{\v\f2 INEQ} -{\f2 package in order -to find minima and maxima for the summation index. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g34} - -${\footnote \pard\plain \sl240 \fs20 $ 3j and 6j symbols} - -+{\footnote \pard\plain \sl240 \fs20 + index:0034} -}{\b\f2 3j and 6j symbols}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb ThreejSymbol operator} -{\v\f2 ThreejSymbol}{\f2 \par -}{\f2 \tab}{\f2\uldb Clebsch_Gordan operator} -{\v\f2 Clebsch_Gordan}{\f2 \par -}{\f2 \tab}{\f2\uldb SixjSymbol operator} -{\v\f2 SixjSymbol}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # HYPERGEOMETRIC} - -${\footnote \pard\plain \sl240 \fs20 $ HYPERGEOMETRIC} - -+{\footnote \pard\plain \sl240 \fs20 + g35:1137} - - K{\footnote \pard\plain \sl240 \fs20 K generalized hypergeometric function;hypergeometric function;HYPERGEOMETRIC operator;operator} - -}{\b\f2 HYPERGEOMETRIC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 Hypergeometric} {\f2 operator provides simplifications for the -generalized hypergeometric functions. -The }{\f3 Hypergeometric} {\f2 operator is included in the package specfn2. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 hypergeometric} {\f4 (,, - ) -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -load specfn2; \par - \par -hypergeometric (\{1/2,1\},\{3/2\},-x^2); \par - \par - \par - atan(x) \par - -------- \par - x \par - \par - \par -hypergeometric (\{\},\{\},z); \par - \par - z \par - e \par - \par -\pard \sl240 }{\f2 The special case where the length of the first list is equal to 2 and -the length of the second list is equal to 1 is often called -``the hypergeometric function'' (notated as 2F1(a1,a2,b;x)). -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MeijerG} - -${\footnote \pard\plain \sl240 \fs20 $ MeijerG} - -+{\footnote \pard\plain \sl240 \fs20 + g35:1138} - - K{\footnote \pard\plain \sl240 \fs20 K MeijerG operator;operator} - -}{\b\f2 MEIJERG}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 MeijerG} {\f2 operator provides simplifications for Meijer's G -function. The simplifications are performed towards polynomials, -elementary or -special functions or (generalized) } -{\f2\uldb hypergeometric}{\v\f2 HYPERGEOMETRIC} -{\f2 functions. -\par -\par -The }{\f3 MeijerG} {\f2 operator is included in the package specfn2. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 MeijerG} {\f4 (,, - ) -\par -\par -}{\f2 \par -The first element of the lists has to be the list containing the -first group (mostly called ``m'' and ``n'') of parameters. This passes -the four parameters of a Meijer's G function implicitly via the -length of the lists. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -load specfn2; \par - \par -MeijerG(\{\{\},1\},\{\{0\}\},x); \par - \par - heaviside(-x+1) \par - \par - \par -MeijerG(\{\{\}\},\{\{1+1/4\},1-1/4\},(x^2)/4) * sqrt pi; \par - \par - \par - \par - 2 \par - sqrt(2)*sin(x)*x \par - ------------------ \par - 4*sqrt(x) \par - \par -\pard \sl240 }{\f2 Many well-known functions can be written as G functions, -e.g. exponentials, logarithms, trigonometric functions, Bessel functions -and hypergeometric functions. -The formulae can be found e.g. in -\par -\par -A.P.Prudnikov, Yu.A.Brychkov, O.I.Marichev: -Integrals and Series, Volume 3: More special functions, -Gordon and Breach Science Publishers (1990). -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Heaviside} - -${\footnote \pard\plain \sl240 \fs20 $ Heaviside} - -+{\footnote \pard\plain \sl240 \fs20 + g35:1139} - - K{\footnote \pard\plain \sl240 \fs20 K Heaviside operator;operator} - -}{\b\f2 HEAVISIDE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par -The }{\f3 Heaviside} {\f2 operator returns the Heaviside function. -\par -\par -Heaviside(~w) => if (w < 0) then 0 else 1 -\par -\par -when numberp w; -\par -\par - \par -syntax: \par -}{\f4 }{\f3 Heaviside} {\f4 () -\par -\par -}{\f2 \par -This operator is often included in the result of the simplification -of a generalized } -{\f2\uldb hypergeometric}{\v\f2 HYPERGEOMETRIC} -{\f2 function or a -} -{\f2\uldb MeijerG}{\v\f2 MeijerG} -{\f2 function. -\par -\par -No simplification is done for this function. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # erfi} - -${\footnote \pard\plain \sl240 \fs20 $ erfi} - -+{\footnote \pard\plain \sl240 \fs20 + g35:1140} - - K{\footnote \pard\plain \sl240 \fs20 K erfi operator;operator} - -}{\b\f2 ERFI}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par -The }{\f3 erfi} {\f2 operator returns the error function of an imaginary argument. -\par -\par -erfi(~x) => 2/sqrt(pi) * defint(e**(t**2),t,0,x); -\par -\par - \par -syntax: \par -}{\f4 }{\f3 erfi} {\f4 () -\par -\par -}{\f2 \par -This operator is sometimes included in the result of the simplification -of a generalized } -{\f2\uldb hypergeometric}{\v\f2 HYPERGEOMETRIC} -{\f2 function or a -} -{\f2\uldb MeijerG}{\v\f2 MeijerG} -{\f2 function. -\par -\par -No simplification is done for this function. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g35} - -${\footnote \pard\plain \sl240 \fs20 $ Miscellaneous} - -+{\footnote \pard\plain \sl240 \fs20 + index:0035} -}{\b\f2 Miscellaneous}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb HYPERGEOMETRIC operator} -{\v\f2 HYPERGEOMETRIC}{\f2 \par -}{\f2 \tab}{\f2\uldb MeijerG operator} -{\v\f2 MeijerG}{\f2 \par -}{\f2 \tab}{\f2\uldb Heaviside operator} -{\v\f2 Heaviside}{\f2 \par -}{\f2 \tab}{\f2\uldb erfi operator} -{\v\f2 erfi}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g24} - -${\footnote \pard\plain \sl240 \fs20 $ Special Functions} - -+{\footnote \pard\plain \sl240 \fs20 + index:0024} -}{\b\f2 Special Functions}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Special Function Package introduction} -{\v\f2 Special_Function_Package}{\f2 \par -}{\f2 \tab}{\f2\uldb Constants concept} -{\v\f2 Constants}{\f2 \par -}{\f2 \tab}{\f2\uldb Bernoulli Euler Zeta} -{\v\f2 g25}{\f2 \par -}{\f2 \tab}{\f2\uldb Bessel Functions} -{\v\f2 g26}{\f2 \par -}{\f2 \tab}{\f2\uldb Airy Functions} -{\v\f2 g27}{\f2 \par -}{\f2 \tab}{\f2\uldb Jacobi's Elliptic Functions and Elliptic Integrals} -{\v\f2 g28}{\f2 \par -}{\f2 \tab}{\f2\uldb Gamma and Related Functions} -{\v\f2 g29}{\f2 \par -}{\f2 \tab}{\f2\uldb Miscellaneous Functions} -{\v\f2 g30}{\f2 \par -}{\f2 \tab}{\f2\uldb Orthogonal Polynomials} -{\v\f2 g31}{\f2 \par -}{\f2 \tab}{\f2\uldb Integral Functions} -{\v\f2 g32}{\f2 \par -}{\f2 \tab}{\f2\uldb Combinatorial Operators} -{\v\f2 g33}{\f2 \par -}{\f2 \tab}{\f2\uldb 3j and 6j symbols} -{\v\f2 g34}{\f2 \par -}{\f2 \tab}{\f2\uldb Miscellaneous} -{\v\f2 g35}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TAYLOR_introduction} - -${\footnote \pard\plain \sl240 \fs20 $ TAYLOR_introduction} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1141} - - K{\footnote \pard\plain \sl240 \fs20 K TAYLOR introduction;introduction} - -}{\b\f2 TAYLOR}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par -\par - -This short note describes a package of REDUCE procedures that allow -Taylor expansion in one or more variables and efficient manipulation -of the resulting Taylor series. Capabilities include basic operations -(addition, subtraction, multiplication and division) and also -application of certain algebraic and transcendental functions. To a -certain extent, Laurent expansion can be performed as well. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylor} - -${\footnote \pard\plain \sl240 \fs20 $ taylor} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1142} - - K{\footnote \pard\plain \sl240 \fs20 K taylor operator;operator} - -}{\b\f2 TAYLOR}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - The }{\f3 taylor} {\f2 operator is used for expanding an expression into a - Taylor series. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 taylor} {\f4 ( - }{\f3 ,} {\f4 }{\f3 ,} {\f4 - }{\f3 ,} {\f4 -\par -\par -\{}{\f3 ,} {\f4 }{\f3 ,} {\f4 - }{\f3 ,} {\f4 \}*) - \par -\par -}{\f2 \par - can be any valid REDUCE algebraic expression. - must be a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 , and is the expansion - variable. The following it denotes the point - about which the expansion is to take place. must be a - non-negative integer and denotes the maximum expansion order. If - more than one triple is specified }{\f3 taylor} {\f2 will expand its - first argument independently with respect to all the variables. - Note that once the expansion has been done it is not possible to - calculate higher orders. -\par -\par -Instead of a } -{\f2\uldb kernel}{\v\f2 KERNEL} -{\f2 , may also be a list of - kernels. In this case expansion will take place in a way so that - the sum/ of the degrees of the kernels does not exceed the - maximum expansion order. If the expansion point evaluates to the - special identifier }{\f3 infinity} {\f2 , }{\f3 taylor} {\f2 tries to expand in - a series in 1/. -\par -\par -The expansion is performed variable per variable, i.e. in the - example above by first expanding - exp(x^2+y^2) - with respect to - }{\f3 x} {\f2 and then expanding every coefficient with respect to }{\f3 y} {\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - taylor(e^(x^2+y^2),x,0,2,y,0,2); \par - \par - \par - 2 2 2 2 2 2 \par - 1 + Y + X + Y *X + O(X ,Y ) \par - \par - \par - taylor(e^(x^2+y^2),\{x,y\},0,2); \par - \par - \par - 2 2 2 2 \par - 1 + Y + X + O(\{X ,Y \}) \par - \par -\pard \sl240 }{\f2 The following example shows the case of a non-analytical function.}{\f4 \pard \tx3420 \par - \par - taylor(x*y/(x+y),x,0,2,y,0,2); \par - \par - \par - ***** Not a unit in argument to QUOTTAYLOR \par - \par -\pard \sl240 }{\f2 -\par -\par -Note that it is not generally possible to apply the standard - reduce operators to a Taylor kernel. For example, } -{\f2\uldb part}{\v\f2 PART} -{\f2 , - } -{\f2\uldb coeff}{\v\f2 COEFF} -{\f2 , or } -{\f2\uldb coeffn}{\v\f2 COEFFN} -{\f2 cannot be used. Instead, the - expression at hand has to be converted to standard form first - using the } -{\f2\uldb taylortostandard}{\v\f2 taylortostandard} -{\f2 operator. -\par -\par -Differentiation of a Taylor expression is possible. If you - differentiate with respect to one of the Taylor variables the - order will decrease by one. -\par -\par -Substitution is a bit restricted: Taylor variables can only be - replaced by other kernels. There is one exception to this rule: - you can always substitute a Taylor variable by an expression that - evaluates to a constant. Note that REDUCE will not always be able - to determine that an expression is constant: an example is - sin(acos(4)). -\par -\par -Only simple taylor kernels can be integrated. More complicated - expressions that contain Taylor kernels as parts of themselves are - automatically converted into a standard representation by means of - the } -{\f2\uldb taylortostandard}{\v\f2 taylortostandard} -{\f2 operator. In this case a suitable - warning is printed. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylorautocombine} - -${\footnote \pard\plain \sl240 \fs20 $ taylorautocombine} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1143} - - K{\footnote \pard\plain \sl240 \fs20 K taylorautocombine switch;switch} - -}{\b\f2 TAYLORAUTOCOMBINE}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - If you set }{\f3 taylorautocombine} {\f2 to }{\f3 on} {\f2 , REDUCE - automatically combines Taylor expressions during the simplification - process. This is equivalent to applying } -{\f2\uldb taylorcombine}{\v\f2 taylorcombine} -{\f2 to - every expression that contains Taylor kernels. Default is - }{\f3 on} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylorautoexpand} - -${\footnote \pard\plain \sl240 \fs20 $ taylorautoexpand} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1144} - - K{\footnote \pard\plain \sl240 \fs20 K taylorautoexpand switch;switch} - -}{\b\f2 TAYLORAUTOEXPAND}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - }{\f3 taylorautoexpand} {\f2 makes Taylor expressions ``contagious'' in - the sense that } -{\f2\uldb taylorcombine}{\v\f2 taylorcombine} -{\f2 tries to Taylor expand all - non-Taylor subexpressions and to combine the result with the rest. - Default is }{\f3 off} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylorcombine} - -${\footnote \pard\plain \sl240 \fs20 $ taylorcombine} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1145} - - K{\footnote \pard\plain \sl240 \fs20 K taylorcombine operator;operator} - -}{\b\f2 TAYLORCOMBINE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - This operator tries to combine all Taylor kernels found in its - argument into one. Operations currently possible are: - \par -\par -\tab Addition, subtraction, multiplication, and division. - \par -\tab Roots, exponentials, and logarithms. - \par -\tab Trigonometric and hyperbolic functions and their inverses. - \par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - hugo := taylor(exp(x),x,0,2); \par - \par - 1 2 3 \par - HUGO := 1 + X + -*X + O(X ) \par - 2 \par - \par - \par - taylorcombine log hugo; \par - \par - 3 \par - X + O(X ) \par - \par - \par - taylorcombine(hugo + x); \par - \par - 1 2 3 \par - (1 + X + -*X + O(X )) + X \par - 2 \par - \par - \par - on taylorautoexpand; \par - \par - taylorcombine(hugo + x); \par - \par - 1 2 3 \par - 1 + 2*X + -*X + O(X ) \par - 2 \par - \par -\pard \sl240 }{\f2 Application of unary operators like }{\f3 log} {\f2 and }{\f3 atan} {\f2 - will nearly always succeed. For binary operations their arguments - have to be Taylor kernels with the same template. This means that - the expansion variable and the expansion point must match. - Expansion order is not so important, different order usually means - that one of them is truncated before doing the operation. -\par -\par -If } -{\f2\uldb taylorkeeporiginal}{\v\f2 taylorkeeporiginal} -{\f2 is set to }{\f3 on} {\f2 and if all - Taylor kernels in its argument have their original expressions - kept }{\f3 taylorcombine} {\f2 will also combine these and store the - result as the original expression of the resulting Taylor kernel. - There is also the switch } -{\f2\uldb taylorautoexpand}{\v\f2 taylorautoexpand} -{\f2 . -\par -\par -There are a few restrictions to avoid mathematically undefined - expressions: it is not possible to take the logarithm of a Taylor - kernel which has no terms (i.e. is zero), or to divide by such a - beast. There are some provisions made to detect singularities - during expansion: poles that arise because the denominator has - zeros at the expansion point are detected and properly treated, - i.e. the Taylor kernel will start with a negative power. (This - is accomplished by expanding numerator and denominator separately - and combining the results.) Essential singularities of the known - functions (see above) are handled correctly. - \par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylorkeeporiginal} - -${\footnote \pard\plain \sl240 \fs20 $ taylorkeeporiginal} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1146} - - K{\footnote \pard\plain \sl240 \fs20 K taylorkeeporiginal switch;switch} - -}{\b\f2 TAYLORKEEPORIGINAL}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - }{\f3 taylorkeeporiginal} {\f2 , if set to }{\f3 on} {\f2 , forces the - } -{\f2\uldb taylor}{\v\f2 taylor} -{\f2 and all Taylor kernel manipulation operators to - keep the original expression, i.e. the expression that was Taylor - expanded. All operations performed on the Taylor kernels are also - applied to this expression which can be recovered using the operator - } -{\f2\uldb taylororiginal}{\v\f2 taylororiginal} -{\f2 . Default is }{\f3 off} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylororiginal} - -${\footnote \pard\plain \sl240 \fs20 $ taylororiginal} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1147} - - K{\footnote \pard\plain \sl240 \fs20 K taylororiginal operator;operator} - -}{\b\f2 TAYLORORIGINAL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - Recovers the original expression (the one that was expanded) from - the Taylor kernel that is given as its argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 taylororiginal} {\f4 () or - }{\f3 taylororiginal} {\f4 - \par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par - hugo := taylor(exp(x),x,0,2); \par - \par - 1 2 3 \par - HUGO := 1 + X + -*X + O(X ) \par - 2 \par - \par - \par - taylororiginal hugo; \par - \par - ***** Taylor kernel doesn't have an original part in TAYLORORIGINAL \par - \par - \par - on taylorkeeporiginal; \par - \par - hugo := taylor(exp(x),x,0,2); \par - \par - 1 2 3 \par - HUGO := 1 + X + -*X + O(X ) \par - 2 \par - \par - \par - taylororiginal hugo; \par - \par - X \par - E \par - \par -\pard \sl240 }{\f2 An error is signalled if the argument is not a Taylor kernel or if - the original expression was not kept, i.e. if - } -{\f2\uldb taylorkeeporiginal}{\v\f2 taylorkeeporiginal} -{\f2 was set }{\f3 off} {\f2 during expansion. - \par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylorprintorder} - -${\footnote \pard\plain \sl240 \fs20 $ taylorprintorder} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1148} - - K{\footnote \pard\plain \sl240 \fs20 K taylorprintorder switch;switch} - -}{\b\f2 TAYLORPRINTORDER}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par -\par - - }{\f3 taylorprintorder} {\f2 , if set to }{\f3 on} {\f2 , causes the remainder - to be printed in big-O notation. Otherwise, three dots are printed. - Default is }{\f3 on} {\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylorprintterms} - -${\footnote \pard\plain \sl240 \fs20 $ taylorprintterms} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1149} - - K{\footnote \pard\plain \sl240 \fs20 K taylorprintterms variable;variable} - -}{\b\f2 TAYLORPRINTTERMS}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - Only a certain number of (non-zero) coefficients are printed. If - there are more, an expression of the form }{\f3 n terms} {\f2 is printed - to indicate how many non-zero terms have been suppressed. The - number of terms printed is given by the value of the shared - algebraic variable }{\f3 taylorprintterms} {\f2 . Allowed values are - integers and the special identifier }{\f3 all} {\f2 . The latter setting - specifies that all terms are to be printed. The default setting is - 5. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - taylor(e^(x^2+y^2),x,0,4,y,0,4); \par - \par - \par - 2 1 4 2 2 2 5 5 \par - 1 + Y + -*Y + X + Y *X + (4 terms) + O(X ,Y ) \par - 2 \par - \par - \par - taylorprintterms := all; \par - \par - TAYLORPRINTTERMS := ALL \par - \par - \par - taylor(e^(x^2+y^2),x,0,4,y,0,4); \par - \par - \par - 2 1 4 2 2 2 1 4 2 1 4 1 2 4 \par - 1 + Y + -*Y + X + Y *X + -*Y *X + -*X + -*Y *X \par - 2 2 2 2 \par - 1 4 4 5 5 \par - + -*Y *X + O(X ,Y ) \par - 4 \par - \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylorrevert} - -${\footnote \pard\plain \sl240 \fs20 $ taylorrevert} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1150} - - K{\footnote \pard\plain \sl240 \fs20 K taylorrevert operator;operator} - -}{\b\f2 TAYLORREVERT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - }{\f3 taylorrevert} {\f2 allows reversion of a Taylor series of a - function f, i.e., to compute the first terms of the expansion of the - inverse of }{\f4 f}{\f2 from the expansion of }{\f4 f}{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 taylorrevert} {\f4 (}{\f3 ,} {\f4 - }{\f3 ,} {\f4 ) - \par -\par -}{\f2 \par -The first argument must evaluate to a Taylor kernel with the second - argument being one of its expansion variables. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - taylor(u - u**2,u,0,5); \par - \par - 2 6 \par - U - U + O(U ) \par - \par - \par - taylorrevert (ws,u,x); \par - \par - 2 3 4 5 6 \par - X + X + 2*X + 5*X + 14*X + O(X ) \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylorseriesp} - -${\footnote \pard\plain \sl240 \fs20 $ taylorseriesp} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1151} - - K{\footnote \pard\plain \sl240 \fs20 K taylorseriesp operator;operator} - -}{\b\f2 TAYLORSERIESP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - This operator may be used to determine if its argument is a Taylor - kernel. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 taylorseriesp} {\f4 () or }{\f3 taylorseriesp} {\f4 - - \par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par - hugo := taylor(exp(x),x,0,2); \par - \par - 1 2 3 \par - HUGO := 1 + X + -*X + O(X ) \par - 2 \par - \par - \par - if taylorseriesp hugo then OK; \par - \par - OK \par - \par - \par - if taylorseriesp(hugo + y) then OK else NO; \par - \par - \par - NO \par - \par -\pard \sl240 }{\f2 Note that this operator is subject to the same restrictions as, - e.g., }{\f3 ordp} {\f2 or }{\f3 numberp} {\f2 , i.e. it may only be used in - boolean expressions in }{\f3 if} {\f2 or }{\f3 let} {\f2 statements. - \par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylortemplate} - -${\footnote \pard\plain \sl240 \fs20 $ taylortemplate} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1152} - - K{\footnote \pard\plain \sl240 \fs20 K taylortemplate operator;operator} - -}{\b\f2 TAYLORTEMPLATE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - The template of a Taylor kernel, i.e. the list of all variables - with respect to which expansion took place together with expansion - point and order can be extracted using -\par -\par - \par -syntax: \par -}{\f4 }{\f3 taylortemplate} {\f4 () or - }{\f3 taylortemplate} {\f4 - \par -\par -}{\f2 \par -This returns a list of lists with the three elements - (VAR,VAR0,ORDER). An error is signalled if the argument is not a - Taylor kernel. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - hugo := taylor(exp(x),x,0,2); \par - \par - 1 2 3 \par - HUGO := 1 + X + -*X + O(X ) \par - 2 \par - \par - \par - taylortemplate hugo; \par - \par - \{\{X,0,2\}\} \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # taylortostandard} - -${\footnote \pard\plain \sl240 \fs20 $ taylortostandard} - -+{\footnote \pard\plain \sl240 \fs20 + g36:1153} - - K{\footnote \pard\plain \sl240 \fs20 K taylortostandard operator;operator} - -}{\b\f2 TAYLORTOSTANDARD}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - This operator converts all Taylor kernels in its argument into - standard form and resimplifies the result. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 taylortostandard} {\f4 () or - }{\f3 taylortostandard} {\f4 -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par - hugo := taylor(exp(x),x,0,2); \par - \par - 1 2 3 \par - HUGO := 1 + X + -*X + O(X ) \par - 2 \par - \par - \par - taylortostandard hugo; \par - \par - 2 \par - X + 2*X + 2 \par - ------------ \par - 2 \par - \par -\pard \sl240 }{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g36} - -${\footnote \pard\plain \sl240 \fs20 $ Taylor series} - -+{\footnote \pard\plain \sl240 \fs20 + index:0036} -}{\b\f2 Taylor series}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb TAYLOR introduction} -{\v\f2 TAYLOR_introduction}{\f2 \par -}{\f2 \tab}{\f2\uldb taylor operator} -{\v\f2 taylor}{\f2 \par -}{\f2 \tab}{\f2\uldb taylorautocombine switch} -{\v\f2 taylorautocombine}{\f2 \par -}{\f2 \tab}{\f2\uldb taylorautoexpand switch} -{\v\f2 taylorautoexpand}{\f2 \par -}{\f2 \tab}{\f2\uldb taylorcombine operator} -{\v\f2 taylorcombine}{\f2 \par -}{\f2 \tab}{\f2\uldb taylorkeeporiginal switch} -{\v\f2 taylorkeeporiginal}{\f2 \par -}{\f2 \tab}{\f2\uldb taylororiginal operator} -{\v\f2 taylororiginal}{\f2 \par -}{\f2 \tab}{\f2\uldb taylorprintorder switch} -{\v\f2 taylorprintorder}{\f2 \par -}{\f2 \tab}{\f2\uldb taylorprintterms variable} -{\v\f2 taylorprintterms}{\f2 \par -}{\f2 \tab}{\f2\uldb taylorrevert operator} -{\v\f2 taylorrevert}{\f2 \par -}{\f2 \tab}{\f2\uldb taylorseriesp operator} -{\v\f2 taylorseriesp}{\f2 \par -}{\f2 \tab}{\f2\uldb taylortemplate operator} -{\v\f2 taylortemplate}{\f2 \par -}{\f2 \tab}{\f2\uldb taylortostandard operator} -{\v\f2 taylortostandard}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GNUPLOT_and_REDUCE} - -${\footnote \pard\plain \sl240 \fs20 $ GNUPLOT_and_REDUCE} - -+{\footnote \pard\plain \sl240 \fs20 + g37:1154} - - K{\footnote \pard\plain \sl240 \fs20 K GNUPLOT and REDUCE introduction;introduction} - -}{\b\f2 GNUPLOT AND REDUCE}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par -\par - -\par -\par -The GNUPLOT system provides easy to use graphics output -for curves or surfaces which are defined by -formulas and/or data sets. GNUPLOT supports -a great variety of output devices -such as X-windows, VGA screen, postscript, picTeX. -The REDUCE GNUPLOT package lets one use the GNUPLOT -graphical output directly from inside REDUCE, either for -the interactive display of curves/surfaces or for the production -of pictures on paper. -\par -\par -Note that this package may not be supported on all system -platforms. -\par -\par -For a detailed description you should read the GNUPLOT -system documentation, available together with the GNUPLOT -installation material from several servers by anonymous FTP. -\par -\par -The REDUCE developers thank the GNUPLOT people for their permission -to distribute GNUPLOT together with REDUCE. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Axes_names} - -${\footnote \pard\plain \sl240 \fs20 $ Axes_names} - -+{\footnote \pard\plain \sl240 \fs20 + g37:1155} - - K{\footnote \pard\plain \sl240 \fs20 K Axes names concept;concept} - -}{\b\f2 AXES NAMES}{\f2 \par -\par - -Inside REDUCE the choice of variable names for a graph is completely -free. For referring to the GNUPLOT axes the names -X and Y for 2 dimensions, X,Y and Z for 3 dimensions are used -in the usual schoolbook sense independent from the variables of -the REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Pointset} - -${\footnote \pard\plain \sl240 \fs20 $ Pointset} - -+{\footnote \pard\plain \sl240 \fs20 + g37:1156} - - K{\footnote \pard\plain \sl240 \fs20 K plot;Pointset type;type} - -}{\b\f2 POINTSET}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par -\par - - \par -\par -A curve can be give as set of precomputed points (a polygon) -in 2 or 3 dimensions. Such a point set is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 -of points, where each point is a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 2 (or 3) -numbers. These numbers are interpreted as }{\f3 (x,y)} {\f2 -(or }{\f3 x,y,z} {\f2 ) coordinates. All points of one set must have -the same dimension. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 Also a surface in 3d can be given by precomputed points, -but only on a logically orthogonal mesh: the surface is defined -by a list of curves (in 3d) which must have a uniform length. -GNUPLOT then will draw an orthogonal mesh by first drawing the -given lines, and second connecting the 1st point of the 1st curve -with the 1st point of the 2nd curve, that one with the 1st point -of the 3rd curve and so on for all curves and for all indexes. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PLOT} - -${\footnote \pard\plain \sl240 \fs20 $ PLOT} - -+{\footnote \pard\plain \sl240 \fs20 + g37:1157} - - K{\footnote \pard\plain \sl240 \fs20 K plot;graphics;PLOT command;command} - -}{\b\f2 PLOT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The command }{\f3 plot} {\f2 is the main entry for drawing a -picture from inside REDUCE. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 plot} {\f4 (,,...) -\par -\par -}{\f2 \par -where is a , a or an ,,,) -\par -\par -}{\f2 \par -, :- matrices. -\par -\par -, :- positive integers. -\par -\par -}{\f3 copy_into} {\f2 copies matrix into with -(1,1) at (,). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -G := mat((0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0)); \par - \par - \par - [0 0 0 0 0] \par - [ ] \par - [0 0 0 0 0] \par - [ ] \par - g := [0 0 0 0 0] \par - [ ] \par - [0 0 0 0 0] \par - [ ] \par - [0 0 0 0 0] \par - \par - \par - \par -copy_into(A,G,1,2); \par - \par - [0 1 2 3 0] \par - [ ] \par - [0 4 5 6 0] \par - [ ] \par - [0 7 8 9 0] \par - [ ] \par - [0 0 0 0 0] \par - [ ] \par - [0 0 0 0 0] \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb augment_columns}{\v\f2 augment_columns} -{\f2 , } -{\f2\uldb extend}{\v\f2 extend} -{\f2 , } -{\f2\uldb matrix_augment}{\v\f2 matrix_augment} -{\f2 , -} -{\f2\uldb matrix_stack}{\v\f2 matrix_stack} -{\f2 , } -{\f2\uldb stack_rows}{\v\f2 stack_rows} -{\f2 , } -{\f2\uldb sub_matrix}{\v\f2 sub_matrix} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # diagonal} - -${\footnote \pard\plain \sl240 \fs20 $ diagonal} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1191} - - K{\footnote \pard\plain \sl240 \fs20 K diagonal operator;operator} - -}{\b\f2 DIAGONAL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 diagonal} {\f4 (\{\}) -\par -\par -}{\f2 \par -(If you are feeling lazy then the braces can be omitted.) -\par -\par - :- each can be either a scalar expression or a -square } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par -}{\f3 diagonal} {\f2 creates a matrix that contains the input on the -diagonal. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -H := mat((66,77),(88,99)); \par - \par - [66 77] \par - h := [ ] \par - [88 99] \par - \par - \par - \par -diagonal(\{A,x,H\}); \par - \par - [1 2 3 0 0 0 ] \par - [ ] \par - [4 5 6 0 0 0 ] \par - [ ] \par - [7 8 9 0 0 0 ] \par - [ ] \par - [0 0 0 x 0 0 ] \par - [ ] \par - [0 0 0 0 66 77] \par - [ ] \par - [0 0 0 0 88 99] \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb jordan_block}{\v\f2 jordan_block} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # extend} - -${\footnote \pard\plain \sl240 \fs20 $ extend} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1192} - - K{\footnote \pard\plain \sl240 \fs20 K extend operator;operator} - -}{\b\f2 EXTEND}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 extend} {\f4 (,,,) -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par -, :- positive integers. -\par -\par - :- algebraic expression or symbol. -\par -\par -}{\f3 extend} {\f2 returns a copy of that has been extended by - rows and columns. The new entries are made equal to -. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -extend(A,1,2,x); \par - \par - [1 2 3 x x] \par - [ ] \par - [4 5 6 x x] \par - [ ] \par - [7 8 9 x x] \par - [ ] \par - [x x x x x] \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb copy_into}{\v\f2 copy_into} -{\f2 , } -{\f2\uldb matrix_augment}{\v\f2 matrix_augment} -{\f2 , } -{\f2\uldb matrix_stack}{\v\f2 matrix_stack} -{\f2 , -} -{\f2\uldb remove_columns}{\v\f2 remove_columns} -{\f2 , } -{\f2\uldb remove_rows}{\v\f2 remove_rows} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # find_companion} - -${\footnote \pard\plain \sl240 \fs20 $ find_companion} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1193} - - K{\footnote \pard\plain \sl240 \fs20 K find_companion operator;operator} - -}{\b\f2 FIND_COMPANION}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 find_companion} {\f4 (,) -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par - :- the variable. -\par -\par -Given a companion matrix, }{\f3 find_companion} {\f2 finds the polynomial -from which it was made. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -C := companion(x^4+17*x^3-9*x^2+11,x); \par - \par - \par - [0 0 0 -11] \par - [ ] \par - [1 0 0 0 ] \par - c := [ ] \par - [0 1 0 9 ] \par - [ ] \par - [0 0 1 -17] \par - \par - \par - \par -find_companion(C,x); \par - \par - 4 3 2 \par - x +17*x -9*x +11 \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb companion}{\v\f2 companion} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # get_columns} - -${\footnote \pard\plain \sl240 \fs20 $ get_columns} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1194} - - K{\footnote \pard\plain \sl240 \fs20 K get_columns operator;operator} - -}{\b\f2 GET_COLUMNS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -Get columns, get rows: -\par -\par - \par -syntax: \par -}{\f4 }{\f3 get_columns} {\f4 (,) -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par - :- either a positive integer or a list of positive - integers. -\par -\par -}{\f3 get_columns} {\f2 removes the columns of specified in - and returns them as a list of column matrices. -\par -\par -}{\f3 get_rows} {\f2 performs the same task on the rows of . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -get_columns(A,\{1,3\}); \par - \par - \{ \par - [1] \par - [ ] \par - [4] \par - [ ] \par - [7] \par - , \par - [3] \par - [ ] \par - [6] \par - [ ] \par - [9] \par - \} \par - \par - \par - \par -get_rows(A,2); \par - \par - \{ \par - [4 5 6] \par - \} \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb augment_columns}{\v\f2 augment_columns} -{\f2 , } -{\f2\uldb stack_rows}{\v\f2 stack_rows} -{\f2 , } -{\f2\uldb sub_matrix}{\v\f2 sub_matrix} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # get_rows} - -${\footnote \pard\plain \sl240 \fs20 $ get_rows} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1195} - - K{\footnote \pard\plain \sl240 \fs20 K get_rows operator;operator} - -}{\b\f2 GET_ROWS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -see: } -{\f2\uldb get_columns}{\v\f2 get_columns} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # gram_schmidt} - -${\footnote \pard\plain \sl240 \fs20 $ gram_schmidt} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1196} - - K{\footnote \pard\plain \sl240 \fs20 K gram_schmidt operator;operator} - -}{\b\f2 GRAM_SCHMIDT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 gram_schmidt} {\f4 (\{\}) -\par -\par -}{\f2 \par -(If you are feeling lazy then the braces can be omitted.) -\par -\par - :- linearly independent vectors. Each vector must be -written as a list, eg:\{1,0,0\}. -\par -\par -}{\f3 gram_schmidt} {\f2 performs the gram_schmidt orthonormalization on -the input vectors. -\par -\par -It returns a list of orthogonal normalized vectors. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -gram_schmidt(\{\{1,0,0\},\{1,1,0\},\{1,1,1\}\}); \par - \par - \par - \{\{1,0,0\},\{0,1,0\},\{0,0,1\}\} \par - \par - \par - \par -gram_schmidt(\{\{1,2\},\{3,4\}\}); \par - \par - \par - 1 2 2*sqrt(5) -sqrt(5) \par - \{\{ ------- , ------- \},\{ --------- , -------- \}\} \par - sqrt(5) sqrt(5) 5 5 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # hermitian_tp} - -${\footnote \pard\plain \sl240 \fs20 $ hermitian_tp} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1197} - - K{\footnote \pard\plain \sl240 \fs20 K hermitian_tp operator;operator} - -}{\b\f2 HERMITIAN_TP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 hermitian_tp} {\f4 () -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par -}{\f3 hermitian_tp} {\f2 computes the hermitian transpose of . -\par -\par -This is a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 in which the (i,j)'th entry is the conjugate -of the (j,i)'th entry of . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -J := mat((i+1,i+2,i+3),(4,5,2),(1,i,0)); \par - \par - \par - [i + 1 i + 2 i + 3] \par - [ ] \par - j := [ 4 5 2 ] \par - [ ] \par - [ 1 i 0 ] \par - \par - \par - \par -hermitian_tp(j); \par - \par - [ - i + 1 4 1 ] \par - [ ] \par - [ - i + 2 5 - i] \par - [ ] \par - [ - i + 3 2 0 ] \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb tp}{\v\f2 TP} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # hessian} - -${\footnote \pard\plain \sl240 \fs20 $ hessian} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1198} - - K{\footnote \pard\plain \sl240 \fs20 K hessian operator;operator} - -}{\b\f2 HESSIAN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 hessian} {\f4 (,) -\par -\par -}{\f2 \par - :- a scalar expression. -\par -\par - :- either a single variable or a list of - variables. -\par -\par -}{\f3 hessian} {\f2 computes the hessian matrix of w.r.t. the -variables in . -\par -\par -This is an n by n matrix where n is the number of variables and the -(i,j)'th entry is } -{\f2\uldb df}{\v\f2 DF} -{\f2 (,(i), -(j)). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -hessian(x*y*z+x^2,\{w,x,y,z\}); \par - \par - [0 0 0 0] \par - [ ] \par - [0 2 z y] \par - [ ] \par - [0 z 0 x] \par - [ ] \par - [0 y x 0] \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb df}{\v\f2 DF} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # hilbert} - -${\footnote \pard\plain \sl240 \fs20 $ hilbert} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1199} - - K{\footnote \pard\plain \sl240 \fs20 K hilbert operator;operator} - -}{\b\f2 HILBERT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 hilbert} {\f4 (,) -\par -\par -}{\f2 \par - :- a positive integer. -\par -\par - :- an algebraic expression. -\par -\par -}{\f3 hilbert} {\f2 computes the square hilbert matrix of dimension -. -\par -\par -This is the symmetric matrix in which the (i,j)'th entry is -1/(i+j-). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -hilbert(3,y+x); \par - \par - [ - 1 - 1 - 1 ] \par - [----------- ----------- -----------] \par - [ x + y - 2 x + y - 3 x + y - 4 ] \par - [ ] \par - [ - 1 - 1 - 1 ] \par - [----------- ----------- -----------] \par - [ x + y - 3 x + y - 4 x + y - 5 ] \par - [ ] \par - [ - 1 - 1 - 1 ] \par - [----------- ----------- -----------] \par - [ x + y - 4 x + y - 5 x + y - 6 ] \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # jacobian} - -${\footnote \pard\plain \sl240 \fs20 $ jacobian} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1200} - - K{\footnote \pard\plain \sl240 \fs20 K jacobian operator;operator} - -}{\b\f2 JACOBIAN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 jacobian} {\f4 (,) -\par -\par -}{\f2 \par - :- either a single algebraic expression or a list - of algebraic expressions. -\par -\par - :- either a single variable or a list of - variables. -\par -\par -}{\f3 jacobian} {\f2 computes the jacobian matrix of -w.r.t. . -\par -\par -This is a matrix whose (i,j)'th entry is } -{\f2\uldb df}{\v\f2 DF} -{\f2 ( -(i),(j)). -\par -\par -The matrix is n by m where n is the number of variables and m the number -of expressions. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -jacobian(\{x^4,x*y^2,x*y*z^3\},\{w,x,y,z\}); \par - \par - \par - [ 3 ] \par - [0 4*x 0 0 ] \par - [ ] \par - [ 2 ] \par - [0 y 2*x*y 0 ] \par - [ ] \par - [ 3 3 2] \par - [0 y*z x*z 3*x*y*z ] \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb hessian}{\v\f2 hessian} -{\f2 , } -{\f2\uldb df}{\v\f2 DF} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # jordan_block} - -${\footnote \pard\plain \sl240 \fs20 $ jordan_block} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1201} - - K{\footnote \pard\plain \sl240 \fs20 K jordan_block operator;operator} - -}{\b\f2 JORDAN_BLOCK}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 jordan_block} {\f4 (,) -\par -\par -}{\f2 \par - :- an algebraic expression or symbol. -\par -\par - :- a positive integer. -\par -\par -}{\f3 jordan_block} {\f2 computes the square jordan block matrix J of -dimension . -\par -\par -The entries of J are: -\par -\par -J(i,i) = for i=1 - ... n, J(i,i+1) = 1 for i=1 - ... n-1, and all other entries are 0. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -jordan\_block(x,5); \par - \par - [x 1 0 0 0] \par - [ ] \par - [0 x 1 0 0] \par - [ ] \par - [0 0 x 1 0] \par - [ ] \par - [0 0 0 x 1] \par - [ ] \par - [0 0 0 0 x] \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb diagonal}{\v\f2 diagonal} -{\f2 , } -{\f2\uldb companion}{\v\f2 companion} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # lu_decom} - -${\footnote \pard\plain \sl240 \fs20 $ lu_decom} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1202} - - K{\footnote \pard\plain \sl240 \fs20 K lu_decom operator;operator} - -}{\b\f2 LU_DECOM}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 lu_decom} {\f4 () -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 containing either numeric entries - or imaginary entries with numeric coefficients. -\par -\par -}{\f3 lu_decom} {\f2 performs LU decomposition on , ie: it -returns \{L,U\} where L is a lower diagonal } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 , U an -upper diagonal } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 and A = LU. -\par -\par -Caution: -\par -\par -The algorithm used can swap the rows of during the -calculation. This means that LU does not equal but a row -equivalent of it. Due to this, }{\f3 lu_decom} {\f2 returns \{L,U,vec\}. -The call }{\f3 convert(meta\{matrix} {\f2 ,vec)\} will return the matrix that has -been decomposed, i.e: LU = convert(,vec). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -K := mat((1,3,5),(-4,3,7),(8,6,4)); \par - \par - \par - [1 3 5] \par - [ ] \par - k := [-4 3 7] \par - [ ] \par - [8 6 4] \par - \par - \par - \par -on rounded; \par - \par -lu := lu_decom(K); \par - \par - lu := \{ \par - [8 0 0 ] \par - [ ] \par - [-4 6.0 0 ] \par - [ ] \par - [1 2.25 1.125] \par - , \par - [1 0.75 0.5] \par - [ ] \par - [0 1 1.5] \par - [ ] \par - [0 0 1 ] \par - , \par - [3 2 3]\} \par - \par - \par - \par -first lu * second lu; \par - \par - [8 6.0 4.0] \par - [ ] \par - [-4 3.0 7.0] \par - [ ] \par - [1 3.0 5.0] \par - \par - \par - \par -convert(K,third lu); \par - \par - P := mat((i+1,i+2,i+3),(4,5,2),(1,i,0)); \tab \par - [i + 1 i + 2 i + 3] \par - [ ] \par - p := [ 4 5 2 ] \par - [ ] \par - [ 1 i 0 ] \par - \par - \par -lu := lu_decom(P); \par - \par - lu := \{ \par - [ 1 0 0 ] \par - [ ] \par - [ 4 - 4*i + 5 0 ] \par - [ ] \par - [i + 1 3 0.414634146341*i + 2.26829268293] \par - , \par - [1 i 0 ] \par - [ ] \par - [0 1 0.19512195122*i + 0.243902439024] \par - [ ] \par - [0 0 1 ] \par - , \par - [3 2 3]\} \par - \par - \par - \par -first lu * second lu; \par - \par - [ 1 i 0 ] \par - [ ] \par - [ 4 5 2.0 ] \par - [ ] \par - [i + 1 i + 2 i + 3.0] \par - \par - \par - \par -convert(P,third lu); \par - \par - [ 1 i 0 ] \par - [ ] \par - [ 4 5 2 ] \par - [ ] \par - [i + 1 i + 2 i + 3] \par - \par -\pard \sl240 }{\f2 -\par -\par -Related functions: } -{\f2\uldb cholesky}{\v\f2 cholesky} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # make_identity} - -${\footnote \pard\plain \sl240 \fs20 $ make_identity} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1203} - - K{\footnote \pard\plain \sl240 \fs20 K make_identity operator;operator} - -}{\b\f2 MAKE_IDENTITY}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 make_identity} {\f4 () -\par -\par -}{\f2 \par - :- a positive integer. -\par -\par -}{\f3 make_identity} {\f2 creates the identity matrix of dimension -. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -make_identity(4); \par - \par - [1 0 0 0] \par - [ ] \par - [0 1 0 0] \par - [ ] \par - [0 0 1 0] \par - [ ] \par - [0 0 0 1] \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb diagonal}{\v\f2 diagonal} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # matrix_augment} - -${\footnote \pard\plain \sl240 \fs20 $ matrix_augment} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1204} - - K{\footnote \pard\plain \sl240 \fs20 K matrix_augment operator;operator} - -}{\b\f2 MATRIX_AUGMENT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -Matrix augment, matrix stack: -\par -\par - \par -syntax: \par -}{\f4 }{\f3 matrix_augment} {\f4 \{\} -\par -\par -}{\f2 \par -(If you are feeling lazy then the braces can be omitted.) -\par -\par - :- matrices. -\par -\par -}{\f3 matrix_augment} {\f2 sticks the matrices in -together horizontally. -\par -\par -}{\f3 matrix_stack} {\f2 sticks the matrices in -together vertically. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -matrix_augment(\{A,A\}); \par - \par - [1 2 3 1 2 3] \par - [ ] \par - [4 5 6 4 5 6] \par - [ ] \par - [7 8 9 7 8 9] \par - \par - \par - \par -matrix_stack(A,A); \par - \par - [1 2 3] \par - [ ] \par - [4 5 6] \par - [ ] \par - [7 8 9] \par - [ ] \par - [1 2 3] \par - [ ] \par - [4 5 6] \par - [ ] \par - [7 8 9] \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb augment_columns}{\v\f2 augment_columns} -{\f2 , } -{\f2\uldb stack_rows}{\v\f2 stack_rows} -{\f2 , } -{\f2\uldb sub_matrix}{\v\f2 sub_matrix} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # matrixp} - -${\footnote \pard\plain \sl240 \fs20 $ matrixp} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1205} - - K{\footnote \pard\plain \sl240 \fs20 K matrixp operator;operator} - -}{\b\f2 MATRIXP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 matrixp} {\f4 () -\par -\par -}{\f2 \par - :- anything you like. -\par -\par -}{\f3 matrixp} {\f2 is a boolean function that returns t if the input is a -matrix and nil otherwise. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -matrixp A; \par - \par - t \par - \par - \par -matrixp(doodlesackbanana); \par - \par - nil \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb squarep}{\v\f2 squarep} -{\f2 , } -{\f2\uldb symmetricp}{\v\f2 symmetricp} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # matrix_stack} - -${\footnote \pard\plain \sl240 \fs20 $ matrix_stack} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1206} - - K{\footnote \pard\plain \sl240 \fs20 K matrix_stack operator;operator} - -}{\b\f2 MATRIX_STACK}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -see: } -{\f2\uldb matrix_augment}{\v\f2 matrix_augment} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # minor} - -${\footnote \pard\plain \sl240 \fs20 $ minor} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1207} - - K{\footnote \pard\plain \sl240 \fs20 K minor operator;operator} - -}{\b\f2 MINOR}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 minor} {\f4 (,,) -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -, :- positive integers. -\par -\par -}{\f3 minor} {\f2 computes the (,)'th minor of . -This is created by removing the 'th row and the 'th -column from . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -minor(A,1,3); \par - \par - [4 5] \par - [ ] \par - [7 8] \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb remove_columns}{\v\f2 remove_columns} -{\f2 , } -{\f2\uldb remove_rows}{\v\f2 remove_rows} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # mult_columns} - -${\footnote \pard\plain \sl240 \fs20 $ mult_columns} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1208} - - K{\footnote \pard\plain \sl240 \fs20 K mult_columns operator;operator} - -}{\b\f2 MULT_COLUMNS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -Mult columns, mult rows: -\par -\par - \par -syntax: \par -}{\f4 }{\f3 mult_columns} {\f4 (,,) -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par - :- a positive integer or a list of positive - integers. -\par -\par - :- an algebraic expression. -\par -\par -}{\f3 mult_columns} {\f2 returns a copy of in which the -columns specified in have been multiplied by -. -\par -\par -}{\f3 mult_rows} {\f2 performs the same task on the rows of . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -mult_columns(A,\{1,3\},x); \par - \par - [ x 2 3*x] \par - [ ] \par - [4*x 5 6*x] \par - [ ] \par - [7*x 8 9*x] \par - \par - \par - \par -mult_rows(A,2,10); \par - \par - [1 2 3 ] \par - [ ] \par - [40 50 60] \par - [ ] \par - [7 8 9 ] \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb add_to_columns}{\v\f2 add_to_columns} -{\f2 , } -{\f2\uldb add_to_rows}{\v\f2 add_to_rows} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # mult_rows} - -${\footnote \pard\plain \sl240 \fs20 $ mult_rows} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1209} - - K{\footnote \pard\plain \sl240 \fs20 K mult_rows operator;operator} - -}{\b\f2 MULT_ROWS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -see: } -{\f2\uldb mult_columns}{\v\f2 mult_columns} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # pivot} - -${\footnote \pard\plain \sl240 \fs20 $ pivot} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1210} - - K{\footnote \pard\plain \sl240 \fs20 K pivot operator;operator} - -}{\b\f2 PIVOT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 pivot} {\f4 (,,) -\par -\par -}{\f2 \par - :- a matrix. -\par -\par -, :- positive integers such that (, - ) neq 0. -\par -\par -}{\f3 pivot} {\f2 pivots about it's (,)'th -entry. - \par -\par -To do this, multiples of the 'th row are added to every other -row in the matrix. -\par -\par -This means that the 'th column will be 0 except for the -(,)'th entry. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -pivot(A,2,3); \par - \par - [ - 1 ] \par - [-1 ------ 0] \par - [ 2 ] \par - [ ] \par - [4 5 6] \par - [ ] \par - [ 1 ] \par - [1 --- 0] \par - [ 2 ] \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb rows_pivot}{\v\f2 rows_pivot} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # pseudo_inverse} - -${\footnote \pard\plain \sl240 \fs20 $ pseudo_inverse} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1211} - - K{\footnote \pard\plain \sl240 \fs20 K pseudo_inverse operator;operator} - -}{\b\f2 PSEUDO_INVERSE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 pseudo_inverse} {\f4 () -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par -}{\f3 pseudo_inverse} {\f2 , also known as the Moore-Penrose inverse, -computes the pseudo inverse of . -\par -\par -Given the singular value decomposition of , i.e: -A = }{\f4 U*P*V^T}{\f2 , then the pseudo inverse }{\f4 A^-1}{\f2 is defined by -}{\f4 A^-1 = V^T*P^-1*U}{\f2 . -\par -\par -Thus * pseudo_inverse(A) = Id. -(Id is the identity matrix). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -R := mat((1,2,3,4),(9,8,7,6)); \par - \par - [1 2 3 4] \par - r := [ ] \par - [9 8 7 6] \par - \par - \par - \par -on rounded; \par - \par -pseudo_inverse(R); \par - \par - [ - 0.199999999996 0.100000000013 ] \par - [ ] \par - [ - 0.0499999999988 0.0500000000037 ] \par - [ ] \par - [ 0.0999999999982 - 5.57825497203e-12] \par - [ ] \par - [ 0.249999999995 - 0.0500000000148 ] \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb svd}{\v\f2 svd} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # random_matrix} - -${\footnote \pard\plain \sl240 \fs20 $ random_matrix} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1212} - - K{\footnote \pard\plain \sl240 \fs20 K random_matrix operator;operator} - -}{\b\f2 RANDOM_MATRIX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 random_matrix} {\f4 (,,) -\par -\par -}{\f2 \par -,, :- positive integers. -\par -\par -}{\f3 random_matrix} {\f2 creates an by matrix with random -entries in the range -limit < entry < limit. -\par -\par -Switches: -\par -\par -}{\f3 imaginary} {\f2 :- if on then matrix entries are x+i*y where -limit < x,y - < . -\par -\par -}{\f3 not_negative} {\f2 :- if on then 0 < entry < . In the imaginary - case we have 0 < x,y < . -\par -\par -}{\f3 only_integer} {\f2 :- if on then each entry is an integer. In the imaginary - case x and y are integers. -\par -\par -}{\f3 symmetric} {\f2 :- if on then the matrix is symmetric. -\par -\par -}{\f3 upper_matrix} {\f2 :- if on then the matrix is upper triangular. -\par -\par -}{\f3 lower_matrix} {\f2 :- if on then the matrix is lower triangular. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -on rounded; \par - \par -random_matrix(3,3,10); \par - \par - [ - 8.11911717343 - 5.71677292768 0.620580830035 ] \par - [ ] \par - [ - 0.032596262422 7.1655452861 5.86742633837 ] \par - [ ] \par - [ - 9.37155438255 - 7.55636708637 - 8.88618627557] \par - \par - \par - \par -on only_integer, not_negative, upper_matrix, imaginary; \par - \par -random_matrix(4,4,10); \par - \par - [70*i + 15 28*i + 8 2*i + 79 27*i + 44] \par - [ ] \par - [ 0 46*i + 95 9*i + 63 95*i + 50] \par - [ ] \par - [ 0 0 31*i + 75 14*i + 65] \par - [ ] \par - [ 0 0 0 5*i + 52 ] \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # remove_columns} - -${\footnote \pard\plain \sl240 \fs20 $ remove_columns} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1213} - - K{\footnote \pard\plain \sl240 \fs20 K remove_columns operator;operator} - -}{\b\f2 REMOVE_COLUMNS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -Remove columns, remove rows: -\par -\par - \par -syntax: \par -}{\f4 }{\f3 remove_columns} {\f4 (,) -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . - :- either a positive integer or a list of positive - integers. -\par -\par -}{\f3 remove_columns} {\f2 removes the columns specified in - from . -\par -\par -}{\f3 remove_rows} {\f2 performs the same task on the rows of . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -remove_columns(A,2); \par - \par - [1 3] \par - [ ] \par - [4 6] \par - [ ] \par - [7 9] \par - \par - \par - \par -remove_rows(A,\{1,3\}); \par - \par - [4 5 6] \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb minor}{\v\f2 minor} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # remove_rows} - -${\footnote \pard\plain \sl240 \fs20 $ remove_rows} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1214} - - K{\footnote \pard\plain \sl240 \fs20 K remove_rows operator;operator} - -}{\b\f2 REMOVE_ROWS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -see: } -{\f2\uldb remove_columns}{\v\f2 remove_columns} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # row_dim} - -${\footnote \pard\plain \sl240 \fs20 $ row_dim} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1215} - - K{\footnote \pard\plain \sl240 \fs20 K row_dim operator;operator} - -}{\b\f2 ROW_DIM}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -see: } -{\f2\uldb column_dim}{\v\f2 column_dim} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # rows_pivot} - -${\footnote \pard\plain \sl240 \fs20 $ rows_pivot} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1216} - - K{\footnote \pard\plain \sl240 \fs20 K rows_pivot operator;operator} - -}{\b\f2 ROWS_PIVOT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 rows_pivot} {\f4 (,,,\{\}) -\par -\par -}{\f2 \par - :- a namerefmatrix. -\par -\par -, :- positive integers such that (, - ) neq 0. -\par -\par - :- positive integer or a list of positive integers. -\par -\par -}{\f3 rows_pivot} {\f2 performs the same task as }{\f3 pivot} {\f2 but applies -the pivot only to the rows specified in . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -N := mat((1,2,3),(4,5,6),(7,8,9),(1,2,3),(4,5,6)); \par - \par - \par - [1 2 3] \par - [ ] \par - [4 5 6] \par - [ ] \par - n := [7 8 9] \par - [ ] \par - [1 2 3] \par - [ ] \par - [4 5 6] \par - \par - \par - \par -rows_pivot(N,2,3,\{4,5\}); \par - \par - [1 2 3] \par - [ ] \par - [4 5 6] \par - [ ] \par - [7 8 9] \par - [ ] \par - [ - 1 ] \par - [-1 ------ 0] \par - [ 2 ] \par - [ ] \par - [0 0 0] \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb pivot}{\v\f2 pivot} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # simplex} - -${\footnote \pard\plain \sl240 \fs20 $ simplex} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1217} - - K{\footnote \pard\plain \sl240 \fs20 K simplex operator;operator} - -}{\b\f2 SIMPLEX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 simplex} {\f4 (,, -\{\}) -\par -\par -}{\f2 \par - :- either max or min (signifying maximize and - minimize). -\par -\par - :- the function you are maximizing or - minimizing. -\par -\par - :- the constraint inequalities. Each one must - be of the form sum of variables ( - <=,=,>=) number. -\par -\par -}{\f3 simplex} {\f2 applies the revised simplex algorithm to find the -optimal(either maximum or minimum) value of the - under the linear inequality constraints. -\par -\par -It returns \{optimal value,\{ values of variables at this optimal\}\}. -\par -\par -The algorithm implies that all the variables are non-negative. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par - simplex(max,x+y,\{x>=10,y>=20,x+y<=25\}); \par - \par - \par - ***** Error in simplex: Problem has no feasible solution \par - \par - \par - \par -simplex(max,10x+5y+5.5z,\{5x+3z<=200,x+0.1y+0.5z<=12, \par -0.1x+0.2y+0.3z<=9, 30x+10y+50z<=1500\}); \par - \par - \par - \{525.0,\{x=40.0,y=25.0,z=0\}\} \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # squarep} - -${\footnote \pard\plain \sl240 \fs20 $ squarep} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1218} - - K{\footnote \pard\plain \sl240 \fs20 K squarep operator;operator} - -}{\b\f2 SQUAREP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 squarep} {\f4 () -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par -}{\f3 squarep} {\f2 is a predicate that returns t if the is -square and nil otherwise. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -squarep(mat((1,3,5))); \par - \par - nil \par - \par - \par -squarep(A); \par -t \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb matrixp}{\v\f2 matrixp} -{\f2 , } -{\f2\uldb symmetricp}{\v\f2 symmetricp} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # stack_rows} - -${\footnote \pard\plain \sl240 \fs20 $ stack_rows} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1219} - - K{\footnote \pard\plain \sl240 \fs20 K stack_rows operator;operator} - -}{\b\f2 STACK_ROWS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -see: } -{\f2\uldb augment_columns}{\v\f2 augment_columns} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # sub_matrix} - -${\footnote \pard\plain \sl240 \fs20 $ sub_matrix} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1220} - - K{\footnote \pard\plain \sl240 \fs20 K sub_matrix operator;operator} - -}{\b\f2 SUB_MATRIX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 sub_matrix} {\f4 (,,) -\par -\par -}{\f2 \par - :- a matrix. -, :- either a positive integer or a - list of positive integers. -\par -\par -namesub_matrix produces the matrix consisting of the intersection of -the rows specified in and the columns specified in -. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -sub_matrix(A,\{1,3\},\{2,3\}); \par - \par - [2 3] \par - [ ] \par - [8 9] \par - \par -\pard \sl240 }{\f2 Related functions: -} -{\f2\uldb augment_columns}{\v\f2 augment_columns} -{\f2 , } -{\f2\uldb stack_rows}{\v\f2 stack_rows} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # svd} - -${\footnote \pard\plain \sl240 \fs20 $ svd} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1221} - - K{\footnote \pard\plain \sl240 \fs20 K singular value decomposition;svd operator;operator} - -}{\b\f2 SVD}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -Singular value decomposition: -\par -\par - \par -syntax: \par -}{\f4 }{\f3 svd} {\f4 () -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 containing only numeric entries. -\par -\par -}{\f3 svd} {\f2 computes the singular value decomposition of . -\par -\par -It returns -\par -\par -\{U,P,V\} -\par -\par -where A = }{\f4 U*P*V^T}{\f2 -\par -\par -and P = diag(sigma(1) ... sigma(n)). -\par -\par -sigma(i) for i= 1 ... n are the singular values of -. -\par -\par -n is the column dimension of . -\par -\par -The singular values of are the non-negative square roots -of the eigenvalues of }{\f4 A^T*A}{\f2 . -\par -\par -U and V are such that }{\f4 U*U^T = V*V^T = V^T*V}{\f2 = Id. -Id is the identity matrix. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -Q := mat((1,3),(-4,3)); \par - \par - [1 3] \par - q := [ ] \par - [-4 3] \par - \par - \par - \par -on rounded; \par - \par -svd(Q); \par - \par - \{ \par - [ 0.289784137735 0.957092029805] \par - [ ] \par - [ - 0.957092029805 0.289784137735] \par - , \par - [5.1491628629 0 ] \par - [ ] \par - [ 0 2.9130948854] \par - , \par - [ - 0.687215403194 0.726453707825 ] \par - [ ] \par - [ - 0.726453707825 - 0.687215403194] \par - \} \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # swap_columns} - -${\footnote \pard\plain \sl240 \fs20 $ swap_columns} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1222} - - K{\footnote \pard\plain \sl240 \fs20 K swap_columns operator;operator} - -}{\b\f2 SWAP_COLUMNS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -Swap columns, swap rows: -\par -\par - \par -syntax: \par -}{\f4 }{\f3 swap_columns} {\f4 (,,) -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par -, :- positive integers. - \par -\par -}{\f3 swap_columns} {\f2 swaps column of with -column . -\par -\par -}{\f3 swap_rows} {\f2 performs the same task on two rows of . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -swap_columns(A,2,3); \par - \par - [1 3 2] \par - [ ] \par - [4 6 5] \par - [ ] \par - [7 9 8] \par - \par - \par - \par -swap_rows(A,1,3); \par - \par - [7 8 9] \par - [ ] \par - [4 5 6] \par - [ ] \par - [1 2 3] \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb swap_entries}{\v\f2 swap_entries} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # swap_entries} - -${\footnote \pard\plain \sl240 \fs20 $ swap_entries} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1223} - - K{\footnote \pard\plain \sl240 \fs20 K swap_entries operator;operator} - -}{\b\f2 SWAP_ENTRIES}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 swap_entries} {\f4 (,\{,\},\{, -\}) -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par -,,, :- positive integers. -\par -\par -}{\f3 swap_entries} {\f2 swaps (,) with -(,). -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -swap_entries(A,\{1,1\},\{3,3\}); \par - \par - [9 2 3] \par - [ ] \par - [4 5 6] \par - [ ] \par - [7 8 1] \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb swap_columns}{\v\f2 swap_columns} -{\f2 , } -{\f2\uldb swap_rows}{\v\f2 swap_rows} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # swap_rows} - -${\footnote \pard\plain \sl240 \fs20 $ swap_rows} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1224} - - K{\footnote \pard\plain \sl240 \fs20 K swap_rows operator;operator} - -}{\b\f2 SWAP_ROWS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -see: } -{\f2\uldb swap_columns}{\v\f2 swap_columns} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # symmetricp} - -${\footnote \pard\plain \sl240 \fs20 $ symmetricp} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1225} - - K{\footnote \pard\plain \sl240 \fs20 K symmetricp operator;operator} - -}{\b\f2 SYMMETRICP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 symmetricp} {\f4 () -\par -\par -}{\f2 \par - :- a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 . -\par -\par -}{\f3 symmetricp} {\f2 is a predicate that returns t if the matrix is symmetric -and nil otherwise. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -symmetricp(make_identity(11)); \par - \par - t \par - \par - \par -symmetricp(A); \par - \par - nil \par - \par -\pard \sl240 }{\f2 Related functions: } -{\f2\uldb matrixp}{\v\f2 matrixp} -{\f2 , } -{\f2\uldb squarep}{\v\f2 squarep} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # toeplitz} - -${\footnote \pard\plain \sl240 \fs20 $ toeplitz} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1226} - - K{\footnote \pard\plain \sl240 \fs20 K toeplitz operator;operator} - -}{\b\f2 TOEPLITZ}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 toeplitz} {\f4 () -\par -\par -}{\f2 \par -(If you are feeling lazy then the braces can be omitted.) -\par -\par - :- list of algebraic expressions. -\par -\par -}{\f3 toeplitz} {\f2 creates the toeplitz matrix from the . -\par -\par -This is a square symmetric matrix in which the first expression is -placed on the diagonal and the i'th expression is placed on the (i-1)'th -sub and super diagonals. -\par -\par -It has dimension n where n is the number of expressions. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -toeplitz(\{w,x,y,z\}); \par - \par - [w x y z] \par - [ ] \par - [x w x y] \par - [ ] \par - [y x w x] \par - [ ] \par - [z y x w] \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # vandermonde} - -${\footnote \pard\plain \sl240 \fs20 $ vandermonde} - -+{\footnote \pard\plain \sl240 \fs20 + g38:1227} - - K{\footnote \pard\plain \sl240 \fs20 K vandermonde operator;operator} - -}{\b\f2 VANDERMONDE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par - \par -syntax: \par -}{\f4 }{\f3 vandermonde} {\f4 (\{\}) -\par -\par -}{\f2 \par -(If you are feeling lazy then the braces can be omitted.) -\par -\par - :- list of algebraic expressions. -\par -\par -}{\f3 vandermonde} {\f2 creates the vandermonde matrix from the -. -\par -\par -This is the square matrix in which the (i,j)'th entry is -}{\f4 (i)^(j-1)}{\f2 . -\par -\par -It has dimension n where n is the number of expressions. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -vandermonde(\{x,2*y,3*z\}); \par - \par - \par - [ 2 ] \par - [1 x x ] \par - [ ] \par - [ 2] \par - [1 2*y 4*y ] \par - [ ] \par - [ 2] \par - [1 3*z 9*z ] \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g38} - -${\footnote \pard\plain \sl240 \fs20 $ Linear Algebra package} - -+{\footnote \pard\plain \sl240 \fs20 + index:0038} -}{\b\f2 Linear Algebra package}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Linear Algebra package introduction} -{\v\f2 Linear_Algebra_package}{\f2 \par -}{\f2 \tab}{\f2\uldb fast_la switch} -{\v\f2 fast_la}{\f2 \par -}{\f2 \tab}{\f2\uldb add_columns operator} -{\v\f2 add_columns}{\f2 \par -}{\f2 \tab}{\f2\uldb add_rows operator} -{\v\f2 add_rows}{\f2 \par -}{\f2 \tab}{\f2\uldb add_to_columns operator} -{\v\f2 add_to_columns}{\f2 \par -}{\f2 \tab}{\f2\uldb add_to_rows operator} -{\v\f2 add_to_rows}{\f2 \par -}{\f2 \tab}{\f2\uldb augment_columns operator} -{\v\f2 augment_columns}{\f2 \par -}{\f2 \tab}{\f2\uldb band_matrix operator} -{\v\f2 band_matrix}{\f2 \par -}{\f2 \tab}{\f2\uldb block_matrix operator} -{\v\f2 block_matrix}{\f2 \par -}{\f2 \tab}{\f2\uldb char_matrix operator} -{\v\f2 char_matrix}{\f2 \par -}{\f2 \tab}{\f2\uldb char_poly operator} -{\v\f2 char_poly}{\f2 \par -}{\f2 \tab}{\f2\uldb cholesky operator} -{\v\f2 cholesky}{\f2 \par -}{\f2 \tab}{\f2\uldb coeff_matrix operator} -{\v\f2 coeff_matrix}{\f2 \par -}{\f2 \tab}{\f2\uldb column_dim operator} -{\v\f2 column_dim}{\f2 \par -}{\f2 \tab}{\f2\uldb companion operator} -{\v\f2 companion}{\f2 \par -}{\f2 \tab}{\f2\uldb copy_into operator} -{\v\f2 copy_into}{\f2 \par -}{\f2 \tab}{\f2\uldb diagonal operator} -{\v\f2 diagonal}{\f2 \par -}{\f2 \tab}{\f2\uldb extend operator} -{\v\f2 extend}{\f2 \par -}{\f2 \tab}{\f2\uldb find_companion operator} -{\v\f2 find_companion}{\f2 \par -}{\f2 \tab}{\f2\uldb get_columns operator} -{\v\f2 get_columns}{\f2 \par -}{\f2 \tab}{\f2\uldb get_rows operator} -{\v\f2 get_rows}{\f2 \par -}{\f2 \tab}{\f2\uldb gram_schmidt operator} -{\v\f2 gram_schmidt}{\f2 \par -}{\f2 \tab}{\f2\uldb hermitian_tp operator} -{\v\f2 hermitian_tp}{\f2 \par -}{\f2 \tab}{\f2\uldb hessian operator} -{\v\f2 hessian}{\f2 \par -}{\f2 \tab}{\f2\uldb hilbert operator} -{\v\f2 hilbert}{\f2 \par -}{\f2 \tab}{\f2\uldb jacobian operator} -{\v\f2 jacobian}{\f2 \par -}{\f2 \tab}{\f2\uldb jordan_block operator} -{\v\f2 jordan_block}{\f2 \par -}{\f2 \tab}{\f2\uldb lu_decom operator} -{\v\f2 lu_decom}{\f2 \par -}{\f2 \tab}{\f2\uldb make_identity operator} -{\v\f2 make_identity}{\f2 \par -}{\f2 \tab}{\f2\uldb matrix_augment operator} -{\v\f2 matrix_augment}{\f2 \par -}{\f2 \tab}{\f2\uldb matrixp operator} -{\v\f2 matrixp}{\f2 \par -}{\f2 \tab}{\f2\uldb matrix_stack operator} -{\v\f2 matrix_stack}{\f2 \par -}{\f2 \tab}{\f2\uldb minor operator} -{\v\f2 minor}{\f2 \par -}{\f2 \tab}{\f2\uldb mult_columns operator} -{\v\f2 mult_columns}{\f2 \par -}{\f2 \tab}{\f2\uldb mult_rows operator} -{\v\f2 mult_rows}{\f2 \par -}{\f2 \tab}{\f2\uldb pivot operator} -{\v\f2 pivot}{\f2 \par -}{\f2 \tab}{\f2\uldb pseudo_inverse operator} -{\v\f2 pseudo_inverse}{\f2 \par -}{\f2 \tab}{\f2\uldb random_matrix operator} -{\v\f2 random_matrix}{\f2 \par -}{\f2 \tab}{\f2\uldb remove_columns operator} -{\v\f2 remove_columns}{\f2 \par -}{\f2 \tab}{\f2\uldb remove_rows operator} -{\v\f2 remove_rows}{\f2 \par -}{\f2 \tab}{\f2\uldb row_dim operator} -{\v\f2 row_dim}{\f2 \par -}{\f2 \tab}{\f2\uldb rows_pivot operator} -{\v\f2 rows_pivot}{\f2 \par -}{\f2 \tab}{\f2\uldb simplex operator} -{\v\f2 simplex}{\f2 \par -}{\f2 \tab}{\f2\uldb squarep operator} -{\v\f2 squarep}{\f2 \par -}{\f2 \tab}{\f2\uldb stack_rows operator} -{\v\f2 stack_rows}{\f2 \par -}{\f2 \tab}{\f2\uldb sub_matrix operator} -{\v\f2 sub_matrix}{\f2 \par -}{\f2 \tab}{\f2\uldb svd operator} -{\v\f2 svd}{\f2 \par -}{\f2 \tab}{\f2\uldb swap_columns operator} -{\v\f2 swap_columns}{\f2 \par -}{\f2 \tab}{\f2\uldb swap_entries operator} -{\v\f2 swap_entries}{\f2 \par -}{\f2 \tab}{\f2\uldb swap_rows operator} -{\v\f2 swap_rows}{\f2 \par -}{\f2 \tab}{\f2\uldb symmetricp operator} -{\v\f2 symmetricp}{\f2 \par -}{\f2 \tab}{\f2\uldb toeplitz operator} -{\v\f2 toeplitz}{\f2 \par -}{\f2 \tab}{\f2\uldb vandermonde operator} -{\v\f2 vandermonde}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Smithex} - -${\footnote \pard\plain \sl240 \fs20 $ Smithex} - -+{\footnote \pard\plain \sl240 \fs20 + g39:1228} - - K{\footnote \pard\plain \sl240 \fs20 K Smithex operator;operator} - -}{\b\f2 SMITHEX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The operator }{\f3 smithex} {\f2 computes the Smith normal form S of a -} -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 A (say). It returns \{S,P,}{\f4 P^-1}{\f2 \} where }{\f4 P*S*P^-1 = A}{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 smithex} {\f4 (,) -\par -\par - :- a rectangular } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f4 of univariate polynomials in - . - :- the variable. -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par - a := mat((x,x+1),(0,3*x^2)); \par - \par - [x x + 1] \par - [ ] \par - a := [ 2 ] \par - [0 3*x ] \par - \par - \par - \par - smithex(a,x); \par - \par - [1 0 ] [1 0] [x x + 1] \par - \{ [ ], [ ], [ ] \} \par - [ 3] [ 2 ] [ ] \par - [0 x ] [3*x 1] [-3 -3 ] \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Smithex\_int} - -${\footnote \pard\plain \sl240 \fs20 $ Smithex_int} - -+{\footnote \pard\plain \sl240 \fs20 + g39:1229} - - K{\footnote \pard\plain \sl240 \fs20 K Smithex_int operator;operator} - -}{\b\f2 SMITHEX\_INT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The operator }{\f3 smithex_int} {\f2 performs the same task as }{\f3 smithex} {\f2 -but on matrices containing only integer entries. Namely, -}{\f3 smithex_int} {\f2 returns \{S,P,}{\f4 P^-1}{\f2 \} where S is the smith normal -form of the input } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 (A say), and }{\f4 P*S*P^-1 = A}{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 smithex_int} {\f4 () -\par -\par - :- a rectangular } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f4 of integer entries. -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par - a := mat((9,-36,30),(-36,192,-180),(30,-180,180)); \par - \par - \par - [ 9 -36 30 ] \par - [ ] \par - a := [-36 192 -180] \par - [ ] \par - [30 -180 180 ] \par - \par - \par - \par - smithex_int(a); \par - \par - [3 0 0 ] [-17 -5 -4 ] [1 -24 30 ] \par - [ ] [ ] [ ] \par - \{ [0 12 0 ], [64 19 15 ], [-1 25 -30] \} \par - [ ] [ ] [ ] \par - [0 0 60] [-50 -15 -12] [0 -1 1 ] \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Frobenius} - -${\footnote \pard\plain \sl240 \fs20 $ Frobenius} - -+{\footnote \pard\plain \sl240 \fs20 + g39:1230} - - K{\footnote \pard\plain \sl240 \fs20 K Frobenius operator;operator} - -}{\b\f2 FROBENIUS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The operator }{\f3 frobenius} {\f2 computes the }{\f3 frobenius} {\f2 normal form F of a -} -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 (A say). It returns \{F,P,}{\f4 P^-1}{\f2 \} where }{\f4 P*F*P^-1 = A}{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 frobenius} {\f4 () -\par -\par - :- a square } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f4 . -\par -\par -}{\f2 \par -Field Extensions: -\par -\par -By default, calculations are performed in the rational numbers. To -extend this field the } -{\f2\uldb arnum}{\v\f2 ARNUM} -{\f2 package can be used. The package must -first be loaded by load_package arnum;. The field can now be extended -by using the defpoly command. For example, defpoly sqrt2**2-2; will -extend the field to include the square root of 2 (now defined by sqrt2). -\par -\par -Modular Arithmetic: -\par -\par -}{\f3 Frobenius} {\f2 can also be calculated in a modular base. To do this -first type on modular;. Then setmod p; (where p is a prime) will set -the modular base of calculation to p. By further typing on balanced_mod -the answer will appear using a symmetric modular representation. See -} -{\f2\uldb ratjordan}{\v\f2 Ratjordan} -{\f2 for an example. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - a := mat((x,x^2),(3,5*x)); \par - \par - [ 2 ] \par - [x x ] \par - a := [ ] \par - [3 5*x] \par - \par - \par - frobenius(a); \par - \par - [ 2] [1 x] [ - x ] \par - \{ [0 - 2*x ], [ ], [1 -----] \} \par - [ ] [0 3] [ 3 ] \par - [1 6*x ] [ ] \par - [ 1 ] \par - [0 --- ] \par - [ 3 ] \par - \par - \par - load\_package arnum; \par - \par - defpoly sqrt2**2-2; \par - \par - a := mat((sqrt2,5),(7*sqrt2,sqrt2)); \par - \par - \par - [ sqrt2 5 ] \par - a := [ ] \par - [7*sqrt2 sqrt2] \par - \par - \par - \par - frobenius(a); \par - \par - [0 35*sqrt2 - 2] [1 sqrt2 ] [ 1 ] \par - \{ [ ], [ ], [1 - --- ] \} \par - [1 2*sqrt2 ] [1 7*sqrt2] [ 7 ] \par - [ ] \par - [ 1 ] \par - [0 ----*sqrt2] \par - [ 14 ] \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Ratjordan} - -${\footnote \pard\plain \sl240 \fs20 $ Ratjordan} - -+{\footnote \pard\plain \sl240 \fs20 + g39:1231} - - K{\footnote \pard\plain \sl240 \fs20 K Ratjordan operator;operator} - -}{\b\f2 RATJORDAN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The operator }{\f3 ratjordan} {\f2 computes the rational Jordan normal form R -of a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 (A say). It returns \{R,P,}{\f4 P^-1}{\f2 \} where }{\f4 P*R*P^-1 = A}{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 ratjordan} {\f4 () -\par -\par - :- a square } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f4 . -\par -\par -}{\f2 \par -Field Extensions: -\par -\par -By default, calculations are performed in the rational numbers. To -extend this field the }{\f3 arnum} {\f2 package can be used. The package must -first be loaded by load_package arnum;. The field can now be extended -by using the defpoly command. For example, defpoly sqrt2**2-2; will -extend the field to include the square root of 2 (now defined by sqrt2). -See } -{\f2\uldb frobenius}{\v\f2 Frobenius} -{\f2 for an example. -\par -\par -Modular Arithmetic: -\par -\par -}{\f3 ratjordan} {\f2 can also be calculated in a modular base. To do this -first type on modular;. Then setmod p; (where p is a prime) will set -the modular base of calculation to p. By further typing on balanced_mod -the answer will appear using a symmetric modular representation. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - a := mat((5,4*x),(2,x^2)); \par - \par - [5 4*x] \par - [ ] \par - a := [ 2 ] \par - [2 x ] \par - \par - \par - \par - ratjordan(a); \par - \par - [0 x*( - 5*x + 8)] [1 5] [ -5 ] \par - \{ [ ], [ ], [1 -----] \} \par - [ 2 ] [0 2] [ 2 ] \par - [1 x + 5 ] [ ] \par - [ 1 ] \par - [0 -----] \par - [ 2 ] \par - \par - \par - on modular; \par - \par - setmod 23; \par - \par - a := mat((12,34),(56,78)); \par - \par - [12 11] \par - a := [ ] \par - [10 9 ] \par - \par - \par - \par - ratjordan(a); \par - \par - [15 0] [16 8] [1 21] \par - \{ [ ], [ ], [ ] \} \par - [0 6] [19 4] [1 4 ] \par - \par - \par - \par - on balanced\_mod; \par - \par - ratjordan(a); \par - \par - [- 8 0] [ - 7 8] [1 - 2] \par - \{ [ ], [ ], [ ] \} \par - [ 0 6] [ - 4 4] [1 4 ] \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Jordansymbolic} - -${\footnote \pard\plain \sl240 \fs20 $ Jordansymbolic} - -+{\footnote \pard\plain \sl240 \fs20 + g39:1232} - - K{\footnote \pard\plain \sl240 \fs20 K Jordansymbolic operator;operator} - -}{\b\f2 JORDANSYMBOLIC}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The operator }{\f3 jordansymbolic} {\f2 computes the Jordan normal form J -of a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 (A say). It returns \{J,L,P,}{\f4 P^-1}{\f2 \} where -}{\f4 P*J*P^-1 = A}{\f2 . L = \{ll,mm\} where mm is a name and ll is a list of -irreducible factors of p(mm). -\par -\par - \par -syntax: \par -}{\f4 }{\f3 jordansymbolic} {\f4 () -\par -\par - :- a square } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f4 . -\par -\par -}{\f2 \par -Field Extensions: -\par -\par -By default, calculations are performed in the rational numbers. To -extend this field the } -{\f2\uldb arnum}{\v\f2 ARNUM} -{\f2 package can be used. The package must -first be loaded by load_package arnum;. The field can now be extended -by using the defpoly command. For example, defpoly sqrt2**2-2; will -extend the field to include the square root of 2 (now defined by sqrt2). -See } -{\f2\uldb frobenius}{\v\f2 Frobenius} -{\f2 for an example. -\par -\par -Modular Arithmetic: -\par -\par -}{\f3 jordansymbolic} {\f2 can also be calculated in a modular base. To do this -first type on modular;. Then setmod p; (where p is a prime) will set -the modular base of calculation to p. By further typing on balanced_mod -the answer will appear using a symmetric modular representation. See -} -{\f2\uldb ratjordan}{\v\f2 Ratjordan} -{\f2 for an example. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par - a := mat((1,y),(2,5*y)); \par - \par - [1 y ] \par - a := [ ] \par - [2 5*y] \par - \par - \par - \par - jordansymbolic(a); \par - \par - \{ \par - [lambda11 0 ] \par - [ ] \par - [ 0 lambda12] \par - , \par - 2 \par - lambda - 5*lambda*y - lambda + 3*y,lambda, \par - [lambda11 - 5*y lambda12 - 5*y] \par - [ ] \par - [ 2 2 ] \par - , \par - [ 2*lambda11 - 5*y - 1 5*lambda11*y - lambda11 - y + 1 ] \par - [---------------------- ---------------------------------] \par - [ 2 2 ] \par - [ 25*y - 2*y + 1 2*(25*y - 2*y + 1) ] \par - [ ] \par - [ 2*lambda12 - 5*y - 1 5*lambda12*y - lambda12 - y + 1 ] \par - [---------------------- ---------------------------------] \par - [ 2 2 ] \par - [ 25*y - 2*y + 1 2*(25*y - 2*y + 1) ] \par - \} \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Jordan} - -${\footnote \pard\plain \sl240 \fs20 $ Jordan} - -+{\footnote \pard\plain \sl240 \fs20 + g39:1233} - - K{\footnote \pard\plain \sl240 \fs20 K Jordan operator;operator} - -}{\b\f2 JORDAN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The operator }{\f3 jordan} {\f2 computes the Jordan normal form J -of a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 (A say). It returns \{J,P,}{\f4 P^-1}{\f2 \} where }{\f4 P*J*P^-1 = A}{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 jordan} {\f4 () -\par -\par - :- a square } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f4 . -\par -\par -}{\f2 \par -Field Extensions: -By default, calculations are performed in the rational numbers. To -extend this field the }{\f3 arnum} {\f2 package can be used. The package must -first be loaded by load_package arnum;. The field can now be extended -by using the defpoly command. For example, defpoly sqrt2**2-2; will -extend the field to include the square root of 2 (now defined by sqrt2). -See } -{\f2\uldb frobenius}{\v\f2 Frobenius} -{\f2 for an example. -\par -\par -Modular Arithmetic: -}{\f3 Jordan} {\f2 can also be calculated in a modular base. To do this -first type on modular;. Then setmod p; (where p is a prime) will set -the modular base of calculation to p. By further typing on balanced_mod -the answer will appear using a symmetric modular representation. See -} -{\f2\uldb ratjordan}{\v\f2 Ratjordan} -{\f2 for an example. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par - a := mat((1,x),(0,x)); \par - \par - [1 x] \par - a := [ ] \par - [0 x] \par - \par - \par - \par - jordan(a); \par - \par - \{ \par - [1 0] \par - [ ] \par - [0 x] \par - , \par - [ 1 x ] \par - [------- --------------] \par - [ x - 1 2 ] \par - [ x - 2*x + 1 ] \par - [ ] \par - [ 1 ] \par - [ 0 ------- ] \par - [ x - 1 ] \par - , \par - [x - 1 - x ] \par - [ ] \par - [ 0 x - 1] \par - \} \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g39} - -${\footnote \pard\plain \sl240 \fs20 $ Matrix Normal Forms} - -+{\footnote \pard\plain \sl240 \fs20 + index:0039} -}{\b\f2 Matrix Normal Forms}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Smithex operator} -{\v\f2 Smithex}{\f2 \par -}{\f2 \tab}{\f2\uldb Smithex\_int operator} -{\v\f2 Smithex\_int}{\f2 \par -}{\f2 \tab}{\f2\uldb Frobenius operator} -{\v\f2 Frobenius}{\f2 \par -}{\f2 \tab}{\f2\uldb Ratjordan operator} -{\v\f2 Ratjordan}{\f2 \par -}{\f2 \tab}{\f2\uldb Jordansymbolic operator} -{\v\f2 Jordansymbolic}{\f2 \par -}{\f2 \tab}{\f2\uldb Jordan operator} -{\v\f2 Jordan}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # Miscellaneous_Packages} - -${\footnote \pard\plain \sl240 \fs20 $ Miscellaneous_Packages} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1234} - - K{\footnote \pard\plain \sl240 \fs20 K Miscellaneous Packages introduction;introduction} - -}{\b\f2 MISCELLANEOUS PACKAGES}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par -\par - -REDUCE includes a large number of packages that have been contributed by -users from various fields. Some of these, together with their relevant -commands, switches and so on (e.g., the NUMERIC package), have -been described elsewhere. This section describes those packages for which -no separate help material exists. Each has its own switches, commands, -and operators, and some redefine special characters to aid in their -notation. However, the brief descriptions given here do not include all -such information. Readers are referred to the general package -documentation in this case, which can be found, along with the source -code, under the subdirectories }{\f3 doc} {\f2 and }{\f3 src} {\f2 in the -}{\f3 reduce} {\f2 directory. The } -{\f2\uldb load_package}{\v\f2 LOAD\_PACKAGE} -{\f2 command is used to -load the files you wish into your system. There will be a short delay -while the package is loaded. A package cannot be unloaded. Once it -is in your system, it stays there until you end the session. Each package -also has a test file, which you will find under its name in the -}{\f3 $reduce/xmpl} {\f2 directory. -\par -\par -Finally, it should be mentioned that such user-contributed packages are -unsupported; any questions or problems should be directed to their -authors. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ALGINT_package} - -${\footnote \pard\plain \sl240 \fs20 $ ALGINT_package} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1235} - - K{\footnote \pard\plain \sl240 \fs20 K integration of square roots;integration;ALGINT package;package} - -}{\b\f2 ALGINT}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: James H. Davenport -\par -\par -The }{\f3 algint} {\f2 package provides indefinite integration of square roots. -This package, which is an extension of the basic integration package -distributed with REDUCE, will analytically integrate a wide range of -expressions involving square roots. The } -{\f2\uldb algint}{\v\f2 ALGINT} -{\f2 switch provides for -the use of the facilities given by the package, and is automatically turned -on when the package is loaded. If you want to return to the standard -integration algorithms, turn } -{\f2\uldb algint}{\v\f2 ALGINT} -{\f2 off. An error message is given -if you try to turn the } -{\f2\uldb algint}{\v\f2 ALGINT} -{\f2 switch on when its package is not -loaded. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # APPLYSYM} - -${\footnote \pard\plain \sl240 \fs20 $ APPLYSYM} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1236} - - K{\footnote \pard\plain \sl240 \fs20 K symmetries;differential equations;APPLYSYM package;package} - -}{\b\f2 APPLYSYM}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Thomas Wolf -\par -\par -This package provides programs APPLYSYM, QUASILINPDE and DETRAFO for -computing with infinitesimal symmetries of differential equations. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ARNUM} - -${\footnote \pard\plain \sl240 \fs20 $ ARNUM} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1237} - - K{\footnote \pard\plain \sl240 \fs20 K algebraic numbers;ARNUM package;package} - -}{\b\f2 ARNUM}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Eberhard Schruefer -\par -\par -This package provides facilities for handling algebraic numbers as polynomial -coefficients in REDUCE calculations. It includes facilities for introducing -indeterminates to represent algebraic numbers, for calculating splitting -fields, and for factoring and finding greatest common divisors in such -domains. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ASSIST} - -${\footnote \pard\plain \sl240 \fs20 $ ASSIST} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1238} - - K{\footnote \pard\plain \sl240 \fs20 K utilities;ASSIST package;package} - -}{\b\f2 ASSIST}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Hubert Caprasse -\par -\par -ASSIST contains a large number of additional general purpose functions -that allow a user to better adapt REDUCE to various calculational -strategies and to make the programming task more straightforward and more -efficient. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # AVECTOR} - -${\footnote \pard\plain \sl240 \fs20 $ AVECTOR} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1239} - - K{\footnote \pard\plain \sl240 \fs20 K dot product;cross product;vector algebra;AVECTOR package;package} - -}{\b\f2 AVECTOR}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: David Harper -\par -\par -This package provides REDUCE with the ability to perform vector algebra -using the same notation as scalar algebra. The basic algebraic operations -are supported, as are differentiation and integration of vectors with -respect to scalar variables, cross product and dot product, component -manipulation and application of scalar functions (e.g. cosine) to a vector -to yield a vector result. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BOOLEAN} - -${\footnote \pard\plain \sl240 \fs20 $ BOOLEAN} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1240} - - K{\footnote \pard\plain \sl240 \fs20 K boolean expressions;BOOLEAN package;package} - -}{\b\f2 BOOLEAN}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Herbert Melenk -\par -\par -This package supports the computation with boolean expressions in the -propositional calculus. The data objects are composed from algebraic -expressions connected by the infix boolean operators and, or, - implies, equiv, and the unary prefix operator not. - Boolean allows you to simplify expressions built from these -operators, and to test properties like equivalence, subset property etc. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CALI} - -${\footnote \pard\plain \sl240 \fs20 $ CALI} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1241} - - K{\footnote \pard\plain \sl240 \fs20 K commutative algebra;Groebner;polynomial;CALI package;package} - -}{\b\f2 CALI}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Hans-Gert Gr"abe -\par -\par -This package contains algorithms for computations in commutative algebra -closely related to the Groebner algorithm for ideals and modules. Its -heart is a new implementation of the Groebner algorithm that also allows -for the computation of syzygies. This implementation is also applicable to -submodules of free modules with generators represented as rows of a matrix. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CAMAL} - -${\footnote \pard\plain \sl240 \fs20 $ CAMAL} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1242} - - K{\footnote \pard\plain \sl240 \fs20 K Fourier series;celestial mechanics;CAMAL package;package} - -}{\b\f2 CAMAL}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: John P. Fitch -\par -\par -This packages implements in REDUCE the Fourier transform procedures of the -CAMAL package for celestial mechanics. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CHANGEVR} - -${\footnote \pard\plain \sl240 \fs20 $ CHANGEVR} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1243} - - K{\footnote \pard\plain \sl240 \fs20 K CHANGEVR package;package} - -}{\b\f2 CHANGEVR}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - -Author: G. Ucoluk -\par -\par -This package provides facilities for changing the independent variables in -a differential equation. It is basically the application of the chain rule. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COMPACT} - -${\footnote \pard\plain \sl240 \fs20 $ COMPACT} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1244} - - K{\footnote \pard\plain \sl240 \fs20 K simplification;COMPACT package;package} - -}{\b\f2 COMPACT}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Anthony C. Hearn -\par -\par -COMPACT is a package of functions for the reduction of a polynomial in the -presence of side relations. COMPACT applies the side relations to the -polynomial so that an equivalent expression results with as few terms as -possible. For example, the evaluation of -\par -\par -\pard \tx3420 }{\f4 \par - compact(s*(1-sin x^2)+c*(1-cos x^2)+sin x^2+cos x^2, \par - \{cos x^2+sin x^2=1\}); \par - \par -\pard \sl240 }{\f2 yields the result -\pard \tx3420 }{\f4 \par - \par - 2 2 \par - SIN(X) *C + COS(X) *S + 1 \par -\pard \sl240 }{\f2 \par -\par -The first argument to the operator }{\f3 compact} {\f2 is the expression -and the second is a list of side relations that can be -equations or simple expressions (implicitly equated to zero). The -kernels in the side relations may also be free variables with the -same meaning as in rules, e.g. -\pard \tx3420 }{\f4 \par - sin_cos_identity := \{cos ~w^2+sin ~w^2=1\}$ \par - compact(u,in_cos_identity); \par -\pard \sl240 }{\f2 \par -\par -Also the full rule syntax with the replacement operator is allowed here. - \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CONTFR} - -${\footnote \pard\plain \sl240 \fs20 $ CONTFR} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1245} - - K{\footnote \pard\plain \sl240 \fs20 K continued fraction;CONTFR package;package} - -}{\b\f2 CONTFR}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Herbert Melenk -\par -\par -This package provides for the simultaneous approximation of a real number -by a continued fraction and a rational number with optional user -controlled precision (an upper bound for the denominator). -\par -\par -To use this package, the }{\f3 misc} {\f2 package should be loaded. One can then -use the operator }{\f3 continued_fraction} {\f2 to approximate the real -number by a continued fraction. This operator has one or two arguments, the -number to be converted and an optional precision. The result is a list of -two elements: the first is the rational value of the approximation and the -second the list of terms of the continued fraction that represent the same -value according to the definition t0 +1/(t1 + 1/(t2 + ...)). The -second optional parameter }{\f3 size} {\f2 is an upper bound on the absolute -value of the result denominator. If omitted, the approximation is performed -up to the current system precision. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -continued\_fraction pi; \par - \par - 1146408 \par - \{---------,\{3,7,15,1,292,1,1,1,2,1\}\} \par - 364913 \par - \par - \par - \par -continued\_fraction(pi,100); \par - \par - 22 \par - \{----,\{3,7\}\} \par - 7 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CRACK} - -${\footnote \pard\plain \sl240 \fs20 $ CRACK} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1246} - - K{\footnote \pard\plain \sl240 \fs20 K differential equation;CRACK package;package} - -}{\b\f2 CRACK}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Authors: Andreas Brand, Thomas Wolf -\par -\par -CRACK is a package for solving overdetermined systems of partial or -ordinary differential equations (PDEs, ODEs). Examples of programs which -make use of CRACK for investigating ODEs (finding symmetries, first -integrals, an equivalent Lagrangian or a ``differential factorization'') are -included. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CVIT} - -${\footnote \pard\plain \sl240 \fs20 $ CVIT} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1247} - - K{\footnote \pard\plain \sl240 \fs20 K Dirac algebra;CVIT package;package} - -}{\b\f2 CVIT}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Authors: V.Ilyin, A.Kryukov, A.Rodionov, A.Taranov -\par -\par -This package provides an alternative method for computing traces of Dirac -gamma matrices, based on an algorithm by Cvitanovich that treats gamma -matrices as 3-j symbols. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DEFINT} - -${\footnote \pard\plain \sl240 \fs20 $ DEFINT} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1248} - - K{\footnote \pard\plain \sl240 \fs20 K definite integration;DEFINT package;package} - -}{\b\f2 DEFINT}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Authors: Kerry Gaskell, Stanley M. Kameny, Winfried Neun -\par -\par -This package finds the definite integral of an expression in a stated -interval. It uses several techniques, including an innovative approach -based on the Meijer G-function, and contour integration. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DESIR} - -${\footnote \pard\plain \sl240 \fs20 $ DESIR} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1249} - - K{\footnote \pard\plain \sl240 \fs20 K differential equation;DESIR package;package} - -}{\b\f2 DESIR}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Authors: C. Dicrescenzo, F. Richard-Jung, E. Tournier -\par -\par -This package enables the basis of formal solutions to be computed for an -ordinary homogeneous differential equation with polynomial coefficients -over Q of any order, in the neighborhood of zero (regular or irregular -singular point, or ordinary point). -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DFPART} - -${\footnote \pard\plain \sl240 \fs20 $ DFPART} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1250} - - K{\footnote \pard\plain \sl240 \fs20 K partial derivative;DFPART package;package} - -}{\b\f2 DFPART}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Herbert Melenk -\par -\par -This package supports computations with total and partial derivatives of -formal function objects. Such computations can be useful in the context -of differential equations or power series expansions. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # DUMMY} - -${\footnote \pard\plain \sl240 \fs20 $ DUMMY} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1251} - - K{\footnote \pard\plain \sl240 \fs20 K dummy variable;DUMMY package;package} - -}{\b\f2 DUMMY}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Alain Dresse -\par -\par -This package allows a user to find the canonical form of expressions -involving dummy variables. In that way, the simplification of -polynomial expressions can be fully done. The indeterminates are general -operator objects endowed with as few properties as possible. In that way -the package may be used in a large spectrum of applications. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EXCALC} - -${\footnote \pard\plain \sl240 \fs20 $ EXCALC} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1252} - - K{\footnote \pard\plain \sl240 \fs20 K differential form;differential calculus;exterior calculus;EXCALC package;package} - -}{\b\f2 EXCALC}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Eberhard Schruefer -\par -\par -The }{\f3 excalc} {\f2 package is designed for easy use by all who are familiar -with the calculus of Modern Differential Geometry. The program is currently -able to handle scalar-valued exterior forms, vectors and operations between -them, as well as non-scalar valued forms (indexed forms). It is thus an ideal -tool for studying differential equations, doing calculations in general -relativity and field theories, or doing simple things such as calculating the -Laplacian of a tensor field for an arbitrary given frame. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FPS} - -${\footnote \pard\plain \sl240 \fs20 $ FPS} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1253} - - K{\footnote \pard\plain \sl240 \fs20 K Laurent-Puiseux series;power series;FPS package;package} - -}{\b\f2 FPS}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Authors: Wolfram Koepf, Winfried Neun -\par -\par -This package can expand a specific class of functions into their -corresponding Laurent-Puiseux series. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FIDE} - -${\footnote \pard\plain \sl240 \fs20 $ FIDE} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1254} - - K{\footnote \pard\plain \sl240 \fs20 K FIDE package;package} - -}{\b\f2 FIDE}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - -Author: Richard Liska -\par -\par -This package performs automation of the process of numerically -solving partial differential equations systems (PDES) by means of -computer algebra. For PDES solving, the finite difference method is applied. -The computer algebra system REDUCE and the numerical programming -language FORTRAN are used in the presented methodology. The main aim of -this methodology is to speed up the process of preparing numerical -programs for solving PDES. This process is quite often, especially for -complicated systems, a tedious and time consuming task. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GENTRAN} - -${\footnote \pard\plain \sl240 \fs20 $ GENTRAN} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1255} - - K{\footnote \pard\plain \sl240 \fs20 K C;FORTRAN;code generation;GENTRAN package;package} - -}{\b\f2 GENTRAN}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Barbara L. Gates -\par -\par -This package is an automatic code GENerator and TRANslator. It constructs -complete numerical programs based on sets of algorithmic specifications and -symbolic expressions. Formatted FORTRAN, RATFOR or C code can be generated -through a series of interactive commands or under the control of a template -processing routine. Large expressions can be automatically segmented into -subexpressions of manageable size, and a special file-handling mechanism -maintains stacks of open I/O channels to allow output to be sent to any -number of files simultaneously and to facilitate recursive invocation of the -whole code generation process. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # IDEALS} - -${\footnote \pard\plain \sl240 \fs20 $ IDEALS} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1256} - - K{\footnote \pard\plain \sl240 \fs20 K ideal;commutative algebra;Groebner;polynomial;IDEALS package;package} - -}{\b\f2 IDEALS}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Herbert Melenk -\par -\par -This package implements the basic arithmetic for polynomial ideals by -exploiting the Groebner bases package of REDUCE. In order to save -computing time all intermediate Groebner bases are stored internally such -that time consuming repetitions are inhibited. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INEQ} - -${\footnote \pard\plain \sl240 \fs20 $ INEQ} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1257} - - K{\footnote \pard\plain \sl240 \fs20 K inequality;INEQ package;package} - -}{\b\f2 INEQ}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Herbert Melenk -\par -\par -This package supports the operator }{\f3 ineq_solve} {\f2 that -tries to solves single inequalities and sets of coupled inequalities. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INVBASE} - -${\footnote \pard\plain \sl240 \fs20 $ INVBASE} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1258} - - K{\footnote \pard\plain \sl240 \fs20 K INVBASE package;package} - -}{\b\f2 INVBASE}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - -Authors: A.Yu. Zharkov and Yu.A. Blinkov -\par -\par -Involutive bases are a new tool for solving problems in connection with -multivariate polynomials, such as solving systems of polynomial equations -and analyzing polynomial ideals. An involutive basis of polynomial ideal -is nothing but a special form of a redundant Groebner basis. The -construction of involutive bases reduces the problem of solving polynomial -systems to simple linear algebra. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LAPLACE} - -${\footnote \pard\plain \sl240 \fs20 $ LAPLACE} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1259} - - K{\footnote \pard\plain \sl240 \fs20 K transform;LAPLACE package;package} - -}{\b\f2 LAPLACE}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Authors: C. Kazasov, M. Spiridonova, V. Tomov -\par -\par -This package can calculate ordinary and inverse Laplace transforms of -expressions. Documentation is in plain text. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LIE} - -${\footnote \pard\plain \sl240 \fs20 $ LIE} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1260} - - K{\footnote \pard\plain \sl240 \fs20 K LIE package;package} - -}{\b\f2 LIE}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - -Authors: Carsten and Franziska Sch"obel -\par -\par -}{\f3 Lie} {\f2 is a package of functions for the classification of real -n-dimensional Lie algebras. It consists of two modules: }{\f3 liendmc1} {\f2 -and }{\f3 lie1234} {\f2 . With the help of the functions in the }{\f3 liendmcl} {\f2 -module, real n-dimensional Lie algebras }{\f4 L}{\f2 with a derived algebra -}{\f4 L^(1)}{\f2 of dimension 1 can be classified. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # MODSR} - -${\footnote \pard\plain \sl240 \fs20 $ MODSR} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1261} - - K{\footnote \pard\plain \sl240 \fs20 K modular polynomial;MODSR package;package} - -}{\b\f2 MODSR}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Herbert Melenk -\par -\par -This package supports solve (M_SOLVE) and roots (M_ROOTS) operators for -modular polynomials and modular polynomial systems. The moduli need not -be primes. M_SOLVE requires a modulus to be set. M_ROOTS takes the -modulus as a second argument. For example: -\par -\par -\pard \tx3420 }{\f4 \par -on modular; setmod 8; \par -m_solve(2x=4); -> \{\{X=2\},\{X=6\}\} \par -m_solve(\{x^2-y^3=3\}); \par - -> \{\{X=0,Y=5\}, \{X=2,Y=1\}, \{X=4,Y=5\}, \{X=6,Y=1\}\} \par -m_solve(\{x=2,x^2-y^3=3\}); -> \{\{X=2,Y=1\}\} \par -off modular; \par -m_roots(x^2-1,8); -> \{1,3,5,7\} \par -m_roots(x^3-x,7); -> \{0,1,6\} \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NCPOLY} - -${\footnote \pard\plain \sl240 \fs20 $ NCPOLY} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1262} - - K{\footnote \pard\plain \sl240 \fs20 K non-commutativity;NCPOLY package;package} - -}{\b\f2 NCPOLY}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Authors: Herbert Melenk, Joachim Apel -\par -\par -This package allows the user to set up automatically a consistent -environment for computing in an algebra where the non--commutativity is -defined by Lie-bracket commutators. The package uses the REDUCE -}{\f3 noncom} {\f2 mechanism for elementary polynomial arithmetic; the commutator -rules are automatically computed from the Lie brackets. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ORTHOVEC} - -${\footnote \pard\plain \sl240 \fs20 $ ORTHOVEC} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1263} - - K{\footnote \pard\plain \sl240 \fs20 K curl;grad;div;dot product;cross product;Taylor;Laplacian;vector calculus;vector algebra;ORTHOVEC package;package} - -}{\b\f2 ORTHOVEC}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: James W. Eastwood -\par -\par -}{\f3 orthovec} {\f2 is a collection of REDUCE procedures and operations which -provide a simple-to-use environment for the manipulation of scalars and -vectors. Operations include addition, subtraction, dot and cross -products, division, modulus, div, grad, curl, laplacian, differentiation, -integration, and Taylor expansion. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PHYSOP} - -${\footnote \pard\plain \sl240 \fs20 $ PHYSOP} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1264} - - K{\footnote \pard\plain \sl240 \fs20 K PHYSOP package;package} - -}{\b\f2 PHYSOP}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - -Author: Mathias Warns -\par -\par -This package has been designed to meet the requirements of theoretical -physicists looking for a computer algebra tool to perform complicated -calculations in quantum theory with expressions containing operators. -These operations consist mainly of the calculation of commutators between -operator expressions and in the evaluations of operator matrix elements in -some abstract space. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PM} - -${\footnote \pard\plain \sl240 \fs20 $ PM} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1265} - - K{\footnote \pard\plain \sl240 \fs20 K pattern matching;PM package;package} - -}{\b\f2 PM}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Kevin McIsaac -\par -\par -PM is a general pattern matcher similar in style to those found in systems -such as SMP and Mathematica, and is based on the pattern matcher described -in Kevin McIsaac, ``Pattern Matching Algebraic Identities'', SIGSAM Bulletin, -19 (1985), 4-13. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RANDPOLY} - -${\footnote \pard\plain \sl240 \fs20 $ RANDPOLY} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1266} - - K{\footnote \pard\plain \sl240 \fs20 K random polynomial;RANDPOLY package;package} - -}{\b\f2 RANDPOLY}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Francis J. Wright -\par -\par -This package is based on a port of the Maple random polynomial -generator together with some support facilities for the generation -of random numbers and anonymous procedures. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # REACTEQN} - -${\footnote \pard\plain \sl240 \fs20 $ REACTEQN} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1267} - - K{\footnote \pard\plain \sl240 \fs20 K chemical reaction;REACTEQN package;package} - -}{\b\f2 REACTEQN}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Herbert Melenk -\par -\par -This package allows a user to transform chemical reaction systems into -ordinary differential equation systems (ODE) corresponding to the laws of -pure mass action. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RESET} - -${\footnote \pard\plain \sl240 \fs20 $ RESET} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1268} - - K{\footnote \pard\plain \sl240 \fs20 K RESET package;package} - -}{\b\f2 RESET}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - -Author: John Fitch -\par -\par -This package defines a command command RESETREDUCE that works through the -history of previous commands, and clears any values which have been -assigned, plus any rules, arrays and the like. It also sets the various -switches to their initial values. It is not complete, but does work for -most things that cause a gradual loss of space. It would be relatively -easy to make it interactive, so allowing for selective resetting. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RESIDUE} - -${\footnote \pard\plain \sl240 \fs20 $ RESIDUE} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1269} - - K{\footnote \pard\plain \sl240 \fs20 K RESIDUE package;package} - -}{\b\f2 RESIDUE}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - -Author: Wolfram Koepf -\par -\par -This package supports the calculation of residues of arbitrary -expressions. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # RLFI} - -${\footnote \pard\plain \sl240 \fs20 $ RLFI} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1270} - - K{\footnote \pard\plain \sl240 \fs20 K TEX;output;RLFI package;package} - -}{\b\f2 RLFI}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Richard Liska -\par -\par -This package -adds LaTeX syntax -to REDUCE. Text generated by REDUCE in this mode can be directly -used in LaTeX source -documents. Various -mathematical constructions are supported by the interface including -subscripts, superscripts, font changing, Greek letters, divide-bars, -integral and sum signs, derivatives, and so on. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SCOPE} - -${\footnote \pard\plain \sl240 \fs20 $ SCOPE} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1271} - - K{\footnote \pard\plain \sl240 \fs20 K optimization;code generation;SCOPE package;package} - -}{\b\f2 SCOPE}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: J.A. van Hulzen -\par -\par -SCOPE is a package for the production of an optimized form of a set of -expressions. It applies an heuristic search for common (sub)expressions -to almost any set of proper REDUCE assignment statements. The output is -obtained as a sequence of assignment statements. }{\f3 gentran} {\f2 is used to -facilitate expression output. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SETS} - -${\footnote \pard\plain \sl240 \fs20 $ SETS} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1272} - - K{\footnote \pard\plain \sl240 \fs20 K SETS package;package} - -}{\b\f2 SETS}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - -Author: Francis J. Wright -\par -\par -The SETS package provides algebraic-mode support for set operations on -lists regarded as sets (or representing explicit sets) and on implicit -sets represented by identifiers. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SPDE} - -${\footnote \pard\plain \sl240 \fs20 $ SPDE} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1273} - - K{\footnote \pard\plain \sl240 \fs20 K Lie symmetry;differential equation;SPDE package;package} - -}{\b\f2 SPDE}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Fritz Schwartz -\par -\par -The package }{\f3 spde} {\f2 provides a set of functions which may be used to -determine the symmetry group of Lie- or point-symmetries of a given system of -partial differential equations. In many cases the determining system is -solved completely automatically. In other cases the user has to provide -additional input information for the solution algorithm to terminate. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # SYMMETRY} - -${\footnote \pard\plain \sl240 \fs20 $ SYMMETRY} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1274} - - K{\footnote \pard\plain \sl240 \fs20 K SYMMETRY package;package} - -}{\b\f2 SYMMETRY}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - -Author: Karin Gatermann -\par -\par -This package computes symmetry-adapted bases and block diagonal forms of -matrices which have the symmetry of a group. The package is the -implementation of the theory of linear representations for small finite -groups such as the dihedral groups. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TPS} - -${\footnote \pard\plain \sl240 \fs20 $ TPS} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1275} - - K{\footnote \pard\plain \sl240 \fs20 K Taylor series;power series;TPS package;package} - -}{\b\f2 TPS}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Authors: Alan Barnes, Julian Padget -\par -\par -This package implements formal Laurent series expansions in one variable -using the domain mechanism of REDUCE. This means that power series -objects can be added, multiplied, differentiated etc., like other first -class objects in the system. A lazy evaluation scheme is used and thus -terms of the series are not evaluated until they are required for printing -or for use in calculating terms in other power series. The series are -extendible giving the user the impression that the full infinite series is -being manipulated. The errors that can sometimes occur using series that -are truncated at some fixed depth (for example when a term in the required -series depends on terms of an intermediate series beyond the truncation -depth) are thus avoided. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TRI} - -${\footnote \pard\plain \sl240 \fs20 $ TRI} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1276} - - K{\footnote \pard\plain \sl240 \fs20 K TEX;output;TRI package;package} - -}{\b\f2 TRI}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Werner Antweiler -\par -\par -This package provides facilities written in REDUCE-Lisp for typesetting -REDUCE formulas -using TeX. The -TeX-REDUCE-Interface incorporates three levels -of TeX output: -without line breaking, with line breaking, and -with line breaking plus indentation. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # TRIGSIMP} - -${\footnote \pard\plain \sl240 \fs20 $ TRIGSIMP} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1277} - - K{\footnote \pard\plain \sl240 \fs20 K simplification;TRIGSIMP package;package} - -}{\b\f2 TRIGSIMP}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Wolfram Koepf -\par -\par -TRIGSIMP is a useful tool for all kinds of trigonometric and hyperbolic -simplification and factorization. There are three procedures included in -TRIGSIMP: }{\f3 trigsimp} {\f2 , }{\f3 trigfactorize} {\f2 and }{\f3 triggcd} {\f2 . The -first is for finding simplifications of trigonometric or hyperbolic -expressions with many options, the second for factorizing them and the -third for finding the greatest common divisor of two trigonometric or -hyperbolic polynomials. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # XCOLOR} - -${\footnote \pard\plain \sl240 \fs20 $ XCOLOR} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1278} - - K{\footnote \pard\plain \sl240 \fs20 K high energy physics;XCOLOR package;package} - -}{\b\f2 XCOLOR}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: A. Kryukov -\par -\par -This package calculates the color factor in non-abelian gauge field -theories using an algorithm due to Cvitanovich. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # XIDEAL} - -${\footnote \pard\plain \sl240 \fs20 $ XIDEAL} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1279} - - K{\footnote \pard\plain \sl240 \fs20 K Groebner basis;XIDEAL package;package} - -}{\b\f2 XIDEAL}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: David Hartley -\par -\par -}{\f3 xideal} {\f2 constructs Groebner bases for solving the left ideal -membership problem: Groebner left ideal bases or GLIBs. For graded -ideals, where each form is homogeneous in degree, the distinction between -left and right ideals vanishes. Furthermore, if the generating forms are -all homogeneous, then the Groebner bases for the non-graded and graded -ideals are identical. In this case, }{\f3 xideal} {\f2 is able to save time by -truncating the Groebner basis at some maximum degree if desired. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # WU} - -${\footnote \pard\plain \sl240 \fs20 $ WU} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1280} - - K{\footnote \pard\plain \sl240 \fs20 K Wu-Wen-Tsun algorithm;polynomial;WU package;package} - -}{\b\f2 WU}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Author: Russell Bradford -\par -\par -This is a simple implementation of the Wu algorithm implemented in REDUCE -working directly from ``A Zero Structure Theorem for -Polynomial-Equations-Solving,'' Wu Wen-tsun, Institute of Systems Science, -Academia Sinica, Beijing. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ZEILBERG} - -${\footnote \pard\plain \sl240 \fs20 $ ZEILBERG} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1281} - - K{\footnote \pard\plain \sl240 \fs20 K summation;ZEILBERG package;package} - -}{\b\f2 ZEILBERG}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - - \par -\par -Authors: Gregor St"olting and Wolfram Koepf -\par -\par -This package is a careful implementation of the Gosper and Zeilberger -algorithms for indefinite and definite summation of hypergeometric terms, -respectively. Extensions of these algorithms are also included that are -valid for ratios of products of powers, -factorials, gamma function -terms, binomial coefficients, and shifted factorials that are -rational-linear in their arguments. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ZTRANS} - -${\footnote \pard\plain \sl240 \fs20 $ ZTRANS} - -+{\footnote \pard\plain \sl240 \fs20 + g40:1282} - - K{\footnote \pard\plain \sl240 \fs20 K ZTRANS package;package} - -}{\b\f2 ZTRANS}{\f2 \tab \tab \tab \tab }{\b\f2 package}{\f2 \par -\par - -Authors: Wolfram Koepf, Lisa Temme -\par -\par -This package is an implementation of the Z-transform of a sequence. -This is the discrete analogue of the Laplace Transform. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g40} - -${\footnote \pard\plain \sl240 \fs20 $ Miscellaneous Packages} - -+{\footnote \pard\plain \sl240 \fs20 + index:0040} -}{\b\f2 Miscellaneous Packages}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb Miscellaneous Packages introduction} -{\v\f2 Miscellaneous_Packages}{\f2 \par -}{\f2 \tab}{\f2\uldb ALGINT package} -{\v\f2 ALGINT_package}{\f2 \par -}{\f2 \tab}{\f2\uldb APPLYSYM package} -{\v\f2 APPLYSYM}{\f2 \par -}{\f2 \tab}{\f2\uldb ARNUM package} -{\v\f2 ARNUM}{\f2 \par -}{\f2 \tab}{\f2\uldb ASSIST package} -{\v\f2 ASSIST}{\f2 \par -}{\f2 \tab}{\f2\uldb AVECTOR package} -{\v\f2 AVECTOR}{\f2 \par -}{\f2 \tab}{\f2\uldb BOOLEAN package} -{\v\f2 BOOLEAN}{\f2 \par -}{\f2 \tab}{\f2\uldb CALI package} -{\v\f2 CALI}{\f2 \par -}{\f2 \tab}{\f2\uldb CAMAL package} -{\v\f2 CAMAL}{\f2 \par -}{\f2 \tab}{\f2\uldb CHANGEVR package} -{\v\f2 CHANGEVR}{\f2 \par -}{\f2 \tab}{\f2\uldb COMPACT package} -{\v\f2 COMPACT}{\f2 \par -}{\f2 \tab}{\f2\uldb CONTFR package} -{\v\f2 CONTFR}{\f2 \par -}{\f2 \tab}{\f2\uldb CRACK package} -{\v\f2 CRACK}{\f2 \par -}{\f2 \tab}{\f2\uldb CVIT package} -{\v\f2 CVIT}{\f2 \par -}{\f2 \tab}{\f2\uldb DEFINT package} -{\v\f2 DEFINT}{\f2 \par -}{\f2 \tab}{\f2\uldb DESIR package} -{\v\f2 DESIR}{\f2 \par -}{\f2 \tab}{\f2\uldb DFPART package} -{\v\f2 DFPART}{\f2 \par -}{\f2 \tab}{\f2\uldb DUMMY package} -{\v\f2 DUMMY}{\f2 \par -}{\f2 \tab}{\f2\uldb EXCALC package} -{\v\f2 EXCALC}{\f2 \par -}{\f2 \tab}{\f2\uldb FPS package} -{\v\f2 FPS}{\f2 \par -}{\f2 \tab}{\f2\uldb FIDE package} -{\v\f2 FIDE}{\f2 \par -}{\f2 \tab}{\f2\uldb GENTRAN package} -{\v\f2 GENTRAN}{\f2 \par -}{\f2 \tab}{\f2\uldb IDEALS package} -{\v\f2 IDEALS}{\f2 \par -}{\f2 \tab}{\f2\uldb INEQ package} -{\v\f2 INEQ}{\f2 \par -}{\f2 \tab}{\f2\uldb INVBASE package} -{\v\f2 INVBASE}{\f2 \par -}{\f2 \tab}{\f2\uldb LAPLACE package} -{\v\f2 LAPLACE}{\f2 \par -}{\f2 \tab}{\f2\uldb LIE package} -{\v\f2 LIE}{\f2 \par -}{\f2 \tab}{\f2\uldb MODSR package} -{\v\f2 MODSR}{\f2 \par -}{\f2 \tab}{\f2\uldb NCPOLY package} -{\v\f2 NCPOLY}{\f2 \par -}{\f2 \tab}{\f2\uldb ORTHOVEC package} -{\v\f2 ORTHOVEC}{\f2 \par -}{\f2 \tab}{\f2\uldb PHYSOP package} -{\v\f2 PHYSOP}{\f2 \par -}{\f2 \tab}{\f2\uldb PM package} -{\v\f2 PM}{\f2 \par -}{\f2 \tab}{\f2\uldb RANDPOLY package} -{\v\f2 RANDPOLY}{\f2 \par -}{\f2 \tab}{\f2\uldb REACTEQN package} -{\v\f2 REACTEQN}{\f2 \par -}{\f2 \tab}{\f2\uldb RESET package} -{\v\f2 RESET}{\f2 \par -}{\f2 \tab}{\f2\uldb RESIDUE package} -{\v\f2 RESIDUE}{\f2 \par -}{\f2 \tab}{\f2\uldb RLFI package} -{\v\f2 RLFI}{\f2 \par -}{\f2 \tab}{\f2\uldb SCOPE package} -{\v\f2 SCOPE}{\f2 \par -}{\f2 \tab}{\f2\uldb SETS package} -{\v\f2 SETS}{\f2 \par -}{\f2 \tab}{\f2\uldb SPDE package} -{\v\f2 SPDE}{\f2 \par -}{\f2 \tab}{\f2\uldb SYMMETRY package} -{\v\f2 SYMMETRY}{\f2 \par -}{\f2 \tab}{\f2\uldb TPS package} -{\v\f2 TPS}{\f2 \par -}{\f2 \tab}{\f2\uldb TRI package} -{\v\f2 TRI}{\f2 \par -}{\f2 \tab}{\f2\uldb TRIGSIMP package} -{\v\f2 TRIGSIMP}{\f2 \par -}{\f2 \tab}{\f2\uldb XCOLOR package} -{\v\f2 XCOLOR}{\f2 \par -}{\f2 \tab}{\f2\uldb XIDEAL package} -{\v\f2 XIDEAL}{\f2 \par -}{\f2 \tab}{\f2\uldb WU package} -{\v\f2 WU}{\f2 \par -}{\f2 \tab}{\f2\uldb ZEILBERG package} -{\v\f2 ZEILBERG}{\f2 \par -}{\f2 \tab}{\f2\uldb ZTRANS package} -{\v\f2 ZTRANS}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ED} - -${\footnote \pard\plain \sl240 \fs20 $ ED} - -+{\footnote \pard\plain \sl240 \fs20 + g41:1283} - - K{\footnote \pard\plain \sl240 \fs20 K ED command;command} - -}{\b\f2 ED}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 ed} {\f2 command invokes a simple line editor for REDUCE input -statements. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 ed} {\f4 or }{\f3 ed} {\f4 -\par -\par -}{\f2 \par -}{\f3 ed} {\f2 called with no argument edits the last input statement. If - is greater than or equal to the current line number, an error -message is printed. Reenter a proper }{\f3 ed} {\f2 command or return to the -top level with a semicolon. -\par -\par -The editor formats REDUCE's version of the desired input statement, -dividing it into lines at semicolons and dollar signs. The statement is -printed at the beginning of the edit session. The editor works on one -line at a time, and has a pointer (shown by }{\f3 ^} {\f2 ) to the current -character of that line. When the session begins, the pointer is at the -left hand side of the first line. The editing prompt is }{\f3 >} {\f2 . -\par -\par -The following commands are available. They may be entered in either upper -or lower case. All commands are activated by the carriage return, which -also prints out the current line after changes. Several commands can be -placed on a single line, except that commands terminated by an }{\f3 ESC} {\f2 -must be the last command before the carriage return. -\par -\par -\tab b -Move pointer to beginning of current line. -\par -\par -\tab d -Delete current character and next (digit-1) characters. An error message -is printed if anything other than a single digit follows d. If there are -fewer than characters left on the line, all but the final -dollar sign or semicolon is removed. To delete a line completely, use the -k command. -\par -\par -\tab e -End the current session, causing the edited expression to be reparsed by -REDUCE. -\par -\par -\tab f -Find the next occurrence of the character to the right of the -pointer on the current line and move the pointer to it. If the character is -not found, an error message is printed and the pointer remains in its -original position. Other lines are not searched. The f command is not -case-sensitive. -\par -\par -\tab i}{\f3 ESC} {\f2 -Insert in front of pointer. The }{\f3 ESC} {\f2 key is your -delimiter for the input string. No other command may follow this one on -the same line. -\par -\par -\tab k -Kill rest of the current line, including the semicolon or dollar sign -terminator. If there are characters remaining on the current line, and it -is the last line of the input statement, a semicolon is added to the line -as a terminator for REDUCE. If the current line is now empty, one of the -following actions is performed: If there is a following line, it becomes -the current line and the pointer is placed at its first character. If the -current line was the final line of the statement, and there is a previous -line, the previous line becomes the current line. If the current line was -the only line of the statement, and it is empty, a single semicolon is -inserted for REDUCE to parse. -\par -\par -\tab l -Finish editing this line and move to the last previous line. An error message -is printed if there is no previous line. -\par -\par -\tab n -Finish editing this line and move to the next line. An error message is -printed if there is no next line. -\par -\par -\tab p -Print out all the lines of the statement. Then a dotted line is printed, and -the current line is reprinted, with the pointer under it. -\par -\par -\tab q -Quit the editing session without saving the changes. If a semicolon is -entered after q, a new line prompt is given, otherwise REDUCE prompts you -for another command. Whatever you type in to the prompt appearing after -the q is entered is stored as the input for the line number in which you -called the edit. Thus if you enter a semicolon, neither } -{\f2\uldb input}{\v\f2 INPUT} -{\f2 -}{\f3 ed} {\f2 will find anything under the current number. -\par -\par -\tab r -Replace the character at the pointer by . -\par -\par -\tab s}{\f3 ESC} {\f2 -Search for the first occurrence of to the right of the -pointer on the current line and move the pointer to its first character. -The }{\f3 ESC} {\f2 key is your delimiter for the input string. The s function -does not search other lines of the statement. If the string is not found, -an error message is printed and the pointer remains in its original -position. The s command is not case-sensitive. No other command may -follow this one on the same line. -\par -\par -\tab x -Move the pointer one character to the right. If the pointer is already at -the end of the line, an error message is printed. -\par -\par -\tab - <(minus)> -Move the pointer one character to the left. If the pointer is already at the -beginning of the line, an error message is printed. -\par -\par -\tab ? -Display the Help menu, showing the commands and their actions. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \pard \sl240 }{\f2 (Line numbers are shown in the following examples)}{\f4 \pard \tx3420 \par - \par -2: >>x**2 + y; \par - \par -X^\{2\} + Y \par - \par -3: >>ed 2; \par - \par - X**2 + Y; \par - \par - ^ \par - \par -For help, type '?' \par - \par -?- (Enter three spaces and \key\{Return\}) \par - \par - X**2 + Y; \par - \par - ^ \par - \par -?- r5 \par - \par - X**5 + Y; \par - \par - ^ \par - \par -?- fY \par - \par - X**5 + Y; \par - \par - ^ \par - \par -?- iabc (Terminate with \key\{ESC\} and \key\{Return\}) \par - \par - X**5 + abcY; \par - \par - ^ \par - \par -?- ---- \par - \par - X**5 + abcY; \par - \par - ^ \par - \par -?- fbd2 \par - \par - X**5 + aY; \par - \par - ^ \par - \par -?- b \par - \par - X**5 + aY; \par - \par - ^ \par - \par -?- e \par - \par -AY + X^\{5\} \par - \par -4: >>procedure dumb(a); \par - \par ->>write a; \par - \par -DUMB \par - \par -5: >>dumb(17); \par - \par -17 \par - \par -6: >>ed 4; \par - \par - PROCEDURE DUMB (A); \par - \par - ^ \par - \par -WRITE A; \par - \par -?- fArBn \par - \par - WRITE A; \par - \par - ^ \par - \par -?- ibegin scalar a; a := b + 10; (Type a space, \key\{ESC\}, and \key\{Return\}) \par - \par - begin scalar a; a := b + 10; WRITE A; \par - \par -?- f;i end \key\{ESC\}, \key\{Return\} \par - \par - begin scalar b; b := a + 10; WRITE A end; \par - \par - ^ \par - \par -?- p \par - \par - PROCEDURE DUMB (B); \par - \par - begin scalar b; b := a + 10; WRITE A end; \par - \par - - - - - - - - - - - \par - \par - begin scalar b; b := a + 10; WRITE A end; \par - \par - ^ \par - \par -?- e \par - \par -DUMB \par - \par -7: >>dumb(17); \par - \par -27 \par - \par -8: >> \par - \par -\pard \sl240 }{\f2 -\par -\par -Note that REDUCE reparsed the procedure }{\f3 dumb} {\f2 and updated the -definition. -\par -\par -Since REDUCE divides the expression to be edited into lines at semicolons or -dollar sign terminators, some lines may occupy more than one line of screen -space. If the pointer is directly beneath the last line of text, it -refers to the top line of text. If there is a blank line between the -last line of text and the pointer, it refers to the second line -of text, and likewise for cases of greater than two lines of text. In other -words, the entire REDUCE statement up to the next terminator is printed, even -if it runs to several lines, then the pointer line is printed. -\par -\par -You can insert new statements which contain semicolons of their own into the -current line. They are run into the current line where you placed them -until you edit the statement again. REDUCE will understand the set of -statements if the syntax is correct. -\par -\par -If you leave out needed closing brackets when you exit the editor, a message -is printed allowing you to redo the edit (you can edit the previous line -number and return to where you were). 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FH(numberp)g(x)i(and)f(numberp)f(y)i(and)f(x>y)664 -4490 y FQ(do)s(es)i(not)g(attempt)h(to)g(mak)m(e)g(the)f -FJ(x>y)f FQ(comparison)g(unless)g FJ(X)g FQ(and)h FJ(Y)f -FQ(are)i(b)s(oth)664 4603 y(v)m(eri\014ed)30 b(to)h(b)s(e)e(n)m(um)m(b) -s(ers.)664 4766 y(Similarly)-8 b(,)31 b(ev)-5 b(aluation)33 -b(of)g(a)h(sequence)g(of)f(b)s(o)s(olean)f(expressions)g(connected)i(b) -m(y)g FJ(OR)664 4879 y FQ(stops)d(as)f(so)s(on)g(as)h(a)g -FD(true)37 b FQ(expression)29 b(is)g(found.)p eop -%%Page: 57 57 -57 56 bop 241 299 a @beginspecial @setspecial @endspecial -FM(3.4.)72 b(EQUA)-8 b(TIONS)2136 b FQ(57)241 555 y(NB:)42 -b(In)e(a)i(b)s(o)s(olean)e(expression,)j(and)e(in)f(a)i(place)f(where)g -(a)g(b)s(o)s(olean)g(expression)241 668 y(is)i(exp)s(ected,)48 -b(the)c(algebraic)f(v)-5 b(alue)43 b(0)i(is)e(in)m(terpreted)g(as)h -FD(false)p FQ(,)k(while)42 b(all)g(other)241 781 y(algebraic)c(v)-5 -b(alues)38 b(are)h(con)m(v)m(erted)h(to)f FD(true)p FQ(.)65 -b(So)38 b(in)g(algebraic)g(mo)s(de)g(a)h(pro)s(cedure)241 -894 y(can)31 b(b)s(e)e(written)h(for)g(direct)g(usage)i(in)d(b)s(o)s -(olean)g(expressions,)h(returning)e(sa)m(y)k(1)f(or)f(0)241 -1007 y(as)g(its)g(v)-5 b(alue)30 b(as)h(in)589 1238 y -FH(procedure)40 b(polynomialp\(u,x\);)720 1337 y(if)j(den\(u\)=1)d(and) -j(deg\(u,x\)>=1)c(then)j(1)h(else)f(0;)241 1568 y FQ(One)30 -b(can)g(then)g(use)g(this)g(in)f(a)i(b)s(o)s(olean)e(construct,)i(suc)m -(h)f(as)589 1774 y FH(if)43 b(polynomialp\(q,z\))37 b(and)43 -b(not)f(polynomialp\(q,y\))37 b(then)42 b(...)241 1993 -y FQ(In)36 b(addition,)h(an)m(y)g(pro)s(cedure)e(that)j(do)s(es)e(not)h -(ha)m(v)m(e)h(a)f(de\014ned)f(return)f(v)-5 b(alue)36 -b(\(for)241 2106 y(example,)f(a)f(blo)s(c)m(k)g(without)f(a)h -FJ(RETURN)e FQ(statemen)m(t)k(in)d(it\))h(has)g(the)g(b)s(o)s(olean)f -(v)-5 b(alue)241 2219 y FD(false)p FQ(.)241 2374 y @beginspecial -@setspecial @endspecial 181 x FE(3.4)135 b(Equations)241 -2808 y FQ(Equations)29 b(are)i(a)g(particular)e(t)m(yp)s(e)h(of)h -(expression)e(with)g(the)h(syn)m(tax)589 3039 y FH()39 -b(=)k(.)241 3283 y FQ(In)37 b(addition)e(to)k(their)d(role) -i(as)f(b)s(o)s(olean)g(expressions,)h(they)g(can)g(also)f(b)s(e)g(used) -g(as)241 3396 y(argumen)m(ts)27 b(to)h(sev)m(eral)f(op)s(erators)g -(\(e.g.,)j FJ(SOLVE)p FQ(\),)c(and)g(can)i(b)s(e)e(returned)g(as)h(v)-5 -b(alues.)241 3558 y(Under)19 b(normal)h(circumstances,)i(the)f(righ)m -(t-hand-side)d(of)j(the)g(equation)f(is)f(ev)-5 b(aluated)241 -3671 y(but)33 b(not)h(the)g(left-hand-side.)49 b(This)32 -b(also)h(applies)f(to)j(an)m(y)f(substitutions)d(made)j(b)m(y)241 -3784 y(the)c FJ(SUB)g FQ(op)s(erator.)41 b(If)30 b(b)s(oth)f(sides)h -(are)g(to)h(b)s(e)f(ev)-5 b(aluated,)31 b(the)f(switc)m(h)g -FJ(EVALLHSEQP)241 3897 y FQ(should)e(b)s(e)i(turned)f(on.)241 -4047 y(T)-8 b(o)34 b(facilitate)g(the)f(handling)e(of)j(equations,)h(t) -m(w)m(o)g(selectors,)g FJ(LHS)67 b FQ(and)33 b FJ(RHS)p -FQ(,)g(whic)m(h)241 4146 y(return)23 b(the)i(left-)g(and)f(righ)m -(t-hand)f(sides)h(of)g(a)h(equation)g(resp)s(ectiv)m(ely)-8 -b(,)26 b(are)f(pro)m(vided.)241 4246 y(F)-8 b(or)31 b(example,)589 -4452 y FH(lhs\(a+b=c\))40 b(->)j(a+b)241 4551 y(and)589 -4651 y(rhs\(a+b=c\))d(->)j(c.)241 4795 y @beginspecial -@setspecial @endspecial eop -%%Page: 58 58 -58 57 bop 664 299 a @beginspecial @setspecial @endspecial -FQ(58)1625 b FM(CHAPTER)30 b(3.)71 b(EXPRESSIONS)664 -555 y FE(3.5)136 b(Prop)t(er)44 b(Statemen)l(ts)j(as)e(Expressions)664 -795 y FQ(Sev)m(eral)33 b(kinds)f(of)h(prop)s(er)e(statemen)m(ts)k -(deliv)m(er)d(an)h(algebraic)g(or)g(n)m(umerical)f(result)664 -894 y(of)42 b(some)h(kind,)g(whic)m(h)e(can)h(in)f(turn)g(b)s(e)g(used) -g(as)h(an)g(expression)f(or)h(part)g(of)g(an)664 994 -y(expression.)c(F)-8 b(or)25 b(example,)h(an)f(assignmen)m(t)g -(statemen)m(t)h(itself)e(has)h(a)g(v)-5 b(alue,)26 b(namely)664 -1094 y(the)31 b(v)-5 b(alue)30 b(assigned.)40 b(So)1013 -1299 y FH(2)j(*)g(\(x)g(:=)g(a+b\))664 1505 y FQ(is)24 -b(equal)h(to)h FJ(2*\(a+b\))p FQ(,)e(as)h(w)m(ell)f(as)i(ha)m(ving)e -(the)i(\\side-e\013ect")g(of)g(assigning)d(the)i(v)-5 -b(alue)664 1605 y FJ(a+b)30 b FQ(to)h FJ(X)p FQ(.)f(In)g(con)m(text,) -1013 1811 y FH(y)43 b(:=)g(2)g(*)g(\(x)g(:=)g(a+b\);)664 -2030 y FQ(sets)31 b FJ(X)f FQ(to)h FJ(a+b)f FQ(and)f -FJ(Y)h FQ(to)h FJ(2*\(a+b\))p FQ(.)664 2193 y(The)g(sections)f(on)h -(the)g(v)-5 b(arious)30 b(prop)s(er)f(statemen)m(t)k(t)m(yp)s(es)e -(indicate)f(whic)m(h)f(of)i(these)664 2306 y(statemen)m(ts)h(are)f -(also)f(useful)f(as)i(expressions.)p eop -%%Page: 59 59 -59 58 bop 241 299 a @beginspecial @setspecial @endspecial -165 x @beginspecial @setspecial @endspecial 764 x FI(Chapter)64 -b(4)241 1693 y FT(Lists)241 2174 y FQ(A)33 b(list)e(is)h(an)g(ob)5 -b(ject)34 b(consisting)d(of)i(a)g(sequence)g(of)g(other)g(ob)5 -b(jects)33 b(\(including)d(lists)241 2274 y(themselv)m(es\),)i -(separated)f(b)m(y)g(commas)h(and)f(surrounded)d(b)m(y)j(braces.)43 -b(Examples)30 b(of)241 2373 y(lists)f(are:)589 2579 y -FH({a,b,c})589 2779 y({1,a-b,c=d})589 2978 y({{a},{{b,c},d},e})o(.)241 -3184 y FQ(The)h(empt)m(y)g(list)f(is)h(represen)m(ted)g(as)589 -3390 y FH({}.)241 3533 y @beginspecial @setspecial @endspecial -193 x FE(4.1)135 b(Op)t(erations)46 b(on)f(Lists)241 -3979 y FQ(Sev)m(eral)31 b(op)s(erators)h(in)f(the)h(system)f(return)g -(their)g(results)f(as)i(lists,)f(and)g(a)h(user)f(can)241 -4091 y(create)38 b(new)e(lists)f(using)h(braces)g(and)g(commas.)61 -b(Alternativ)m(ely)-8 b(,)38 b(one)f(can)g(use)f(the)241 -4204 y(op)s(erator)30 b(LIST)e(to)i(construct)g(a)g(list.)39 -b(An)29 b(imp)s(ortan)m(t)g(class)g(of)h(op)s(erations)f(on)g(lists)241 -4317 y(are)h(MAP)g(and)e(SELECT)g(op)s(erations.)40 b(F)-8 -b(or)30 b(details,)f(please)h(refer)f(to)h(the)g(c)m(hapters)241 -4430 y(on)e(MAP)-8 b(,)30 b(SELECT)d(and)h(the)h(F)m(OR)g(command.)40 -b(See)29 b(also)g(the)g(do)s(cumen)m(tation)g(on)241 -4543 y(the)h(ASSIST)f(pac)m(k)-5 b(age.)241 4706 y(T)d(o)40 -b(facilitate)f(the)g(use)g(of)h(lists,)g(a)g(n)m(um)m(b)s(er)e(of)h(op) -s(erators)h(are)g(also)f(a)m(v)-5 b(ailable)39 b(for)241 -4819 y(manipulating)34 b(them.)61 b FJ(PART\(,n\))33 -b FQ(for)k(example)g(will)d(return)i(the)h FL(n)2986 -4786 y FF(th)3093 4819 y FQ(ele-)241 4932 y(men)m(t)30 -b(of)f(a)h(list.)39 b FJ(LENGTH)28 b FQ(will)e(return)j(the)g(length)g -(of)g(a)h(list.)39 b(Sev)m(eral)30 b(op)s(erators)f(are)1690 -5187 y(59)p eop -%%Page: 60 60 -60 59 bop 664 299 a @beginspecial @setspecial @endspecial -FQ(60)2013 b FM(CHAPTER)29 b(4.)72 b(LISTS)664 555 y -FQ(also)26 b(de\014ned)f(uniquely)e(for)j(lists.)38 b(F)-8 -b(or)26 b(those)h(familiar)d(with)g(them,)j(these)g(op)s(erators)664 -668 y(in)33 b(fact)h(mirror)e(the)i(op)s(erations)f(de\014ned)g(for)g -(Lisp)g(lists.)49 b(These)34 b(op)s(erators)f(are)i(as)664 -781 y(follo)m(ws:)664 908 y @beginspecial @setspecial -@endspecial 166 x FR(4.1.1)113 b(LIST)664 1282 y FQ(The)36 -b(op)s(erator)g(LIST)f(is)g(an)h(alternativ)m(e)g(to)h(the)f(usage)h -(of)f(curly)f(brac)m(k)m(ets.)59 b(LIST)664 1382 y(accepts)42 -b(an)f(arbitrary)e(n)m(um)m(b)s(er)g(of)i(argumen)m(ts)g(and)f(returns) -g(a)h(list)e(of)i(its)f(argu-)664 1482 y(men)m(ts.)54 -b(This)33 b(op)s(erator)h(is)g(useful)f(in)g(cases)j(where)e(op)s -(erators)g(ha)m(v)m(e)i(to)f(b)s(e)f(passed)664 1581 -y(as)d(argumen)m(ts.)41 b(E.g.,)664 1787 y FH(list\(a,list\(list\()o -(b,)o(c\))o(,d\))o(,e)o(\);)299 b(->)86 b({{a},{{b,c},d},e)o(})664 -1926 y @beginspecial @setspecial @endspecial 154 x FR(4.1.2)113 -b(FIRST)664 2302 y FQ(This)41 b(op)s(erator)h(returns)f(the)i(\014rst)e -(mem)m(b)s(er)h(of)g(a)h(list.)75 b(An)42 b(error)g(o)s(ccurs)f(if)h -(the)664 2415 y(argumen)m(t)31 b(is)e(not)i(a)g(list,)e(or)h(the)h -(list)e(is)g(empt)m(y)-8 b(.)664 2560 y @beginspecial -@setspecial @endspecial 148 x FR(4.1.3)113 b(SECOND)664 -2929 y FJ(SECOND)23 b FQ(returns)g(the)h(second)g(mem)m(b)s(er)g(of)g -(a)g(list.)38 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-(reverse\({a,b,c}\))343 b(->)217 b({c,b,a})589 2460 y -(reverse\({{a,b,c},)o(d}\))168 b(->)217 b({d,{a,b,c}}.)241 -2594 y @beginspecial @setspecial @endspecial 155 x FR(4.1.9)112 -b(List)37 b(Argumen)m(ts)f(of)i(Other)f(Op)s(erators)241 -2970 y FQ(If)29 b(an)g(op)s(erator)h(other)g(than)f(those)i(sp)s -(eci\014cally)c(de\014ned)h(for)i(lists)e(is)g(giv)m(en)i(a)g(single) -241 3083 y(argumen)m(t)23 b(that)g(is)f(a)h(list,)g(then)f(the)h -(result)e(of)i(this)e(op)s(eration)h(will)e(b)s(e)i(a)h(list)e(in)h -(whic)m(h)241 3196 y(that)30 b(op)s(erator)g(is)f(applied)f(to)i(eac)m -(h)h(elemen)m(t)f(of)g(the)g(list.)39 b(F)-8 b(or)31 -b(example,)f(the)g(result)241 3309 y(of)g(ev)-5 b(aluating)30 -b FJ(log)p FP(f)p FJ(a,b,c)p FP(g)f FQ(is)h(the)g(expression)f -FP(f)p FJ(LOG\(A\),LOG\(B\),LOG\(C\))p FP(g)p FQ(.)241 -3472 y(There)f(are)h(t)m(w)m(o)h(w)m(a)m(ys)g(to)f(inhibit)d(this)h(op) -s(erator)i(distribution.)37 b(Firstly)-8 b(,)28 b(the)h(switc)m(h)241 -3585 y FJ(LISTARGS)p FQ(,)34 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-i(second)f(argumen)m(t)h(to)h(CONS)d(\(a)j("dotted)f(pair")241 -4959 y(in)29 b(LISP)g(terms\))i(is)e(not)i(allo)m(w)m(ed)f(and)g -(causes)h(an)f("in)m(v)-5 b(alid)28 b(as)j(list")e(error.)p -eop -%%Page: 62 62 -62 61 bop 664 299 a @beginspecial @setspecial @endspecial -FQ(62)2013 b FM(CHAPTER)29 b(4.)72 b(LISTS)708 555 y -FH(a)43 b(:=)g(17)f(.)i(4;)664 754 y(*****)e(17)g(4)i(invalid)c(as)j -(list)664 960 y FQ(Also,)29 b(the)f(initialization)d(of)k(a)f(scalar)g -(v)-5 b(ariable)27 b(is)h(not)g(the)h(empt)m(y)f(list)f({)i(one)f(has)g -(to)664 1060 y(set)j(list)e(t)m(yp)s(e)i(v)-5 b(ariables)29 -b(explicitly)-8 b(,)28 b(as)j(in)e(the)i(follo)m(wing)d(example:)708 -1266 y FH(load_package)38 b(assist;)708 1465 y(procedure)i(lotto)h -(\(n,m\);)751 1565 y(begin)h(scalar)f(list_1_n,)f(luckies,)g(hit;)882 -1664 y(list_1_n)g(:=)j({};)882 1764 y(luckies)e(:=)i({};)882 -1864 y(for)f(k:=1:n)g(do)g(list_1_n)f(:=)h(k)i(.)f(list_1_n;)882 -1963 y(for)f(k:=1:m)g(do)969 2063 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3662 y(system)j(to)i(system,)g(but)e(is)f(normally) -g(the)p 1807 3574 314 4 v 1807 3687 4 113 v 63 w FG(Return)p -2117 3687 V 1807 3690 314 4 v 63 w FQ(k)m(ey)i(on)f(an)h(ASCI)s(I)e -(terminal.)241 3775 y(Sp)s(eci\014c)c(systems)h(ma)m(y)h(also)f(use)g -(additional)f(k)m(eys)i(as)f(statemen)m(t)i(terminators.)241 -3938 y(If)e(a)g(statemen)m(t)j(is)c(a)i(prop)s(er)e(statemen)m(t,)j -(the)f(appropriate)e(action)i(tak)m(es)h(place.)241 4100 -y(Dep)s(ending)i(on)i(the)g(nature)f(of)h(the)g(prop)s(er)e(statemen)m -(t)k(some)e(result)f(or)g(resp)s(onse)241 4213 y(ma)m(y)f(or)f(ma)m(y)h -(not)g(b)s(e)e(prin)m(ted)g(out,)j(and)d(the)i(resp)s(onse)e(ma)m(y)i -(or)g(ma)m(y)g(not)f(dep)s(end)241 4326 y(on)d(the)h(terminator)f -(used.)241 4489 y(If)h(a)h(statemen)m(t)h(is)d(an)i(expression,)e(it)h -(is)g(ev)-5 b(aluated.)44 b(If)31 b(the)h(terminator)f(is)f(a)i(semi-) -241 4602 y(colon,)j(the)f(result)g(is)f(prin)m(ted.)50 -b(If)34 b(the)g(terminator)g(is)f(a)i(dollar)e(sign,)h(the)g(result)f 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b(\\p)s(olarit)m(y")g(is)g(the)g(opp)s(osite)g(of)g(that) -h(in)e FJ(WHILE)47 b(...DO.)664 3045 y FQ(As)31 b(an)f(example,)g(w)m -(e)h(rewrite)f(the)g(example)g(from)g(the)h FJ(WHILE)46 -b(...DO)29 b FQ(section:)1013 3251 y FH(ex:=0;)41 b(term:=1;)1013 -3351 y(repeat)g(<>)1144 3451 y(until)e(num\(term)f(-)k(1/1000\))c(<)j(0;) -1013 3550 y(ex;)664 3769 y FQ(In)31 b(this)g(case,)i(the)f(answ)m(er)g -(will)d(b)s(e)i(the)h(same)g(as)g(b)s(efore,)g(b)s(ecause)g(in)e -(neither)h(case)664 3882 y(is)f(a)g(term)h(added)e(to)j -FJ(EX)d FQ(whic)m(h)g(is)h(less)g(than)g(1/1000.)664 -4042 y @beginspecial @setspecial @endspecial 177 x FE(5.7)136 -b(Comp)t(ound)44 b(Statemen)l(ts)664 4471 y FQ(Often)g(the)g(desired)f -(pro)s(cess)g(can)i(b)s(est)e(\(or)i(only\))e(b)s(e)h(describ)s(ed)e -(as)i(a)g(series)g(of)664 4584 y(steps)f(to)g(b)s(e)f(carried)f(out)i -(one)g(after)g(the)g(other.)77 b(In)42 b(man)m(y)h(cases,)j(this)c(can) -h(b)s(e)664 4697 y(ac)m(hiev)m(ed)33 b(b)m(y)e(use)g(of)h(the)g(group)f 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FQ(men)m(ts\).)43 b(The)30 b(last)h(statemen)m(t)h(m)m(ust)f -(also)g(b)s(e)f(the)h(command)f FJ(RETURN)g FQ(follo)m(w)m(ed)g(b)m(y) -664 668 y(the)38 b(v)-5 b(ariable)37 b(or)h(expression)f(whose)h(v)-5 -b(alue)37 b(is)g(to)i(b)s(e)e(the)i(v)-5 b(alue)37 b(returned)g(b)m(y)h -(the)664 781 y(pro)s(cedure.)57 b(If)36 b(the)g FJ(RETURN)f -FQ(is)g(omitted)h(\(or)h(nothing)e(is)g(written)g(after)i(the)f(w)m -(ord)664 894 y FJ(RETURN)p FQ(\))e(the)i(pro)s(cedure)e(will)f(ha)m(v)m -(e)k(no)e(v)-5 b(alue)35 b(or)h(the)f(v)-5 b(alue)35 -b(zero,)j(dep)s(ending)33 b(on)664 1007 y(ho)m(w)d(it)f(is)g(used)g -(\(and)g FJ(NIL)g FQ(in)f(sym)m(b)s(olic)g(mo)s(de\).)41 -b(Remem)m(b)s(er)29 b(to)h(put)f(a)h(terminator)664 1120 -y(after)h(the)g FJ(END)p FQ(.)664 1282 y FD(Example:)664 -1432 y FQ(Giv)m(en)45 b(a)f(previously)f(assigned)g(in)m(teger)i(v)-5 -b(alue)44 b(for)g FJ(N)p FQ(,)h(the)g(follo)m(wing)e(blo)s(c)m(k)h -(will)664 1532 y(compute)31 b(the)f(Legendre)h(p)s(olynomial)d(of)i -(degree)h FJ(N)f FQ(in)f(the)i(v)-5 b(ariable)29 b FJ(X)p -FQ(:)1013 1737 y FH(begin)41 b(scalar)g(seed,deriv,top,fa)o(ct;)1144 -1837 y(seed:=1/\(y^2)d(-)43 b(2*x*y)f(+1\)^\(1/2\);)1144 -1937 y(deriv:=df\(seed,)o(y,n)o(\);)1144 2036 y(top:=sub\(y=0,de)o(riv) -o(\);)1144 2136 y(fact:=for)d(i:=1:n)j(product)e(i;)1144 -2236 y(return)h(top/fact)1013 2335 y(end;)664 2474 y -@beginspecial @setspecial @endspecial 154 x FR(5.7.1)113 -b(Comp)s(ound)38 b(Statemen)m(ts)f(with)f(GO)h(TO)664 -2850 y FQ(It)30 b(is)e(p)s(ossible)f(to)j(ha)m(v)m(e)h(more)f -(complicated)f(structures)f(inside)g(the)h FJ(BEGIN)47 -b(...END)664 2963 y FQ(brac)m(k)m(ets)34 b(than)e(indicated)f(in)f(the) -j(previous)e(example.)46 b(That)32 b(the)g(individual)c(lines)664 -3076 y(of)41 b(the)h(program)e(need)h(not)g(b)s(e)g(assignmen)m(t)f -(statemen)m(ts,)46 b(but)40 b(could)g(b)s(e)h(almost)664 -3189 y(an)m(y)30 b(other)g(kind)e(of)i(statemen)m(t)i(or)d(command,)h -(needs)f(no)h(explanation.)40 b(F)-8 b(or)30 b(exam-)664 -3302 y(ple,)h(conditional)e(statemen)m(ts,)k(and)e FJ(WHILE)e -FQ(and)i FJ(REPEAT)60 b FQ(constructions,)31 b(ha)m(v)m(e)h(an)664 -3414 y(ob)m(vious)e(role)g(in)f(de\014ning)f(more)j(in)m(tricate)f(blo) -s(c)m(ks.)664 3577 y(If)35 b(these)h(structured)e(constructs)h(don't)g -(su\016ce,)i(it)d(is)h(p)s(ossible)d(to)k(use)f(lab)s(els)f(and)664 -3690 y FJ(GO)40 b(TO)p FQ(s)g(within)e(a)j(comp)s(ound)e(statemen)m(t,) -45 b(and)40 b(also)h(to)g(use)f FJ(RETURN)80 b FQ(in)39 -b(places)664 3803 y(within)31 b(the)j(blo)s(c)m(k)f(other)h(than)f -(just)g(b)s(efore)g(the)h FJ(END)p FQ(.)f(The)g(follo)m(wing)f -(subsections)664 3916 y(discuss)38 b(these)h(matters)h(in)e(detail.)67 -b(F)-8 b(or)40 b(man)m(y)g(readers)f(the)g(follo)m(wing)f(example,)664 -4029 y(presen)m(ting)29 b(one)h(p)s(ossible)c(de\014nition)i(of)h(a)h -(pro)s(cess)f(to)h(calculate)g(the)f(factorial)h(of)f -FJ(N)664 4142 y FQ(for)h(preassigned)f FJ(N)h FQ(will)e(su\016ce:)664 -4291 y FD(Example:)1013 4497 y FH(begin)41 b(scalar)g(m;)1187 -4597 y(m:=1;)1056 4696 y(l:)i(if)g(n=0)f(then)g(return)f(m;)1187 -4796 y(m:=m*n;)1187 4896 y(n:=n-1;)p eop -%%Page: 73 73 -73 72 bop 241 299 a @beginspecial @setspecial @endspecial -FM(5.7.)72 b(COMPOUND)30 b(ST)-8 b(A)g(TEMENTS)1471 b -FQ(73)764 555 y FH(go)43 b(to)f(l)589 655 y(end;)241 -794 y @beginspecial @setspecial @endspecial 154 x FR(5.7.2)112 -b(Lab)s(els)38 b(and)h(GO)e(TO)h(Statemen)m(ts)241 1170 -y FQ(Within)c(a)j FJ(BEGIN)46 b(...END)34 b FQ(comp)s(ound)h(statemen)m -(t)j(it)e(is)f(p)s(ossible)f(to)i(lab)s(el)f(state-)241 -1282 y(men)m(ts,)e(and)e(transfer)g(to)i(them)f(out)g(of)g(sequence)g -(using)f FJ(GO)g(TO)g FQ(statemen)m(ts.)47 b(Only)241 -1395 y(statemen)m(ts)28 b(on)f(the)h(top)f(lev)m(el)f(inside)f(comp)s -(ound)h(statemen)m(ts)i(can)g(b)s(e)e(lab)s(eled,)g(not)241 -1508 y(ones)k(inside)e(subsidiary)f(constructions)i(lik)m(e)h -FL(<<)f FQ(.)16 b(.)f(.)h FL(>>)p FQ(,)30 b FJ(IF)f FQ(.)15 -b(.)h(.)f FJ(THEN)29 b FQ(.)16 b(.)f(.)h(,)30 b FJ(WHILE)241 -1621 y FQ(.)15 b(.)h(.)f FJ(DO)30 b FQ(.)15 b(.)h(.)f(,)31 -b(etc.)241 1771 y(Lab)s(els)e(and)h FJ(GO)47 b(TO)30 -b FQ(statemen)m(ts)i(ha)m(v)m(e)f(the)g(syn)m(tax:)589 -1976 y FH()c(::=)j(GO)h(TO)g(>; 46 -max(-5,-10,-a); -5 - -\endsection -\item[MCD] -MCD (pages 123, 125, 126) - -When MCD is on, sums and differences of rational expressions are put -on a common denominator. Default is ON. - -Examples: 5*A + B*X + B -a/(x+1) + b/5; --------------- - 5*(X + 1) -off mcd; - -1 -a/(x+1) + b/5; (X + 1) *A + 1/5*B - -1/6 + 1/7; 13/42 - -Even with MCD off, rational expressions involving only numbers are -still put over a common denominator. - -Turning MCD off is useful when explicit negative powers are needed, or -if no greatest common divisor calculations are desired, or when -differentiating complicated rational expressions. Results when MCD is -off are no longer in canonical form, and expressions equivalent to -zero may not simplify to 0. Some operations, such as factoring cannot -be done while MCD is off. This option should therefore be used with -some caution. Turning MCD off is most valuable in intermediate parts -of a complicated calculation, and should be turned back on for the -last stage. - -\endsection -\xitem[MEIJERG] -Meijer's G function (page 187) - -The MEIJERG operator provides simplifications for Meijer's G -function. The simplifications are performed towards polynomials, -elementary or -special functions or (generalised) HYPERGEOMETRIC functions. - -The MEIJERG operator is included in the package specfn2. - -MEIJERG(list of parameters,list of parameters,argument) -The first element of the lists has to be the list containing the -first group (mostly called "m" and "n") of parameters. This passes -the four parameters of a Meijer's G function implicitly via the -length of the lists. - -Examples: -load specfn2; -MeijerG({{},1},{{0}},x); & heaviside(-x+1) -MeijerG({{}},{{1+1/4},1-1/4},(x^2)/4) * sqrt pi; - & \rfrac{sqrt(2)*sin(x)*x^2}{4*sqrt(x)} - -Many well-known functions can be written as G functions, -e.g. exponentials, logarithms, trigonometric functions, Bessel functions -and hypergeometric functions. -The formulae can be found e.g. in -A.P.Prudnikov, Yu.A.Brychkov, O.I.Marichev: -Integrals and Series, Volume 3: More special functions, -Gordon and Breach Science Publishers (1990). - -\endsection -\xitem[METRIC command] -METRIC command (page 271) - -\endsection -\xitem[metric structure] -metric structure (page 262) - -\endsection -\item[MIN] -MIN (page 73) - -The operator MIN is an n-ary prefix operator, which returns the -smallest value in its arguments. - - MIN(expression{,expression}) - -expression must evaluate to a number. MIN of an empty list -returns 0. - -Examples: -min(-3,0,17,2); -3 -<>; 16 -min(5,10,a); 5 - -\endsection -\xitem[Minimum] -Minimum (page 182) - -\endsection -\item[MKID] -MKID (page 83) - -The MKID command constructs an identifier, given a stem and an identifier -or an integer. - - MKID(stem,leaf) - -stem can be any valid REDUCE identifier that does not include escaped -special characters. leaf may be an integer, including one given by a -local variable in a FOR loop, or any other legal group of characters. - -Examples: -mkid(x,3); X3 -factorize(x^15 - 1); {X - 1, - - 2 - X + X + 1, - - 4 3 2 - X + X + X + X + 1, - - 8 7 5 4 3 - X - X + X - X + X - X + 1} - -for i := 1:length ws do write set(mkid(f,i),part(ws,i)); - X - 1 - - 2 - X + X + 1 - - 4 3 2 - X + X + X + X + 1 - - 8 7 5 4 3 - X - X + X - X + X - X + 1 - -You can use MKID to construct identifiers from inside procedures. This -allows you to handle an unknown number of factors, or deal with variable -amounts of data. It is particularly helpful to attach identifiers to the -answers returned by FACTORIZE and SOLVE. - -\endsection -\item[MKPOLY] -MKPOLY (page 370) - -Given a roots list as returned by ROOTS, the operator MKPOLY -constructs a polynomial which has these numbers as roots. - - MKPOLY rl - -where rl is a LIST with equations, which all have the same KERNEL on -their left-hand sides and numbers as right-hand sides. - -Examples: - 4 3 2 - mkpoly{x=1,x=-2,x=i,x=-i}; X + X - X + X - 2 - - -Note that this polynomial is unique only up to a numeric factor. - -\endsection -\xitem[MM] -MM (page 379) - -\endsection -\xitem[Mode] -Mode (page 68) - -\endsection -\xitem[Mode communication] -Mode communication (page 197) - -\endsection -\item[MODULAR] -MODULAR (page 134) - -When MODULAR is on, polynomial coefficients are reduced by the -modulus set by SETMOD. If no modulus has been set, MODULAR -has no effect. - -Examples: -setmod 2; 1 -on modular; - 2 2 -(x+y)**2; X + Y - 2 -145*x**2 + 20*x**3 + 17 + 15*x*y; X + X*Y + 1 - -Modular operations are only conducted on the coefficients, not the -exponents. The modulus is not restricted to being prime. When the -modulus is prime, division by a number not relatively prime to the -modulus results in a Zero divisor error message. When the modulus is -a composite number, division by a power of the modulus results in an -error message, but division by an integer which is a factor of the -modulus does not. The representation of modular number can be -influenced by BALANCED_MOD. - -\endsection -\xitem[Modular coefficient] -Modular coefficient (page 134) - -\endsection -\item[MSG] -MSG (page 218) - -When MSG is off, the printing of warning messages is suppressed. Error -messages are still printed. - -Warning messages include those about redimensioning an ARRAY or declaring -an OPERATOR where one is expected. - -\endsection -\item[MSHELL] -MSHELL (page 210) - -The MSHELL command puts particles on the mass shell in high-energy -physics calculations. - MSHELL vector-var{,vector-var} - -vector-var must have had a mass attached to it by a MASS -declaration. - -Examples: -vector v1,v2; -mass v1=m,v2=q; -mshell v1; - 2 -v1.v1; M -v2.v2; V2.V2 -mshell v2; - 2 2 -v1.v1*v2.v2; M *Q - -Even though a mass is attached to a vector variable representing a -particle, the replacement does not take place until the MSHELL -declaration is given for that vector variable. - -\endsection -\xitem[Multiple assignment statement] -Multiple assignment statement (page 54) - -\endsection -\item[MULTIPLICITIES] -MULTIPLICITIES (page 86) - -When MSG is off, the printing of warning messages is suppressed. Error -messages are still printed. - -Warning messages include those about redimensioning an ARRAY or declaring -an OPERATOR where one is expected. - -\endsection -\xitem[MULTIROOT] -MULTIROOT (page 373) - -\endsection -\item[NAT] -NAT (page 111, 259) - -When NAT is on, output is printed to the screen in natural form, with -raised exponents. NAT should be turned off when outputting expressions -to a file for future input. Default is ON. - -Examples: 3 2 2 3 -(x + y)**3; X + 3*X *Y + 3*X*Y + Y -off nat; -(x + y)**3; X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3$ -on fort; -(x + y)**3; ANS=X**3+3.*X**2*Y+3.*X*Y**2+Y**3 - -With NAT off, a dollar sign is printed at the end of each expression. -An output file written with NAT off is ready to be read into REDUCE -using the command IN. - -\endsection -\item[NEARESTROOT] -NEARESTROOT (pages 370, 372) - -The operator NEARESTROOT finds one root of a polynomial with an -iteration using a given starting point. - - NEARESTROOT(p,pt) - -where p is a univariate polynomial and pt is a number. - -Example: - - nearestroot(x^2+2,2); {X=1.41421*I} - -The minimal accuracy of the result values is controlled by ROOTACC. - -\endsection -\xitem[NEARESTROOTS] -NEARESTROOTS (page 370) - -\endsection -\xitem[NEGATIVE] -NEGATIVE (page 368) - -\endsection -\item[NERO] -NERO (page 108) - -When NERO is on, zero assignments (such as matrix elements) are not -printed. - -Examples: -matrix a; -a := mat((1,0),(0,1)); A(1,1) := 1 - A(1,2) := 0 - A(2,1) := 0 - A(2,2) := 1 -on nero; -a; MAT(1,1) := 1 - MAT(2,2) := 1 -a(1,2); {nothing is printed.} -b := 0; {nothing is printed.} -off nero; -b := 0; B := 0 - -NERO is often used when dealing with large sparse matrices, to avoid -being overloaded with zero assignments. - -\endsection -\xitem[Newton's method] -Newton's method (page 182) - -\endsection -\item[NEXTPRIME] -NEXTPRIME (page 74) - - NEXTPRIME(expression) - -If the argument of NEXTPRIME is an integer, the least prime greater -than that argument is returned. Otherwise, a type error results. - -Examples: -nextprime 5001; 5003 -nextprime(10^30); 1000000000000000000000000000057 -nextprime a; ***** A invalid as integer - -\endsection -\xitem[NN] -NN (page 379) - -\endsection -\item[NOARG] -NOARG - -When DFPRINT is on, expressions in the differentiation operator -DF are printed in a more ``natural'' notation, with the -differentiation variables appearing as subscripts. When NOARG -is on (the default), the arguments of the differentiated operator are also -suppressed. - -Examples: -operator f; -df(f x,x); DF(F(X),X); -on dfprint; -ws; F - X -df(f(x,y),x,y); F - X,Y -off noarg; -ws; F(X) - X - -\endsection -\item[NODEPEND] -NODEPEND (page 95) - -The NODEPEND declaration removes the dependency declared with DEPEND. - - NODEPEND dep-kernel{,kernel} - -dep-kernel -must be a kernel that has had a dependency declared upon -the one or more other kernels that are its other arguments. - -Examples: -depend y,x,z; -df(sin y,x); COS(Y)*DF(Y,X) -df(sin y,x,z); COS(Y)*DF(Y,X,Z) - DF(Y,X)*DF(Y,Z)*SIN(Y) -nodepend y,z; -df(sin y,x); COS(Y)*DF(Y,X) -df(sin y,x,z); 0 - -A warning message is printed if the dependency had not been declared by -DEPEND. - -\endsection -\xitem[NOETHER function] -NOETHER function (pages 258, 271) - -\endsection -\xitem[Non-commuting operator] -Non-commuting operator (page 92) - -\endsection -\item[NOLNR] -NOLNR - -When NOLNR is on, the linear properties of the integration operator -INT are suppressed if the integral cannot be found in closed terms. - - -REDUCE uses the linear properties of integration to attempt to break down -an integral into manageable pieces. If an integral cannot be found in -closed terms, these pieces are returned. When the NOLNR switch is off, -as many of the pieces as possible are integrated. When it is on, if any piece -fails, the rest of them remain unevaluated. - -\endsection -\item[NONCOM] -NONCOM (page 92) - -NONCOM declares that already-declared operators are noncommutative -under multiplication. - - NONCOM operator{,operator} - -operator must have been declared an OPERATOR, or a warning message is -given. - -Examples: -operator f,h; -noncom f; -f(a)*f(b) - f(b)*f(a); F(A)*F(B) - F(B)*F(A) -h(a)*h(b) - h(b)*h(a); 0 -operator comm; -for all x,y such that x neq y and ordp(x,y) - let f(x)*f(y) = f(y)*f(x) + comm(x,y); -f(1)*f(2); F(1)*F(2) -f(2)*f(1); COMM(2,1) + F(1)*F(2) - -The last example introduces the commutator of f(x) and f(y) for all x -and y. The equality check is to prevent an infinite loop. The -operator f can have other functionality attached to it if desired, or -it can remain an indeterminate operator. - -\endsection -\item[NONZERO] -NONZERO (page 90) - - NONZERO identifier{,identifier} - -If an operator F is declared ODD, then F(0) is replaced by zero unless -F is also declared non zero by the declaration NONZERO. - -Examples: - odd f; - f(0) 0 - nonzero f; - f(0) F(0) - -\endsection -\item[NOSPLIT] -NOSPLIT (page 103) - -Under normal circumstances, the printing routines try to break an expression -across lines at a natural point. This is a fairly expensive process. If -you are not overly concerned about where the end-of-line breaks come, you -can speed up the printing of expressions by turning off the switch -NOSPLIT. This switch is normally on. - -\endsection -\item[NOSPUR] -NOSPUR (page 210) - -The NOSPUR declaration prevents the trace calculation over the given -line identifiers in high-energy physics calculations. - NOSPUR line-id{,line-id} - - -line-id is a scalar identifier that will be used as a line identifier. - -Examples: -vector a1,b1,c1; -g(line1,a1,b1)*g(line2,b1,c1); A1.B1*B1.C1 -nospur line2; -g(line1,a1,b1)*g(line2,b1,c1); A1.B1*G(LINE2,B1,C1) - -Nospur declarations can be removed by making the declaration SPUR. - -\endsection -\xitem[NOSUM command] -NOSUM command (pages 262, 271) - -\endsection -\xitem[NOSUM switch] -NOSUM switch (page 262) - -\endsection -\item[NOXPND @] -NOXPND @ (pages 254, 271) -NOXPND D (pages 253, 271) - -(Part of the EXCALC package) - -There are two forms of the NOXPND command, which controls the use of -the product rule for the d operator and the expansion into partial -derivatives. The default for both these is OFF. - - noxpnd d; - noxpnd @; - -Example: - load_package excalc; *** ^ redefined - (excalc) - pform x=0,y=k,z=m; - K - d(y^z); ( - 1) *Y^d Z + d Y^Z - - noxpnd d; - d(y^z); d(Y^Z) - - -See also XPND -\endsection -\xitem[NS dummy variable] -NS dummy variable (page 260) - -\endsection -\item[NULLSPACE] -NULLSPACE (page 166) - -NULLSPACE(matrix_expression) - -nullspace calculates for its MATRIX argument, A, a list of linear -independent vectors (a basis) whose linear combinations satisfy the -equation A x = 0. The basis is provided in a form such that as many -upper components as possible are isolated. - -Examples: -nullspace mat((1,2,3,4),(5,6,7,8)); { - [ 1 ] - [ ] - [ 0 ] - [ ] - [ - 3] - [ ] - [ 2 ] - , - [ 0 ] - [ ] - [ 1 ] - [ ] - [ - 2] - [ ] - [ 1 ] - } - -Note that with B := NULLSPACE A, the expression LENGTH B is the -nullity of A, and that SECOND LENGTH A - LENGTH B calculates the rank -of A. The rank of a matrix expression can also be found more directly -by the RANK operator. - -In addition to the REDUCE matrix form, NULLSPACE accepts as input a -matrix given as a LIST of lists, that is interpreted as a row matrix. If -that form of input is chosen, the vectors in the result will be -represented by lists as well. This additional input syntax facilitates -the use of NULLSPACE in applications different from classical linear -algebra. - -\endsection -\item[NUM] -NUM (page 131) -The NUM operator returns the numerator of its argument. - - NUM(expression) or NUM simple_expression - -expression can be any valid REDUCE scalar expression. - -Examples: -num(100/6); 50 -num(a/5 + b/6); 6*A + 5*B -num(sin(x)); SIN(X) - -NUM returns the numerator of the expression after it has been simplified -by REDUCE. As seen in the examples, this includes putting sums of rational -expressions over a common denominator, and reducing common factors where -possible. If the expression is not a rational expression, it is returned -unchanged. - -\endsection -\item[NUMVAL] -NUMVAL - -With ROUNDED on, elementary functions with numerical arguments -will return a numerical answer where appropriate. If you wish to inhibit -this evaluation, NUMVAL should be turned off. It is normally on. - -Examples: - on rounded; - cos 3.4; - 0.966798192579 - off numval; - cos 3.4; COS(3.4) - -\endsection -\item[NUM_INT] -NUM_INT (page 182) - -For the numerical evaluation of univariate integrals over a finite -interval the following strategy is used: If INT finds a formal -antiderivative which is bounded in the integration interval, this is -evaluated and the end points and the difference is returned. -Otherwise a Chebyshev fit is computed, starting with order 20, -eventually up to order 80. If that is recognized as sufficiently -convergent it is used for computing the integral by directly -integrating the coefficient sequence. If none of these methods is -successful, an adaptive multilevel quadrature algorithm is used. - -For multivariate integrals only the adaptive quadrature is used. This -algorithm tolerates isolated singularities. The value ITERATIONS here -limits the number of local interval intersection levels. a is a -measure for the relative total discretization error (comparison of -order 1 and order 2 approximations). - -NUM_INT(exp,var=(l .. u) [,var=(l .. u),...] [,accuracy=a][,iterations=i]) - -where exp is the function to be integrated, var are the integration -variables, l are the lower bounds, u are the upper bounds. - -Result is the value of the integral. - -Example: - on rounded; - num_int(sin x,x=(0 .. pi)); 2.0 - -\endsection -\item[NUM_MIN] -NUM_MIN (page 182) - -The Fletcher Reeves version of the STEEPEST_DESCENT algorithms is used -to find the minimum of a function of one or more variables. The -function must have continuous partial derivatives with respect to all -variables. The starting point of the search can be specified; if not, -random values are taken instead. The steepest descent algorithms in -general find only local minima. - -Syntax: - - NUM_MIN(exp, var[=val] [,var[=val] ... [,accuracy=a] [,iterations=i]) -NUM_MIN(exp, {var[=val] [,var[=val} ...] } [,accuracy=a] [,iterations=i]) - -where exp is a function expression, var are the variables in exp and -val are the (optional) start values. For a and i see NUMERIC_ACCURACY. - -NUM_MIN tries to find the next local minimum along the descending path -starting at the given point. The result is a LIST with the minimum -function value as first element followed by a list of equations, where -the variables are equated to the coordinates of the result point. - -Examples: - load numeric; - num_min(sin(x)+x/5, x); { - 0.0775892231689,{x=4.51200216375}} - num_min(sin(x)+x/5, x=0); { - 1.33416631212,{x= - 1.78326532423}} - -\endsection -\item[NUM_ODESOLVE] -NUM_ODESOLVE (page 182) - -The Runge-Kutta method of order 3 finds an approximate graph for the -solution of real ODE initial value problem. - -NUM_ODESOLVE(exp,depvar=start, indep=(from .. to) [,accuracy=a][,iterations=i]) -NUM_ODESOLVE({exp,exp,...},{depvar=start,depvar=start,...} indep=(from .. to) - [,accuracy=a][,iterations=i]) - -where depvar and start specify the dependent variable(s) and the -starting point value (vector), indep, from and to specify the -independent variable and the integration interval (starting point and -end point), exp are equations or expressions which contain the first -derivative of the independent variable with respect to the dependent -variable. - -The ODEs are converted to an explicit form, which then is used for a -Runge Kutta iteration over the given range. The number of steps is -controlled by the value of i (default: 20). If the steps are too -coarse to reach the desired accuracy in the neighborhood of the -starting point, the number is increased automatically. - -Result is a list of pairs, each representing a point of the -approximate solution of the ODE problem. - -Example: -num_odesolve(df(y,x)=y,y=1,x=(0 .. 1), iterations=5); - - {{0.0,1.0},{0.2,1.2214},{0.4,1.49181796},{0.6,1.8221064563}, - {0.8,2.2255208258},{1.0,2.7182511366}} - -If in exp the differential is not isolated on the left-hand side, -please ensure that the dependent variable is explicitly declared using -a DEPEND otherwise the formal derivative will be computed to zero by -REDUCE. - -The operator SOLVE is used to convert the form into an explicit -ODE. If that process fails or has no unique result, the evaluation is -stopped with an error message. - -\endsection -\item[NUM_SOLVE] -NUM_SOLVE (page 182) - -An adaptively damped Newton iteration is used to find an approximative -root of a function (function vector) or the solution of an EQUATION -(equation system). The expressions must have continuous derivatives -for all variables. A starting point for the iteration can be -given. If not given random values are taken instead. When the number -of forms is not equal to the number of variables, the Newton method -cannot be applied. Then the minimum of the sum of absolute squares is -located instead. - -With COMPLEX on, solutions with imaginary parts can be found, if -either the expression(s) or the starting point contain a nonzero -imaginary part. - - NUM_SOLVE(exp, var[=val][,accuracy=a][,iterations=i]) - NUM_SOLVE({exp,...,exp}, var[=val],...,var[=val] [,accuracy=a][,iterations=i]) - NUM_SOLVE({exp,...,exp}, {var[=val],...,var[=val]} - [,accuracy=a][,iterations=i]) - -where exp are function expressions, - var are the variables, - val are optional start values. -For a and i see NUMERIC_ACCURACY. - -NUM_SOLVE tries to find a zero/solution of the expression(s). Result -is a list of equations, where the variables are equated to the -coordinates of the result point. - -The Jacobian matrix is stored as side effect the shared jacobian. - -Examples: -num_solve({sin x=cos y, x + y = 1},{x=1,y=2}); - {X= - 52.1216769476,Y=53.1216769476} - [COS(X) SIN(Y)] -jacobian; [ ] - [ 1 1 ] -\endsection -\xitem[Number] -Number (pages 34, 35) - -\endsection -\item[NUMBERP] -NUMBERP (page 46) -The NUMBERP operator returns TRUE if its argument is a number, -and NIL otherwise. - - NUMBERP(expression) or NUMBERP expression - -expression can be any REDUCE scalar expression. - -Examples: -cc := 15.3; CC := 15.3 -if numberp(cc) then write "number" else write "nonnumber"; number -if numberp(cb) then write "number" else write "nonnumber"; nonnumber - -Logical operators can only be used in conditional expressions, such as -IF...THEN...ELSE and WHILE...DO. - -\endsection -\item[NUMERIC package] -NUMERIC package (page 337) - -The numeric package supplies algorithms based on approximation -techniques of numerical mathematics. The algorithms use the ROUNDED -mode arithmetic of REDUCE, including the variable precision feature -which is exploited in some algorithms in an adaptive manner in order -to reach the desired accuracy. - -\endsection -\xitem[Numerical operator] -Numerical operator (page 71) - -\endsection -\xitem[Numerical precision] -Numerical precision (page 36) - -\endsection -\item[ODD] -ODD (page 90) - - ODD identifier{,identifier} - -This declaration is used to declare an operator odd in its first -argument. Expressions involving an operator declared in this manner -are transformed if the first argument contains a minus sign. Any -other arguments are not affected. - -Examples: - odd f; - f(-a) -F(A) - f(-a,-b) -F(A,-B) - f(a,-b) F(A,-B) - -If say F is declared odd, then F(0) is replaced by zero unless F is -also declared non zero by the declaration NONZERO. - -\endsection -\xitem[ODEDEGREE] -ODEDEGREE (page 350) - -\endsection -\xitem[ODELINEARITY] -ODELINEARITY (page 350) - -\endsection -\xitem[ODEORDER] -ODEORDER (page 350) - -\endsection -\item[ODESOLVE] -ODESOLVE (pages 183, 349) - -Main Author: Malcolm A.H. MacCallum -Other contributors: Francis Wright, Alan Barnes - -Ordinary Differential Equations Solver. - -The ODESOLVE package is a solver for ordinary differential -equations. At the present time it has very limited capabilities. -It can handle only a single scalar equation presented as an -algebraic expression or equation, and it can solve only first- -order equations of simple types, linear equations with constant -coefficients and Euler equations. These solvable types are exactly -those for which Lie symmetry techniques give no useful information. - -For example, the evaluation of - depend(y,x); - odesolve(df(y,x)=x**2+e**x,y,x); -yields the result - X 3 - 3*E + 3*ARBCONST(1) + X - {Y=---------------------------} - 3 - -\endsection -\item[OFF] -OFF (pages 68, 69) - -The OFF command is used to turn switches off. - - OFF switch{,switch} - -switch can be any SWITCH name. There is no problem if the switch is -already off. If the switch name is mistyped, an error message is -given. - -\endsection -\item[ON] -ON (pages 68, 69) - -The ON command is used to turn switches on. - - ON switch{,switch} - -switch can be any SWITCH name. There is no problem if the switch is -already on. If the switch name is mistyped, an error message is -given. - -\endsection -\item[ONE_OF] -ONE_OF (page 86) -The operator ONE_OF is used to represent an indefinite choice -of one element from a finite set of objects. - -Example: - x=one_of{1,2,5} - -This equation encodes that x can take one of the values 1,2 or 5 - -REDUCE generates a ONE_OF form in cases when an implicit ROOT_OF -expression could be converted to an explicit solution set. A ONE_OF -form can be converted to a SOLVE solution using EXPAND_CASES. See -ROOT_OF. - -\endsection -\item[OPERATOR] -OPERATOR (page 202) - -Use the OPERATOR declaration to declare your own operators. - - OPERATOR identifier{,identifier} - -identifier can be any valid REDUCE identifier, which is not the name -of a MATRIX, ARRAY, scalar variable or previously-defined operator. - -Examples: -operator dis,fac; -let dis(~x,~y) = sqrt(x^2 + y^2); -dis(1,2); SQRT(5) - 2 -dis(a,10); SQRT(A + 100) -on rounded; -dis(1.5,7.2); 7.35459040329 -let fac(~n) = - if n=0 then 1 - else if not(fixp n and n>0) - then rederr "choose non-negative integer" - else for i := 1:n product i; - -fac(5); 120 -fac(-2); ***** choose non-negative integer - -The first operator is the Euclidean distance metric, the distance of -point (x,y) from the origin. The second operator is the factorial. - -Operators can have various properties assigned to them; they can be -declared INFIX, LINEAR, SYMMETRIC, ANTISYMMETRIC, or NONCOMmutative. -The default operator is prefix, nonlinear, and commutative. -Precedence can also be assigned to operators using the declaration -PRECEDENCE. - -Functionality is assigned to an operator by a LET statement or a -FORALL...LET statement, (or possibly by a procedure with the name of -the operator). Be careful not to redefine a system operator by -accident. REDUCE permits you to redefine system operators, giving you -a warning message that the operator was already defined. This -flexibility allows you to add mathematical rules that do what you want -them to do, but can produce odd or erroneous behaviour if you are not -careful. - -You can declare operators from inside PROCEDUREs, as long as they are -not local variables. Operators defined inside procedures are global. -A formal parameter may be declared as an operator, and has the effect -of declaring the calling variable as the operator. - -\endsection -\xitem[Operator precedence] -Operator precedence (page 39, 41) - -\endsection -\item[ORDER] -ORDER (pages 101, 114) - -The ORDER declaration changes the order of precedence of kernels for -display purposes only. - - ORDER identifier{,identifier} - -kernel must be a valid KERNEL or OPERATOR name complete with argument. - -Examples: -x + y + z + cos(a); COS(A) + X + Y + Z -order z,y,x,cos(a); -x + y + z + cos(a); Z + Y + X + COS(A) - 2 2 -(x + y)**2; Y + 2*Y*X + X -order nil; - 2 2 -(z + cos(z))**2; COS(Z) + 2*COS(Z)*Z + Z - -ORDER affects the printing order of the identifiers only; internal -order is unchanged. Change internal order of evaluation with the -declaration KORDER. You can use ORDER to feature variables or -functions you are particularly interested in. - -Declarations made with ORDER are cumulative: kernels in new order -declarations are ordered behind those in previous declarations, and -previous declarations retain their relative order. Of course, -specific kernels named in new declarations are removed from previous -ones and given the new priority. Return to the standard canonical -printing order with the statement ORDER NIL. - -The print order specified by ORDER commands is not in effect if the -switch PRI is off. - -\endsection -\xitem[ordering exterior form] -ordering - exterior form (page 268) - -\endsection -\xitem[ordinary differential equations] -ordinary differential equations (page 349) - -\endsection -\item[ORDP] -ORDP (pages 46, 92) - -The ORDP logical operator returns TRUE if its first argument is -ordered ahead of its second argument in canonical internal ordering, -or is identical to it. - - ORDP(expression1,expression2) - -expression1 and expression2 can be any valid REDUCE scalar expression. - -Examples: -if ordp(x**2 + 1,x**3 + 3) then write "yes" else write "no"; no -if ordp(101,100) then write "yes" else write "no"; yes -if ordp(x,x) then write "yes" else write "no"; yes - -Logical operators can only be used in conditional expressions, such as -IF...THEN...ELSE and WHILE...DO. - -\endsection -\item[ORTHOVEC] -ORTHOVEC (pages 184, 353) - -Author: James W. Eastwood - -A Package for the Manipulation of Scalars and Vectors. - -ORTHOVEC is a collection of REDUCE procedures and operations which -provide a simple-to-use environment for the manipulation of scalars -and vectors. Operations include addition, subtraction, dot and -cross products, division, modulus, div, grad, curl, laplacian, -differentiation, integration, and Taylor expansion. - -\endsection -\item[OUT] -OUT (pages 153, 154) - -The OUT command directs output to the filename that is its argument, -until another OUT changes the output file, or SHUT closes it. - OUT filename or OUT "pathname " or OUT T - -filename must be in the current directory, or be a valid complete -file description for your system. If the file name is not -in the current directory, quote marks are needed around the file name. -If the file already exists, a message is printed allowing you to decide -whether to supersede the contents of the file with new material. - - -To restore output to the terminal, type OUT T, or SHUT the -file. When you use OUT T, the file remains available, and if you -open it again (with another OUT), new material is appended rather -than overwriting. - -To write a file using OUT that can be input at a later time, the -switch NAT must be turned off, so that the standard linear form -is saved that can be read in by IN. If NAT is on, exponents -are printed on the line above the expression, which causes trouble -when REDUCE tries to read the file. - -There is a slight complication if you are using the OUT command from -inside a file to create another file. The ECHO switch is normally -off at the top-level and on while reading files (so you can see what is -being read in). If you create a file using OUT at the top-level, -the result lines are printed into the file as you want them. But if you -create such a file from inside a file, the ECHO switch is on, and -every line is echoed, first as you typed it, then as REDUCE parsed it, and -then once more for the file. Therefore, when you create a file from -a file, you need to turn ECHO off explicitly before the OUT -command, and turn it back on when you SHUT the created file, so your -executing file echoes as it should. This behaviour also means that as you -watch the file execute, you cannot see the lines that are being put into -the OUT file. As soon as you turn ECHO on, you can see -output again. - -\endsection -\item[OUTPUT] -OUTPUT (page 100) - -When OUTPUT is off, no output is printed from any REDUCE calculation. -The calculations have their usual effects other than printing. Default is -ON. - - -Turn output OFF if you do not wish to see output when executing -large files, or to save the time REDUCE spends formatting large expressions -for display. Results are still available with WS, or in their -assigned variables. - -\endsection -\xitem[Output] -Output (pages 105, 110) - -\endsection -\xitem[Output declaration] -Output declaration (pages 100, 101) - -\endsection -\item[OVERVIEW] -OVERVIEW - -When OVERVIEW is on, the amount of detail reported by the factoriser -switches TRFAC and TRALLFAC is reduced. - - -\endsection -\item[PART] -PART (pages 49, 113, 116) -The operator PART permits the extraction of various parts or -operators of expressions and LISTS. - - PART(expression,integer{,integer}) - -expression can be any valid REDUCE expression or a list, integer may -be an expression that evaluates to a positive or negative integer or -0. A positive integer n picks up the nth term, counting from the -first term toward the end. A negative integer n picks up the nth -term, counting from the back toward the front. The integer 0 picks up -the operator (which is LIST when the expression is a list). - -Examples: - 2 3 -part((x + y)**5,4); 10*X *Y - - 2 -part((x + y)**5,4,2); X - -part((x + y)**5,4,2,1); X -part((x + y)**5,0); PLUS - 4 -part((x + y)**5,-5); 5*x *y - - 5 4 3 2 4 5 -part((x + y)**5,4) := sin(x); x + 5*x *y + 10*x *y + sin(x) + 5*x*y + y - -alist := {x,y,{aa,bb,cc},x**2*sqrt(y)}; - ALIST := {X, - Y, - {AA,BB,CC}, - 2 - SQRT(Y)*X } -part(alist,3,2); BB -part(alist,4,0); TIMES - -Additional integer arguments after the first one examine the terms -recursively, as shown above. In the third line, the fourth term is -picked from the original polynomial, 10x^2y^3, then the second term -from that, x^2, and finally the first component, x. If an integer's -absolute value is too large for the appropriate expression, a message -is given. - -PART works on the form of the expression as printed, or as it would -have been printed at that point of the calculation, bearing in mind -the current switch settings. It is important to realise that the -switch settings change the operation of PART. PRI must be on when -PART is used. - -When PART is used on a polynomial expression that has minus signs, the -+ is always returned as the top-level operator. The minus is found as -a unary operator attached to the negative term. - -PART can also be used to change the relevant part of the expression or -list as shown in the sixth example line. The PART operator returns the -changed expression, though original expression is not changed. You can -also use PART to change the operator. - -\endsection -\xitem[partial differentiation] -partial differentiation (page 251) - -\endsection -\item[PAUSE] -PAUSE (page 160)) -The PAUSE command, given in an interactive file, stops operation and -asks if you want to continue or not. - -Examples: -An interactive file is running, and at some point you see the -question - Cont? (Y or N) -If you type y {Return} -the file continues to run until the next pause or the end. -If you type n {Return} - -you will get a numbered REDUCE prompt, and be allowed to enter and -execute any REDUCE statements. If you later wish to continue with the -file, type - cont; -and the file resumes. - -To use PAUSE in your own interactive files, type - -PAUSE; - -in the file wherever you want it. - -PAUSE does not allow you to continue without typing either Y or N. -Its use is to slow down scrolling of interactive files, or to let you -change parameters or switch settings for the calculations. - -If you have stopped an interactive file at a PAUSE, and do not wish to -resume the file, type END;. This does not end the REDUCE session, but -stops input from the file. A second END; ends the REDUCE session. -However, if you have pauses from more than one file stacked up, an -END; brings you back to the top level, not the file directly above. - -A PAUSE typed from the terminal has no effect. - -\endsection -\xitem[PCLASS] -PCLASS (pages 379, 380, 383) - -\endsection -\xitem[Percent sign] -Percent sign (page 38) - -\endsection -\item[PERIOD] -PERIOD (page 111) - -When PERIOD is on, periods are added after integers in -Fortran-compatible output (when FORT is on). There is no effect -when FORT is off. Default is ON. - -\endsection -\item[PF] -PF (page 83) - - PF(expression,variable) - -PF transforms expression into a LIST of partial fractions with respect -to the main variable, variable. PF does a complete partial fraction -decomposition, and as the algorithms used are fairly unsophisticated -(factorisation and the extended Euclidean algorithm), the code may be -unacceptably slow in complicated cases. - -Examples: - 2 - 2 2 -pf(2/((x+1)^2*(x+2)),x); {-------,-------,--------------} - x + 2 x + 1 2 - x + 2*x + 1 -off exp; -pf(2/((x+1)^2*(x+2)),x); - 2 - 2 2 -pf(2/((x+1)^2*(x+2)),x); {-------,-------,----------} - x + 2 x + 1 2 - (x + 1) - - 2 -for each j in ws sum j; ------------------ - 2 - (x + 2)*(x + 1) - -If you want the denominators in factored form, turn EXP off, as shown -in the second example above. As shown in the final example, the FOR -EACH construct can be used to recombine the terms. Alternatively, one -can use the operations on lists to extract any desired term. - -\endsection -\xitem[PFORM command] -PFORM command (page 271) - -\endsection -\xitem[PFORM statement] -PFORM statement (page 249) - -\endsection -\item[PI] -PI (page 37) - -The identifier PI is reserved for use as the circular constant. -Its value is given by 3.14159265358..., which REDUCE gives to the current -decimal precision when REDUCE is in a floating-point mode. - - -PI may be used as a looping variable in a FOR statement, -or as a local variable in a PROCEDURE. Its value in such cases will be -taken from the local environment. - -\endsection -\xitem[PLOT] -PLOT (page 181) - -\endsection -\item[POCHHAMMER] -POCHHAMMER (pages 185, 394) - -The POCHHAMMER operator implements the Pochhammer notation -(shifted factorial). - - POCHHAMMER(expression,expression) - -Examples: -load_package specfn; (SPECFN) -pochhammer(17,4); 116280 - - FACTORIAL(2*Z) -pochhammer(1/2,z); ------------------- - 2*Z - 2 *FACTORIAL(Z) - -A number of complex rules for POCHHAMMER are inactive, because they -cause a huge system load in algebraic mode. If one wants to use more -rules for the simplification of Pochhammer's notation, one can do: - let special!*pochhammer!*rules; - -\endsection -\item[POLYGAMMA] -POLYGAMMA (pages 185, 395) - -The POLYGAMMA operator returns the Polygamma function. - - Polygamma(n,x) := df(Psi(z),z,n); - - POLYGAMMA(integer,expression) - -Examples: - load_package specfn; (SPECFN) - PI - 6 - Polygamma(1,2); --------- - 6 - on rounded; - Polygamma(1,2.35); 0.52849689109 - -The POLYGAMMA function is used for simplification of the ZETA function -for some arguments. - -\endsection -\xitem[Polynomial] -Polynomial (page 119) - -\endsection -\xitem[Polynomial equations] -Polynomial equations (page 181) - -\endsection -\xitem[POSITIVE] -POSITIVE (page 368) - -\endsection -\xitem[power series] -power series (page 413) - -\endsection -\xitem[power series arithmetic] -power series - arithmetic (page 422) - composition (page 420) - differentiation (page 422) - of integral (page 415) - of user defined function (page 415) - -\endsection -\item[PRECEDENCE] -PRECEDENCE (page 94) - -The PRECEDENCE declaration attaches a precedence to an infix operator. - - PRECEDENCE operator, known_operator - -operator should have been declared an operator but may be a REDUCE -identifier that is not already an operator, array, or matrix. -known_operator must be a system infix operator or have had its -precedence already declared. - -Examples: -operator f,h; -precedence f,+; -precedence h,*; -a + f(1,2)*c; (1 F 2)*C + A -a + h(1,2)*c; 1 H 2*C + A -a*1 f 2*c; A F 2*C -a*1 h 2*c; 1 H 2*A*C - -The operator whose precedence is being declared is inserted into the -infix operator precedence list at the next higher place than -known-operator. - -Attaching a precedence to an operator has the side effect of declaring -the operator to be infix. If the identifier argument for PRECEDENCE -has not been declared to be an operator, an attempt to use it causes -an error message. After declaring it to be an operator, it becomes an -infix operator with the precedence previously given. Infix operators -may be used in prefix form; if they are used in infix form, a space -must be left on each side of the operator to avoid ambiguity. -Declared infix operators are always binary. - -To see the infix operator precedence list, enter symbolic mode and -type PRECLIS!*;. The lowest precedence operator is listed first. - -All prefix operators have precedence higher than infix operators. - -\endsection -\item[PRECISE] -PRECISE (page 78) - -When the PRECISE switch is on, simplification of roots of even -powers returns absolute values, a more precise answer mathematically. - -Examples: -sqrt(x**2); X -(x**2)**(1/4); SQRT(X) -on precise; -sqrt(x**2); ABS(X) -(x**2)**(1/4); SQRT(ABS(X)) - -In many types of mathematical work, simplification of powers and surds -can proceed by the fastest means of simplifying the exponents -arithmetically. When it is important to you that the positive root be -returned, turn PRECISE on. One situation where this is important is -when graphing square-root expressions such as sqrt(x^2+y^2) to avoid a -spike caused by REDUCE simplifying sqrt(y^2) to y when x is zero. - -\endsection -\item[PRECISION] -PRECISION (pages 132, 374) - -The PRECISION declaration sets the number of decimal places used when -ROUNDED is on. Default is system dependent, and normally about 12. - - PRECISION(integer) or PRECISION integer - -integer must be a positive integer. When integer is 0, the current -precision is displayed, but not changed. There is no upper limit, but -precision of greater than several hundred causes unpleasantly slow -operation on numeric calculations. - -Examples: -on rounded; -7/9; 0.777777777778 -precision 20; 20 -7/9; 0.77777777777777777778 -sin(pi/4); 0.7071067811865475244 - -Trailing zeroes are dropped, so sometimes fewer than 20 decimal places -are printed as in the last example. Turn on the switch FULLPREC if -you want to print all significant digits. The ROUNDED mode carries -calculations to two more places than given by PRECISION, and rounds -off. - -\endsection -\item[PREDUCE] -PREDUCE (page 308) - - PREDUCE(p, {exp, ... }[,vars]) - -where p is an expression, and {exp, ... } is a list of expressions or -equations and vars is an optional list of variables (see IDEAL -parameters). - -PREDUCE computes the remainder of EXP modulo the given set of -polynomials resp. equations. This result is unique (canonical) only -if the given set is a GROEBNER basis under the current TERM order. - -see also: PREDUCET operator. - -\endsection -\item[PREDUCET] -PREDUCET (page 311) - - PREDUCE(p,{v=exp...}[,vars]) - -where p is an expression, v are kernels (simple or indexed variables), -EXP are polynomials and optional vars is a variable list (see IDEAL -parameters). - -PREDUCET computes the remainder of p modulo {exp,...} similar to -PREDUCE, but the result is an equation which expresses the remainder -as combination of the polynomials. - -Example: - - gb2 := {g1=2*x - y + 1,g2=9*y**2 - 2*y - 199}$ - preducet(q=x**2,gb2); - - - 16*Y + 208= - 18*X*G1 - 9*Y*G1 + 36*Q + 9*G1 - G2 - - -\endsection -\xitem[Prefix] -Prefix (pages 71, 93, 95) - -\endsection -\xitem[Prefix operator] -Prefix operator (page 38, 39) - -\endsection -\item[PRET] -PRET (pages 217, 218) - -When PRET is on, input is printed in standard REDUCE format and then -evaluated. - -Examples: -on pret; -(x+1)^3; (x + 1)**3; - 3 2 - X + 3*X + 3*X + 1 - -procedure fac(n); - if not (fixp(n) and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n-1 product i+1; - procedure fac n; - if not (fixp n and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n - 1 product i + 1; - - FAC - -fac 5; fac 5; - 120 - -Note that all input is converted to lower case except strings (which -keep the same case) all operators with a single argument have had the -parentheses removed, and all infix operators have had a space added on -each side. In addition, syntactical constructs like IF...THEN...ELSE -are printed in a standard format. - -\endsection -\xitem[PRETTYPRINT] -PRETTYPRINT (page 218) - -\endsection -\xitem[Prettyprinting] -Prettyprinting (pages 217, 218) - -\endsection -\xitem[PRGEN] -PRGEN (page 378) - -\endsection -\item[PRI] -PRI (page 101) - -When PRI is on, the declarations ORDER and FACTOR can -be used, and the switches ALLFAC, DIV, RAT, -and REVPRI take effect when they are on. Default is ON. - - -Printing of expressions is faster with PRI off. The expressions are -then returned in one standard form, without any of the display options that -can be used to feature or display various parts of the expression. You can -also gain insight into REDUCE's representation of expressions with -PRI off. - -\endsection -\item[PRIMEP] -PRIMEP (page 46) - - PRIMEP(expression) or PRIMEP simple_expression - -If expression evaluates to a integer, PRIMEP returns TRUE if -expression is a prime number and NIL otherwise. If expression does -not have an integer value, a type error occurs. - -Examples: -if primep 3 then write "yes" else write "no"; YES -if primep a then 1; ***** A invalid as integer - -\endsection -\item[PRINT_PRECISION] -PRINT_PRECISION (page 133) - - PRINT_PRECISION(integer) or PRINT_PRECISION integer - -In ROUNDED mode, numbers are normally printed to the specified -precision. If the user wishes to print such numbers with less -precision, the printing precision can be set by the declaration -PRINT_PRECISION. - -Examples: -on rounded; -1/3; 0.333333333333 -print_precision 5; -1/3 0.33333 - -\endsection -\item[PROCEDURE] -PROCEDURE (page 169) - -The PROCEDURE command allows you to define a mathematical operation as a -function with arguments. - PROCEDURE identifier (arg{,arg});body - -The option may be ALGEBRAIC or SYMBOLIC, indicating the mode under -which the procedure is executed, or REAL or INTEGER, indicating the -type of answer expected. The default is algebraic. Real or integer -procedures are subtypes of algebraic procedures; type-checking is done -on the results of integer procedures, but not on real procedures (in -the current REDUCE release). identifier may be any valid REDUCE -identifier that is not already a procedure name, operator, ARRAY or -MATRIX. arg is a formal parameter that may be any valid REDUCE -identifier. body is a single statement (a GROUP or BLOCK statement -may be used) with the desired activities in it. - -Examples: - -procedure fac(n); - if not (fixp(n) and n>=0) - then rederr "Choose nonneg. integer only" - else for i := 0:n-1 product i+1; - FAC -fac(0); 1 -fac(5); 120 -fac(-5); ***** choose nonneg. integer only - -Procedures are automatically declared as operators upon definition. -When REDUCE has parsed the procedure definition and successfully -converted it to a form for its own use, it prints the name of the -procedure. Procedure definitions cannot be nested. Procedures can -call other procedures, or can recursively call themselves. Procedure -identifiers can be cleared as you would clear an operator. Unlike LET -statements, new definitions under the same procedure name replace the -previous definitions completely. - -Be careful not to use the name of a system operator for your own -procedure. REDUCE may or may not give you a warning message. If you -redefine a system operator in your own procedure, the original -function of the system operator is lost for the remainder of the -REDUCE session. - -Procedures may have none, one, or more than one parameter. A REDUCE -parameter is a formal parameter only; the use of x as a parameter in a -PROCEDURE definition has no connection with a value of x in the REDUCE -session, and the results of calling a procedure have no effect on the -value of x. If a procedure is called with x as a parameter, the -current value of x is used as specified in the computation, but is not -changed outside the procedure. Making an assignment statement by := -with a formal parameter on the left-hand side only changes the value -of the calling parameter within the procedure. - -Using a LET statement inside a procedure always changes the value -globally: a LET with a formal parameter makes the change to the -calling parameter. LET statements cannot be made on local variables -inside BEGIN...END BLOCKS. When CLEAR statements are used on formal -parameters, the calling variables associated with them are cleared -globally too. The use of LET or CLEAR statements inside procedures -should be done with extreme caution. - -Arrays and operators may be used as parameters to procedures. The -body of the procedure can contain statements that appropriately -manipulate these arguments. Changes are made to values of the calling -arrays or operators. Simple expressions can also be used as -arguments, in the place of scalar variables. Matrices may not be used -as arguments to procedures. - -A procedure that has no parameters is called by the procedure name, -immediately followed by empty parentheses. The empty parentheses may -be left out when writing a procedure with no parameters, but must -appear in a call of the procedure. If this is a nuisance to you, use -a LET statement on the name of the procedure (i.e., LET NOARGS = -NOARGS()) after which you can call the procedure by just its name. - -Procedures that have a single argument can leave out the parentheses -around it both in the definition and procedure call. (You can use the -parentheses if you wish.) Procedures with more than one argument must -use parentheses, with the arguments separated by commas. - -Procedures often have a BEGIN...END block in them. Inside the block, -local variables are declared using SCALAR, REAL or INTEGER -declarations. The declarations must be made immediately after the -word BEGIN, and if more than one type of declaration is made, they are -separated by semicolons. REDUCE currently does no type checking on -local variables; REAL and INTEGER are treated just like SCALAR. -Actions take place as specified in the statements inside the block -statement. Any identifiers that are not formal parameters or local -variables are treated as global variables, and activities involving -these identifiers are global in effect. - -If a return value is desired from a procedure call, a specific RETURN -command must be the last statement executed before exiting from the -procedure. If no RETURN is used, a procedure returns a zero or no -value. - -Procedures are often written in a file using an editor, then the file -is input using the command IN. This method allows easy changes in -development, and also allows you to load the named procedures whenever -you like, by loading the files that contain them. - -\endsection -\xitem[Procedure body] -Procedure body (pages 171--173) - -\endsection -\xitem[Procedure heading] -Procedure heading (page 170) - -\endsection -\item[PROD] -PROD operator (page 403) - -The operator PROD returns -the indefinite or definite product of a given expression. - - -PROD(expr,k[,lolim [,uplim ]]) - - -where expr is the expression to be multiplied, k is the -control variable (a KERNEL), and lolim and uplim -uplim are the optional lower and upper limits. If uplim is -not supplied the upper limit is taken as k. The GOSPER -algorithm is used. If there is no closed form solution, the operator -returns the input unchanged. - -Examples: -prod(k/(k-2),k); k*( - k + 1) - -\endsection -\item[PRODUCT] -PRODUCT (page 57, 58) - -See the FOR loop construction. - -\endsection -\xitem[Program] -Program (page 38) - -\endsection -\xitem[Program structure] -Program structure (page 33) - -\endsection -\xitem[Proper statement] -Proper statement (pages 48, 53, 54) - -\endsection -\xitem[PRSYS] -PRSYS (pages 378, 382) - -\endsection -\xitem[PS] -PS (page 188) - -\endsection -\xitem[PS operator] -PS operator (page 414) - -\endsection -\xitem[PSCHANGEVAR operator] -PSCHANGEVAR operator (page 418) - -\endsection -\xitem[PSCOMPOSE operator] -PSCOMPOSE operator (page 420) - -\endsection -\xitem[PSDEPVAR operator] -PSDEPVAR operator (page 418) - -\endsection -\xitem[PSEXPANSIONPT operator] -PSEXPANSIONPT operator (page 418) - -\endsection -\xitem[PSEXPLIM operator] -PSEXPLIM operator (pages 414, 416) - -\endsection -\xitem[PSFUNCTION operator] -PSFUNCTION operator (page 418) - -\endsection -\item[PSI] -PSI (pages 185, 395) - -The PSI operator returns the Psi (or DiGamma) function. - - Psi(x) := df(Gamma(z),z)/ Gamma (z) - - GAMMA(expression) - -Examples: - load_package specfn; - 1 - 2*LOG(2) + PSI(---) + PSI(1) + 3 - 2 - Psi(3); ---------------------------------- - 2 - - on rounded; - - Psi(1); 0.577215664902 - -Euler's constant can be found as - Psi(1). - -\endsection -\xitem[PSINTCONST (shared)] -PSINTCONST (shared) (page 415) - -\endsection -\xitem[PSORDER operator] -PSORDER operator (page 417) - -\endsection -\xitem[PSORDLIM operator] -PSORDLIM operator (page 416) - -\endsection -\xitem[PSREVERSE operator] -PSREVERSE operator (page 419) - -\endsection -\xitem[PSSETORDER operator] -PSSETORDER operator (page 417) - -\endsection -\xitem[PSSUM operator] -PSSUM operator (page 421) - -\endsection -\xitem[PSTERM operator] -PSTERM operator (page 417) - -\endsection -\xitem[Puiseux expansion] -Puiseux expansion (page 419) - -\endsection -\xitem[PUTCSYSTEM command] -PUTCSYSTEM command (page 235) - -\endsection -\xitem[Quadrature] -Quadrature (page 182) - -\endsection -\item[QUIT] -QUIT (page 70) - -The QUIT command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are -at the top level, the QUIT command exits REDUCE. BYE is a synonym for -QUIT. - -\endsection -\xitem[QUOTE] -QUOTE (page 193) - -\endsection -\xitem[RANDOM] -RANDOM (page 74) - -\endsection -\xitem[RANDOM_NEW_SEED] -RANDOM_NEW_SEED (page 75) - -\endsection -\item[RANK] -RANK (page 167) - - RANK(matrix_expression) -RANK calculates the rank of its matrix argument. - -Examples: - rank mat((a,b,c),(d,e,f)); 2 - -The argument to RANK can also be a LIST of lists, interpreted either -as a row matrix or a set of equations. If that form of input is -chosen, the vectors in the result will be represented by lists as -well. This additional input syntax facilitates the use of RANK in -applications different from classical linear algebra. - -\endsection -\item[RAT] -RAT (page 104) - -When the RAT switch is on, and kernels have been selected to display -with the FACTOR declaration, the denominator is printed with each -term rather than one common denominator at the end of an expression. - -Examples: 3 - SIN(Y)*X + SIN(Y) + X -(x+1)/x + x**2/sin y; ------------------------ - SIN(Y)*X -factor x; - 3 - X + X*SIN(Y) + SIN(Y) -(x+1)/x + x**2/sin y; ------------------------ - X*SIN(Y) -on rat; - 2 - X -1 -(x+1)/x + x**2/sin y; -------- + 1 + X - SIN(Y) - -The RAT switch only has effect when the PRI switch is on. -When PRI is off, regardless of the setting of RAT, the -printing behaviour is as if RAT were off. RAT only has -effect upon the display of expressions, not their internal form. - -\endsection -\item[RATARG] -RATARG (pages 115, 128) - -When RATARG is on, rational expressions can be given to operators -such as COEFF and LTERM that normally require -polynomials in one of their arguments. When RATARG is off, rational -expressions cause an error message. - -Examples: 3 2 3 - X + X*Y + Y -aa := x/y**2 + 1/x + y/x**2; AA := ---------------- - 2 2 - X *Y - 3 2 3 - X + X*Y + Y -coeff(aa,x); ***** ---------------- invalid as POLYNOMIAL - 2 2 - X *Y -on ratarg; - Y 1 1 -coeff(aa,x); {----,----,0,-------} - 2 2 2 2 - X X X *Y - -\endsection -\item[RATIONAL] -RATIONAL (page 132) - -When RATIONAL is on, polynomial expressions with rational coefficients -are produced. - -Examples: - 2*X + 3*Y -x/2 + 3*y/4; ----------- - 4 - 2 - X + 5*X + 17 -(x**2 + 5*x + 17)/2; --------------- - 2 -on rational; - 1 3 -x/2 + 3y/4; ---*(X + ---*Y) - 2 2 - - 1 2 -(x**2 + 5*x + 17)/2; ---*(X + 5*X + 17) - 2 - -By using RATIONAL, polynomial expressions with rational coefficients -can be used in some commands that expect polynomials. With RATIONAL -off, such a polynomial becomes a rational expression, with denominator -the least common multiple of the denominators of the rational number -coefficients. - -\endsection -\xitem[Rational coefficient] -Rational coefficient (page 132) - -\endsection -\xitem[Rational function] -Rational function (page 119) - -\endsection -\item[RATIONALIZE] -RATIONALIZE (page 135) - -When the RATIONALIZE switch is on, denominators of rational expressions -that contain complex numbers or root expressions are simplified by -multiplication by their conjugates. - -Examples: - SQRT(3) + 1 -qq := (1+sqrt(3))/(sqrt(3)-7); QQ := ------------- - SQRT(3) - 7 -on rationalize; - - 4*SQRT(3) - 5 -qq; ------------------ - 23 - 2/3 1/3 - 6 - 4*6 + 16 -2/(4 + 6**(1/3)); -------------------- - 35 -on complex; - 1 - 2*i -(i-1)/(i+3); --------- - 5 - - -\endsection -\item[RATPRI] -RATPRI (page 104) - -When the RATPRI switch is on, rational expressions and fractions are -printed as two lines separated by a fraction bar, rather than in a linear -style. Default is ON. - -Examples: - 3 -3/17; ---- - 17 - 3*B + 2*Y -2/b + 3/y; ----------- - B*Y -off ratpri; -3/17; 3/17 -2/b + 3/y; (3*B + 2*Y)/(B*Y) - -\endsection -\xitem[RATROOT] -RATROOT (page 373) - -\endsection -\item[REAL] -REAL (page 61) - -The REAL declaration must be made immediately after a BEGIN (or other -variable declaration such as INTEGER and SCALAR) and declares local -integer variables. They are initialised to zero. - - REAL identifier{,identifier} - -identifier may be any valid REDUCE identifier, except T or NIL. - -Real variables remain local, and do not share values with variables of -the same name outside the BEGIN...END block. When the block is -finished, the variables are removed. You may use the words INTEGER or -SCALAR in the place of REAL. REAL does not indicate type-checking by -the current REDUCE; it is only for your own information. Declaration -statements must immediately follow the BEGIN, without a semicolon -between BEGIN and the first variable declaration. - -Any variables used inside a BEGIN...END BLOCK that were not declared -SCALAR, REAL or INTEGER are global, and any change made to them inside -the block affects their global value. Any array or matrix declared -inside a block is always global. - -\endsection -\xitem[Real] -Real (pages 34, 35) - -\endsection -\xitem[Real coefficient] -Real coefficient (page 132) - -\endsection -\item[REALROOTS] -REALROOTS (pages 369, 370) - -The operator REALROOTS finds that real roots of a polynomial to an -accuracy that is sufficient to separate them and which is a minimum of -6 decimal places. - - REALROOTS(p) - REALROOTS(p,from,to) - -where p is a univariate polynomial. The optional parameters from and -to classify an interval: if given, exactly the real roots in this -interval will be returned. from and to can also take the values -INFINITY or -INFINITY. If omitted all real roots will be returned. -Result is a LIST of equations which represent the roots of the -polynomial at the given accuracy. - -Examples: - realroots(x^5-2); {X=1.1487} - realroots(x^3-104*x^2+403*x-300,2,infinity); {X=3.0,X=100.0} - realroots(x^3-104*x^2+403*x-300,-infinity,2); {X=1} - -The minimal accuracy of the result values is controlled by ROOTACC. - -\endsection -\xitem[REDERR] -REDERR (page 173) - -\endsection -\item[REDUCT] -REDUCT (page 131) -The REDUCT operator returns the remainder of its expression after the -leading term is removed. - - REDUCT(expression,kernel) - -expression is ordinarily a polynomial. If RATARG is on, a rational -expression may also be used, otherwise an error results. kernel must -be a KERNEL. - -Examples: - 3 -reduct((x+y)**3,x); (x + y) - -reduct(x + sin(x)**3,sin(x)); x - 3 -reduct(x + sin(x)**3,y); sin(x) + x - -If the expression does not contain the kernel, REDUCT returns the -expression. - -\endsection -\xitem[side relations] -relations - side (page 241) - -\endsection -\item[REMAINDER] -REMAINDER (page 126) -The REMAINDER operator returns the remainder after its first -argument is divided by its second argument. - - REMAINDER(expression,expression) - -expression can be any valid REDUCE polynomial, and is not limited -to numeric values. - -Examples: -remainder(13,6); 1 -remainder(x**2 + 3*x + 2,x+1); 0 -remainder(x**3 + 12*x + 4,x**2 + 1); 11*X + 4 -remainder(sin(2*x),x*y); SIN(2*X) - -If the first argument to REMAINDER contains a denominator not equal to -1, an error occurs. - -\endsection -\item[REMFAC] -REMFAC (page 102) - -The REMFAC declaration removes the special factoring treatment of its -arguments that was declared with FACTOR. - -REMFAC kernel{,kernel} - -kernel must be a KERNEL or OPERATOR name that was declared as special -with the FACTOR declaration. - -\endsection -\xitem[REMFORDER command] -REMFORDER command (pages 268, 271) - -\endsection -\item[REMIND] -REMIND (page 206) - -The REMIND declaration removes the special status of its arguments -as indices, which was set in the INDEX declaration, in -high-energy physics calculations. - REMIND identifier{,identifier} - -identifier must have been declared to be of type INDEX. - -\endsection -\xitem[RENOSUM command] -RENOSUM command (pages 262, 271) - -\endsection -\item[REPART] -REPART (pages 72, 73, 75) - - REPART(expression) or REPART simple_expression - -This operator returns the real part of an expression, if that argument -has an numerical value. A non-numerical argument is returned as an -expression in the operators REPART and IMPART. - -Examples: -repart(1+i); 1 -repart(a+i*b); REPART(A) - IMPART(B) - -\endsection -\item[REPEAT] -REPEAT (pages 60, 61, 63, 65) - -The REPEAT command causes repeated execution of a statement UNTIL -the given condition is found to be true. The statement is always executed -at least once. - REPEAT statement UNTIL condition - -statement can be a single statement, GROUP statement, or -a BEGIN...END BLOCK. condition must be a logical -operator that evaluates to rue or nil. - -Examples: -<> until m = 0>>; - 400*X - 300*X - 200*X - 100*X - -<> until m <= 0>>; - -1 - -REPEAT must always be followed by an UNTIL with a condition. Be -careful not to generate an infinite loop with a condition that is -never true. In the second example, if the condition had been M = 0, -it would never have been true since M already had value -2 when the -condition was first evaluated. - -\endsection -\xitem[Reserved variable] -Reserved variable (pages 36, 37) - -\endsection -\item[REST] -REST (page 50) - -The REST operator returns a LIST containing all but the first element of -the list it is given. - REST(list) or REST list - - -list must be a non-empty list, but need not have more than one element. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D}; -rest alist; {B,C,D} -blist := {x,y,{aa,bb,cc},z}; BLIST := {X,Y,{AA,BB,CC},Z} -second rest blist; {AA,BB,CC} -clist := {c}; CLIST := C -rest clist; {} - -\endsection -\xitem[RESULT] -RESULT (page 378) - -\endsection -\item[RESULTANT] -RESULTANT (page 126) -The RESULTANT operator computes the resultant of two polynomials with -respect to a given variable. If the resultant is 0, the polynomials -have a root in common. - RESULTANT(expression,expression,kernel) - -expression must be a polynomial containing kernel ; -kernel must be a KERNEL. - -Examples: -resultant(x**2 + 2*x + 1,x+1,x); 0 -resultant(x**2 + 2*x + 1,x-3,x); 16 -resultant(z**3 + z**2 + 5*z + 5, - z**4 - 6*z**3 + 16*z**2 - 30*z + 55, - z); 0 -resultant(x**3*y + 4*x*y + 10,y**2 + 6*y + 4,y); - 6 4 3 2 - 4*(x + 8*x - 15*x + 16*x - 60*x + 25) - -The resultant is the determinant of the Sylvester matrix, formed from the -coefficients of the two polynomials in the following way: - -Given two polynomials: - - n n-1 - a x + a1 x + ... + an - -and - m m-1 - b x + b1 x + ... + bm - -form the (m+n)x(m+n-1) Sylvester matrix by the following means: - - 0.......0 a a1 .......... an - 0....0 a a1 .......... an 0 - . . . . - a0 a1 .......... an 0.......0 - 0.......0 b b1 .......... bm - 0....0 b b1 .......... bm 0 - . . . . - b b1 .......... bm 0.......0 - -If the determinant of this matrix is 0, the two polynomials have a -common root. Finding the resultant of large expressions is -time-consuming, due to the time needed to find a large determinant. - -The sign conventions RESULTANT uses are those given in the article, -``Computing in Algebraic Extensions,'' by R. Loos, appearing in -Computer Algebra--Symbolic and Algebraic Computation, 2nd ed., edited -by B. Buchberger, G.E. Collins and R. Loos, and published by - -Springer-Verlag, 1983. - -These are: - resultant(p(x),q(x),x) = (-1)^{deg p(x)*deg q(x)} * resultant(q(x),p(x),x), - resultant(a,p(x),x) = a^{deg p(x)}, - resultant(a,b,x) = 1 - -where p(x) and q(x) are polynomials which have x as a variable, and -a and b are free of x. - -Error messages are given if RESULTANT is given a non-polynomial -expression, or a non-kernel variable. - -\endsection -\item[RETRY] -RETRY (page 157) -The RETRY command allows you to retry the latest statement that resulted -in an error message. - -Examples: -matrix a; -det a; ***** Matrix A not set -a := mat((1,2),(3,4)); A(1,1) := 1 - A(1,2) := 2 - A(2,1) := 3 - A(2,2) := 4 -retry; -2 - -RETRY remembers only the most recent statement that resulted in an -error message. It allows you to stop and fix something obvious, then -continue on your way without retyping the original command. - -\endsection -\item[RETURN] -RETURN (pages 62--64) - -The RETURN command causes a value to be returned from inside a -BEGIN...END BLOCK. - BEGIN statements RETURN (expression) - END - -statements can be any valid REDUCE statements. The value of -expression is returned. - -Examples: -begin write "yes"; return a end; yes - A -procedure dumb(a); - begin if numberp(a) then return a - else return 10 end; - DUMB -dumb(x); 10 -dumb(-5); -5 -procedure dumb2(a); - begin c := a**2 + 2*a + 1; - d := 17; c*d; return end; - DUMB2 -dumb2(4); -c; 25 -d; 17 - -Note in DUMB2 above that the assignments were made as requested, but -the product C*D cannot be accessed. Changing the procedure to read -RETURN C*D would remedy this problem. - -The RETURN statement is always the last statement executed before -leaving the block. If RETURN has no argument, the block is exited but -no value is returned. A block statement does not need a RETURN ; the -statements inside terminate in their normal fashion without one. In -that case no value is returned, although the specified actions inside -the block take place. - -The RETURN command can be used inside <<...>> GROUP statements and -IF...THEN...ELSE commands that are inside BEGIN...END BLOCKs. It is -not valid in these constructions that are not inside a BEGIN...END -block. It is not valid inside FOR, REPEAT...UNTIL or WHILE...DO loops -in any construction. To force early termination from loops, the GO -TO(GOTO) command must be used. When you use nested block statements, -a RETURN from an inner block exits returning a value to the -next-outermost block, rather than all the way to the outside. - -\endsection -\item[REVERSE] -REVERSE (page 51) - -The REVERSE operator returns a LIST that is the reverse of the list it -is given. - REVERSE(list) or REVERSE list - -list must be a LIST. - -Examples: - 2 3 -aa := {c,b,a,{x**2,z**3},y}; AA := {C,B,A,{X ,Z },Y} - 2 3 -reverse aa; {Y,{X ,Z},A,B,C} - 2 3 -reverse(q . reverse aa); {C,B,A,{X ,Z },Y,Q} - -REVERSE and CONS can be used together to add a new element to the end -of a list (. adds its new element to the beginning). The REVERSE -operator uses a noticeable amount of system resources, especially if -the list is long. If you are doing much heavy-duty list manipulation, -you should probably design your algorithms to avoid much reversing of -lists. A moderate amount of list reversing is no problem. - -\endsection -\item[REVGRADLEX] -REVGRADLEX (page 293) - -The terms are ordered first with their total degree (degree sum), and -if the total degree is identical the comparison is the inverse of LEX -term order. With GROEBNER and GROEBNERF calculations this term order -is similar to GRADLEX term order; it is known as most efficient -ordering with respect to computing time. - -\endsection -\item[REVPRI] -REVPRI (page 105) - -When the REVPRI switch is on, terms are printed in reverse order from -the normal printing order. - -Examples: - 5 2 -x**5 + x**2 + 18 + sqrt(y); SQRT(Y) + X + X + 18 - -a + b + c + w; A + B + C + W - -on revpri; - 2 5 -x**5 + x**2 + 18 + sqrt(y); 17 + X + X + SQRT(Y) - -a + b + c + w; W + C + B + A - -Turn REVPRI on when you want to display a polynomial in ascending -rather than descending order. - -\endsection -\item[RHS] -RHS (page 47) -The RHS operator returns the right-hand side of an EQUATION, such as -those returned in a LIST by SOLVE. - - RHS(equation) or RHS equation - -equation must be an equation of the form left-hand side = right-hand side. - -Examples: - roots := solve(x**2 + 6*x*y + 5x + 3y**2,x); - - 2 - - (SQRT(24*Y + 60*Y + 25) + 6*Y + 5) - ROOTS := {X=----------------------------------------, - 2 - - 2 - SQRT(24*Y + 60*Y + 25) - 6*Y - 5 - X=-----------------------------------} - 2 - -root1 := rhs first roots; - - (SQRT(24*Y + 60*Y + 25) + 6*Y + 5) - ROOT1 := ---------------------------------------- - 2 -root2 := rhs second roots; - 2 - SQRT(24*Y + 60*Y + 25) - 6*Y - 5 - ROOT2 := ----------------------------------- - 2 - -An error message is given if RHS is applied to something other than an -equation. - -\endsection -\xitem[Riemann Zeta Function] -Riemann Zeta Function (pages 185, 395) - -\endsection -\xitem[RIEMANNCONX command] -RIEMANNCONX command (pages 267, 271) - -\endsection -\xitem[Riemannian Connections] -Riemannian Connections (page 267) - -\endsection -\xitem[Rlisp] -Rlisp (page 213) - -\endsection -\item[RLISP88] -RLISP88 (page 204) - -Rlisp '88 is a superset of the Rlisp that has been traditionally used -for the support of REDUCE. It is fully documented in the book Marti, -J.B., ``RLISP '88: An Evolutionary Approach to Program Design and -Reuse'', World Scientific, Singapore (1993). It supports different -looping constructs from the traditional Rlisp, and treats ``-'' as a -letter unless separated by spaces. Turning on the switch RLISP88 -converts to Rlisp '88 parsing conventions in symbolic mode, and -enables the use of Rlisp '88 extensions. Turning off the switch -reverts to the traditional Rlisp and the previous mode (SYMBOLIC or -ALGEBRAIC) in force before RLISP88 was turned on. - -\endsection -\item[RLROOTNO] -RLROOTNO (page 369) - -The function RLROOTNO computes the number of real roots of p in the -specified region, but does not find the roots. - - RLROOTNO(expression) - RLROOTNO(expression, POSITIVE) - RLROOTNO(expression, NEGATIVE) - RLROOTNO(expression, lo, hi) - -For more details on the specification of an interval, see ISOLATER. - -Examples: - load_package roots; - rlrootno (x^3-3x^2+2x+10); 1 - rlrootno(x^3-3x^2+2x+10,positive); 0 -\endsection -\xitem[root finding] -root finding (page 367) - -\endsection -\item[ROOT_OF] -ROOT_OF (pages 85, 86) - -When the operator SOLVE is unable to find an explicit solution or if -that solution would be too complicated, the result is presented as -formal root expression using the internal operator ROOT_OF and a new -local variable. An expression with a top level ROOT_OF is implicitly a -list with an unknown number of elements since we can't always know how -many solutions an equation has. If a substitution is made into such an -expression, closed form solutions can emerge. If this occurs, the -ROOT_OF construct is replaced by an operator ONE_OF. At this point it -is of course possible to transform the result if the original SOLVE -operator expression into a standard SOLVE solution. To effect this, -the operator EXPAND_CASES can be used. - -Examples: 7 2 -solve(a*x^7-x^2+1,x); {x=root_of(a*x_ - x_ + 1,x_)} -sub(a=0,ws); {x=one_of(1,-1)} -expand_cases ws; {x=1,x=-1} - -The components of ROOT_OF and ONE_OF expressions can be processed as -usual with operators ARGLENGTH and PART. - -\endsection -\item[ROOT_MULTIPLICITES] -ROOT_MULTIPLICITES - -The ROOT_MULTIPLICITIES variable is set to the list of the -multiplicities of the roots of an equation by the SOLVE operator. - - -SOLVE returns its solutions in a list. The multiplicities of -each solution are put in the corresponding locations of the list -ROOT_MULTIPLICITIES. - -\endsection -\xitem[ROOT_VAL] -ROOT_VAL (page 370) - -\endsection -\item[ROOTACC] -ROOTACC (page 373) - -The operator ROOTACC allows you to set the accuracy up to which the -roots package computes its results. - - ROOTACC(n) - -Here n is an integer value. The internal accuracy of the ROOTS package -is adjusted to a value of MAX(6,N). The default value is 6. - -\endsection -\xitem[ROOTMSG] -ROOTMSG (page 373) - -\endsection -\xitem[ROOTPREC] -ROOTPREC (page 374) - -\endsection -\item[ROOTS] -ROOTS (pages 184, 369, 370) - -The operator ROOTS is the main top level function of the roots -package. It will find all roots, real and complex, of the polynomial -p to an accuracy that is sufficient to separate them and which is a -minimum of 6 decimal places. - - ROOTS(p) - -where p is a univariate polynomial. Result is a LIST of equations -which represent the roots of the polynomial at the given accuracy. In -addition, ROOTS stores separate lists of real roots and complex roots -in the global variables ROOTSREAL and ROOTSCOMPLEX. - -Examples: - - roots(x^5-2); {X=-0.929316 + 0.675188*I, - X=-0.929316 - 0.675188*I, - X=0.354967 + 1.09248*I, - X=0.354967 - 1.09248*I, - X=1.1487} - -The minimal accuracy of the result values is controlled by -ROOTACC. - -\endsection -\xitem[ROOTS package] -ROOTS package (page 367) - -\endsection -\xitem[ROOTS_AT_PREC] -ROOTS_AT_PREC (page 370) - -\endsection -\item[ROOTSCOMPLEX] -ROOTSCOMPLEX (page 369) - -When the operator ROOTS is called the complex roots are collected in -the global variable ROOTSCOMPLEX as LIST. - -\endsection -\item[ROOTSREAL] -ROOTSREAL (page 369) - -When the operator ROOTS is called the real roots are collected in the -global variable ROOTREAL as LIST. - -\endsection -\item[ROUND] -ROUND (page 75) - - ROUND(expression) - -If its argument has a numerical value, ROUND rounds it to the nearest -integer. For non-numeric arguments, the value is an expression in the -original operator. - -Examples: -round 3.4; 3 -round 3.5; 4 -round a; ROUND(A) - -\endsection -\item[ROUNDALL] -ROUNDALL (page 133) - -In ROUNDED mode, rational numbers are normally converted to a -floating point representation. If ROUNDALL is off, this conversion -does not occur. ROUNDALL is normally ON. - -Examples: -on rounded; -1/2; 0.5 -off roundall; - 1 -1/2; --- - 2 - -\endsection -\item[ROUNDBF] -ROUNDBF (page 133) - -When ROUNDED is on, the normal defaults cause underflows to be -converted to zero. If you really want the small number that results -in such cases, ROUNDBF can be turned on. - -Examples: -on rounded; -exp(-100000.1^2); 0 -on roundbf; -exp(-100000.1^2); 1.18441281937E-4342953505 - -If a polynomial is input in ROUNDED mode at the default precision into -any ROOTS function, and it is not possible to represent any of the -coefficients of the polynomial precisely in the system floating point -representation, the switch ROUNDBF will be automatically turned on. -All rounded computation will use the internal bigfloat representation -until the user subsequently turns ROUNDBF off. (A message is output to -indicate that this condition is in effect.) - -\endsection -\item[ROUNDED] -ROUNDED (pages 36, 44, 78, 108, 132, 372) - -When ROUNDED is on, floating-point arithmetic is enabled, with -precision initially at a system default value, which is usually 12 -digits. The precise number can be found by the command PRECISION(0). - -Examples: -pi; PI - - 5 -35/217; ---- - 31 -on rounded; -pi; 3.14159265359 - -35/217; 0.161 - -sqrt(3); 1.73205080756 - -If more than the default number of decimal places are required, use the -PRECISION command to set the required number. - -\endsection -\item[Rule lists] -Rule lists (page 147) - -A RULE is an instruction to replace an algebraic expression -or a part of an expression by another one. - lhs => rhs or - lhs => rhs WHEN cond -lhs is an algebraic expression used as search pattern and -rhs is an algebraic expression which replaces matches of -rhs. => is the operator REPLACE. - -lsh can contain free variables which are preceded by a tilde ~ in -their leftmost position in lhs. If a rule has a WHEN cond part it -will fire only if the evaluation of cond has a result TRUE. cond may -contain references to free variables of lhs. - -Rules can be collected in a LIST which then forms a RULE LIST. RULE -LISTS can be used to collect algebraic knowledge for a specific -evaluation context. - -RULES and RULE LISTS are globally activated and deactivated by LET, -FORALL, CLEARRULES. For a single evaluation they can be locally -activate by WHERE. The active rules for an operator can be visualised -by SHOWRULES. - -Examples: -operator f,g,h; -let f(x) => x^2; - 2 -f(x); X -g_rules:={g(~n,~x)=>h(n/2,x) when evenp n, -g(~n,~x)=>h((1-n)/2,x) when not evenp n}$ -let g_rules; -g(3,x); H(-1,X) - -\endsection -\item[SAVEAS] -SAVEAS (page 99)) -The SAVEAS command saves the current workspace under the name of its -argument. - - SAVEAS identifier - -identifier can be any valid REDUCE identifier. - -Examples: - -(The numbered prompts are shown below, unlike in most examples) -1: solve(x^2-3); {x=sqrt(3),x= - sqrt(3)} -2: saveas rts(0)$ -3: rts(0); {x=sqrt(3),x= - sqrt(3)} - -SAVEAS works only for the current workspace, the last algebraic -expression produced by REDUCE. This allows you to save a result that -you did not assign to an identifier when you originally typed the -input. For access to previous output use WS. - -\endsection -\xitem[savesfs] -savesfs (page 393) - -\endsection -\item[SAVESTRUCTR] -SAVESTRUCTR (page 113) - -When SAVESTRUCTR is on, results of the STRUCTR command are returned as -a list whose first element is the representation for the expression -and the remaining elements are equations showing the relationships of -the generated variables. - -Examples: -off exp; - -structr((x+y)^3 + sin(x)^2); ANS3 - where - 3 2 - ANS3 := ANS1 + ANS2 - - ANS2 := SIN(X) - - ANS1 := X + Y - -ans3; ANS3 -on savestructr; - 3 2 -structr((x+y)^3 + sin(x)^2); {ANS3,ANS3=ANS1 + ANS2 ,ANS2=SIN(X),ANS1=X + Y} - 3 2 -ans3 where rest ws; (X + Y) + SIN(X) - -In normal operation, STRUCTR is only a display command. With -SAVESTRUCTR on, you can access the various parts of the expression -produced by STRUCTR. - -The generic system names use the stem ANS. You can change this to your -own stem by the command VARNAME. REDUCE adds integers to this stem -to make unique identifiers. - -\endsection -\xitem[Saving an expression] -Saving an expression (page 111) - -\endsection -\item[SCALAR] -SCALAR (pages 61, 62) - -The SCALAR declaration must be made immediately after a BEGIN (or -other variable declaration such as INTEGER and REAL) and declares -local scalar variables. They are initialised to 0. - - SCALAR identifier{,identifier} - -identifier may be any valid REDUCE identifier, except T or NIL. - -Scalar variables remain local, and do not share values with variables -of the same name outside the BEGIN...END BLOCK. When the block is -finished, the variables are removed. You may use the words REAL or -INTEGER in the place of SCALAR. REAL and INTEGER do not indicate -type-checking by the current REDUCE; they are only for your own -information. Declaration statements must immediately follow the -BEGIN, without a semicolon between BEGIN and the first variable -declaration. - -Any variables used inside BEGIN...END blocks that were not declared -SCALAR, REAL or INTEGER are global, and any change made to them inside -the block affects their global value. Arrays declared inside a block -are always global. - -\endsection -\xitem[Scalar] -Scalar (page 43) - -\endsection -\xitem[SCALEFACTORS operator] -SCALEFACTORS operator (page 234) - -\endsection -\item[SCIENTIFIC_NOTATION] -SCIENTIFIC_NOTATION (page 34) - - SCIENTIFIC_NOTATION(m) or SCIENTIFIC_NOTATION(m,n) - -m and n are positive integers. SCIENTIFIC_NOTATION controls the -output format of floating point numbers. At the default settings, any -number with five or less digits before the decimal point is printed in -a fixed-point notation, e.g., 12345.6. Numbers with more than five -digits are printed in scientific notation, e.g., 1.234567E+5. -Similarly, by default, any number with eleven or more zeros after the -decimal point is printed in scientific notation. - -When SCIENTIFIC_NOTATION is called with the numerical argument m a -number with more than m digits before the decimal point, or m or more -zeros after the decimal point, is printed in scientific notation. -When SCIENTIFIC_NOTATION is called with a list {m, n}, a number with -more than m digits before the decimal point, or n or more zeros after -the decimal point is printed in scientific notation. - -Examples: - -on rounded; -12345.6; 12345.6 - -123456.5; 1.234565e+5 - -0.00000000000000012; 1.2e-16 - -scientific_notation 20; {5,11} - -5: 123456.7; 123456.7 - -0.00000000000000012; 0.00000000000000012 - -\endsection -\item[SCOPE] -SCOPE (page 185) - -Author: J.A. van Hulzen - -REDUCE Source Code Optimization Package. - -SCOPE is a package for the production of an optimised form of a -set of expressions. It applies an heuristic search for common -(sub)expressions to almost any set of proper REDUCE assignment -statements. The output is obtained as a sequence of assignment -statements. GENTRAN is used to facilitate expression output. - -\endsection -\xitem[SDER(I)] -SDER(I) (page 379) - -\endsection -\item[SEC] -SEC (pages 76, 78) - -The SEC operator returns the secant of its argument. - - SEC(expression) or SEC simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -sec abc; SEC(ABC) -sec(pi); -1 -sec 4; SEC(4) -on rounded; -sec(4); - 1.52988565647 -sec log 5; - 25.8852966005 - -SEC returns a numeric value only if ROUNDED is on. Then the secant is -calculated to the current degree of floating point precision. - -\endsection -\item[SECH] -SECH (pages 76, 78) - -The SECH operator returns the hyperbolic secant of its argument. - - SECH(expression) or SECH simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -sech abc; SECH(ABC) -sech(0); 1 -sech 4; SECH(4) -on rounded; -sech(4); 0.0366189934737 -sech log 5; 0.384615384615 - -SECH returns a numeric value only if ROUNDED is on. Then the -expression is calculated to the current degree of floating point -precision. - -\endsection -\item[SECOND] -SECOND (page 50) - -The SECOND operator returns the second element of a list. - SECOND(list) or SECOND list - -list must be a list with at least two elements, to avoid an error -message. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D} -second alist; B -blist := {x,{aa,bb,cc},z}; BLIST := {X,{AA,BB,CC},Z} -second second blist; BB - -\endsection -\xitem[Selector] -Selector (page 198) - -\endsection -\xitem[Semicolon] -Semicolon (page 53) - -\endsection -\item[SET] -SET (pages 55, 83) - -The SET operator is used for assignments when you want both sides of -the assignment statement to be evaluated. - - SET(restricted_expression,expression) - -expression can be any REDUCE expression; restricted_expression -must be an identifier or an expression that evaluates to an identifier. - -Examples: -a := y; A := Y - 2 -set(a,sin(x^2)); SIN(X ) - 2 -a; SIN(X ) - 2 -y; SIN(X ) - -a := b + c; A := B + C - -set(a-c,z); Z - -b; Z - -Using an ARRAY or MATRIX reference as the first argument to SET has -the result of setting the contents of the designated element to SET's -second argument. You should be careful to avoid unwanted side effects -when you use this facility. - -\endsection -\item[SETMOD] -SETMOD (page 134) - -The SETMOD command sets the modulus value for subsequent MODULAR -arithmetic. - - SETMOD integer - -integer must be positive, and greater than 1. It need not be a prime -number. - -Examples: -setmod 6; 1 -on modular; -16; 4 - 2 -x^2 + 5x + 7; X + 5*X + 1 - X -x/3; --- - 3 -setmod 2; 6 - 4 -(x+1)^4; X + 1 -x/3; X - -SETMOD returns the previous modulus, or 1 if none has been set before. -SETMOD only has effect when MODULAR is on. - -Modular operations are done only on numbers such as coefficients of -polynomials, not on the exponents. The modulus need not be prime. -Attempts to divide by a power of the modulus produces an error -message, since the operation is equivalent to dividing by 0. However, -dividing by a factor of a non-prime modulus does not produce an error -message. - -\endsection -\xitem[SGN indeterminate sign] -SGN - indeterminate sign (page 257) - -\endsection -\item[SHARE] -SHARE (page 197) - -The SHARE declaration allows access to its arguments by both -algebraic and symbolic modes. - - SHARE identifier{,identifier} - -identifier can be any valid REDUCE identifier. - -Programming in SYMBOLIC as well as algebraic mode allows you a wider -range of techniques than just algebraic mode alone. Expressions do -not cross the boundary since they have different representations, -unless the SHARE declaration is used. For more information on using -symbolic mode, see the REDUCE User's Manual, and the Standard Lisp -Report. - -You should be aware that a previously-declared array is destroyed by -the SHARE declaration. Scalar variables retain their values. You can -share a declared MATRIX that has not yet been dimensioned so that it -can be used by both modes. Values that are later put into the matrix -are accessible from symbolic mode too, but not by the usual matrix -reference mechanism. In symbolic mode, a matrix is stored as a list -whose first element is MAT, and whose next elements are the rows of -the matrix stored as lists of the individual elements. Access in -symbolic mode is by the operators FIRST, SECOND, THIRD and REST. - -\endsection -\item[SHOWRULES] -SHOWRULES (page 150) - - SHOWRULES(expression) or SHOWRULES simple_expression - -SHOWRULES returns in RULE-LIST form any OPERATOR rules associated with -its argument. - -Examples: -showrules log; {log(e) => 1, - - log(1) => 0, - - ~x - log(e ) => ~x, - - 1 - df(log(~x),~x) => ----} - ~x - -Such rules can then be manipulated further as with any LIST. For example -RHS FIRST WS; has the value 1. - -An operator may have properties that cannot be displayed in such a form, -such as the fact it is an odd function, or has a definition defined as a -procedure. - -\endsection -\item[SHOWTIME] -SHOWTIME (page 70) - -The SHOWTIME command prints the elapsed system time since the last -call of this command or since the beginning of the session, if it has -not been called before. - -Examples: -showtime; Time: 1020 ms - 2 -factorize(x^4 - 8x^4 + 8x^2 - 136x - 153); {X - 9,X + 17,X + 1} -showtime; Time: 920 ms - -The time printed is either the elapsed cpu time or the elapsed wall -clock time, depending on your system. SHOWTIME allows you to see the -system time resources REDUCE uses in its calculations. Your time -readings will of course vary from this example according to the system -you use. - -\endsection -\item[SHUT] -SHUT (pages 153--155) - -The SHUT command closes output files. - SHUT filename{,filename} - -filename must have been a file opened by OUT. - - -A file that has been opened by OUT must be SHUT before it is -brought in by IN. Files that have been opened by OUT should -always be SHUT before the end of the REDUCE session, to avoid either -loss of information or the printing of extraneous information into the file. -In most systems, terminating a session by BYE closes all open -output files. - -\endsection -\xitem[Side effect] -Side effect (page 48) - -\endsection -\xitem[side relations] -side relations (page 241) - -\endsection -\item[SIGN] -SIGN (page 75) - - SIGN expression - -SIGN tries to evaluate the sign of its argument. If this is possible -SIGN returns one of 1, 0 or -1. Otherwise, the result is the original -form or a simplified variant. - -Examples: - sign(-5) -1 - sign(-a^2*b) -SIGN(B) - -Even powers of formal expressions are assumed to be positive only as long -as the switch COMPLEX is off. - -\endsection -\xitem[SIGNATURE command] -SIGNATURE command (page 271) - -\endsection -\xitem[Simplification] -Simplification (pages 44, 97) - -\endsection -\xitem[SIMPSYS] -SIMPSYS (pages 378, 380, 383) - -\endsection -\item[SIN] -SIN (pages 76, 78) - -The SIN operator returns the sine of its argument. - - SIN(expression) or SIN simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -sin aa; SIN(AA) -sin(pi/2); 1 -on rounded; -sin 3; 0.14112000806 -sin(pi/2); 1.0 - -SIN returns a numeric value only if ROUNDED is on. Then the sine is -calculated to the current degree of floating point precision. The -argument in this case is assumed to be in radians. - -\endsection -\item[SINH] -SINH (pages 76, 78) - -The SINH operator returns the hyperbolic sine of its argument. The -derivative of SINH and some simple transformations are known to the -system. - - SINH(expression) or SINH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -sinh b; SINH(B) -sinh(0); 0 - 2 -df(sinh(x**2),x); 2*COSH(X )*X - COSH(4*X) -int(sinh(4*x),x); ----------- - 4 -on rounded; -sinh 4; 27.2899171971 - - -You may attach further functionality by defining its inverse (see -ASINH). A numeric value is not returned by SINH unless the switch -ROUNDED is on and its argument evaluates to a number. - -\endsection -\xitem[SMACRO] -SMACRO (page 196) - -\endsection -\item[SOLVE] -SOLVE (pages 84, 85, 90, 181) - -The SOLVE operator solves a single algebraic EQUATION or a system of -simultaneous equations. - - SOLVE(expression [ , kernel]) or - - SOLVE({expression,...} [ ,{ kernel ,...}] ) - -If the number of equations equals the number of distinct kernels, the -optional kernel argument(s) may be omitted. expression is either a -scalar expression or an EQUATION. When more than one expression is -given, the LIST of expressions is surrounded by curly braces. The -optional list of KERNELs follows, also in curly braces. - -Examples: -sss := solve(x^2 + 7); Unknown: X - SSS := {X= - SQRT(7)*I, - X=SQRT(7)*I} -rhs first sss; - SQRT(7)*I -solve(sin(x^2*y),y); - PI*(2*ARBINT(1) + 1) - {Y=----------------------, - 2 - X - - 2*ARBINT(1)*PI - Y=----------------} - 2 - X - -off allbranch; -solve(sin(x**2*y),y); {Y=0} -solve({3x + 5y = -4,2*x + y = -10},{x,y}); - 46 22 - {{x=-------,y=----}} - 7 7 -solve({x + a*y + z,2x + 5},{x,y}); - 5 - 2*z + 5 - {{x=------,y=------------}} - 2 2*a -ab := (x+2)^2*(x^6 + 17x + 1); - 8 7 6 3 2 - ab := x + 4*x + 4*x + 17*x + 69*x + 72*x + 4 - - 6 -www := solve(ab,x); {X=ROOT_OF(X_ + 17*X_ + 1),X=-2} -root_multiplicities; {1,2} - -Results of the SOLVE operator are returned as EQUATIONS in a LIST. -You can use the usual list access methods (FIRST, SECOND, THIRD, REST -and PART) to extract the desired equation, and then use the operators -RHS and LHS to access the right-hand or left-hand expression of the -equation. When SOLVE is unable to solve an equation, it returns the -unsolved part as the argument of ROOT_OF, with the variable renamed to -avoid confusion, as shown in the last example above. - -For one equation, SOLVE uses square-free factorisation, roots of -unity, and the known inverses of the LOG, SIN, COS, ACOS, ASIN, and -exponentiation operators. The quadratic, cubic and quartic formulas -are used if necessary, but these are applied only when the switch -FULLROOTS is set on; otherwise or when no closed form is available the -result is returned as ROOT_OF expression. The switch TRIGFORM -determines which type of cubic and quartic formula is used. The -multiplicity of each solution is given in a list as the system -variable ROOT_MULTIPLICITIES. For systems of simultaneous linear -equations, matrix inversion is used. For nonlinear systems, the -Groebner basis method is used. - -Linear equation system solving is influenced by the switch CRAMER. - -Singular systems can be solved when the switch SOLVESINGULAR is on, -which is the default setting. A message is given if the system of -equations is inconsistent. - -Related: ALLBRANCH switch, FULLROOTS switch, ROOTS operator, ROOT_OF -operator, TRIGFORM switch. - -\endsection -\item[SOLVESINGULAR] -SOLVESINGULAR (page 89) - -When SOLVESINGULAR is on, singular or under determined systems of -linear equations are solved, using arbitrary real, complex or integer -variables in the answer. Default is ON. - -Examples: - - ARBCOMPLEX(1) -solve({2x + y,4x + 2y},{x,y}); {{X=------------------,Y=ARBCOMPLEX(1)}} - 2 - - 8*arbcomplex(2) -solve({7x + 15y - z,x - y - z},{x,y,z});{{x=-----------------, - 11 - - - 3*ARBCOMPLEX(2) - Y=--------------------, - 11 - - Z=ARBCOMPLEX(2)}} - -off solvesingular; -solve({2x + y,4x + 2y},{x,y}); ***** SOLVE given singular equations -solve({7x + 15y - z,x - y - z},{x,y,z});***** SOLVE given singular equations - -The integer following the identifier ARBCOMPLEX above is assigned by -the system, and serves to identify the variable uniquely. It has no other -significance. - -\endsection -\xitem[SORTOUTODE] -SORTOUTODE (page 350) - -\endsection -\xitem[SPACEDIM command] -SPACEDIM command (pages 251, 271) - -\endsection -\item[SPDE] -SPDE (page 185) - -Author: Fritz Schwartz - -The package SPDE provides a set of functions which may be used to -determine the symmetry group of Lie- or point-symmetries of a given system of -partial differential equations. In many cases the determining system is -solved completely automatically. In other cases the user has to provide -additional input information for the solution algorithm to terminate. - - -\endsection -\xitem[SPECFN] -SPECFN (page 185) - -\endsection -\xitem[SPECFN package] -SPECFN package (page 391) - -\endsection -\xitem[SPECFN2] -SPECFN2 (page 187) - -\endsection -\xitem[spherical coordinates] -spherical coordinates (pages 265, 355) - -\endsection -\item[SPLIT_FIELD] -SPLIT_FIELD function (page 227) - -SPLIT_FIELD is part of the ARNUM package for algebraic numbers. It -calculates a primitive element of minimal degree for which a given -polynomial splits into linear factors. The algorithm as described by -Trager. - -Example: - load arnum; - split!_field(x**3-3*x+7); - - *** Splitting field is generated by: - - 6 4 2 - A5 - 18*A5 + 81*A5 + 1215 - - - - 4 2 - {1/126*A5 - 5/42*A5 - 1/2*A5 + 2/7, - - - 4 2 - - (1/63*A5 - 5/21*A5 + 4/7), - - - 4 2 - 1/126*A5 - 5/42*A5 + 1/2*A5 + 2/7} - - - for each j in ws product (x-j); - - 3 - X - 3*X + 7 - - -\endsection -\item[SPUR] -SPUR (page 210) - -The SPUR declaration removes the special exemption from trace -calculations that was declared by NOSPUR, in high-energy physics -calculations. - SPUR line-id{,line-id} - -line-id must be a line-identifier that has previously been declared -NOSPUR. - -\endsection -\xitem[SQFRF] -SQFRF (page 373) - -\endsection -\item[SQRT] -SQRT (pages 76, 78) - -The SQRT operator returns the square root of its argument. - - SQRT(expression) - -expression can be any REDUCE scalar expression. - -Examples: -sqrt(16*a^3); 4*SQRT(A)*A -sqrt(17); SQRT(17) -on rounded; -sqrt(17); 4.12310562562 -off rounded; 2 -sqrt(a*b*c^5*d^3*27); 3*SQRT(D)*SQRT(C)*SQRT(B)*SQRT(A)*SQRT(3)*C *D - -SQRT checks its argument for squared factors and removes them. - -Numeric values for square roots that are not exact integers are given -only when ROUNDED is on. - -Please note that SQRT(A**2) is given as A, which may be incorrect if A -eventually has a negative value. If you are programming a calculation -in which this is a concern, you can turn on the PRECISE switch, which -causes the absolute value of the square root to be returned. - -\endsection -\xitem[Standard form] -Standard form (page 198) - -\endsection -\xitem[Standard quotient] -Standard quotient (page 198) - -\endsection -\xitem[Statement] -Statement (page 53) - -\endsection -\xitem[Stirling Numbers] -Stirling Numbers (page 185, 394) - -\endsection -\item[STIRLING1] -STIRLING1 (pages 185, 394) - -The STIRLING1 operator returns the Stirling Numbers S(n,m) of the -first kind, i.e. the number of permutations of n symbols which have -exactly m cycles (divided by (-1)**(n-m)). - - STIRLING1(integer,integer) - -Examples: - load_package specfn; (SPECFN) - Stirling1 (17,4); -87077748875904 - - GAMMA(N + 1) - Stirling1 (n,n-1); ----------------- - 2*GAMMA(N - 1) - -The operator STIRLING1 evaluates the Stirling numbers of the first -kind by rulesets for special cases or by a computing the closed form, -which is a series involving the operators BINOMIAL and STIRLING2. - -\endsection -\item[STIRLING2] -STIRLING2 (pages 185, 394) - -The STIRLING1 operator returns the Stirling Numbers S(n,m) of the -second kind, i.e. the number of ways of partitioning a set of n elements -into m non-empty subsets. - - STIRLING2(integer,integer) - -Examples: - load_package specfn; (SPECFN) - Stirling2 (17,4); 694337290 - GAMMA(N + 1) - Stirling2 (n,n-1); ---------------- - 2*GAMMA(N - 1) - -The operator STIRLING2 evaluates the Stirling numbers of the second -kind by rulesets for special cases or by a computing the closed form. - -\endsection -\item[String] -String (page 37)) -A STRING is any collection of characters enclosed in double quotation -marks ("). It may be used as an argument for a variety of commands -and operators, such as IN, REDERR and WRITE. -Examples: -write "this is a string"; this is a string -write a, " ", b, " ",c,"!"; A B C! - -\endsection -\item[STRUCTR] -STRUCTR (pages 112, 113) - -The STRUCTR operator breaks its argument expression into named -subexpressions. - - STRUCTR(expression [,identifier[,identifier ...]]) - -expression may be any valid REDUCE scalar expression. identifier may -be any valid REDUCE IDENTIFIER. The first identifier is the stem for -subexpression names, the second is the name to be assigned to the -structured expression. - -Examples: -structr(sqrt(x**2 + 2*x) + sin(x**2*z)); ANS1*ANS3 + ANS2 - - WHERE - - 1/2 - ANS3 := X - - 2 - ANS2 := SIN(X *Z) - - 1/2 - ANS1 := (X + 2) - -ans3; ANS3 -on fort; -structr((x+1)**5 + tan(x*y*z),var,aa); - VAR1=TAN(X*Y*Z) - AA=VAR1+X**5+5.*X**4+10.*X**3+10.*X**2+5.*X+1. - -The second argument to STRUCTR is optional. If it is not given, the -default stem ANS is used by REDUCE to construct names for the -subexpression. The names are only for display purposes: REDUCE does -not store the names and their values unless the switch SAVESTRUCTR is -on. - -If a third argument is given, the structured expression as a whole is -named by this argument, when FORT is on. The expression is not stored -under this name. You can send these structured Fortran expressions to -a file with the OUT command. - -\endsection -\xitem[Structuring] -Structuring (page 97) - -\endsection -\xitem[Struve Functions] -Struve Functions (pages 185, 397) - -\endsection -\item[STRUVEH] -STRUVEH (pages 185, 397) - -The STRUVEH operator returns Struve's H function. - - STRUVEH(order,argument) - -Examples: -load_package specfn; (SPECFN) - - 3 - - BESSELJ(---,X) - 2 -struveh(-3/2,x); ------------------- - I - - -There is currently no numeric support for the operator STRUVEH. - -\endsection -\item[STRUVEL] -STRUVEL (pages 185, 397) - -The STRUVEL operator returns the modified Struve L function . - - STRUVEL(order,argument) - -Examples: - load_package specfn; (SPECFN); - 3 - struvel(-3/2,x); BESSELI(---,X) - 2 - -There is currently no numeric support for the operator STRUVEL. - -\endsection -\xitem[Sturm Sequences] -Sturm Sequences (page 369) - -\endsection -\item[SUB] -SUB (page 137) - -The SUB operator substitutes a new expression for a kernel in an -expression. - - SUB(kernel=expression {,kernel=expression} expression) - or - SUB({kernel=expression, kernel=EXPRESSION},expression}) - -kernel must be a KERNEL, expression can be any REDUCE scalar -expression. - -Examples: -sub(x=3,y=4,(x+y)**3); 343 -x; X -sub({cos=sin,sin=cos},cos a+sin b} COS(B) + SIN(A) - -Note in the second example that operators can be replaced using the -SUB operator. - -\endsection -\xitem[SUCH THAT] -SUCH THAT (page 142) - -\endsection -\item[SUM] -SUM (pages 57, 58, 187) - -The operator SUM returns -the indefinite or definite summation of a given expression. - - -SUM(expr,k[,lolim [,uplim ]]) - - -where expr is the expression to be added, k is the -control variable (a KERNEL), and lolim and uplim -uplim are the optional lower and upper limits. If uplim is -not supplied the upper limit is taken as k. The GOSPER -algorithm is used. If there is no closed form solution, the operator -returns the input unchanged. - -Examples: - 2 2 -sum(4n**3,n); N *(N + 2*N + 1) - -sum(2a+2k*r,k,0,n-1); N*(2*A + N*R - R) - -\endsection -\xitem[SUM-SQ] -SUM-SQ (page 404) - -\endsection -\xitem[SVEC] -SVEC (page 355) - -\endsection -\xitem[Switch] -Switch (pages 68, 69) - -\endsection -\item[SYMBOLIC] -SYMBOLIC (page 191) - -The SYMBOLIC command changes REDUCE's mode of operation to symbolic. -When SYMBOLIC is followed by an expression, that expression is -evaluated in symbolic mode, but REDUCE's mode is not changed. It is -equivalent to the LISP command. - -Examples: -symbolic; NIL -cdr '(a b c); (B C) -algebraic; -x + symbolic car '(y z); X + Y - -\endsection -\xitem[Symbolic mode] -Symbolic mode (pages 191, 193, 197, 198) - -\endsection -\xitem[Symbolic procedure] -Symbolic procedure (page 196) - -\endsection -\item[SYMMETRIC] -SYMMETRIC (page 93) - -When an operator is declared SYMMETRIC, its arguments are reordered -to conform to the internal ordering of the system. - - SYMMETRIC identifier{,identifier} - -identifier is an identifier that has been declared an operator. - -Examples: -operator m,n; -symmetric m,n; -m(y,a,sin(x)); M(SIN(X),A,Y) -n(z,m(b,a,q)); N(M(A,B,Q),Z) - -If identifier has not been declared to be an operator, the flag -SYMMETRIC is still attached to it. When identifier is subsequently -used as an operator, the message - DECLARE identifier OPERATOR ? (Y OR N) -is printed. If the user replies Y, the symmetric property of the -operator is used. - -\endsection -\xitem[system precision] -system precision (page 374) - -\endsection -\item[T] -T (page 37) - -The constant T stands for the truth value true. It cannot be used as -a scalar variable in a BLOCK, as a looping variable in a FOR statement -or as an OPERATOR name. - -\endsection -\item[TAN] -TAN (pages 76, 78, 81) - -The TAN operator returns the tangent of its argument. - - TAN(expression) or TAN simple_expression - -expression is any valid scalar REDUCE expression, simple_expression is -a single identifier or begins with a prefix operator name. - -Examples: -tan a; TAN(A) -tan(pi/3); SQRT(3) -on rounded; -tan(pi/3); 1.73205080757 - -TAN returns a numeric value only if ROUNDED is on. Then the tangent -is calculated to the current degree of floating point accuracy. - -When ON ROUNDED is in force, no check is made to see if the argument -to TAN is a multiple of pi/2, for which the tangent goes to positive -or negative infinity. (Of course, since REDUCE uses a fixed-point -representation of pi/2, it produces a large but not infinite number). -You need to make a check for multiples of pi/ in any program you use -that might possibly ask for the tangent of such a quantity. - -\endsection -\xitem[tangent vector] -tangent vector (page 252) - -\endsection -\item[TANH] -TANH (pages 76, 78) - -The TANH operator returns the hyperbolic tangent of its argument. The -derivative of TANH and some simple transformations are known to the -system. - - TANH(expression) or TANH simple_expression - -expression may be any scalar REDUCE expression, not an array, matrix -or vector expression. simple_expression must be a single identifier or -begin with a prefix operator name. - -Examples: -tanh b; TANH(B) -tanh(0); 0 - 2 -df(tanh(x*y),x); Y*( - TANH(X*Y) + 1) - 2*X -int(tanh(x),x); LOG(E + 1) - X -on rounded; -tanh 2; 0.964027580076 - -You may attach further functionality by defining its inverse (see -ATANH). A numeric value is not returned by TANH unless the switch -ROUNDED is on and its argument evaluates to a number. - -\endsection -\item[TAYLOR] -TAYLOR (page 188, 406) - -The TAYLOR operator is used for expanding an expression into a Taylor -series. - -TAYLOR(expression, var, expression, number) -TAYLOR(expression, var, expression, number {,var, expression, number}) - -expression can be any valid REDUCE algebraic expression. var must be -a KERNEL, and is the expansion variable. The expression following it -denotes the point about which the expansion is to take place. number -must be a non-negative integer and denotes the maximum expansion -order. If more than one triple is specified TAYLOR will expand its -first argument independently with respect to all the variables. Note -that once the expansion has been done it is not possible to calculate -higher orders. - -Instead of a KERNEL, var may also be a list of kernels. In this case -expansion will take place in a way so that the sum of the degrees of -the kernels does not exceed the maximum expansion order. If the -expansion point evaluates to the special identifier INFINITY, TAYLOR -tries to expand in a series in 1/var. - -The expansion is performed variable per variable, i.e. in the example -below by first expanding exp(x^2+y^2) with respect to x and then -expanding every coefficient with respect to y. - -Examples: - 2 2 2 2 3 3 -taylor(e^(x^2+y^2),x,0,2,y,0,2); 1 + Y + X + Y *X + O(X ,Y ) - - 2 2 3 -taylor(e^(x^2+y^2),{x,y},0,2); 1 + Y + X + O({X,Y} ) - -taylor(x*y/(x+y),x,0,2,y,0,2); ***** Not a unit in argument to quottaylor - -Note that it is not generally possible to apply the standard REDUCE -operators to a Taylor kernel. For example, PART, COEFF, or COEFFN -cannot be used. Instead, the expression at hand has to be converted -to standard form first using the TAYLORTOSTANDARD operator. - -Differentiation of a Taylor expression is possible. If you -differentiate with respect to one of the Taylor variables the order -will decrease by one. - -Substitution is a bit restricted: Taylor variables can only be -replaced by other kernels. There is one exception to this rule: you -can always substitute a Taylor variable by an expression that -evaluates to a constant. Note that REDUCE will not always be able to -determine that an expression is constant: an example is sin(acos(4)). - -Only simple taylor kernels can be integrated. More complicated -expressions that contain Taylor kernels as parts of themselves are -automatically converted into a standard representation by means of the -TAYLORTOSTANDARD operator. In this case a suitable warning is -printed. - -\endsection -\xitem[TAYLOR package] -TAYLOR package (page 405) - -\endsection -\xitem[Taylor series arithmetic] -Taylor series - arithmetic (page 407) - differentiation (page 408) - integration (page 408) - reversion (page 408) - substitution (page 408) - -\endsection -\item[TAYLORAUTOCOMBINE] -TAYLORAUTOCOMBINE switch (page 408) - -If you set TAYLORAUTOCOMBINE to ON, REDUCE automatically combines -Taylor expressions during the simplification process. This is -equivalent to applying TAYLORCOMBINE to every expression that contains -Taylor kernels. Default is ON. - -\endsection -\item[TAYLORAUTOEXPAND] -TAYLORAUTOEXPAND switch (pages 408, 409) - -TAYLORAUTOEXPAND makes Taylor expressions ``contagious'' in the sense -that TAYLORCOMBINE tries to Taylor expand all non-Taylor -subexpressions and to combine the result with the rest. Default is -OFF. - -\endsection -\item[TAYLORCOMBINE] -TAYLORCOMBINE (page 407) - -This operator tries to combine all Taylor kernels found in its -argument into one. Operations currently possible are: - -Addition, subtraction, multiplication, and division. -Roots, exponentials, and logarithms. -Trigonometric and hyperbolic functions and their inverses. - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 - 3 -taylorcombine log hugo; X + O(X ) - - 1 2 3 -taylorcombine(hugo + x); (1 + X + ---*X + O(X )) + X - 2 -on taylorautoexpand; - 1 2 3 -taylorcombine(hugo + x); 1 + 2*X + ---*X + O(X ) - 2 - -Application of unary operators like LOG and ATAN will nearly always -succeed. For binary operations their arguments have to be Taylor -kernels with the same template. This means that the expansion -variable and the expansion point must match. Expansion order is not -so important, different order usually means that one of them is -truncated before doing the operation. - -If TAYLORKEEPORIGINAL is set to ON and if all Taylor kernels in its -argument have their original expressions kept TAYLORCOMBINE will also -combine these and store the result as the original expression of the -resulting Taylor kernel. There is also the switch TAYLORAUTOEXPAND. - -There are a few restrictions to avoid mathematically undefined -expressions: it is not possible to take the logarithm of a Taylor -kernel which has no terms (i.e. is zero), or to divide by such a -beast. There are some provisions made to detect singularities during -expansion: poles that arise because the denominator has zeros at the -expansion point are detected and properly treated, i.e. the Taylor -kernel will start with a negative power. (This is accomplished by -expanding numerator and denominator separately and combining the -results.) Essential singularities of the known functions (see above) -are handled correctly. - -\endsection -\item[TAYLORKEEPORIGINAL] -TAYLORKEEPORIGINAL (pages 406, 407, 409, 411) - -TAYLORKEEPORIGINAL, if set to ON, forces the TAYLOR and all Taylor -kernel manipulation operators to keep the original expression, -i.e. the expression that was Taylor expanded. All operations -performed on the Taylor kernels are also applied to this expression -which can be recovered using the operator TAYLORORIGINAL. Default is -OFF. - -\endsection -\item[TAYLORORIGINAL] -TAYLORORIGINAL (pages 411, 412) - -TAYLORORINAL can recover the original expression (the one that was -expanded) from the Taylor kernel that is given as its argument. - - TAYLORORIGINAL(expression) - TAYLORORIGINAL simple_expression - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 -taylororiginal hugo; - ***** Taylor kernel doesn't have an original part in taylororiginal - -on taylorkeeporiginal; - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 - X -taylororiginal hugo; E - -An error is signalled if the argument is not a Taylor kernel or if the -original expression was not kept, i.e. if TAYLORKEEPORIGINAL was set -OFF during expansion. - -\endsection -\item[TAYLORPRINTORDER] -TAYLORPRINTORDER switch (page 409) - -TAYLORPRINTORDER, if set to ON, causes the remainder to be printed in -big-O notation. Otherwise, three dots are printed. Default is -ON. - -\endsection -\item[TAYLORPRINTTERMS] -TAYLORPRINTTERMS (pages 406, 412) - -Only a certain number of (non-zero) coefficients are printed. If there -are more, an expression of the form N TERMS is printed to indicate how -many non-zero terms have been suppressed. The number of terms printed -is given by the value of the shared algebraic variable -TAYLORPRINTTERMS. Allowed values are integers and the special -identifier ALL. The latter setting specifies that all terms are to be -printed. The default setting is 5. - -Examples: -taylor(e^(x^2+y^2),x,0,4,y,0,4); - 2 1 4 2 2 2 5 5 - 1 + Y + ---*Y + X + Y *X + (4 TERMS) + O(X ,Y ) - 2 -taylorprintterms := all; - ALL -taylor(e^(x^2+y^2),x,0,4,y,0,4); - 2 1 4 2 2 2 1 4 2 1 4 1 2 4 - 1 + y + ---*y + x + y *x + ---*y *x + ---*x + ---*y *x - 2 2 2 2 - - 1 4 4 5 5 - + ---*y *x + O(x ,y ) - 4 - -\endsection -\item[TAYLORREVERT] -TAYLORREVERT (page 411) - -TAYLORREVERT allows reversion of a Taylor series of a function f, -i.e., to compute the first terms of the expansion of the inverse of f -from the expansion of f. - - TAYLORREVERT(expression, var, var) - -The first argument must evaluate to a Taylor kernel with the second -argument being one of its expansion variables. - -Examples: - 2 6 -taylor(u - u**2,u,0,5); U - U + O(U ) - 2 3 4 5 6 -taylorrevert(ws,u,x); X + X + 2*X + 5*X + 14*X + O(X ) - -\endsection -\item[TAYLORSERIESP] -TAYLORSERIESP (page 407) - -The TAYLORSERIESP operator may be used to determine if its argument is -a Taylor kernel. - - TAYLORSERIESP(expression) - TAYLORSERIESP simple_expression - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 -if taylorseriesp hugo then OK; OK -if taylorseriesp(hugo + y) then OK else NO; NO - -Note that this operator is subject to the same restrictions as, e.g., -ORDP or NUMBERP, i.e. it may only be used in boolean expressions in IF -or LET statements. -\endsection -\item[TAYLORTEMPLATE] -TAYLORTEMPLATE (pages 407, 412) - -The template of a Taylor kernel, i.e. the list of all variables with -respect to which expansion took place together with expansion point -and order can be extracted using - - TAYLORTEMPLATE(expression) - TAYLORTEMPLATE simple_expression - -The operator returns a list of lists with the three elements -(VAR,VAR0,ORDER). An error is signalled if the argument is not a -Taylor kernel. - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 -taylortemplate hugo; {{X,0,2}} - -\endsection -\item[TAYLORTOSTANDARD] -TAYLORTOSTANDARD (page 407) - -The TAYLORTOSTANDARD operator converts all Taylor kernels in its -argument into standard form and resimplifies the result. - - TAYLORTOSTANDARD(expression) - TAYLORTOSTANDARD simple_expression - -Examples: - 1 2 3 -hugo := taylor(exp(x),x,0,2); HUGO := 1 + X + ---*X + O(X ) - 2 - 2 - X + 2*X + 2 -taylortostandard hugo; -------------- - 2 -\endsection -\xitem[Terminator] -Terminator (page 53) - -\endsection -\item[THIRD] -THIRD (page 50) - -The THIRD operator returns the third item of a LIST. - THIRD(list) or THIRD list - - - -list must be a list containing at least three items to avoid an error -message. - -Examples: -alist := {a,b,c,d}; ALIST := {A,B,C,D} -third alist; C -blist := {x,{aa,bb,cc},y,z}; BLIST := {X,{AA,BB,CC},Y,Z}; -third second blist; CC -third blist; Y - -\endsection -\item[TIME] -TIME (page 68) - -When TIME is on, the system time used in executing each REDUCE -statement is printed after the answer is printed. - -Examples: -on time; Time: 4940 ms - 2 -df(sin(x**2 + y),y); COS(X + Y ) - Time: 180 ms -solve(x**2 - 6*y,x); {X= - SQRT(Y)*SQRT(6), - X=SQRT(Y)*SQRT(6)} - Time: 320 ms - -When TIME is first turned on, the time since the beginning of the -REDUCE session is printed. After that, the time used in computation, -(usually in milliseconds, though this is system dependent) is printed -after the results of each command. Idle time or time spent typing in -commands is not counted. If TIME is turned off, the first reading -after it is turned on again gives the time elapsed since it was turned -off. The time printed is CPU or wall clock time, depending on the -system. - -\endsection -\item[TORDER] -TORDER (pages 296, 315, 316) - -The operator TORDER sets the actual term order. - -1. simple term order: - TORDER m - -where m is the name of a term order mode LEX term order, GRADLEX term -order, REVGRADLEX term order or another implemented parameterless -mode. - -2. stepped term order: - TORDER m,n - TORDER {m,n} - -where m is the name of a two step term order, one of GRADLEXGRADLEX -term order, GRADLEXREVGRADLEX term order, LEXGRADLEX term order or -LEXREVGRADLEX term order, and n is a positive integer. - -3. weighted term order - TORDER WEIGHTED, n,n,... - TORDER WEIGHTED, {n,n,...} - -where the n are positive integers, see weighted term order. - -TORDER sets the term order mode. The default mode is LEX. The -previous order mode is returned. - -\endsection -\item[TP] -TP (page 165) - -The TP operator returns the transpose of its MATRIX - argument. - TP identifier or TP(identifier) - -identifier must be a matrix, which either has had its dimensions set -in its declaration, or has had values put into it by MAT. - -Examples: -matrix m,n; -m := mat((1,2,3),(4,5,6))$ -n := tp m; N(1,1) := 1 - N(1,2) := 4 - N(2,1) := 2 - N(2,2) := 5 - N(3,1) := 3 - N(3,2) := 6 - -In an assignment statement involving TP, the matrix identifier on the -left-hand side is redimensioned to the correct size for the transpose. - -\endsection -\item[TPS] -TPS (pages 188, 330) - -Authors: Alan Barnes and Julian Padget - -A Truncated Power Series Package. - -This package implements formal Laurent series expansions in one -variable using the domain mechanism of REDUCE. This means that power -series objects can be added, multiplied, differentiated etc., like -other first class objects in the system. A lazy evaluation scheme -is used and thus terms of the series are not evaluated until they -are required for printing or for use in calculating terms in other -power series. The series are extendible giving the user the -impression that the full infinite series is being manipulated. The -errors that can sometimes occur using series that are truncated at -some fixed depth (for example when a term in the required series -depends on terms of an intermediate series beyond the truncation -depth) are thus avoided. - -\endsection -\xitem[TRA] -TRA (page 178) - -\endsection -\item[TRACE] -TRACE (page 166) - -The TRACE operator finds the trace of its MATRIX argument. - TRACE(expression) or TRACE simple_expression - -expression or simple_expression must evaluate to a square -matrix. - -Examples: -matrix a; -a := mat((x1,y1),(x2,y2))$ -trace a; X1 + Y2 - -The trace is the sum of the entries along the diagonal of a square matrix. -Given a non-matrix expression, or a non-square matrix, TRACE returns -an error message. - -\endsection -\xitem[tracing EXCALC] -tracing - EXCALC (page 266) - ODESOLVE (page 351) - ROOTS package (page 373) - SPDE package (page 380) - SUM package (page 404) - -\endsection -\item[TRALLFAC] -TRALLFAC - -When TRALLFAC is on, a more detailed trace of factoriser calls is -generated. - - -The TRALLFAC switch takes precedence over TRFAC if they are -both on. TRFAC gives a factorisation trace with less detail in it. -When the FACTOR switch is on also, all input polynomials are sent to -the factoriser automatically and trace information is generated. The -OUT command saves the results of the factoring, but not the trace. - - -\endsection -\item[TRFAC] -TRFAC (page 122) - -When TRFAC is on, a narrative trace of any calls to the factoriser is -generated. Default is OFF. - - -When the switch FACTOR is on, and TRFAC is on, every input -polynomial is sent to the factoriser, and a trace generated. With -FACTOR off, only polynomials that are explicitly factored with the -command FACTORIZE generate trace information. - -The OUT command saves the results of the factoring, but not -the trace. The TRALLFAC switch gives trace information to a -greater level of detail. - -\endsection -\item[TRGROEB] -TRGROEB (pages 299, 303) - -If TRGROEB is on, intermediate H polynomials are printed during a -GROEBNER or GROEBNERF calculation. - -\endsection -\xitem[TRGROEB1] -TRGROEB1 (pages 299, 303) - -\endsection -\xitem[TRGROEBR] -TRGROEBR (page 304) - -\endsection -\item[TRGROEBS] -TRGROEBS (pages 299, 303) - -If TRGROEBS is on, intermediate H and S polynomials are printed during -a GROEBNER or GROEBNERF calculation. - -\endsection -\item[TRIGFORM] -TRIGFORM (page 87) - -When FULLROOTS is on, SOLVE will compute the -roots of a cubic or quartic polynomial is closed form. When -TRIGFORM is on, the roots will be expressed by trigonometric -forms. Otherwise nested surds are used. Default is ON. - -\endsection -\item[TRINT] -TRINT (page 178) - -When TRINT is on, a narrative tracing various steps in the -integration process is produced. - -The OUT command saves the results of the integration, but not the -trace. - -\endsection -\item[TRNONLNR] -TRNONLNR - -When TRNONLNR is on, a narrative tracing various steps in -the process for solving non-linear equations is produced. - - -TRNONLNR can only be used after the solve package has been loaded -(e.g., by an explicit call of LOAD_PACKAGE). The OUT -command saves the results of the equation solving, but not the trace. - -\endsection -\xitem[TRODE] -TRODE (page 351) - -\endsection -\xitem[TRROOT] -TRROOT (page 373) - -\endsection -\xitem[TRSUM] -TRSUM (page 404) - -\endsection -\xitem[truncated power series] -truncated power series (page 413) - -\endsection -\xitem[TVECTOR command] -TVECTOR command (pages 249, 271) - -\endsection -\xitem[U(ALFA)] -U(ALFA) (page 379) - -\endsection -\xitem[U(ALFA] -U(ALFA,I) (page 379) - -\endsection -\item[UNTIL] -UNTIL (page 57) - -See the FOR loop construction. -\endsection -\xitem[User packages] -User packages (page 177) - -\endsection -\xitem[VARDF] -VARDF (pages 257, 271) - -\endsection -\xitem[Variable] -Variable (page 36) - -\endsection -\xitem[Variable elimination] -Variable elimination (page 181) - -\endsection -\xitem[variational derivative] -variational derivative (page 257) - -\endsection -\item[VARNAME] -VARNAME (pages 111, 112) - -The declaration VARNAME instructs REDUCE to use its argument as the -default Fortran (when FORT is on) or STRUCTR identifier and identifier -stem, rather than using ANS. - - VARNAME identifier - -identifier can be any combination of one or more alphanumeric -characters. Try to avoid REDUCE reserved words. - -Examples: -varname ident; IDENT -on fort; -x**2 + 1; IDENT=X**2+1. - -off fort,exp; 3 -structr(((x+y)**2 + z)**3); IDENT2 - where - 2 - IDENT2 := IDENT1 + Z - IDENT1 := X + Y - -EXP was turned off so that STRUCTR could show the structure. If EXP -had been on, the expression would have been expanded into a -polynomial. - -\endsection -\xitem[VDF] -VDF (page 359) - -\endsection -\xitem[VEC command] -VEC command (page 232) - -\endsection -\item[VECDIM] -VECDIM (page 212) - -The SPUR declaration removes the special exemption from trace -calculations that was declared by NOSPUR, in high-energy physics -calculations. - SPUR line-id{,line-id} - -line-id must be a line-identifier that has previously been declared -NOSPUR. - -\endsection -\item[VECTOR] -VECTOR (High Energy Physics) (page 208) - -The VECTOR declaration declares that its arguments are of type VECTOR. - VECTOR identifier{,identifier} - -identifier must be a valid REDUCE identifier. It may have already -been used for a matrix, array, operator or scalar variable. After an -identifier has been declared to be a vector, it may not be used as a -scalar variable. - -Vectors are special entities for high-energy physics calculations. -You cannot put values into their coordinates; they do not have -coordinates. They are legal arguments for the high-energy physics -operators EPS, G and . (dot). Vector variables are used to represent -gamma matrices and gamma matrices contracted with Lorentz 4-vectors, -since there are no Dirac variables per se in the system. Vectors do -follow the usual vector rules for arithmetic operations: + and - -operate upon two or more vectors, producing a vector; * and / cannot -be used between vectors; the scalar product is represented by the -. operator; and the product of a scalar and vector expression is well -defined, and is a vector. - -You can represent components of vectors by including representations -of unit vectors in your system. For instance, letting E0 represent -the unit vector (1,0,0,0), the command - -V1.E0 := 0; - -would set up the substitution of zero for the first component of the -vector V1. - -Identifiers that are declared by the INDEX and MASS declarations are -automatically declared to be vectors. - -The following errors can occur in calculations using the high energy -physics package: - -A REPRESENTS ONLY GAMMA5 IN VECTOR EXPRESSIONS -You have tried to use A in some way other than gamma5 in a high-energy -physics expression. - -GAMMA5 NOT ALLOWED UNLESS VECDIM IS 4 -You have used gamma_5 in a high-energy physics computation involving a -vector dimension other than 4. - -ID HAS NO MASS -One of the arguments to MSHELL has had no mass assigned to it, in -high-energy physics calculations. - -MISSING ARGUMENTS FOR G OPERATOR -A line symbol is missing in a gamma matrix expression in high-energy physics -calculations. - -UNMATCHED INDEX list -The parser has found unmatched indices during the evaluation of a -gamma matrix expression in high-energy physics calculations. - -\endsection -\xitem[vector] -vector , integration 233 - addition (page 356) - cross product (page 357) - differentiation (page 233) - division (page 357) - dot product (page 357) - exponentiation (page 357) - inner product (page 357) - modulus (page 357) - multiplication (page 357) - subtraction (page 356) - -\endsection -\xitem[vector algebra] -vector algebra (page 231) - -\endsection -\xitem[VECTORADD] -VECTORADD (page 356) - -\endsection -\xitem[VECTORCROSS] -VECTORCROSS (page 357) - -\endsection -\xitem[VECTORDIFFERENCE] -VECTORDIFFERENCE (page 356) - -\endsection -\xitem[VECTOREXPT] -VECTOREXPT (page 357) - -\endsection -\xitem[VECTORMINUS] -VECTORMINUS (page 356) - -\endsection -\xitem[VECTORPLUS] -VECTORPLUS (page 356) - -\endsection -\xitem[VECTORQUOTIENT] -VECTORQUOTIENT (page 357) - -\endsection -\xitem[VECTORRECIP] -VECTORRECIP (page 357) - -\endsection -\xitem[VECTORTIMES] -VECTORTIMES (page 357) - -\endsection -\xitem[VERBOSELOAD switch] -VERBOSELOAD switch (page 409) - -\endsection -\xitem[VINT] -VINT (page 360) - -\endsection -\xitem[VMOD] -VMOD (page 357) - -\endsection -\xitem[VMOD operator] -VMOD operator (page 233) - -\endsection -\xitem[VOLINT] -VOLINT (page 360) - -\endsection -\xitem[VOLINTEGRAL function] -VOLINTEGRAL function (page 237) - -\endsection -\xitem[VOLINTORDER vector] -VOLINTORDER vector (page 237) - -\endsection -\xitem[VORDER] -VORDER (page 359) - -\endsection -\xitem[VOUT] -VOUT (page 355) - -\endsection -\xitem[VSTART] -VSTART (page 354) - -\endsection -\xitem[VTAYLOR] -VTAYLOR (page 359) - -\endsection -\xitem[wedge] -wedge (page 271) - -\endsection -\item[WEIGHT] -WEIGHT (page 152) - -The WEIGHT command is used to attach weights to kernels for asymptotic -constraints. - - WEIGHT kernel = number - -kernel must be a REDUCE KERNEL, number must be a positive integer, not -0. - -Examples: 4 3 2 2 3 4 -a := (x+y)**4; A := X + 4*X *Y + 6*X *Y + 4*X*Y + Y -weight x=2,y=3; -wtlevel 8; - 4 -a; X -wtlevel 10; - 2 2 2 -a; X *(6*Y + 4*X*Y + X ) -int(x**2,x); ***** X invalid as KERNEL - -Weights and WTLEVEL are used for asymptotic constraints, where -higher-order terms are considered insignificant. - -Weights are originally equivalent to 0 until set by a WEIGHT command. -To remove a weight from a kernel, use the CLEAR command. Weights once -assigned cannot be changed without clearing the identifier. Once a -weight is assigned to a kernel, it is no longer a kernel and cannot be -used in any REDUCE commands or operators that require kernels, until -the weight is cleared. Note that terms are ordered by greatest -weight. - -The weight level of the system is set by WTLEVEL, initially at 2. -Since no kernels have weights, no effect from WTLEVEL can be seen. -Once you assign weights to kernels, you must set WTLEVEL correctly for -the desired operation. When weighted variables appear in a term, -their weights are summed for the total weight of the term (powers of -variables multiply their weights). When a term exceeds the weight -level of the system, it is discarded from the result expression. - -\endsection -\xitem[weighted ordering] -weighted ordering (page 316) - -\endsection -\item[WHEN] -WHEN (page 147) - -The WHEN operator is used inside a RULE to make the -execution of the rule depend on a boolean condition which is -evaluated at execution time. For the use see RULE. - -\endsection -\item[WHERE] -WHERE (page 148) - -The WHERE operator provides an infix notation for one-time -substitutions for kernels in expressions. - - expression WHERE kernel = expression{,kernel = expression} - -expression can be any REDUCE scalar expression, kernel must be a -KERNEL. Alternatively a RULE or a RULE LIST can be a member of the -right-hand part of a WHERE expression. - -Examples: -x**2 + 17*x*y + 4*y**2 where x=1,y=2; - 51 -for i := 1:5 collect x**i*q where q= for j := 1:i product j; - 2 3 4 5 - {X,2*X ,6*X ,24*X ,120*X } - 2 3 -x**2 + y + z where z=y**3,y=3; X + Y + 3 - -Substitution inside a WHERE expression has no effect upon the values -of the kernels outside the expression. The WHERE operator has the -lowest precedence of all the infix operators, which are lower than -prefix operators, so that the substitutions apply to the entire -expression preceding the WHERE operator. However, WHERE is applied -before command keywords such as THEN, REPEAT, or DO. - -A RULE or a RULE SET in the right-hand part of the WHERE expression -act as if the rules were activated by LET immediately before the -evaluation of the expression and deactivated by CLEARRULES immediately -afterwards. - -WHERE gives you a natural notation for auxiliary variables in -expressions. As the second example shows, the substitute expression -can be a command to be evaluated. The substitute assignments are made -in parallel, rather than sequentially, as the last example shows. The -expression resulting from the first round of substitutions is not -reexamined to see if any further such substitutions can be made. -WHERE can also be used to define auxiliary variables in PROCEDURE -definitions. - -\endsection -\item[WHILE] -WHILE (pages 59, 61, 63, 65) - -The WHILE command causes a statement to be repeatedly executed until a -given condition is true. If the condition is initially false, the -statement is not executed at all. - - WHILE condition DO statement - -condition is given by a logical operator, statement must be a single -REDUCE statement, or a GROUP (<<...>>) or BEGIN...END block. - -Examples: -a := 10; A := 10 -while a <= 12 do <>; 10 - 11 - 12 -while a < 5 do <>; .... nothing is printed - -\endsection -\xitem[WHITTAKERM] -WHITTAKERM (pages 185, 397) - -\endsection -\item[WHITTAKERW] -WHITTAKERW (pages 185, 397) - -The WHITTAKERW operator returns Whittaker's W function. - - WHITTAKERW(parameter,parameter,argument) - -Examples: -load_package specfn; (SPECFN) - 1 - 4*SQRT(2)*KUMMERU(---,5,2) - 2 -WhittakerW(2,2,2); ---------------------------- - E - -Whittaker's W function is one of the Confluent Hypergeometric functions. -For reference see the HYPERGEOMETRIC operator. - -\endsection -\xitem[Workspace] -Workspace (page 99) - -\endsection -\item[WRITE] -WRITE (page 105)) - -The WRITE command explicitly writes its arguments to the output device -(terminal or file). - - WRITE item{,item} - -item can be an expression, an assignment or a STRING enclosed in -double quotation marks ("). - -Examples: -write a, sin x, "this is a string"; ASIN(X)this is a string -write a," ",sin x," this is a string"; A SIN(X) this is a string -if not numberp(a) then write "the symbol ",a; the symbol A -array m(10); -for i := 1:5 do write m(i) := 2*i; - M(1) := 2 - M(2) := 4 - M(3) := 6 - M(4) := 8 - M(5) := 10 -m(4); 8 - -The items specified by a single WRITE statement print on a single line -unless they are too long. A printed line is always ended with a carriage -return, so the next item printed starts a new line. - -When an assignment statement is printed, the assignment is also made. -This allows you to get feedback on filling slots in an array with a -FOR statement, as shown in the last example above. - -\endsection -\item[WS] -WS (pages 29, 158) - -The WS operator alone returns the last result; WS with a number -argument returns the results of the REDUCE statement executed after -that numbered prompt. - - WS or WS(number) - -number must be an integer between 1 and the current REDUCE prompt number. - -Examples: -(In the following examples, unlike most others, the numbered -prompt is shown.) -1: df(sin y,y); COS(Y) - 2 -2: ws^2; COS(Y) - -3: df(ws 1,y); -SIN(Y) - -WS and WS(number) can be used anywhere the expression they stand for -can be used. Calling a number for which no result was produced, such -as a switch setting, will give an error message. - -The current workspace always contains the results of the last REDUCE -command that produced an expression, even if several input statements -that do not produce expressions have intervened. For example, if you -do a differentiation, producing a result expression, then change -several switches, the operator WS; returns the results of the -differentiation. The current workspace (WS) can also be used inside -files, though the numbered workspace contains only the IN command that -input the file. - -There are three history lists kept in your REDUCE session. The first -stores raw input, suitable for the statement editor. The second -stores parsed input, ready to execute and accessible by INPUT. The -third stores results, when they are produced by statements, which are -accessible by the WS n operator. If your session is very long, -storage space begins to fill up with these expressions, so it is a -good idea to end the session once in a while, saving needed -expressions to files with the SAVEAS and OUT commands. - -An error message is given if a reference number has not yet been used. - -\endsection -\item[WTLEVEL] -WTLEVEL (page 152) - -In conjunction with WEIGHT, WTLEVEL is used to implement asymptotic -constraints. Default value is 2. - - WTLEVEL integer - -integer is a positive integer that is the greatest weight term to be -retained in expressions involving kernels with weight assignments. - -Examples: 4 3 2 2 3 4 -(x+y)**4; X + 4*X *Y + 6*X *Y + 4*X*Y + Y -weight x=2,y=3; -wtlevel 8; - 4 -(x+y)**4; X -wtlevel 10; - 2 2 2 -(x+y)**4; X *(6*Y + 4*X*Y + X ) -int(x**2,x); ***** X invalid as KERNEL - -WTLEVEL is used in conjunction with the command WEIGHT to enable -asymptotic constraints. Weight of a term is computed by multiplying -the weights of each variable in it by the power to which it has been -raised, and adding the resulting weights for each variable. If the -weight of the term is greater than WTLEVEL, the term is dropped from -the expression, and not used in any further computation involving the -expression. - -Once a weight has been attached to a KERNEL, it is no longer -recognised by the system as a kernel, though still a variable. It -cannot be used in REDUCE commands and operators that need kernels. -The weight attachment can be undone with a CLEAR command. WTLEVEL can -be changed as desired. - -\endsection -\xitem[X(I)] -X(I) (page 379) - -\endsection -\xitem[XI(I)] -XI(I) (page 379) - -\endsection -\item[XPND command] -XPND command (pages 253, 254, 271) - -(Part of the EXCALC package) - -There are two forms of the XPND command, which controls the use of the -product rule for the d operator and the expansion into partial -derivatives. The default for both these is ON. - - xpnd d; - xpnd @; - -Example: - load_package excalc; *** ^ redefined - (excalc) - pform x=0,y=k,z=m; - K - d(y^z); ( - 1) *Y^d Z + d Y^Z - - noxpnd d; - d(y^z); d(Y^Z) - - -See also NOXPND - -\endsection -\item[ZETA] -ZETA (pages 185, 395) - -The ZETA operator returns Riemann's Zeta function, - - Zeta (z) := sum(1/(k**z),k,1,infinity) - - ZETA(expression) - -Examples: - load_package specfn; (SPECFN) - 2 - PI - Zeta(2); ----- - 6 - on rounded; - Zeta 1.01; 100.577943338 - -Numerical computation for the Zeta function for arguments close to 1 -are tedious, because the series is converging very slowly. In this -case a formula (e.g. found in Bender/Orzag: Advanced Mathematical -Methods for Scientists and Engineers, McGraw-Hill) is used. - -No numerical approximation for complex arguments is done. - -\endsection -\xitem[ZETA(ALFA,I)] -ZETA(ALFA,I) (page 379) - -\endsection DELETED r37/packages/arnum/arnum.dvi Index: r37/packages/arnum/arnum.dvi ================================================================== --- r37/packages/arnum/arnum.dvi +++ r37/packages/arnum/arnum.dvi cannot compute difference between binary files DELETED r37/packages/arnum/arnum.ps Index: r37/packages/arnum/arnum.ps ================================================================== --- r37/packages/arnum/arnum.ps +++ r37/packages/arnum/arnum.ps @@ -1,2490 +0,0 @@ -%!PS-Adobe-2.0 -%%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software -%%Title: arnum.dvi -%%CreationDate: Wed Jul 09 08:51:48 2003 -%%Pages: 6 -%%PageOrder: Ascend -%%BoundingBox: 0 0 596 842 -%%DocumentFonts: CMR17 CMR12 CMR10 CMMI10 CMR8 CMMI8 CMSY8 CMSY10 CMBX10 -%%+ CMTI10 CMTT10 -%%DocumentPaperSizes: a4 -%%EndComments -%DVIPSWebPage: (www.radicaleye.com) -%DVIPSCommandLine: dvips arnum -%DVIPSParameters: dpi=600, compressed -%DVIPSSource: TeX output 2003.07.09:0851 -%%BeginProcSet: texc.pro -%! 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-3396 3069 M --105 3 R --24 86 R -6 -4 R -102 -72 V -316 -315 R -142 -101 V --77 24 R -18 -4 R -0 -4 V -6 0 R -53 -16 V --230 138 R -6 0 R -0 -4 V -6 0 R -0 -4 V -6 0 R -70 -29 V --424 391 R -0 4 R -16 65 V --186 38 R -30 82 R -6 -4 R -150 -116 V -307 -371 R -90 -84 V -6 0 R -0 -4 V -6 0 R -0 -4 V --89 40 R --109 102 R -6 0 R -0 -4 V -6 0 R -0 -4 V -6 0 R -0 -4 V -6 0 R -72 -38 V -243 -190 R -6 0 R -178 -110 V --62 9 R -12 0 R -0 -4 V -6 0 R -44 -5 V --261 134 R -471 -489 R -60 -164 R -23 -64 V --83 228 R -267 101 R -4310 1973 M -276 84 V -12 0 R -0 4 V --632 72 R -26 -225 V --26 225 R -84 24 V -177 44 R -3992 1908 M -318 64 V -4073 3308 M --156 -94 R -16 26 R -36 24 R -6 0 V -0 4 R -120 76 V -0 4 R -6 0 V -LT0 -3018 2827 M -110 -30 R -62 84 R -292 60 R -0 -4 R -6 -20 V -7 -15 R -12 4 R -52 32 V -12 44 V -LT1 -3523 2933 M -12 -16 R -48 -56 V -6 0 R -0 -4 V --95 49 R --68 89 R -LT0 -3571 2981 M --12 -43 R -6 0 R -44 27 V -24 44 V -6 0 R -0 4 V -LT1 -4017 2556 M -6 0 R -221 -116 V --39 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-24 -28 V -4762 2435 M -6 0 R -258 167 V -271 -228 V -4967 2205 M -6 0 R -324 169 V -3587 2885 M -61 -66 V --58 37 R -24 -16 R -0 -4 V -6 0 R -0 -4 V -6 0 R -0 -4 V -6 0 R -16 -9 V --125 114 R -18 -12 R -0 -4 V -6 0 R -40 -32 V -3403 1875 M -79 215 V -234 8 V --28 -222 R --285 -1 R -282 4 V --37 940 R -0 -4 R -94 -74 V --51 26 R -6 0 R -0 -4 V -6 0 R -0 -4 V -6 0 R -0 -4 V -6 0 R -27 -14 V --152 115 R -339 -528 R -24 -124 R -18 -72 V --42 196 R -209 70 R -3966 2133 M -172 265 R -108 48 R -105 53 V -108 -152 V -6 0 R -0 -4 V -4227 2201 M --826 925 R -6 4 R -31 35 V --155 62 R -53 50 R -6 -4 R -96 -108 V -LT0 -3559 2938 M -0 -4 R -0 -28 V -2 -4 R -6 4 R -27 19 V -12 40 V --111 -63 R -24 -32 R -40 68 R -LT1 -4314 3510 M -4103 3348 M --16 29 R -12 -28 V --12 28 R -228 132 V -3553 2939 M -18 -32 R -6 -12 V --54 38 R --34 63 R -LT0 -3690 3039 M --48 -55 R -14 8 R -6 4 R -0 4 V -6 0 R -0 8 V -6 0 R -0 4 V -6 0 R -0 8 V -6 0 R -6 12 V -6 0 R -6 12 V -6 0 R -0 4 V -6 0 R -0 4 V -LT1 -3742 2741 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R -18 12 R -36 28 V -136 -7 R -6 4 R -72 94 V -126 -108 V --115 -99 R --279 76 R -55 47 R --33 -132 R -375 108 R -6 4 R -104 124 V -180 -132 V -4209 2664 M --504 288 R -13 91 R -6 0 R -100 -56 V --43 -94 R -0 8 R -43 86 V -LT0 -3628 2818 M -32 -45 R --26 56 R --6 -11 R -LT1 -3619 2813 M -32 25 R --29 -86 R --311 698 R --149 167 R -6 0 R -174 30 V -114 -180 V -325 -574 R -43 94 R -6 -4 R -114 -76 V --79 -98 R -502 -14 R -6 4 R -142 157 V -252 -160 V -4561 2631 M -LT0 -2639 2723 M -12 0 R -198 -25 V -138 48 V --154 28 R -LT1 -3598 3182 M --55 131 R -6 0 R -142 -18 V -18 -116 V -3369 2291 M -0 4 R -79 148 V -114 -8 V --22 -160 R --171 16 R -349 752 R --3 114 R -90 -32 R -49 -18 V --30 -120 R -0 4 R -30 116 V -LT0 -2991 2747 M -12 0 R -138 -24 V -0 -4 R -6 0 V -7 0 R --26 78 R -LT1 -3943 2907 M -0 4 R -67 120 V -174 -104 V -4072 2799 M -474 908 R -48 -52 V -4314 3510 M --57 34 R -54 -36 V --54 36 R -289 163 V -3824 2987 M -30 120 R -6 0 R -150 -76 V --67 -124 R -822 -111 R -0 4 R -192 194 V -324 -188 V -5026 2602 M -3448 2443 M -0 4 R -24 44 V -35 76 R -73 -134 R --132 10 R -2907 1968 M -172 197 V -186 -52 V -3141 1906 M --234 62 R -234 -64 V -21 1713 R --190 192 R -6 0 R -208 43 V -156 -208 V -201 -331 R --83 154 R -6 0 R -175 -7 V -54 -168 V -20 -135 R --24 138 R -6 0 R -168 -42 V --11 -146 R -0 4 R -6 60 V -6 60 R --1 22 V -317 -326 R -0 4 R -98 154 V -234 -128 V -4367 2795 M -915 10 R -0 4 R -241 234 V -410 -221 V -5620 2579 M -6 4 R -307 239 V -4257 3544 M -4087 3377 M --63 17 R -66 -16 V --66 16 R -234 152 V -3854 3107 M -11 146 R -6 0 R -24 -8 V -54 -20 R -114 -42 V --53 -152 R -0 4 R -53 148 V -3079 2165 M -0 4 R -142 165 V -150 -44 V -3268 2113 M --189 52 R -931 866 R -53 152 R -6 0 R -210 -96 V --97 -160 R -3221 2334 M -0 4 R -114 140 V -114 -36 V --80 -151 R --148 43 R -114 144 R -0 4 R -54 64 V -36 50 R -23 -153 R --113 35 R -125 989 R --118 180 R -6 0 R -213 3 V -78 -188 V -385 -68 R -3909 3255 M --67 2 R -24 0 R -6 -4 V -6 0 R -30 0 V --66 4 R -6 4 R -174 132 V --331 -98 R --50 165 R -6 0 R -207 -34 V -12 -148 V -649 -322 R -0 4 R -134 190 V -312 -156 V -4765 2796 M --923 461 R --68 -111 R --68 -12 R -159 119 R --11 173 R -6 0 R -150 -40 V -53 -203 R -0 4 R -18 112 V -199 -214 R -0 4 R -81 183 V -288 -124 V -4515 2956 M -LT0 -3154 2719 M -162 -28 R --70 77 R -LT1 -4063 3183 M -31 181 R -84 -28 R -180 -64 V --78 -187 R --938 562 R --156 205 R -6 0 R -252 16 V -114 -220 V -3248 2537 M -66 60 V -51 51 R --30 -170 R --87 59 R -84 -56 V -1625 513 R -0 4 R -175 229 V -390 -184 V -5282 2805 M -3641 3460 M --80 190 R -6 0 R -247 -24 V -42 -200 V -2706 2058 M -215 186 V -162 -80 V -2907 1968 M --201 90 R -198 -88 V -197 434 R -150 128 V --3 5 R --27 -203 R --120 70 R -120 -72 V -LT0 -2849 2698 M -12 0 R -201 -27 V -84 44 V --155 32 R -LT1 -2921 2244 M -0 4 R -180 156 V --22 -239 R --158 79 R -LT0 -2438 2676 M -12 0 R -256 -24 V -132 44 V --199 27 R -LT1 -3854 3426 M --40 200 R -6 0 R -281 -52 V --7 -210 R -0 80 R -7 130 V -548 -424 R -0 4 R -114 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M - dup stringwidth pop neg vshift R show } def -/Cshow { currentpoint stroke M - dup stringwidth pop -2 div vshift R show } def -/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def - /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def -/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } - {pop pop pop Solid {pop []} if 0 setdash} ifelse } def -/BL { stroke gnulinewidth 2 mul setlinewidth } def -/AL { stroke gnulinewidth 2 div setlinewidth } def -/UL { gnulinewidth mul /userlinewidth exch def } def -/PL { stroke userlinewidth setlinewidth } def -/LTb { BL [] 0 0 0 DL } def -/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def -/LT0 { PL [] 0 1 0 DL } def -/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def -/LT2 { PL [2 dl 3 dl] 1 0 0 DL } def -/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def -/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def -/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def -/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def -/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def -/LT8 { PL [2 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L -420 -477 V -728 -275 V -841 0 V -728 275 V -420 477 V -0 551 V --420 477 V --728 275 V --841 0 V -4384 3653 L -3964 3176 L -0 -551 V -728 -275 V -840 0 V -728 275 V -421 477 V -0 551 V --421 476 V --728 276 V --840 0 V -3964 4129 L -3544 3653 L -0 -551 V -420 -477 V -stroke -grestore -end -showpage -%%Trailer -%%DocumentFonts: Helvetica -%%Pages: 1 - - DELETED r37/packages/plot/eg5a.ps Index: r37/packages/plot/eg5a.ps ================================================================== --- r37/packages/plot/eg5a.ps +++ r37/packages/plot/eg5a.ps @@ -1,357 +0,0 @@ -%!PS-Adobe-2.0 -%%Title: eg5a.ps -%%Creator: gnuplot 3.5 (pre 3.6) patchlevel beta 340 -%%CreationDate: Tue Nov 17 14:13:40 1998 -%%DocumentFonts: (atend) -%%BoundingBox: 50 50 554 770 -%%Orientation: Landscape -%%Pages: (atend) -%%EndComments -/gnudict 120 dict def -gnudict begin -/Color false def -/Solid false def -/gnulinewidth 5.000 def -/userlinewidth gnulinewidth def -/vshift -46 def -/dl {10 mul} def -/hpt_ 31.5 def -/vpt_ 31.5 def -/hpt hpt_ def -/vpt vpt_ def -/M {moveto} bind def -/L {lineto} bind def -/R {rmoveto} bind def -/V {rlineto} bind def -/vpt2 vpt 2 mul def -/hpt2 hpt 2 mul def -/Lshow { currentpoint stroke M - 0 vshift R show } def -/Rshow { currentpoint stroke M - dup stringwidth pop neg vshift R show } def -/Cshow { currentpoint stroke M - dup stringwidth pop -2 div vshift R show } def -/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def - /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def -/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } - {pop pop pop Solid {pop []} if 0 setdash} ifelse } def -/BL { stroke gnulinewidth 2 mul setlinewidth } def -/AL { stroke gnulinewidth 2 div setlinewidth } def -/UL { gnulinewidth mul /userlinewidth exch def } def -/PL { stroke userlinewidth setlinewidth } def -/LTb { BL [] 0 0 0 DL } def -/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def -/LT0 { PL [] 0 1 0 DL } def -/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def -/LT2 { PL [2 dl 3 dl] 1 0 0 DL } def -/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def -/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def -/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def -/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def -/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def -/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def -/Pnt { stroke [] 0 setdash - gsave 1 setlinecap M 0 0 V stroke grestore } def -/Dia { stroke [] 0 setdash 2 copy vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke - Pnt } def -/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V - currentpoint stroke M - hpt neg vpt neg R hpt2 0 V stroke - } def -/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke - Pnt } def -/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M - hpt2 vpt2 neg V currentpoint stroke M - hpt2 neg 0 R hpt2 vpt2 V stroke } def -/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M - hpt neg vpt 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closepath fill grestore } def -/Circle { stroke [] 0 setdash 2 copy - hpt 0 360 arc stroke Pnt } def -/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def -/C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def -/C1 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - vpt 0 360 arc closepath } bind def -/C2 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C3 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C4 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C5 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc - 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc } bind def -/C6 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C7 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C8 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C9 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 450 arc closepath fill - vpt 0 360 arc closepath } bind def -/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill - 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C11 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C12 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C13 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C14 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 360 arc closepath fill - vpt 0 360 arc } bind def -/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto - neg 0 rlineto closepath } bind def -/Square { dup Rec } bind def -/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def -/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def -/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def -/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def -/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def -/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def -/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill - exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def -/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def -/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill - 2 copy vpt Square fill - Bsquare } bind def -/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def -/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def -/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill - Bsquare } bind def -/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill - Bsquare } bind def -/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def -/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy vpt Square fill Bsquare } bind def -/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def -/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def -/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def -/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def -/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def -/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def -/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def -/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def -/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def -/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def -/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def -/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def -/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def -/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def -/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def -/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def -/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def -/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def -/DiaE { stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke } def -/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke } def -/TriUE { stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke } def -/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath stroke } def -/PentE { stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath stroke grestore } def -/CircE { stroke [] 0 setdash - hpt 0 360 arc stroke } def -/BoxFill { gsave Rec 1 setgray fill grestore } def -end -%%EndProlog -%%Page: 1 1 -gnudict begin -gsave -50 50 translate -0.100 0.100 scale -90 rotate -0 -5040 translate -0 setgray -newpath -(Helvetica) findfont 140 scalefont setfont -LTb -728 560 M -63 0 V -6325 0 R --63 0 V -644 560 M -(-10) Rshow -728 1248 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(0) Rshow -728 1937 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(10) Rshow -728 2625 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(20) Rshow -728 3313 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(30) Rshow -728 4002 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(40) Rshow -728 4690 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(50) Rshow -728 560 M -0 63 V -0 4067 R -0 -63 V -728 420 M -(0) Cshow -1793 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(20) Cshow -2857 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(40) Cshow -3922 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(60) Cshow -4987 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(80) Cshow -6051 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(100) Cshow -7116 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(120) Cshow -LTb -728 560 M -6388 0 V -0 4130 V --6388 0 V -728 560 L -140 2625 M -currentpoint gsave translate 90 rotate 0 0 M -(points) Cshow -grestore -3922 210 M -(x) Cshow -3922 4900 M -(REDUCE Plot) Cshow -1.000 UL -LT0 -728 1248 M -994 560 L -533 4130 V -2059 560 L -266 688 V -2591 560 L -533 4130 V -3656 560 L -266 688 V -4188 560 L -533 4130 V -5253 560 L -266 688 V -5785 560 L -533 4130 V -6850 560 L -266 688 V -stroke -grestore -end -showpage -%%Trailer -%%DocumentFonts: Helvetica -%%Pages: 1 - - DELETED r37/packages/plot/eg5b.ps Index: r37/packages/plot/eg5b.ps ================================================================== --- r37/packages/plot/eg5b.ps +++ r37/packages/plot/eg5b.ps @@ -1,360 +0,0 @@ -%!PS-Adobe-2.0 -%%Title: eg5b.ps -%%Creator: gnuplot 3.5 (pre 3.6) patchlevel beta 340 -%%CreationDate: Tue Nov 17 14:16:54 1998 -%%DocumentFonts: (atend) -%%BoundingBox: 50 50 554 770 -%%Orientation: Landscape -%%Pages: (atend) -%%EndComments -/gnudict 120 dict def -gnudict begin -/Color false def -/Solid false def -/gnulinewidth 5.000 def -/userlinewidth gnulinewidth def -/vshift -46 def -/dl {10 mul} def -/hpt_ 31.5 def -/vpt_ 31.5 def -/hpt hpt_ def -/vpt vpt_ def -/M {moveto} bind def -/L {lineto} bind def -/R {rmoveto} bind def -/V {rlineto} bind def -/vpt2 vpt 2 mul def -/hpt2 hpt 2 mul def -/Lshow { currentpoint stroke M - 0 vshift R show } def -/Rshow { currentpoint stroke M - dup stringwidth pop neg vshift R show } def -/Cshow { currentpoint stroke M - dup stringwidth pop -2 div vshift R show } def -/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def - /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def -/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } - {pop pop pop Solid {pop []} if 0 setdash} ifelse } def -/BL { stroke gnulinewidth 2 mul setlinewidth } def -/AL { stroke gnulinewidth 2 div setlinewidth } def -/UL { gnulinewidth mul /userlinewidth exch def } def -/PL { stroke userlinewidth setlinewidth } def -/LTb { BL [] 0 0 0 DL } def -/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def -/LT0 { PL [] 0 1 0 DL } def -/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def -/LT2 { PL [2 dl 3 dl] 1 0 0 DL } def -/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def -/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def -/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def -/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def -/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def -/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def -/Pnt { stroke [] 0 setdash - gsave 1 setlinecap M 0 0 V stroke grestore } def -/Dia { stroke [] 0 setdash 2 copy vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke - Pnt } def -/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V - currentpoint stroke M - hpt neg vpt neg R hpt2 0 V stroke - } def -/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke - Pnt } def -/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M - hpt2 vpt2 neg V currentpoint stroke M - hpt2 neg 0 R hpt2 vpt2 V stroke } def -/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke - Pnt } def -/Star { 2 copy Pls Crs } def -/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath fill } def -/TriUF { stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath fill } def -/TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath stroke - Pnt } def -/TriDF { stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath fill} def -/DiaF { stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath fill } def -/Pent { stroke [] 0 setdash 2 copy gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath stroke grestore Pnt } def -/PentF { stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath fill grestore } def -/Circle { stroke [] 0 setdash 2 copy - hpt 0 360 arc stroke Pnt } def -/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def -/C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def -/C1 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - vpt 0 360 arc closepath } bind def -/C2 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C3 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C4 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C5 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc - 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc } bind def -/C6 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C7 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C8 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C9 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 450 arc closepath fill - vpt 0 360 arc closepath } bind def -/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill - 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C11 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C12 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C13 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C14 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 360 arc closepath fill - vpt 0 360 arc } bind def -/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto - neg 0 rlineto closepath } bind def -/Square { dup Rec } bind def -/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def -/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def -/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def -/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def -/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def -/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def -/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill - exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def -/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def -/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill - 2 copy vpt Square fill - Bsquare } bind def -/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def -/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def -/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill - Bsquare } bind def -/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill - Bsquare } bind def -/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def -/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy vpt Square fill Bsquare } bind def -/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def -/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def -/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def -/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def -/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def -/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def -/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def -/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def -/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def -/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def -/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def -/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def -/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def -/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def -/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def -/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def -/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def -/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def -/DiaE { stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke } def -/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke } def -/TriUE { stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke } def -/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath stroke } def -/PentE { stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath stroke grestore } def -/CircE { stroke [] 0 setdash - hpt 0 360 arc stroke } def -/BoxFill { gsave Rec 1 setgray fill grestore } def -end -%%EndProlog -%%Page: 1 1 -gnudict begin -gsave -50 50 translate -0.100 0.100 scale -90 rotate -0 -5040 translate -0 setgray -newpath -(Helvetica) findfont 140 scalefont setfont -LTb -812 560 M -63 0 V -6241 0 R --63 0 V -728 560 M -(-150) Rshow -812 1386 M -63 0 V -6241 0 R --63 0 V --6325 0 R -(-100) Rshow -812 2212 M -63 0 V -6241 0 R --63 0 V --6325 0 R -(-50) Rshow -812 3038 M -63 0 V -6241 0 R --63 0 V --6325 0 R -(0) Rshow -812 3864 M -63 0 V -6241 0 R --63 0 V --6325 0 R -(50) Rshow 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-126 165 V -252 1652 V -stroke -grestore -end -showpage -%%Trailer -%%DocumentFonts: Helvetica -%%Pages: 1 - - DELETED r37/packages/plot/eg6a.ps Index: r37/packages/plot/eg6a.ps ================================================================== --- r37/packages/plot/eg6a.ps +++ r37/packages/plot/eg6a.ps @@ -1,868 +0,0 @@ -%!PS-Adobe-2.0 -%%Title: eg6a.ps -%%Creator: gnuplot 3.5 (pre 3.6) patchlevel beta 340 -%%CreationDate: Tue Nov 17 14:18:38 1998 -%%DocumentFonts: (atend) -%%BoundingBox: 50 50 554 770 -%%Orientation: Landscape -%%Pages: (atend) -%%EndComments -/gnudict 120 dict def -gnudict begin -/Color false def -/Solid false def -/gnulinewidth 5.000 def -/userlinewidth gnulinewidth def -/vshift -46 def -/dl {10 mul} def -/hpt_ 31.5 def -/vpt_ 31.5 def -/hpt hpt_ def -/vpt vpt_ def -/M {moveto} bind def -/L {lineto} bind def -/R {rmoveto} bind def -/V {rlineto} bind def -/vpt2 vpt 2 mul def -/hpt2 hpt 2 mul def -/Lshow { currentpoint stroke M - 0 vshift R show } def -/Rshow { currentpoint stroke M - dup stringwidth pop neg vshift R show } def -/Cshow { currentpoint stroke M - dup stringwidth pop -2 div vshift R show } def -/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def - /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def -/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } - {pop pop pop Solid {pop []} if 0 setdash} ifelse } def -/BL { stroke gnulinewidth 2 mul setlinewidth } def -/AL { stroke gnulinewidth 2 div setlinewidth } def -/UL { gnulinewidth mul /userlinewidth exch def } def -/PL { stroke userlinewidth setlinewidth } def -/LTb { BL [] 0 0 0 DL } def -/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def -/LT0 { PL [] 0 1 0 DL } def -/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def -/LT2 { PL [2 dl 3 dl] 1 0 0 DL } def -/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def -/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def -/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def -/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def -/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def -/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def -/Pnt { stroke [] 0 setdash - gsave 1 setlinecap M 0 0 V stroke grestore } def -/Dia { stroke [] 0 setdash 2 copy vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke - Pnt } def -/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V - currentpoint stroke M - hpt neg vpt neg R hpt2 0 V stroke - } def -/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke - Pnt } def -/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M - hpt2 vpt2 neg V currentpoint stroke M - hpt2 neg 0 R hpt2 vpt2 V stroke } def -/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke - Pnt } def -/Star { 2 copy Pls Crs } def -/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath fill } def -/TriUF { stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath fill } def -/TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath stroke - Pnt } def -/TriDF { stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath fill} def -/DiaF { stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath fill } def -/Pent { stroke [] 0 setdash 2 copy gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath stroke grestore Pnt } def -/PentF { stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath fill grestore } def -/Circle { stroke [] 0 setdash 2 copy - hpt 0 360 arc stroke Pnt } def -/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def -/C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def -/C1 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - vpt 0 360 arc closepath } bind def -/C2 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C3 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C4 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C5 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc - 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc } bind def -/C6 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C7 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C8 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C9 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 450 arc closepath fill - vpt 0 360 arc closepath } bind def -/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill - 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C11 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C12 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C13 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C14 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 360 arc closepath fill - vpt 0 360 arc } bind def -/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto - neg 0 rlineto closepath } bind def -/Square { dup Rec } bind def -/Bsquare { vpt sub exch 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fill - Bsquare } bind def -/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill - Bsquare } bind def -/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def -/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy vpt Square fill Bsquare } bind def -/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def -/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def -/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def -/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def -/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def -/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def -/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def -/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def -/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def -/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def -/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def -/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def -/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def -/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def -/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def -/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def -/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def -/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def -/DiaE { stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke } def -/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke } def -/TriUE { stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 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-%%DocumentFonts: Helvetica -%%Pages: 1 - - DELETED r37/packages/plot/eg7.ps Index: r37/packages/plot/eg7.ps ================================================================== --- r37/packages/plot/eg7.ps +++ r37/packages/plot/eg7.ps @@ -1,407 +0,0 @@ -%!PS-Adobe-2.0 -%%Title: eg7.ps -%%Creator: gnuplot 3.5 (pre 3.6) patchlevel beta 340 -%%CreationDate: Tue Nov 17 14:22:44 1998 -%%DocumentFonts: (atend) -%%BoundingBox: 50 50 554 770 -%%Orientation: Landscape -%%Pages: (atend) -%%EndComments -/gnudict 120 dict def -gnudict begin -/Color false def -/Solid false def -/gnulinewidth 5.000 def -/userlinewidth gnulinewidth def -/vshift -46 def -/dl {10 mul} def -/hpt_ 31.5 def -/vpt_ 31.5 def -/hpt hpt_ def -/vpt vpt_ def -/M {moveto} bind def -/L {lineto} bind def -/R {rmoveto} bind def -/V {rlineto} bind def -/vpt2 vpt 2 mul def -/hpt2 hpt 2 mul def -/Lshow { currentpoint stroke M - 0 vshift R show } def -/Rshow { currentpoint stroke M - dup stringwidth pop neg vshift R show } def -/Cshow { currentpoint stroke M - dup stringwidth pop -2 div vshift R show } def -/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def - /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def -/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } - {pop pop pop Solid {pop []} if 0 setdash} ifelse } def -/BL { stroke gnulinewidth 2 mul setlinewidth } def -/AL { stroke gnulinewidth 2 div setlinewidth } def -/UL { gnulinewidth mul /userlinewidth exch def } def -/PL { stroke userlinewidth setlinewidth } def -/LTb { BL [] 0 0 0 DL } def -/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def -/LT0 { PL [] 0 1 0 DL } def -/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def -/LT2 { PL [2 dl 3 dl] 1 0 0 DL } def -/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def -/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def -/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def -/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def -/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def -/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def -/Pnt { stroke [] 0 setdash - gsave 1 setlinecap M 0 0 V stroke grestore } def -/Dia { stroke [] 0 setdash 2 copy vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke - Pnt } def -/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V - currentpoint stroke M - hpt neg vpt neg R hpt2 0 V stroke - } def -/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke - Pnt } def -/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M - hpt2 vpt2 neg V currentpoint stroke M - hpt2 neg 0 R hpt2 vpt2 V stroke } def -/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke - Pnt } def -/Star { 2 copy Pls Crs } def -/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath fill } def -/TriUF { stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 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bind def -/C2 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C3 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C4 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C5 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc - 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc } bind def -/C6 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C7 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C8 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C9 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 450 arc closepath fill - vpt 0 360 arc closepath } bind def -/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill - 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C11 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C12 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C13 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C14 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 360 arc closepath fill - vpt 0 360 arc } bind def -/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto - neg 0 rlineto closepath } bind def -/Square { dup Rec } bind def -/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def -/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def -/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def -/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def -/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def -/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def -/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill - exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def -/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def -/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill - 2 copy vpt Square fill - Bsquare } bind def -/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def -/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def -/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill - Bsquare } bind def -/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill - Bsquare } bind def -/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def -/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy vpt Square fill Bsquare } bind def -/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def -/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def -/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def -/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def -/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def -/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def -/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def -/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def -/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def -/D7 { gsave translate 45 rotate 0 0 S7 stroke 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sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath stroke } def -/PentE { stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath stroke grestore } def -/CircE { stroke [] 0 setdash - hpt 0 360 arc stroke } def -/BoxFill { gsave Rec 1 setgray fill grestore } def -end -%%EndProlog -%%Page: 1 1 -gnudict begin -gsave -50 50 translate -0.100 0.100 scale -90 rotate -0 -5040 translate -0 setgray -newpath -(Helvetica) findfont 140 scalefont setfont -LTb -728 560 M -63 0 V -6325 0 R --63 0 V -644 560 M -(0) Rshow -728 1386 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(20) Rshow -728 2212 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(40) Rshow -728 3038 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(60) Rshow -728 3864 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(80) Rshow -728 4690 M -63 0 V -6325 0 R --63 0 V --6409 0 R -(100) Rshow -728 560 M -0 63 V -0 4067 R -0 -63 V -728 420 M -(-60) Cshow -1793 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(-40) Cshow -2857 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(-20) Cshow -3922 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(0) Cshow -4987 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(20) Cshow -6051 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(40) Cshow -7116 560 M -0 63 V -0 4067 R -0 -63 V -0 -4207 R -(60) Cshow -LTb -728 560 M -6388 0 V -0 4130 V --6388 0 V -728 560 L -140 2625 M -currentpoint gsave translate 90 rotate 0 0 M -(points) Cshow -grestore -3922 210 M -(x) Cshow -3922 4900 M -(REDUCE Plot) Cshow -1.000 UL -LT0 -3922 560 M -2998 4627 L -4818 746 L -2157 4323 L -5578 1159 L -1500 3814 L -6111 1749 L -1108 3162 L -6351 2444 L --5324 0 V -5243 718 V -1268 1749 L -5878 3814 L -1800 1159 L -5222 4323 L -2560 746 L -4381 4627 L -3456 560 L -0 4130 V -4381 623 L -2560 4504 L -5222 927 L -1800 4091 L -5878 1436 L -1268 3501 L -6270 2088 L -1027 2806 L -5324 0 V -1108 2088 L -6111 3501 L -1500 1436 L -5578 4091 L -2157 927 L -4818 4504 L -2998 623 L -924 4067 V -0 -4130 V -2998 4627 L -4818 746 L -2157 4323 L -5578 1159 L -1500 3814 L -6111 1749 L -1108 3162 L -6351 2444 L --5324 0 V -5243 718 V -1268 1749 L -5878 3814 L -1800 1159 L -5222 4323 L -2560 746 L -4381 4627 L -3456 560 L -0 4130 V -4381 623 L -2560 4504 L -5222 927 L -1800 4091 L -5878 1436 L -1268 3501 L -6270 2088 L -1027 2806 L -5324 0 V -1108 2088 L -6111 3501 L -1500 1436 L -5578 4091 L -2157 927 L -4818 4504 L -2998 623 L -924 4067 V -0 -4130 V -stroke -grestore -end -showpage -%%Trailer -%%DocumentFonts: Helvetica -%%Pages: 1 - - DELETED r37/packages/plot/eg8.ps Index: r37/packages/plot/eg8.ps ================================================================== --- r37/packages/plot/eg8.ps +++ r37/packages/plot/eg8.ps @@ -1,641 +0,0 @@ -%!PS-Adobe-2.0 -%%Title: eg8.ps -%%Creator: gnuplot 3.5 (pre 3.6) patchlevel beta 340 -%%CreationDate: Tue Nov 17 14:26:42 1998 -%%DocumentFonts: (atend) -%%BoundingBox: 50 50 554 770 -%%Orientation: Landscape -%%Pages: (atend) -%%EndComments -/gnudict 120 dict def -gnudict begin -/Color false def -/Solid false def -/gnulinewidth 5.000 def -/userlinewidth gnulinewidth def -/vshift -46 def -/dl {10 mul} def -/hpt_ 31.5 def -/vpt_ 31.5 def -/hpt hpt_ def -/vpt vpt_ def -/M {moveto} bind def -/L {lineto} bind def -/R {rmoveto} bind def -/V {rlineto} bind def -/vpt2 vpt 2 mul def -/hpt2 hpt 2 mul def -/Lshow { currentpoint stroke M - 0 vshift R show } def -/Rshow { currentpoint stroke M - dup stringwidth pop neg vshift R show } def -/Cshow { currentpoint stroke M - dup stringwidth pop -2 div vshift R show } def -/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def - /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def -/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } - {pop pop pop Solid {pop []} if 0 setdash} ifelse } def -/BL { stroke gnulinewidth 2 mul setlinewidth } def -/AL { stroke gnulinewidth 2 div setlinewidth } def -/UL { gnulinewidth mul /userlinewidth exch def } def -/PL { stroke userlinewidth setlinewidth } def -/LTb { BL [] 0 0 0 DL } def -/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def -/LT0 { PL [] 0 1 0 DL } def -/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def -/LT2 { PL [2 dl 3 dl] 1 0 0 DL } def -/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def -/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def -/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def -/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def -/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def -/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def -/Pnt { stroke [] 0 setdash - gsave 1 setlinecap M 0 0 V stroke grestore } def -/Dia { stroke [] 0 setdash 2 copy vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke - Pnt } def -/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V - currentpoint stroke M - hpt neg vpt neg R hpt2 0 V stroke - } def -/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke - Pnt } def -/Crs { stroke [] 0 setdash exch 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DELETED r37/packages/plot/eg9.ps Index: r37/packages/plot/eg9.ps ================================================================== --- r37/packages/plot/eg9.ps +++ r37/packages/plot/eg9.ps @@ -1,786 +0,0 @@ -%!PS-Adobe-2.0 -%%Title: eg9.ps -%%Creator: gnuplot 3.5 (pre 3.6) patchlevel beta 340 -%%CreationDate: Tue Nov 17 14:31:58 1998 -%%DocumentFonts: (atend) -%%BoundingBox: 50 50 554 770 -%%Orientation: Landscape -%%Pages: (atend) -%%EndComments -/gnudict 120 dict def -gnudict begin -/Color false def -/Solid false def -/gnulinewidth 5.000 def -/userlinewidth gnulinewidth def -/vshift -46 def -/dl {10 mul} def -/hpt_ 31.5 def -/vpt_ 31.5 def -/hpt hpt_ def -/vpt vpt_ def -/M {moveto} bind def -/L {lineto} bind def -/R {rmoveto} bind def -/V {rlineto} bind def -/vpt2 vpt 2 mul def -/hpt2 hpt 2 mul def -/Lshow { currentpoint stroke M - 0 vshift R show } def -/Rshow { currentpoint stroke M - dup stringwidth pop neg vshift R show } def -/Cshow { currentpoint stroke M - dup stringwidth pop -2 div vshift R show } def -/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def - /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def -/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } - {pop pop pop Solid {pop []} if 0 setdash} ifelse } def -/BL { stroke gnulinewidth 2 mul setlinewidth } def -/AL { stroke gnulinewidth 2 div setlinewidth } def -/UL { gnulinewidth mul /userlinewidth exch def } def -/PL { stroke userlinewidth setlinewidth } def -/LTb { BL [] 0 0 0 DL } def -/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def -/LT0 { PL [] 0 1 0 DL } def -/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def -/LT2 { PL [2 dl 3 dl] 1 0 0 DL } def -/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def -/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def -/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def -/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def -/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def -/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def -/Pnt { stroke [] 0 setdash - gsave 1 setlinecap M 0 0 V stroke grestore } def -/Dia { stroke [] 0 setdash 2 copy vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke - Pnt } def -/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V - currentpoint stroke M - hpt neg vpt neg R hpt2 0 V stroke - } def -/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke - Pnt } def -/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M - hpt2 vpt2 neg V currentpoint stroke M - hpt2 neg 0 R hpt2 vpt2 V stroke } def -/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke - Pnt } def -/Star { 2 copy Pls Crs } def -/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath fill } def -/TriUF { stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath 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showpage end -%%Trailer - -%%EndDocument -temp-save restore -end -showpage -%%Trailer -%End-of-file DELETED r38/doc/util/r38.dvi Index: r38/doc/util/r38.dvi ================================================================== --- r38/doc/util/r38.dvi +++ r38/doc/util/r38.dvi cannot compute difference between binary files DELETED r38/doc/util/r38.log Index: r38/doc/util/r38.log ================================================================== --- r38/doc/util/r38.log +++ r38/doc/util/r38.log @@ -1,1055 +0,0 @@ -This is TeX, Version 3.141592 (MiKTeX 2.3) (preloaded format=latex 2000.11.28) 17 APR 2004 17:30 -**r38 -(r38.tex -LaTeX2e <2001/06/01> -Babel and hyphenation patterns for english, french, german, ngerman, du -mylang, nohyphenation, loaded. -(D:\texmf\tex\latex\base\book.cls -Document Class: book 2001/04/21 v1.4e Standard LaTeX document class -(D:\texmf\tex\latex\base\bk11.clo -File: bk11.clo 2001/04/21 v1.4e Standard LaTeX file (size option) -) -\c@part=\count79 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Checking defaults for OML/cmm/m/it on input line 103. -LaTeX Font Info: ... okay on input line 103. -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 103. -LaTeX Font Info: ... okay on input line 103. -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 103. -LaTeX Font Info: ... okay on input line 103. -LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 103. -LaTeX Font Info: ... okay on input line 103. -LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 103. -LaTeX Font Info: ... okay on input line 103. -LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 103. -LaTeX Font Info: ... okay on input line 103. -LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 103. -LaTeX Font Info: ... okay on input line 103. -Package hyperref Info: Link coloring OFF on input line 103. - (D:\texmf\tex\latex\hyperref\nameref.sty -Package: nameref 2001/01/27 v2.19 Cross-referencing by name of section -\c@section@level=\count94 -) -LaTeX Info: Redefining \ref on input line 103. -LaTeX Info: Redefining \pageref on input line 103. - (r38.out) (r38.out) -[1 - -] -LaTeX Font Info: Try loading font information for OMS+cmr on input line 133. - - (D:\texmf\tex\latex\base\omscmr.fd -File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions -) -LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10.95> not available -(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 133. - [2] -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <10.95> on input line 146. -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <8> on input line 146. -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <6> on input line 146. - [3] [4 - -] (r38.toc [5] [6] [7] -[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] -[23] [24] [25] [26]) -\tf@toc=\write5 - [27] [28 - -] [29] [30] [31] [32] [33 - -] [34 - -] [35] [36 - -] -Chapter 1. -[37] [38] [39] [40] [41] [42 - -] -Chapter 2. -[43] [44] [45] [46] [47] [48] [49] [50] [51] [52 - -] -Chapter 3. -[53] [54] [55] [56] [57] [58] -Chapter 4. -[59 - -] -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <12> on input line 1152. - - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `math shift' on input line 1152. - - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `math shift' on input line 1152. - -[60] -LaTeX Font Info: Try loading font information for OMS+cmtt on input line 118 -5. -LaTeX Font Info: No file OMScmtt.fd. on input line 1185. - - -LaTeX Font Warning: Font shape `OMS/cmtt/m/n' undefined -(Font) using `OMS/cmsy/m/n' instead -(Font) for symbol `textbraceleft' on input line 1185. - -[61] [62] -Chapter 5. -[63 - -] [64] [65] [66] -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <7> on input line 1429. -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <5> on input line 1429. - [67] [68] [69] [70] [71] [72] [73] [74] -Chapter 6. -[75 - -] [76] [77] [78] -Chapter 7. -[79 - -] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] -[94] [95] [96] [97] [98] [99] -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <9> on input line 2991. - [100] [101] [102] [103] [104] [105] [106] -[107] [108] [109] [110 - -] -Chapter 8. -[111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] -[123] [124] [125] [126] [127] [128] [129] [130] [131] [132 - -] -Chapter 9. -[133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] -[145] [146] [147] [148] [149] [150] -Chapter 10. -[151 - -] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] -[163] [164] [165] [166] [167] [168 - -] -Chapter 11. -[169] [170] [171] [172 - -] -Chapter 12. -[173] -Underfull \hbox (badness 1132) in paragraph at lines 5856--5857 -[]\OT1/cmr/m/n/10.95 move pointer to next oc-cur-rence of - [] - -[174] [175] [176] -Chapter 13. -[177 - -] [178] [179] [180] [181] [182] [183] [184] -Chapter 14. -[185 - -] [186] [187] [188] [189] [190] [191] [192] -Chapter 15. -[193 - -] - -LaTeX Warning: Reference `CONTFR' on page 194 undefined on input line 6720. - -[194] [195] [196] -Chapter 16. -[197 - -] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208] -[209] [210] -Chapter 17. -[211 - -] [212] [213] [214] [215] [216] [217] [218] -Chapter 18. -[219 - -] [220] [221] [222] [223] [224] -Chapter 19. -[225 - -] -\c@examplectr=\count95 - [226] [227 - -] [228] [229] [230 - -] -Chapter 20. -{ALGINT: Integration of square roots} -[231] [232] [233] [234 - -] -Chapter 21. -[APPLYSYM: Infinitesimal symmetries] -[235] [236] [237] [238 - -] -Chapter 22. -{ARNUM: An algebraic number package} -[239] [240] [241] [242 - -] -Chapter 23. -{ASSIST: Various Useful Utilities} -[243] [244] [245] [246] [247] [248] [249] [250] [251] [252] [253] [254] -[255] [256] [257] [258] [259] [260] [261] [262] [263] [264 - -] -Chapter 24. -{ATENSOR: Package for Tensor Simplification} -[265] [266] [267] [268 - -] -Chapter 25. -{AVECTOR: Vector Algebra} -[269] [270] [271] [272] [273] [274] [275] [276] -Chapter 26. -{BOOLEAN: A package for boolean algebra} -[277 - -] [278] [279] [280] [281] [282 - -] -Chapter 27. -{CALI: Computational Commutative Algebra} -[283] [284] -Chapter 28. -{CAMAL: Calculations in Celestial Mechanics} -[285 - -] [286] [287] [288 - -] -Chapter 29. -{CGB: Comprehensive Gr\"obner Bases} -[289] [290] [291] [292] -Overfull \hbox (16.29929pt too wide) has occurred while \output is active -[]\OT1/cmr/m/sl/10.95 29.7. SWITCH CGBREAL: COMPUTING OVER THE REAL NUMBERS \O -T1/cmr/m/n/10.95 293 - [] - -[293] [294] -Chapter 30. -[CHANGEVR: Change of Variables in DEs] -[295 - -] [296] -Chapter 31. -{COMPACT: Package for compacting expressions} -[297 - -] [298] -Chapter 32. -[CRACK: Overdetermined systems of DEs] -[299 - -] [300] [301] [302] -Chapter 33. -[CVIT:Dirac gamma matrix traces] -[303 - -] [304] -Chapter 34. -{DEFINT: Definite Integration for REDUCE} -[305 - -] [306] [307] [308] -Chapter 35. -[DESIR: Linear Homogeneous DEs] -[309 - -] [310] [311] [312 - -] -Chapter 36. -{DFPART: Derivatives of generic functions} -[313] [314] [315] [316] [317] [318 - -] -Chapter 37. -[DUMMY: Expressions with dummy variables] -[319] [320] [321] [322 - -] -Chapter 38. -{EDS: Exterior differential systems} -[323] [324] [325] -Overfull \hbox (28.56746pt too wide) in paragraph at lines 12120--12132 -\OT1/cmr/m/n/10.95 A sim-ple [] is con-structed us-ing the \OT1/cmtt/m/n/10.95 -EDS \OT1/cmr/m/n/10.95 op-er-a-tor where the [] - 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[] - -[330] [331] [332 - -] -Chapter 39. -{EXCALC: A differential geometry package} -[333] [334] [335] [336] [337] [338] [339] [340] [341] [342] [343] [344 - -] -Chapter 40. -[FIDE: Finite differences for PDEs] -[345] [346] [347] [348] -Chapter 41. -[FPS: Formal power series] -[349 - -] [350] -Chapter 42. -{GENTRAN: A code generation package} -[351 - -] [352] [353] [354] [355] [356] [357] [358] [359] [360] [361] [362 - -] -Chapter 43. -{GEOMETRY: Mechanized (Plane) Geometry Manipulations} -[363] -Underfull \hbox (badness 10000) in paragraph at lines 13711--13794 - - [] - -[364] [365] [366] -Underfull \hbox (badness 10000) in paragraph at lines 13800--13804 - - [] - -[367] -Underfull \hbox (badness 10000) in paragraph at lines 13811--13813 - - [] - -[368] [369] [370 - -] -Chapter 44. -{GNUPLOT: Display of functions and surfaces} -[371] [372] [373] [374 - -] -Chapter 45. -{GROEBNER: A Gr\"obner basis package} -[375] [376] -LaTeX Font Info: Font shape `OMS/cmr/m/it' in size <10.95> not available -(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 14062 -. - [377] - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `math shift' on input line 14132. - - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `\mathop' on input line 14132. - - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `\mathgroup' on input line 14132. - - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `\symoperators' on input line 14132. - - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `\nolimits' on input line 14132. - - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `math shift' on input line 14132. - -[378] [379] [380] [381] [382] [383] -Overfull \hbox (5.98795pt too wide) in paragraph at lines 14424--14424 -[]\OT1/cmr/bx/n/14.4 Ideal De-com-po-si-tion & Equat-ion Sys-tem Solv- - [] - -[384] -Chapter 46. -{IDEALS: Arithmetic for polynomial ideals} -[385 - -] [386] -Chapter 47. -{INEQ: Support for solving inequalities} -[387 - -] [388] -Chapter 48. -{INVBASE: A package for computing involutive bases} -[389 - -] [390] [391] [392 - -] -Chapter 49. -{LAPLACE: Laplace and inverse Laplace transforms} -[393] [394] [395] [396 - -] -Chapter 50. -{LIE: Functions for the classification of real n-dimensional Lie algebras} -[397] [398] -Chapter 51. -{LIMITS: A package for finding limits} -[399 - -] [400] [401] [402 - -] -Chapter 52. -{LINALG: Linear algebra package} -[403] [404] [405] [406] [407] [408] [409] [410] [411] [412] [413] [414 - -] -Chapter 53. -{MATHML : MathML Interface for REDUCE} -[415] [416] [417] [418 - -] -Chapter 54. -{MODSR: Modular solve and roots} -[419] [420 - -] -Chapter 55. -{MRVLIMIT: Package for Computing Limits of "Exp-Log" Functions} -[421] -Underfull \hbox (badness 10000) in paragraph at lines 15926--15932 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 15926--15932 - - [] - -[422] [423] [424 - -] -Chapter 56. -{NCPOLY: Non--commutative polynomial ideals} -[425] [426] [427] [428] [429] [430] -Chapter 57. -{NORMFORM: Computation of matrix normal forms} -[431 - -] [432] [433] -Underfull \hbox (badness 10000) in paragraph at lines 16388--16389 - - [] - -[434] [435] [436 - -] -Chapter 58. -{NUMERIC: Solving numerical problems} -[437] [438] [439] [440] [441] [442] [443] [444] [445] [446] [447] [448 - -] -Chapter 59. -[ODESOLVE: Ordinary differential equations solver] -[449] [450] [451] [452 - -] -Chapter 60. -{ORTHOVEC: Three-dimensional vector analysis} -[453] [454] -Underfull \hbox (badness 10000) in paragraph at lines 17139--17148 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 17149--17158 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 17159--17169 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 17170--17185 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 17186--17198 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 17199--17205 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 17206--17214 - - [] - -[455] -Underfull \hbox (badness 10000) in paragraph at lines 17215--17221 - - [] - -[456] -Underfull \hbox (badness 10000) in paragraph at lines 17290--17296 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 17319--17323 - - [] - -[457] [458] [459] [460 - -] -Chapter 61. -{PHYSOP: Operator calculus in quantum theory} - -Underfull \hbox (badness 10000) in paragraph at lines 17430--17435 - - [] - -[461] [462] [463] -Underfull \hbox (badness 10000) in paragraph at lines 17574--17576 - - [] - -[464] [465] -Underfull \hbox (badness 10000) in paragraph at lines 17686--17695 - - [] - -[466] [467] [468] -Chapter 62. -{PM: A REDUCE pattern matcher} -[469 - -] [470] -This is not true -[471] [472] [473] [474 - -] -Chapter 63. - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `\-command' on input line 17991. - - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `\f@shape' on input line 17991. - - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `\-command' on input line 17991. - 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-[475] -LaTeX Font Info: Font shape `OT1/cmtt/bx/n' in size <14.4> not available -(Font) Font shape `OT1/cmtt/m/n' tried instead on input line 18105 -. - [476] [477] -Underfull \hbox (badness 10000) in paragraph at lines 18184--18187 - - [] - -[478] [479] [480 - -] -Chapter 64. -{RANDPOLY: A random polynomial generator} -[481] [482] [483] [484] [485] [486 - -] -Chapter 65. -{RATAPRX : Rational Approximations Package} -[487] -Underfull \hbox (badness 10000) in paragraph at lines 18463--18465 - - [] - -[488] [489] [490] [491] [492] -Chapter 66. -{REACTEQN: Support for chemical reaction equations} -[493 - -] [494] -Chapter 67. -{REDLOG: Logic System} - -LaTeX Font Warning: Font shape `OT1/cmr/bx/sc' undefined -(Font) using `OT1/cmr/bx/n' instead on input line 18668. - -[495 - -] -Underfull \hbox (badness 10000) in paragraph at lines 18725--18726 - - [] - -[496] -Overfull \hbox (31.73804pt too wide) in paragraph at lines 18754--18766 - [] - [] - -[497] [498] [499] [500] [501] [502] [503] [504 - -] -Chapter 68. -{RESET: Code to reset REDUCE to its initial state} -[505] [506 - -] -Chapter 69. -{RESIDUE: A residue package} -[507] [508] -Chapter 70. -{RLFI: REDUCE LaTeX formula interface} -[509 - -] [510] [511] [512 - -] -Chapter 71. -{ROOTS: A REDUCE root finding package} -[513] [514] [515] [516] [517] [518 - -] -Chapter 72. -[RSOLVE: Rational polynomial solver] -[519] -LaTeX Font Info: Font shape `OT1/cmtt/bx/n' in size <10.95> not available -(Font) Font shape `OT1/cmtt/m/n' tried instead on input line 19478 -. - -Underfull \hbox (badness 10000) in paragraph at lines 19478--19480 -[]\OT1/cmr/m/n/10.95 assign the mul-ti-plic-ity list to the global vari-able - [] - -[520] -Chapter 73. -{SCOPE: REDUCE source code optimisation package} -LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10> not available -(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 19549 -. - -Overfull \hbox (11.1949pt too wide) in paragraph at lines 19542--19557 -[] - [] - -[521 - -] [522] [523] [524 - -] -Chapter 74. -{SETS: A basic set theory package} -[525] [526] [527] [528] [529] [530] -Chapter 75. -{SPARSE: Sparse Matrices} -[531 - -] [532] [533] [534 - -] -Chapter 76. -{SPDE: A package for finding symmetry groups of {PDE}'s} -[535] [536] [537] [538 - -] -Chapter 77. -{SPECFN: Package for special functions} -[539] [540] [541] [542] -Overfull \hbox (9.72835pt too wide) in paragraph at lines 20244--20245 - [] - [] - -[543] -Overfull \vbox (3.89938pt too high) has occurred while \output is active [] - - -[544] -Chapter 78. -{SPECFN2: Package for special special functions} -[545 - -] [546] -Chapter 79. -{SUM: A package for series summation} -[547 - -] [548] [549] [550 - -] -Chapter 80. -{SUSY2: Super Symmetry} - -Underfull \hbox (badness 10000) in paragraph at lines 20470--20474 - - [] - -[551] [552] [553] [554] [555] [556 - -] -Chapter 81. -{SYMMETRY: Operations on symmetric matrices} -[557] [558] [559] [560 - -] -Chapter 82. -{TAYLOR: Manipulation of Taylor series} -[561] [562] [563] [564] [565] [566 - -] -Chapter 83. -{TPS: A truncated power series package} -[567] [568] [569] [570] [571] [572] [573] [574] [575] [576 - -] -Chapter 84. -{TRI: TeX REDUCE interface} -[577] [578] [579] [580] [581] [582 - -] -Chapter 85. -{TRIGSIMP: Simplification and factorisation of trigonometric and hyperbolic fun -ctions} -[583] [584] [585] [586] -Chapter 86. -{WU: Wu algorithm for polynomial systems} -[587 - -] [588] -Chapter 87. -{XCOLOR: Calculation of the color factor in non-abelian gauge field theories} -[589 - -] [590] [591] [592 - -] -Chapter 88. -{XIDEAL: Gr\"obner Bases for exterior algebra} -[593] [594] [595] [596] [597] [598 - -] -Chapter 89. -{ZEILBERG: A package for indefinite and definite summation} -[599] [600] [601] [602] [603] [604] [605] [606 - -] -Chapter 90. - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `math shift' on input line 22398. - - -Package hyperref Warning: Token not allowed in a PDFDocEncoded string, -(hyperref) removing `math shift' on input line 22398. - -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <24.88> on input line 22398. -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <20.74> on input line 22398. -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <17.28> on input line 22398. -{ZTRANS: $Z$-transform package} -[607] [608] [609] [610] [611 - -] [612] -Chapter 91. -{The Standard Lisp Report} -\argwidth=\skip46 -\dewidth=\skip47 -[613 - -] [614] [615] [616] [617] [618] [619] [620] [621] [622] [623] [624] -[625] [626] [627] [628] [629] [630] [631] [632] [633] [634] [635] [636] -[637] [638] [639] [640] [641] [642] [643] [644] [645] [646] [647] [648] -[649] [650] [651] -Underfull \hbox (badness 10000) in paragraph at lines 24307--24307 -[]\OT1/cmtt/m/n/10.95 Spread the actual parameters in ARGS - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 24307--24307 -\OT1/cmtt/m/n/10.95 following the conventions: for calling - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 24307--24307 -\OT1/cmtt/m/n/10.95 functions, transfer to the entry point - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 24307--24307 -\OT1/cmtt/m/n/10.95 of the function, and return the value - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 24307--24307 -[]\OT1/cmtt/m/n/10.95 Bind the actual parameters in ARGS to - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 24307--24307 -\OT1/cmtt/m/n/10.95 the formal parameters of the lambda - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 24307--24307 -\OT1/cmtt/m/n/10.95 expression. If the two lists are not - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 24307--24307 -\OT1/cmtt/m/n/10.95 of equal length then ERROR(000, "Number - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 24307--24307 -\OT1/cmtt/m/n/10.95 of parameters do not match"); The value - [] - -[652] -Underfull \hbox (badness 10000) in paragraph at lines 24337--24337 -[]\OT1/cmtt/m/n/10.95 U is an id. Return the value most - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 24337--24337 -\OT1/cmtt/m/n/10.95 currently bound to U or if there - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 24337--24337 -\OT1/cmtt/m/n/10.95 is no such binding: ERROR(000, - [] - -[653] [654] [655] [656] [657] [658] [659] [660] [661] [662] [663] [664] -[665] [666 - -] [667] [668] -Appendix A. -[669 - -] [670] (r38.bbl [671 - -] [672]) (r38.ind [673] [674 - -] [675] [676] [677] -[678] [679] [680] [681] [682] [683] [684] [685] [686] [687] [688] [689] -[690] [691] [692] [693] [694] [695] [696] [697] [698 - -]) (r38.aux) - -LaTeX Font Warning: Some font shapes were not available, defaults substituted. - - -LaTeX Warning: There were undefined references. - - ) -Here is how much of TeX's memory you used: - 2701 strings out of 96052 - 33499 string characters out of 1197190 - 142858 words of memory out of 1139380 - 5526 multiletter control sequences out of 35000 - 23519 words of font info for 83 fonts, out of 500000 for 1000 - 36 hyphenation exceptions out of 607 - 25i,15n,39p,206b,462s stack positions out of 1500i,500n,5000p,200000b,32768s - -Output written on r38.dvi (698 pages, 1739516 bytes). DELETED r38/help/HC.zip Index: r38/help/HC.zip ================================================================== --- r38/help/HC.zip +++ r38/help/HC.zip cannot compute difference between binary files DELETED r38/help/allfiles.html Index: r38/help/allfiles.html ================================================================== --- r38/help/allfiles.html +++ r38/help/allfiles.html @@ -1,34858 +0,0 @@ - - - -Top -INDEX

-Top

-
  • Concepts

    -

  • Variables

    -

  • Syntax

    -

  • Arithmetic Operations

    -

  • Boolean Operators

    -

  • General Commands

    -

  • Algebraic Operators

    -

  • Declarations

    -

  • Input and Output

    -

  • Elementary Functions

    -

  • General Switches

    -

  • Matrix Operations

    -

  • Groebner package

    -

  • High Energy Physics

    -

  • Numeric Package

    -

  • Roots Package

    -

  • Special Functions

    -

  • Taylor series

    -

  • Gnuplot package

    -

  • Linear Algebra package

    -

  • Matrix Normal Forms

    -

  • Miscellaneous Packages

    -

  • Outmoded Operations

    -

    •  - - -IDENTIFIER -INDEX

    - - - -IDENTIFIER _ _ _ _ _ _ _ _ _ _ _ _ type

    -

    - -Identifiers in REDUCE consist of one or more alphanumeric characters, of -which the first must be alphabetical. The maximum number of characters -allowed is system dependent, but is usually over 100. However, printing -is simplified if they are kept under 25 characters. -

    -

    -You can also use special characters in your identifiers, but each must be -preceded by an exclamation point ! as an escape character. Useful -special characters are # $ % ^ & * - + = ? < > ~ | / ! and -the space. Note that the use of the exclamation point as a special -character requires a second exclamation point as an escape character. -The underscore _ is special in this regard. It must be preceded -by an escape character in the first position in an identifier, but is -treated like a normal letter within an identifier. -

    -

    -Other characters, such as ( ) # ; ` ' " can also be used if -preceded by a !, but as they have special meanings to the Lisp -reader it is best to avoid them to avoid confusion. -

    -

    -Many system identifiers have * before or after their names, or - between -words. If you accidentally pick one of these names for your own identifier, -it could have disastrous effects. For this reason it is wise not to include -* or - anywhere in your identifiers. -

    -

    -You will notice that REDUCE does not use the escape characters when it prints -identifiers containing special characters; however, you still must use them -when you refer to these identifiers. Be careful when editing statements -containing escaped special characters to treat the character and its escape -as an inseparable pair. -

    -

    -Identifiers are used for variable names, labels for go to statements, -and names of arrays, matrices, operators, and procedures. Once an identifier is - -used as a matrix, array, scalar or operator identifier, it may not be used -again as a matrix, array or operator. An operator or array identifier may -later be used as a scalar without problems, but a matrix identifier cannot be -used as a scalar. All procedures are entered into the system as operators, so -the name of a procedure may not be used as a matrix, array, or operator -identifier either. -

    -

    - - - -KERNEL -INDEX

    - - - -KERNEL _ _ _ _ _ _ _ _ _ _ _ _ type

    -

    - -A kernel is a form that cannot be modified further by the REDUCE -canonical simplifier. Scalar variables are always kernels. The -other important class of kernels are operators with their arguments. -Some examples should help clarify this concept: -

    -

    -

    
-        Expression                     Kernel?
-
-          x                              Yes
-          varname                        Yes
-          cos(a)                         Yes
-          log(sin(x**2))                 Yes
-          a*b                            No
-          (x+y)**4                       No
-          matrix-identifier              No
-

    Many REDUCE operators expect kernels among their arguments. Error -messages -result from attempts to use non-kernel expressions for these arguments. -

    -

    - - - -STRING -INDEX

    - - - -STRING _ _ _ _ _ _ _ _ _ _ _ _ type

    -

    - -A string is any collection of characters enclosed in double quotation -marks ("). It may be used as an argument for a variety of commands - -and operators, such as in, rederr and write. -

    -examples:

    -

    
-write "this is a string"; 
-
-  this is a string 
-
-
-write a, " ", b, " ",c,"!"; 
-
-  A B C!
-
-

    -

    - - - -Concepts -INDEX

    -Concepts

    -
  • IDENTIFIER type

    -

  • KERNEL type

    -

  • STRING type

    -

    •  - - -assumptions -INDEX

    - - - -ASSUMPTIONS _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -After solving a linear or polynomial equation system -with parameters, the variable assumptions contains a list -of side relations for the parameters. The solution is valid only -as long as none of these expression is zero. -

    -examples:

    -

    
-solve({a*x-b*y+x,y-c},{x,y});
-
-       b*c
-  {{x=-----,y=c}} 
-      a + 1
-
-
-assumptions; 
-
-  {a + 1}
-
-

    -

    - - - -CARD_NO -INDEX

    - - - -CARD\_NO _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -card_nosets the total number of cards allowed in a Fortran -output statement when fort is on. Default is 20. -

    -

    -

    -examples:

    -

    
-on fort; 
-
-card_no := 4; 
-
-  CARD_NO=4. 
-
-
-z := (x + y)**15; 
-
-        ANS1=5005.*X**6*Y**9+3003.*X**5*Y**10+1365.*X**4*Y**
-       . 11+455.*X**3*Y**12+105.*X**2*Y**13+15.*X*Y**14+Y**15
-        Z=X**15+15.*X**14*Y+105.*X**13*Y**2+455.*X**12*Y**3+ 
-       . 1365.*X**11*Y**4+3003.*X**10*Y**5+5005.*X**9*Y**6+
-       . 6435.*X**8*Y**7+6435.*X**7*Y**8+ANS1
-
-

    Twenty total cards means 19 continuation cards. You may set it for - more -if your Fortran system allows more. Expressions are broken apart in a -Fortran-compatible way if they extend for more than card_no -continuation cards. -

    -

    -

    - - - -E -INDEX

    - - - -E _ _ _ _ _ _ _ _ _ _ _ _ constant

    -

    - -The constant e is reserved for use as the base of the natural -logarithm. Its value is approximately 2.71828284590, which REDUCE gives -to the current decimal precision when the switch -rounded is on. -

    -

    -emay be used as an iterative variable in a -for statement, -or as a local variable or a -procedure. If e is defined -as a local -variable inside the procedure, the normal definition as the base of the -natural logarithm would be suspended inside the procedure. -

    -

    -

    - - - -EVAL_MODE -INDEX

    - - - -EVAL\_MODE _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -The system variable eval_mode contains the current mode, either - -algebraic or -symbolic. -

    -

    -

    -examples:

    -

    
-EVAL_MODE; 
-
-  ALGEBRAIC
-
-

    Some commands do not behave the same way in algebraic and symbolic - modes. -

    -

    -

    - - - -FORT_WIDTH -INDEX

    - - - -FORT\_WIDTH _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -The fort_width variable sets the number of characters in a line of -Fortran-compatible output produced when the -fort switch is on. -Default is 70. -

    -

    -

    -examples:

    -

    
-fort_width := 30; 
-
-  FORT_WIDTH := 30  
-
-
-on fort; 
-
-df(sin(x**3*y),x); 
-
-        ANS=3.*COS(X
-       . **3*Y)*X**2*
-       . Y
-
-

    fort_widthincludes the usually blank characters at the be -ginning -of the card. As you may notice above, it is conservative and makes the -lines even shorter than it was told. -

    -

    -

    - - - -HIGH_POW -INDEX

    - - - -HIGH\_POW _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -The variable high_pow is set by -coeff to the highest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. -

    -

    -

    -examples:

    -

    
-coeff((x+1)^5*(x*(y+3)^2)^2,x); 
-
-  {0,
-   0,
-    4       3       2
-   Y  + 12*Y  + 54*Y  + 108*Y + 81,
-       4       3       2
-   5*(Y  + 12*Y  + 54*Y  + 108*Y + 81),
-        4       3       2
-   10*(Y  + 12*Y  + 54*Y  + 108*Y + 81),
-        4       3       2
-   10*(Y  + 12*Y  + 54*Y  + 108*Y + 81),
-       4       3       2
-   5*(Y  + 12*Y  + 54*Y  + 108*Y + 81),
-    4       3       2
-   Y  + 12*Y  + 54*Y  + 108*Y + 81}
-
-
-high_pow; 
-
-  7
-
-

    - - -I -INDEX

    - - - -I _ _ _ _ _ _ _ _ _ _ _ _ constant

    -

    - -

    -

    -REDUCE knows i is the square root of -1, - and that i^2 = -1. -

    -

    -

    -examples:

    -

    
-(a + b*i)*(c + d*i); 
-
-  A*C + A*D*I + B*C*I - B*D 
-
-
-i**2; 
-
-  -1
-
-

    icannot be used as an identifier. It is all right to use -i -as an index variable in a for loop, or as a local (scalar) -variable inside a begin...end block, but it loses its definition as -the square root of -1 inside the block in that case. -

    -

    -Only the simplest properties of i are known by REDUCE unless -the switch -complex is turned on, which implements full complex -arithmetic in factoring, simplification, and functional values. -complex is ordinarily off. -

    -

    -

    - - - -INFINITY -INDEX

    - - - -INFINITY _ _ _ _ _ _ _ _ _ _ _ _ constant

    -

    - -The name infinity is used to represent the infinite positive number. -However, at the present time, arithmetic in terms of this operator reflects -finite arithmetic, rather than true operations on infinity. -

    -

    - - - -LOW_POW -INDEX

    - - - -LOW\_POW _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -The variable low_pow is set by -coeff to the lowest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. -

    -

    -

    -examples:

    -

    
-coeff((x+2*y)**6,y); 
-
-    6
-  {X ,
-       5
-   12*X ,
-       4
-   60*X ,
-        3
-   160*X ,
-        2
-   240*X ,
-   192*X,
-   64}
-
-
-low_pow; 
-
-  0 
-
-
-coeff(x**2*(x*sin(y) + 1),x); 
-			 
-
-
-  {0,0,1,SIN(Y)} 
-
-
-low_pow; 
-
-  2
-
-

    - - -NIL -INDEX

    - - - -NIL _ _ _ _ _ _ _ _ _ _ _ _ constant

    -

    - -

    -

    -nilrepresents the truth value false in symbolic mode, and is -a synonym for 0 in algebraic mode. It cannot be used for any other -purpose, even inside procedures or -for loops. -

    -

    - - - -PI -INDEX

    - - - -PI _ _ _ _ _ _ _ _ _ _ _ _ constant

    -

    - -The identifier pi is reserved for use as the circular constant. -Its value is given by 3.14159265358..., which REDUCE gives to the current -decimal precision when REDUCE is in a floating-point mode. -

    -

    -pimay be used as a looping variable in a -for statement, -or as a local variable in a -procedure. Its value in such cases -will be taken from the local environment. -

    -

    -

    - - - -requirements -INDEX

    - - - -REQUIREMENTS _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -After an attempt to solve an inconsistent equation system -with parameters, the variable requirements contains a list -of expressions. These expressions define a set of conditions implicitly -equated with zero. Any solution to this system defines a setting for -the parameters sufficient to make the original system consistent. -

    -examples:

    -

    
-solve({x-a,x-y,y-1},{x,y}); 
-
-  {}
-
-
-requirements;
-
-  {a - 1}
-
-

    -

    - - - -ROOT_MULTIPLICITIES -INDEX

    - - - -ROOT\_MULTIPLICITIES _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -The root_multiplicities variable is set to the list of the -multiplicities of the roots of an equation by the -solve operator. -

    -

    - -solvereturns its solutions in a list. The multipliciti -es of -each solution are put in the corresponding locations of the list -root_multiplicities. -

    -

    -

    - - - -T -INDEX

    - - - -T _ _ _ _ _ _ _ _ _ _ _ _ constant

    -

    - -The constant t stands for the truth value true. It cannot be used -as a scalar variable in a -block, as a looping variable in a - -for statement or as an -operator name. -

    -

    - - - -Variables -INDEX

    -Variables

    -
  • assumptions variable

    -

  • CARD\_NO variable

    -

  • E constant

    -

  • EVAL\_MODE variable

    -

  • FORT\_WIDTH variable

    -

  • HIGH\_POW variable

    -

  • I constant

    -

  • INFINITY constant

    -

  • LOW\_POW variable

    -

  • NIL constant

    -

  • PI constant

    -

  • requirements variable

    -

  • ROOT\_MULTIPLICITIES variable

    -

  • T constant

    -

    •  - - -semicolon -INDEX

    - - - -; _ _ _ SEMICOLON _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The semicolon is a statement delimiter, indicating results are to be printed -when used in interactive mode. -

    -

    -

    -examples:

    -

    
-(x+1)**2; 
-
-   2
-  X  + 2*X + 1 
-
-
-df(x**2 + 1,x); 
-
-  2*X
-
-

    Entering a Return without a semicolon or dollar sign resu -lts in a -prompt on the following line. A semicolon or dollar sign can be -added at this point to execute the statement. In interactive mode, a -statement that is ended with a semicolon and Return has its results -printed on the screen. -

    -

    -Inside a group statement <<...>> -or a begin...end block, a -semicolon or dollar sign separates individual REDUCE statements. Since -results are not printed from a block without a specific return -statement, there is no difference between using the semicolon or dollar -sign. In a group statement, the last value produced is the value -returned by the group statement. Thus, if a semicolon or dollar sign is -placed between the last statement and the ending brackets, the group -statement returns the value 0 or nil, rather than the value of the -last statement. -

    -

    -

    - - - -dollar -INDEX

    - - - -$ _ _ _ DOLLAR _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The dollar sign is a statement delimiter, indicating results are not to be -printed when used in interactive mode. -

    -

    -

    -examples:

    -

    
-
-(x+1)**2$

    The workspace is set to x^2 + 2x + 1 - but nothing shows on the screen

    
-
-
-ws; 
-
-   2
-  X   + 2*X + 1
-
-

    -

    -

    -Entering a Return without a semicolon or dollar sign results in a -prompt on the following line. A semicolon or dollar sign can -be added at this point to execute the statement. In interactive mode, a -statement that ends with a dollar sign $ and a Return is -executed, but the results not printed. -

    -

    -Inside a -group statement <<...>> - -or a begin...end -block, a -semicolon or dollar sign separates individual REDUCE statements. Since -results are not printed from a -block without a specific - -return

    -

    -statement, there is no difference between using the semicolon or dollar -sign. -

    -

    -In a group statement, the last value produced is the value returned by the -group statement. Thus, if a semicolon or dollar sign is placed between the -last statement and the ending brackets, the group statement returns the -value 0 or nil, rather than the value of the last statement. -

    -

    -

    -

    - - - -percent -INDEX

    - - - -% _ _ _ PERCENT _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The percent sign is used to precede comments; everything from a percent -to the end of the line is ignored. -

    -

    -

    -examples:

    -

    
-
-df(x**3 + y,x);% This is a comment key{Return} 
-
-
-     2
-  3*X  
-
-
-int(3*x**2,x) %This is a comment; key{Return} 
-

    A prompt is given, waiting for the semicolon that was not -detected in the comment

    -

    -

    -Statement delimiters ; and $ are not detected between a -percent sign and the end of the line. -

    -

    -

    - - - -dot -INDEX

    - - - -. _ _ _ DOT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The . (dot) infix binary operator adds a new item to the beginning of an -existing -list. In high energy physics expressions, -it can also be used -to represent the scalar product of two Lorentz four-vectors. -

    -

    -

    -syntax:

    -<item> . <list> -

    -

    -

    -<item> can be any REDUCE scalar expression, including a list; -<list> must be a -list to avoid producing an error message. -The dot operator is right associative. -

    -

    -

    -examples:

    -

    
-
-liss := a . {}; 
-
-  LISS := {A} 
-
-
-liss := b . liss; 
-
-  LISS := {B,A} 
-
-
-newliss := liss . liss; 
-
-  NEWLISS := {{B,A},B,A} 
-
-
-firstlis := a . b . {c}; 
-
-  FIRSTLIS := {A,B,C} 
-
-
-secondlis := x . y . {z}; 
-
-  SECONDLIS := {X,Y,Z} 
-
-
-for i := 1:3 sum part(firstlis,i)*part(secondlis,i);
- 
-
-
-  A*X + B*Y + C*Z
-
-

    - - -assign -INDEX

    - - - -:= _ _ _ ASSIGN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The := is the assignment operator, assigning the value on the right-han -d -side to the identifier or other valid expression on the left-hand side. -

    -

    -

    -syntax:

    -<restricted\_expression> := <expression> -

    -

    -

    -<restricted\_expression> is ordinarily a single identifier, though simple - -expressions may be used (see Comments below). <expression> is any -valid REDUCE expression. If <expression> is a -matrix -identifier, then -<restricted\_expression> can be a matrix identifier (redimensioned if -necessary) which has each element set to the corresponding elements -of the identifier on the right-hand side. -

    -

    -

    -examples:

    -

    
-a := x**2 + 1; 
-
-        2
-  A := X   + 1 
-
-
-a; 
-
-   2
-  X  + 1 
-
-
-first := second := third; 
-
-  FIRST := SECOND := THIRD 
-
-
-first; 
-
-  THIRD 
-
-
-second; 
-
-  THIRD 
-
-
-b := for i := 1:5 product i; 
-
-  B := 120 
-
-
-b; 
-
-  120 
-
-
-w + (c := x + 3) + z; 
-
-  W + X + Z + 3 
-
-
-c; 
-
-  X + 3 
-
-
-y + b := c; 
-
-  Y + B := C 
-
-
-y; 
-
-  - (B - C)
-
-

    The assignment operator is right associative, as shown in the seco -nd and -third examples. A string of such assignments has all but the last -item set to the value of the last item. Embedding an assignment statement -in another expression has the side effect of making the assignment, as well -as causing the given replacement in the expression. -

    -

    -Assignments of values to expressions rather than simple identifiers (such as in - -the last example above) can also be done, subject to the following remarks: -

    -

    - _ _ _ (i) -If the left-hand side is an identifier, an operator, or a power, the -substitution rule is added to the rule table. -

    -

    - _ _ _ (ii) -If the operators - + / appear on the left-hand side, all but the first - -term of the expression is moved to the right-hand side. -

    -

    - _ _ _ (iii) -If the operator * appears on the left-hand side, any constant terms are - -moved to the right-hand side, but the symbolic factors remain. -

    -

    -Assignment is valid for -array elements, but not for entire arrays. -The assignment operator can also be used to attach functionality to operators. -

    -

    -A recursive construction such as a := a + b is allowed, but when -a is referenced again, the process of resubstitution continues -until the expression stack overflows (you get an error message). -Recursive assignments can be done safely inside controlled loop -expressions, such as -for... or -repeat...until. -

    -

    -

    -

    - - - -equalsign -INDEX

    - - - -= _ _ _ EQUALSIGN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The = operator is a prefix or infix equality comparison operator. -

    -

    -

    -syntax:

    -=(<expression>,<expression>) - or - <expression> = <expression> -

    -

    -

    -<expression> can be any REDUCE scalar expression. -

    -

    -

    -examples:

    -

    
-a := 4; 
-
-  A := 4 
-
-
-if =(a,10) then write "yes" else write "no";
- 
-
-
-  no 
-
-
-b := c; 
-
-  B := C 
-
-
-if b = c then write "yes" else write "no";
- 
-
-
-  yes 
-
-
-on rounded; 
-
-if 4.0 = 4 then write "yes" else write "no";
- 
-
-
-  yes
-
-

    This logical equality operator can only be used inside a condition -al -statement, such as -if...then...else -or -repeat...until. In other places the equal -sign establishes an algebraic object of type -equation. -

    -

    -

    -

    - - - -replace -INDEX

    - - - -=> _ _ _ REPLACE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The => operator is a binary operator used in -rule lists to -denote replacements. -

    -

    -

    -examples:

    -

    
-operator f; 
-
-let f(x) => x^2; 
-
-f(x); 
-
-   2
-  x
-
-

    - - -plussign -INDEX

    - - - -+ _ _ _ PLUSSIGN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The + operator is a prefix or infix n-ary addition operator. -

    -

    -

    -syntax:

    -<expression> { +<expression>}+ -

    -

    -or +(<expression> {,<expression>}+) -

    -

    -

    -<expression> may be any valid REDUCE expression. -

    -

    -

    -examples:

    -

    
-x**4 + 4*x**2 + 17*x + 1; 
-
-   4      2
-  X  + 4*X  + 17*X + 1 
-
-
-14 + 15 + x; 
-
-  X + 29 
-
-
-+(1,2,3,4,5); 
-
-  15
-
-

    +is also valid as an addition operator for -matrix variables -that are of the same dimensions and for -equations. -

    -

    -

    - - - -minussign -INDEX

    - - - -- _ _ _ MINUSSIGN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The - operator is a prefix or infix binary subtraction operator, as wel -l -as the unary minus operator. -

    -

    -

    -syntax:

    -<expression> - <expression> -or -(<expression>,<expression>) -

    -

    -

    -<expression> may be any valid REDUCE expression. -

    -

    -

    -examples:

    -

    
-15 - 4; 
-
-  11 
-
-
-x*(-5); 
-
-  - 5*X 
-
-
-a - b - 15; 
-
-  A - B - 15 
-
-
--(a,4); 
-
-  A - 4
-
-

    The subtraction operator is left associative, so that a - b - c is - equivalent -to (a - b) - c, as shown in the third example. The subtraction operator is -also valid with -matrix expressions of the correct dimensions -and with -equations. -

    -

    -

    - - - -asterisk -INDEX

    - - - -* _ _ _ ASTERISK _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The * operator is a prefix or infix n-ary multiplication operator. -

    -

    -

    -syntax:

    -<expression> { * <expression>}+ -

    -

    -or *(<expression> {,<expression>}+) -

    -

    -

    -<expression> may be any valid REDUCE expression. -

    -

    -

    -examples:

    -

    
-15*3; 
-
-  45 
-
-
-24*x*yvalue*2; 
-
-  48*X*YVALUE 
-
-
-*(6,x); 
-
-  6*X 
-
-
-on rounded; 
-
-3*1.5*x*x*x; 
-
-       3
-  4.5*X  
-
-
-off rounded; 
-
-2x**2; 
-
-     2
-  2*X
-
-

    REDUCE assumes you are using an implicit multiplication operator w -hen an -identifier is preceded by a number, as shown in the last line above. Since -no valid identifiers can begin with numbers, there is no ambiguity in -making this assumption. -

    -

    -The multiplication operator is also valid with -matrix expressions -of the -proper dimensions: matrices A and B -can be multiplied if -A is n x m and B is -m x p. Matrices and -equations can also be -multiplied by scalars: the -result is as if each element was multiplied by the scalar. -

    -

    -

    - - - -slash -INDEX

    - - - -/ _ _ _ SLASH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The / operator is a prefix or infix binary division operator or -prefix unary -reciprocal operator. -

    -

    -

    -syntax:

    -<expression>/<expression> or - /<expression> -

    -

    -or /(<expression>,<expression>) -

    -

    -

    -<expression> may be any valid REDUCE expression. -

    -

    -

    -examples:

    -

    
-20/5; 
-
-  4  
-
-
-100/6; 
-
-  50
-  -- 
-  3
-
-
-16/2/x; 
-
-  8
-  - 
-  X
-
-
-/b; 
-
-  1
-  - 
-  B
-
-
-/(y,5); 
-
-  Y
-  - 
-  5
-
-
-on rounded; 
-
-35/4; 
-
-  8.75 
-
-
-/20; 
-
-  0.05
-
-

    The division operator is left associative, so that a/b/c -is equivalent -to (a/b)/c. The division operator is also valid with square - -matrix expressions of the same dimensions: With A and - -B both n x n matrices and B -invertible, A/B is -given by A*B^-1. -Division of a matrix by a scalar is defined, with the results being the -division of each element of the matrix by the scalar. Division of a -scalar by a matrix is defined if the matrix is invertible, and has the -effect of multiplying the scalar by the inverse of the matrix. When -/ is used as a reciprocal operator for a matrix, the inverse of -the matrix is returned if it exists. -

    -

    -

    - - - -power -INDEX

    - - - -** _ _ _ POWER _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The ** operator is a prefix or infix binary exponentiation operator. -

    -

    -

    -syntax:

    -<expression> **<expression> - or **(<expression>,<expression>) -

    -

    -

    -<expression> may be any valid REDUCE expression. -

    -

    -

    -examples:

    -

    
-x**15; 
-
-   15
-  X   
-
-
-x**y**z; 
-
-   Y*Z
-  X    
-
-
-x**(y**z); 
-
-    Z
-   Y
-  X   
-
-
- **(y,4); 
-
-   4
-  Y  
-
-
-on rounded; 
-
-2**pi; 
-
-  8.82497782708
-
-

    The exponentiation operator is left associative, so that a**b* -*c is -equivalent to (a**b)**c, as shown in the second example. Note -that this is not a**(b**c), which would be right associative. -

    -

    -When -nat is on (the default), REDUCE output produces raised - -exponents, as shown. The symbol ^, which is the upper-case 6 on -most keyboards, may be used in the place of **. -

    -

    -A square -matrix may also be raised to positive and negative pow -ers -with the exponentiation operator (negative powers require the matrix to be -invertible). Scalar expressions and -equations may be raised to -fractional and floating-point powers. -

    -

    -

    - - - -caret -INDEX

    - - - -^ _ _ _ CARET _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The ^ operator is a prefix or infix binary exponentiation operator. -It is equivalent to -power or **. -

    -

    -

    -syntax:

    -<expression> ^<expression> - or ^(<expression>,<expression>) -

    -

    -

    -<expression> may be any valid REDUCE expression. -

    -

    -

    -examples:

    -

    
-x^15; 
-
-   15
-  X   
-
-
-x^y^z; 
-
-   Y*Z
-  X    
-
-
-x^(y^z); 
-
-    Z
-   Y
-  X   
-
-
-^(y,4); 
-
-   4
-  Y  
-
-
-on rounded; 
-
-2^pi; 
-
-  8.82497782708
-
-

    The exponentiation operator is left associative, so that a^b^c - is -equivalent to (a^b)^c, as shown in the second example. Note -that this is <not> a^(b^c), which would be right associative. -

    -

    -When -nat is on (the default), REDUCE output produces raised - -exponents, as shown. -

    -

    -A square -matrix may also be raised to positive -and negative powers with -the exponentiation operator (negative powers require the matrix to be -invertible). Scalar expressions and -equations -may be raised to fractional and floating-point powers. -

    -

    -

    - - - -geqsign -INDEX

    - - - ->= _ _ _ GEQSIGN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - ->= is an infix binary comparison operator, which returns true if -its first argument is greater than or equal to its second argument. -

    -

    -

    -syntax:

    -<expression> >= <expression> -

    -

    -

    -<expression> must evaluate to an integer or floating-point number. -

    -

    -

    -examples:

    -

    
-if (3 >= 2) then yes; 
-
-  yes 
-
-
-a := 15; 
-
-  A := 15 
-
-
-if a >= 20 then big else small;
- 
-
-  small 
-
-

    The binary comparison operators can only be used for comparisons b -etween -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as -if...then...else -or -repeat...until or -while...do. -

    -

    -

    - - - -greater -INDEX

    - - - -> _ _ _ GREATER _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The > is an infix binary comparison operator that returns - true if its first argument is strictly greater than its second. -

    -

    -

    -syntax:

    -<expression> > <expression> -

    -

    -

    -<expression> must evaluate to a number, e.g., integer, rational or -floating point number. -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-if 3.0 > 3 then write "different" else write "same"; 
-
-
-  same 
-
-
-off rounded; 
-
-a := 20; 
-
-  A := 20 
-
-
-if a > 20 then write "bigger" else write "not bigger"; 
-
-
-  not bigger 
-
-

    The binary comparison operators can only be used for comparisons b -etween -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as -if...then...else or - -repeat...until or -while...do. -

    -

    -

    - - - -leqsign -INDEX

    - - - -<= _ _ _ LEQSIGN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -<= is an infix binary comparison operator that returns - true if its first argument is less than or equal to its second argument. -

    -

    -

    -syntax:

    -<expression> <= <expression> -

    -

    -

    -<expression> must evaluate to a number, e.g., integer, rational or -floating point number. -

    -

    -

    -examples:

    -

    
-a := 10; 
-
-  A := 10 
-
-
-if a <= 10 then true; 
-
-  true
-
-

    The binary comparison operators can only be used for comparisons b -etween -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as -if...then...else or - -repeat...until or -while...do. -

    -

    -

    - - - -less -INDEX

    - - - -< _ _ _ LESS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -< is an infix binary logical comparison operator that -returns true if its first argument is strictly less than its second -argument. -

    -

    -

    -syntax:

    -<expression> < <expression> -

    -

    -

    -<expression> must evaluate to a number, e.g., integer, rational or -floating point number. -

    -

    -

    -examples:

    -

    
-f := -3; 
-
-  F := -3 
-
-
-if f < -3 then write "yes" else write "no"; 
-
-
-  no
-
-

    The binary comparison operators can only be used for comparisons b -etween -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as -if...then...else -or -repeat...until or - -while...do. -

    -

    -

    - - - -tilde -INDEX

    - - - -~ _ _ _ TILDE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The ~ is used as a unary prefix operator in the left-hand -sides of -rules to mark -free variables. A double tilde -marks an optional -free variable. -

    -

    - - - -group -INDEX

    - - - -<< _ _ _ GROUP _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The <<...>> command is a group statement, -used to group statements -together where REDUCE expects a single statement. -

    -

    -

    -syntax:

    -<<<statement>{; <statement> or - <statement>}* >> -

    -

    -

    -<statement> may be any valid REDUCE statement or expression. -

    -

    -

    -examples:

    -

    
-a := 2; 
-
-  A := 2 
-
-
-if a < 5 then <<b := a + 10; write b>>; 
-
-
-  12 
-
-
-<<d := c/15; f := d + 3; f**2>>;
- 
-
-   2
-  C  + 90*C + 202
-  ----------------
-        225
-
-

    The value returned from a group statement is the value of the last - -individual statement executed inside it. Note that when a semicolon is -placed between the last statement and the closing brackets, 0 or - nil is returned. Group statements are often used in the -consequence portions of -if...then, - -repeat...until, and - -while...do -clauses. They may also be used in interactive -operation to execute several statements at one time. Statements inside -the group statement are separated by semicolons or dollar signs. -

    -

    -

    -

    - - - -AND -INDEX

    - - - -AND _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The and binary logical operator returns true if both of its -arguments are true. -

    -

    -

    -syntax:

    -<logical\_expression> and <logical\_expression> -

    -

    -

    -<logical\_expression> must evaluate to true or nil. -

    -

    -

    -examples:

    -

    
-a := 12; 
-
-  A := 12 
-
-
-if numberp a and a < 15 then write a**2 else write "no";
- 
-
-
-  144 
-
-
-clear a; 
-
-if numberp a and a < 15 then write a**2 else write "no";
- 
-
-
-  no
-
-

    Logical operators can only be used inside conditional statements, -such as - -while...do or - -if...then...else. and exami -nes each of -its arguments in order, and quits, returning nil, on finding an -argument that is not true. An error results if it is used in other -contexts. -

    -

    -andis left associative: x and y and z is equivalent to -(x and y) and z. -

    -

    -

    - - - -BEGIN -INDEX

    - - - -BEGIN _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -begin is used to start a -block statement, which is closed with -end. -

    -

    -

    -syntax:

    -begin<statement>{; <statement>}* end -

    -

    -

    -<statement> is any valid REDUCE statement. -

    -

    -

    -examples:

    -

    
-begin for i := 1:3 do write i end; 
-
-
-  1
-  2
-  3     
-
-
-begin scalar n;n:=1;b:=for i:=1:4 product(x-i);return n end;
- 
-
-
-  1 
-
-
-b; 
-
-   4        3        2
-  X   - 10*X   + 35*X   - 50*X  + 24
-
-

    A begin...end block can do actions (such as -write), but -does not -return a value unless instructed to by a -return statement, which must -be the last statement executed in the block. It is unnecessary to insert -a semicolon before the end. -

    -

    -Local variables, if any, are declared in the first statement immediately -after begin, and may be defined as scalar, integer, or -real. -array variables declared -within a begin...end block -are global in every case, and -let statements have global -effects. A -let statement involving a formal parameter affects -the calling parameter that corresponds to it. -let statements -involving local variables make global assignments, overwriting outside -variables by the same name or creating them if they do not exist. You -can use this feature to affect global variables from procedures, but be -careful that you do not do it inadvertently. -

    -

    -

    -

    - - - -block -INDEX

    - - - -BLOCK _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -A block is a sequence of statements enclosed by -commands -begin and -end. -

    -

    -

    -syntax:

    -begin<statement>{; <statement>}* end -

    -

    -

    -For more details see -begin. -

    -

    - - - -COMMENT -INDEX

    - - - -COMMENT _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -Beginning with the word comment, all text until the next statement -terminator (; or $) is ignored. -

    -

    -

    -examples:

    -

    
-
-x := a**2 comment--a is the velocity of the particle;;
- 
-
-
-        2
-  X := A
-
-

    Note that the first semicolon ends the comment and the second one - -terminates the original REDUCE statement. -

    -

    -Multiple-line comments are often needed in interactive files. The -comment command allows a normal-looking text to accompany the -REDUCE statements in the file. -

    -

    -

    - - - -CONS -INDEX

    - - - -CONS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The cons operator adds a new element to the beginning of a - -list. Its -operation is identical to the symbol -dot (dot). It can be used -infix or prefix. -

    -

    -

    -syntax:

    -cons(<item>,<list>) or <item> cons <list -> -

    -

    -

    -<item> can be any REDUCE scalar expression, including a list; <list> - -must be a list. -

    -

    -

    -examples:

    -

    
-
-liss := cons(a,{b}); 
-
-  {A,B} 
-
-
-
-liss := c cons liss; 
-
-  {C,A,B} 
-
-
-
-newliss := for each y in liss collect cons(y,list x);
- 
-
-
-  NEWLISS := {{C,X},{A,X},{B,X}} 
-
-
-
-for each y in newliss sum (first y)*(second y);
- 
-
-
-  X*(A + B + C)
-
-

    If you want to use cons to put together two elements into - a new list, -you must make the second one into a list with curly brackets or the list - -command. You can also start with an empty list created by {}. -

    -

    -The cons operator is right associative: a cons b cons c is val -id -if c is a list; b need not be a list. The list produced is -{a,b,c}. -

    -

    -

    -

    - - - -END -INDEX

    - - - -END _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The command end has two main uses: -

    -

    - _ _ _ (i) -as the ending of a -begin...end -block; and -

    - _ _ _ (ii) -to end input from a file. -

    -

    -In a begin...end -block, there need not be a delimiter -(; or $) before the end, though there must be one -after it, or a right bracket matching an earlier left bracket. -

    -

    -Files to be read into REDUCE should end with end;, which must be -preceded by a semicolon (usually the last character of the previous line). -The additional semicolon avoids problems with mistakes in the files. If -you have suspended file operation by answering n to a pause -command, you are still, technically speaking, ``in" the file. Use -end to exit the file. -

    -

    -An end at the top level of a program is ignored. -

    -

    -

    - - - -EQUATION -INDEX

    - - - -EQUATION _ _ _ _ _ _ _ _ _ _ _ _ type

    -

    - -

    -

    -An equation is an expression where two algebraic expressions -are connected by the (infix) operator -equal or by =. -For access to the components of an equation the operators - -lhs, -rhs or -part can be used. The -evaluation of the left-hand side of an equation is controlled -by the switch -evallhseqp, while the right-hand side is -evaluated unconditionally. When an equation is part of a -logical expression, e.g. in a -if or -while statement, -the equation is evaluated by subtracting both sides can comparing -the result with zero. -

    -

    -Equations occur in many contexts, e.g. as arguments of the -sub -operator and in the arguments and the results -of the operator -solve. An equation can be member of a -list -and you may assign an equation to a variable. Elementary arithmetic is supported - -for equations: if -evallhseqp is on, you may add and subtract -equations, and you can combine an equation with a scalar expression by -addition, subtraction, multiplication, division and raise an equation -to a power. -

    -examples:

    -

    
-on evallhseqp;
-
-u:=x+y=1$
-
-v:=2x-y=0$
-
-2*u-v; 
-
-  - 3*y=-2
-
-
-ws/3; 
-
-    2
-  y=--
-    3
-
-

    -

    -Important: the equation must occur in the leftmost term of such an expression. -For other operations, e.g. taking function values of both sides, use the - -map operator. -

    -

    - - - -FIRST -INDEX

    - - - -FIRST _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The first operator returns the first element of a -list. -

    -syntax:

    -

    -

    -first(<list>) or first <list> -

    -

    -

    -<list> must be a non-empty list to avoid an error message. -

    -

    -

    -examples:

    -

    
-alist := {a,b,c,d}; 
-
-  ALIST := {A,B,C,D} 
-
-
-first alist; 
-
-  A 
-
-
-blist := {x,y,{ww,aa,qq},z}; 
-
-  BLIST := {X,Y,{WW,AA,QQ},Z} 
-
-
-first third blist; 
-
-  WW
-
-

    - - -FOR -INDEX

    - - - -FOR _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The for command is used for iterative loops. There are many -possible forms it can take. -

    -

    -

    
-                   /                   
-   /               |STEP <number> UNTIL|        
-   |<var>:=<number>|                   |<number>|
-FOR|               |         :         |        |<action> <exprn>
-   |                                  /        |
-   |EACH <var> IN <list>                        |
-                                               /
-
- where <action> ::= DO|PRODUCT|SUM|COLLECT|JOIN.
-

    <var> can be any valid REDUCE identifier except t o -r -nil, <inc>, <start> and <stop> can be any expression - -that evaluates to a positive or negative integer. <list> must be a -valid -list structure. -The action taken must be one of the actions shown -above, each of which is followed by a single REDUCE expression, statement -or a -group (<<...>>) or -block -( -begin... -end) statement. -

    -

    -

    -examples:

    -

    
-for i := 1:10 sum i;                                    
- 
-
-
-  55 
-
-
-for a := -2 step 3 until 6 product a;
-							
-
-
-  -8 
-
-
-a := 3; 
-
-  A := 3 
-
-
-for iter := 4:a do write iter; 
-
-m := 0; 
-
-  M := 0 
-
-
-for s := 10 step -1 until 3 do <<d := 10*s;m := m + d>>; 
-
-m; 
-
-  520 
-
-
-for each x in {q,r,s} sum x**2; 
-
-   2    2    2
-  Q  + R  + S  
-
-
-for i := 1:4 collect 1/i;                              
- 
-
-
-     1 1 1
-  {1,-,-,-} 
-     2 3 4
-
-
-for i := 1:3 join list solve(x**2 + i*x + 1,x);         
- 
-
-
-        SQRT(3)*I + 1
-  {{X= --------------,
-              2
-        SQRT(3)*I - 1
-    X= --------------}
-              2
-   {X=-1},
-         SQRT(5) + 3   SQRT(5) - 3
-   {X= - -----------,X=-----------}}
-              2             2
-
-

    The behavior of each of the five action words follows: -

    -

    -

    
-                           Action Word Behavior
-Keyword   Argument Type                    Action
-   do    statement, command, group   Evaluates its argument once
-         or block                    for each iteration of the loop,
-                                     not saving results
-collect expression, statement,       Evaluates its argument once for
-        command, group, block, list  each iteration of the loop,
-                                     storing the results in a list
-                                     which is returned by the for
-                                     statement when done
- join   list or an operator which    Evaluates its argument once for
-        produces a list              each iteration of the loop,
-                                     appending the elements in each
-                                     individual result list onto the
-                                     overall result list
-product expression, statement,       Evaluates its argument once for
-        command, group or block      each iteration of the loop,
-                                     multiplying the results together
-                                     and returning the overall product
-  sum   expression, statement,       Evaluates its argument once for
-        command, group or block      each iteration of the loop,
-                                     adding the results together and
-                                     returning the overall sum
-

    For number-driven for statements, if the ending limit is -smaller -than the beginning limit (larger in the case of negative steps) the action -statement is not executed at all. The iterative variable is local to the -for statement, and does not affect the value of an identifier with -the same name. For list-driven for statements, if the list is -empty, the action statement is not executed, but no error occurs. -

    -

    -You can use nested for statements, with the inner for -statement after the action keyword. You must make sure that your inner -statement returns an expression that the outer statement can handle. -

    -

    -

    - - - -FOREACH -INDEX

    - - - -FOREACH _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -foreachis a synonym for the for each variant of the - -for construct. It is designed to iterate down a list, -and an -error will occur if a list is not used. The use of for each is -preferred to foreach. -

    -

    -

    -syntax:

    -foreach<variable> in <list> <action> <expression -> -

    -

    -where <action> ::= do | product | sum | collect | join -

    -

    -

    -

    -examples:

    -

    
-foreach x in {q,r,s} sum x**2; 
-
-   2    2    2
-  Q  + R  + S
-
-

    - - -GEQ -INDEX

    - - - -GEQ _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The geq operator is a binary infix or prefix logical operator. It -returns true if its first argument is greater than or equal to its second -argument. As an infix operator it is identical with >=. -

    -syntax:

    -

    -

    -geq(<expression>,<expression>) or <expression> -geq <expression> -

    -

    -

    -<expression> can be any valid REDUCE expression that evaluates to a -number. -

    -

    -

    -examples:

    -

    
-a := 20; 
-
-  A := 20 
-
-
-if geq(a,25) then write "big" else write "small";
-			 
-
-
-  small 
-
-
-if a geq 20 then write "big" else write "small";
-			 
-
-
-  big  
-
-
-if (a geq 18) then write "big" else write "small";
-			 
-
-
-  big
-
-

    Logical operators can only be used in conditional statements such -as -

    -

    - -if...then...else or - -repeat...until. -

    -

    -

    - - - -GOTO -INDEX

    - - - -GOTO _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -Inside a begin...end -block, goto, or -preferably, go to, transfers flow of control to a labeled statement. -

    -syntax:

    -

    -

    -go to<labeled_statement> or goto <labeled_statement -> -

    -

    -

    -<labeled_statement> is of the form <label> :<statement -> -

    -

    -

    -examples:

    -

    
-     procedure dumb(a);
-        begin scalar q;
-           go to lab;
-           q := df(a**2 - sin(a),a);
-           write q;
-      lab: return a
-        end;
- 
-
-  DUMB 
-
-
-
-dumb(17); 
-
-  17
-
-

    go tocan only be used inside a begin...end - - -block, and inside -the block only statements at the top level can be labeled, not ones inside -<<...>>, -while...do, etc. -

    -

    -

    - - - -GREATERP -INDEX

    - - - -GREATERP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The greaterp logical operator returns true if its first argument is -strictly greater than its second argument. As an infix operator it is -identical with >. -

    -syntax:

    -

    -

    -greaterp(<expression>,<expression>) or <expression> -greaterp <expression> -

    -

    -

    -<expression> can be any valid REDUCE expression that evaluates to a -number. -

    -

    -

    -examples:

    -

    
-
-a := 20; 
-
-  A := 20 
-
-
-if greaterp(a,25) then write "big" else write "small";
-			 
-
-
-  small 
-
-
-if a greaterp 20 then write "big" else write "small";
-			 
-
-
-  small 
-
-
-if (a greaterp 18) then write "big" else write "small";
-			 
-
-
-  big
-
-

    Logical operators can only be used in conditional statements such -as -

    -

    - -if...then...else -or -repeat... -while. -

    -

    -

    - - - -IF -INDEX

    - - - -IF _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The if command is a conditional statement that executes a statement -if a condition is true, and optionally another statement if it is not. -

    -syntax:

    -

    -

    -if<condition> then <statement> - _ _ _ option(else <statement>) -

    -

    -

    -<condition> must be a logical or comparison operator that evaluates to -a -boolean value. -<statement> must be a single REDUCE statement or a - -group (<<...>>) or - -block (begin...end) statement. -

    -

    -

    -examples:

    -

    
-if x = 5 then a := b+c else a := d+f;
-			 
-
-
-  D + F 
-
-
-x := 9; 
-
-  X := 9 
-
-
-if numberp x and x<20 then y := sqrt(x) else write "illegal";
-			 
-
-
-  3  
-
-
-clear x; 
-
-if numberp x and x<20 then y := sqrt(x) else write "illegal";
-			 
-
-
-  illegal 
-
-
-x := 12; 
-
-  X := 12 
-
-
-a := if x < 5 then 100 else 150;
-			 
-
-
-  A := 150 
-
-
-b := u**(if x < 10 then 2);
-			 
-
-  B := 1 
-
-
-bb := u**(if x > 10 then 2);
-			 
-
-         2
-  BB := U
-
-

    An if statement may be used inside an assignment statemen -t and sets -its value depending on the conditions, or used anywhere else an -expression would be valid, as shown in the last example. If there is no - else clause, the value is 0 if a number is expected, and nothing -otherwise. -

    -

    -The else clause may be left out if no action is to be taken if the -condition is false. -

    -

    -The condition may be a compound conditional statement using -and or - -or. If a non-conditional statement, such as a constant -, is used by -accident, it is assumed to have value true. -

    -

    -Be sure to use -group or -block statements after - then or else. -

    -

    -The if operator is right associative. The following constructions are -examples: -

    -

    - _ _ _ (1) -

    -syntax:

    -

    -

    -if<condition> then if <condition> the -n - <action> else <action> -

    -

    -

    -which is equivalent to -

    -syntax:

    -

    -

    -if<condition> then (if <condition> - then <action> else <action>); -

    -

    -

    - _ _ _ (2) -

    -syntax:

    -

    -

    -if<condition> then <action> else if - <condition> then <action> else <action> -

    -

    -

    -which is equivalent to -

    -syntax:

    -

    -

    -if<condition> then <action> else -

    -

    -(if <condition> then <action> - else <action>). -

    -

    -

    -

    -

    -

    - - - -LIST -INDEX

    - - - -LIST _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The list operator constructs a list from its arguments. -

    -syntax:

    -

    -

    -list(<item> {,<item>}*) or - list() to construct an empty list. -

    -

    -

    -<item> can be any REDUCE scalar expression, including another list. -Left and right curly brackets can also be used instead of the operator -list to construct a list. -

    -

    -

    -examples:

    -

    
-liss := list(c,b,c,{xx,yy},3x**2+7x+3,df(sin(2*x),x));
-	 
-
-
-                            2
-  LISS := {C,B,C,{XX,YY},3*X  + 7*X + 3,2*COS(2*X)} 
-
-
-length liss; 
-
-  6 
-
-
-liss := {c,b,c,{xx,yy},3x**2+7x+3,df(sin(2*x),x)};
-	 
-
-
-                            2
-  LISS := {C,B,C,{XX,YY},3*X  + 7*X + 3,2*COS(2*X)} 
-
-
-emptylis := list(); 
-
-  EMPTYLIS := {} 
-
-
-a . emptylis; 
-
-  {A}
-
-

    Lists are ordered, hierarchical structures. The elements stay wher -e you -put them, and only change position in the list if you specifically change -them. Lists can have nested sublists to any (reasonable) level. The - -part operator can be used to access elements anywhere -within a list -hierarchy. The -length operator counts the -number of top-level elements -of its list argument; elements that are themselves lists still only -count as one element. -

    -

    -

    - - - -OR -INDEX

    - - - -OR _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The or binary logical operator returns true if either one or -both of its arguments is true. -

    -syntax:

    -

    -

    -<logical expression> or <logical expression> -

    -

    -

    -<logical expression> must evaluate to true or nil. -

    -

    -

    -examples:

    -

    
-a := 10; 
-
-  A := 10 
-
-
-if a<0 or a>140 then write "not a valid human age" else
-   write "age = ",a;
- 
-
-
-
-  age = 10 
-
-
-a := 200; 
-
-  A := 200 
-
-
-if a < 0 or a > 140 then write "not a valid human age";
-			 
-
-
-  not a valid human age
-
-

    The or operator is left associative: x or y or z - is equivalent to -(x or y) or z. -

    -

    -Logical operators can only be used in conditional expressions, such as -

    -

    - -if...then...else -and -while...do. -or evaluates its arguments in order and quits, returning true, -on finding the first true statement. -

    -

    -

    - - - -PROCEDURE -INDEX

    - - - -PROCEDURE _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The procedure command allows you to define a mathematical operation as -a -function with arguments. -

    -syntax:

    -

    -

    - _ _ _ <option> procedure <identifier> - (<arg>{,<arg>}+);<body> -

    -

    -

    -The <option> may be -algebraic or -symbolic, -indicating the -mode under which the procedure is executed, or -real or - -integer, indicating the type of answer expected. The d -efault is -algebraic. Real or integer procedures are subtypes of algebraic -procedures; type-checking is done on the results of integer procedures, but -not on real procedures (in the current REDUCE release). <identifier> -may be any valid REDUCE identifier that is not already a procedure name, -operator, -array or -matrix. -<arg> is a formal parameter that may be any -valid REDUCE identifier. <body> is a single statement (a -group -or -block statement may be used) with the desired activiti -es in it. -

    -

    -

    -examples:

    -

    
-procedure fac(n);
-   if not (fixp(n) and n>=0)
-     then rederr "Choose nonneg. integer only"
-    else for i := 0:n-1 product i+1;
-
-			 
-
-  FAC 
-
-
-fac(0); 
-
-  1 
-
-
-fac(5); 
-
-  120 
-
-
-fac(-5); 
-
-  ***** choose nonneg. integer only
-
-

    Procedures are automatically declared as operators upon definition -. When -REDUCE has parsed the procedure definition and successfully converted it to -a form for its own use, it prints the name of the procedure. Procedure -definitions cannot be nested. Procedures can call other procedures, or can -recursively call themselves. Procedure identifiers can be cleared as you -would clear an operator. Unlike -let statements, new definitions -under the same procedure name replace the previous definitions completely. -

    -

    -Be careful not to use the name of a system operator for your own procedure. -REDUCE may or may not give you a warning message. If you redefine a system -operator in your own procedure, the original function of the system operator -is lost for the remainder of the REDUCE session. -

    -

    -Procedures may have none, one, or more than one parameter. A REDUCE -parameter is a formal parameter only; the use of x as a parameter in -a procedure definition has no connection with a value of x in -the REDUCE session, and the results of calling a procedure have no effect -on the value of x. If a procedure is called with x as a -parameter, the current value of x is used as specified in the -computation, but is not changed outside the procedure. -Making an assignment statement by := with a -formal parameter on the left-hand side only changes the value of the -calling parameter within the procedure. -

    -

    -Using a -let statement inside a procedure always changes the va -lue -globally: a let with a formal parameter makes the change to the calling - -parameter. let statements cannot be made on local variables inside - -begin...end -blocks. -When -clear statements are used on formal -parameters, the calling variables associated with them are cleared globally too. - -The use of let or clear statements inside procedures -should be done with extreme caution. -

    -

    -Arrays and operators may be used as parameters to procedures. The body of the -procedure can contain statements that appropriately manipulate these -arguments. Changes are made to values of the calling arrays or operators. -Simple expressions can also be used as arguments, in the place of scalar -variables. Matrices may not be used as arguments to procedures. -

    -

    -A procedure that has no parameters is called by the procedure name, -immediately followed by empty parentheses. The empty parentheses may be left -out when writing a procedure with no parameters, but must appear in a call of -the procedure. If this is a nuisance to you, use a -let statement on -the name of the procedure (i.e., let noargs = noargs()) after which -you can call the procedure by just its name. -

    -

    -Procedures that have a single argument can leave out the parentheses around -it both in the definition and procedure call. (You can use the parentheses if -you wish.) Procedures with more than one argument must use parentheses, with -the arguments separated by commas. -

    -

    -Procedures often have a begin...end block in them. Inside the - -block, local variables are declared using scalar, real or -integer declarations. -The declarations must be made immediately after the word -begin, and if more than one type of declaration is made, they are -separated by semicolons. REDUCE currently does no type checking on local -variables; real and integer are treated just like scalar -. -Actions take place as specified in the statements inside the block statement. -Any identifiers that are not formal parameters or local variables are treated -as global variables, and activities involving these identifiers are global in -effect. -

    -

    -If a return value is desired from a procedure call, a specific - -return command must be the last statement executed bef -ore exiting -from the procedure. If no return is used, a procedure returns a -zero or no value. -

    -

    -Procedures are often written in a file using an editor, then the file -is input using the command -in. This method allows easy changes in -development, and also allows you to load the named procedures whenever -you like, by loading the files that contain them. -

    -

    -

    - - - -REPEAT -INDEX

    - - - -REPEAT _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The -repeat command causes repeated execution of a statemen -t - until

    -

    -the given condition is found to be true. The statement is always executed -at least once. -

    -syntax:

    -

    -

    -repeat<statement> until <condition> -

    -

    -

    -<statement> can be a single statement, -group statement, or -a begin...end -block. <condition> must be -a logical operator that evaluates to true or nil. -

    -

    -

    -examples:

    -

    
-<<m := 4; repeat <<write 100*x*m;m := m-1>> until m = 0>
->;
-			 
-
-
-  400*X
-  300*X
-  200*X
-  100*X
-
-
-
-<<m := -1; repeat <<write m; m := m-1>> until m <= 0>
->;
-			 
-
-
-  -1
-
-

    repeatmust always be followed by an until with a - condition. -Be careful not to generate an infinite loop with a condition that is never -true. In the second example, if the condition had been m = 0, it -would never have been true since m already had value -2 when the -condition was first evaluated. -

    -

    -

    - - - -REST -INDEX

    - - - -REST _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The rest operator returns a -list containing all but the first -element of the list it is given. -

    -syntax:

    -

    -

    -rest(<list>) or rest <list> -

    -

    -

    -

    -<list> must be a non-empty list, but need not have more than one element. - -

    -

    -

    -examples:

    -

    
-alist := {a,b,c,d}; 
-
-  ALIST := {A,B,C,D}; 
-
-
-rest alist; 
-
-  {B,C,D} 
-
-
-blist := {x,y,{aa,bb,cc},z}; 
-
-  BLIST := {X,Y,{AA,BB,CC},Z} 
-
-
-second rest blist; 
-
-  {AA,BB,CC} 
-
-
-clist := {c}; 
-
-  CLIST := C 
-
-
-rest clist; 
-
-  {}
-
-

    - - -RETURN -INDEX

    - - - -RETURN _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The return command causes a value to be returned from inside a -begin...end -block. -

    -

    -

    -syntax:

    -begin<statements> return <(expression)> - end

    -

    -

    -

    -<statements> can be any valid REDUCE statements. The value of -<expression> is returned. -

    -

    -

    -examples:

    -

    
-begin write "yes"; return a end; 
-
-  yes
-  A
-
-
-procedure dumb(a);
-  begin if numberp(a) then return a else return 10 end;
-
-						 
-
-  DUMB 
-
-
-dumb(x); 
-
-  10 
-
-
-dumb(-5); 
-
-  -5  
-
-
-procedure dumb2(a);
-  begin c := a**2 + 2*a + 1; d := 17; c*d; return end;
-		 
-
-  DUMB2 
-
-
-dumb2(4); 
-
-c; 
-
-  25 
-
-
-d; 
-
-  17
-
-

    Note in dumb2 above that the assignments were made as req -uested, but -the product c*d cannot be accessed. Changing the procedure to read -return c*d would remedy this problem. -

    -

    -The return statement is always the last statement executed before -leaving the block. If return has no argument, the block is exited but -no value is returned. A block statement does not need a return ; -the statements inside terminate in their normal fashion without one. -In that case no value is returned, although the specified actions inside the -block take place. -

    -

    -The return command can be used inside <<...>> - - -group statements and - -if...then...else commands that -are inside begin...end -blocks. -It is not valid in these constructions that are not inside -a begin...end - block. It is not valid inside -for, - -repeat...until or -while...do - loops in any construction. To force early termination from loops, the -go to( -goto) command must be used. -When you use nested block statements, a -return from an inner block exits returning a value to the next-outermos -t -block, rather than all the way to the outside. -

    -

    -

    - - - -REVERSE -INDEX

    - - - -REVERSE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The reverse operator returns a -list that is the reverse of the -list it is given. -

    -syntax:

    -

    -

    -reverse(<list>) or reverse <list> -

    -

    -

    -<list> must be a -list. -

    -

    -

    -examples:

    -

    
-aa := {c,b,a,{x**2,z**3},y}; 
-
-                 2  3
-  AA := {C,B,A,{X ,Z },Y} 
-
-
-reverse aa; 
-
-       2  3
-  {Y,{X ,Z },A,B,C} 
-
-
-reverse(q . reverse aa); 
-
-           2  3
-  {C,B,A,{X ,Z },Y,Q}
-
-

    reverseand -cons can be used together to add a new element to -the end of a list (. adds its new element to the beginning). The -reverse operator uses a noticeable amount of system resources, -especially if the list is long. If you are doing much heavy-duty list -manipulation, you should probably design your algorithms to avoid much -reversing of lists. A moderate amount of list reversing is no problem. -

    -

    -

    - - - -RULE -INDEX

    - - - -RULE _ _ _ _ _ _ _ _ _ _ _ _ type

    -

    - -

    -

    -A rule is an instruction to replace an algebraic expression -or a part of an expression by another one. -

    -syntax:

    -

    -

    -<lhs> => <rhs> or -<lhs> => <rhs> when <cond> -

    -

    -

    -<lhs> is an algebraic expression used as search pattern and -<rhs> is an algebraic expression which replaces matches of -<rhs>. => is the operator -replace. -

    -

    -<lhs> can contain -free variables which are -symbols preceded by a tilde ~ in their leftmost position -in <lhs>. -A double tilde marks an -optional free variable. -If a rule has a when <cond> -part it will fire only if the evaluation of <cond> has a -result -true. <cond> may contain references to -free variables of <lhs>. -

    -

    -Rules can be collected in a -list which then forms a - rule list. Rule lists can be used to collect -algebraic knowledge for a specific evaluation context. -

    -

    -Rulesand rule lists are globally activated and -deactivated by -let, -forall, -clearrules. -For a single evaluation they can be locally activate by -where. -The active rules for an operator can be visualized by -showrules. -

    -

    -

    -examples:

    -

    
-operator f,g,h; 
-
-let f(x) => x^2; 
-
-f(x); 
-
-   2
-  x
-
-
-g_rules:={g(~n,~x)=>h(n/2,x) when evenp n,
-
-g(~n,~x)=>h((1-n)/2,x) when not evenp n}$
-
-let g_rules;
-
-g(3,x); 
-
-  h(-1,x)
-
-

    - - -Free_Variable -INDEX

    - - - -FREE VARIABLE _ _ _ _ _ _ _ _ _ _ _ _ type

    -

    - -

    -

    -A variable preceded by a tilde is considered as free variable -and stands for an arbitrary part in an algebraic form during -pattern matching. Free variables occur in the left-hand sides -of -rules, in the side relations for -compact -and in the first arguments of -map and -select -calls. See -rule for examples. -

    -

    -In rules also -optional free variables may occur. -

    -

    - - - -Optional_Free_Variable -INDEX

    - - - -OPTIONAL FREE VARIABLE _ _ _ _ _ _ _ _ _ _ _ _ type

    -

    - -

    -

    -A variable preceded by a double tilde is considered as -optional free variable

    -

    -and stands for an arbitrary part part in an algebraic form during -pattern matching. In contrast to ordinary -free variables -an operator pattern with an optional free variable -matches also if the operand for the variable is missing. In such -a case the variable is bound to a neutral value. -Optional free variables can be used as -

    -

    -term in a sum: set to 0 if missing, -

    -

    -factor in a product: set to 1 if missing, -

    -

    -exponent: set to 1 if missing -

    -

    -

    -examples:

    -

    Optional free variables are allowed only in the left-h -and sides -of -rules. -

    -

    - - - -SECOND -INDEX

    - - - -SECOND _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The second operator returns the second element of a list. -

    -syntax:

    -

    -

    -second(<list>) or second <list> -

    -

    -

    -

    -<list> must be a list with at least two elements, to avoid an error -message. -

    -

    -

    -examples:

    -

    
-alist := {a,b,c,d}; 
-
-  ALIST := {A,B,C,D} 
-
-
-second alist; 
-
-  B 
-
-
-blist := {x,{aa,bb,cc},z}; 
-
-  BLIST := {X,{AA,BB,CC},Z} 
-
-
-second second blist; 
-
-  BB
-
-

    - - -SET -INDEX

    - - - -SET _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The set operator is used for assignments when you want both sides of -the assignment statement to be evaluated. -

    -syntax:

    -

    -

    -set(<restricted\_expression>,<expression>) -

    -

    -

    -<expression> can be any REDUCE expression; <restricted\_expression> - -must be an identifier or an expression that evaluates to an identifier. -

    -

    -

    -examples:

    -

    
-a := y; 
-
-  A := Y 
-
-
-set(a,sin(x^2)); 
-
-       2
-  SIN(X ) 
-
-
-a; 
-
-       2
-  SIN(X ) 
-
-
-y; 
-
-       2
-  SIN(X ) 
-
-
-a := b + c; 
-
-  A := B + C 
-
-
-set(a-c,z); 
-
-  Z 
-
-
-b; 
-
-  Z
-
-

    Using an -array or -matrix reference as the first -argument to set has -the result of setting the contents of the designated element to -set's second argument. You should be careful to avoid unwanted -side effects when you use this facility. -

    -

    -

    - - - -SETQ -INDEX

    - - - -SETQ _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The setq operator is an infix or prefix binary assignment operator. -It is identical to :=. -

    -syntax:

    -

    -

    -setq(<restricted\_expression>,<expression>) or -

    -

    -<restricted\_expression> setq <expression> -

    -

    -

    -<restricted expression> is ordinarily a single identifier, though -simple expressions may be used (see Comments below). <expression> can -be any valid REDUCE expression. If <expression> is a -matrix -identifier, then <restricted\_expression> can be a matrix identifier -(redimensioned if necessary), which has each element set to the -corresponding elements of the identifier on the right-hand side. -

    -

    -

    -examples:

    -

    
-setq(b,6); 
-
-  B := 6 
-
-
-c setq sin(x); 
-
-  C := SIN(X) 
-
-
-w + setq(c,x+3) + z; 
-
-  W + X + Z + 3 
-
-
-c; 
-
-  X + 3 
-
-
-setq(a1 + a2,25); 
-
-  A1 + A2 := 25 
-
-
-a1; 
-
-  - (A2 - 25)
-
-

    Embedding a setq statement in an expression has the side -effect of making -the assignment, as shown in the third example above. -

    -

    -Assignments are generally done for identifiers, but may be done for simple -expressions as well, subject to the following remarks: -

    -

    - _ _ _ (i) -If the left-hand side is an identifier, an operator, or a power, the rule -is added to the rule table. -

    -

    - _ _ _ (ii) -If the operators - + / appear on the left-hand side, all but the first - -term of the expression is moved to the right-hand side. -

    -

    - _ _ _ (iii) -If the operator * appears on the left-hand side, any constant terms are - -moved to the right-hand side, but the symbolic factors remain. -

    -

    -Be careful not to make a recursive setq assignment that is not -controlled inside a loop statement. The process of resubstitution -continues until you get a stack overflow message. setq can be used -to attach functionality to operators, as the := does. -

    -

    -

    - - - -THIRD -INDEX

    - - - -THIRD _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The third operator returns the third item of a -list. -

    -syntax:

    -

    -

    -third(<list>) or third <list> -

    -

    -

    -

    -<list> must be a list containing at least three items to avoid an error -message. -

    -

    -

    -examples:

    -

    
-alist := {a,b,c,d}; 
-
-  ALIST := {A,B,C,D} 
-
-
-third alist; 
-
-  C 
-
-
-blist := {x,{aa,bb,cc},y,z}; 
-
-  BLIST := {X,{AA,BB,CC},Y,Z}; 
-
-
-third second blist; 
-
-  CC 
-
-
-third blist; 
-
-  Y
-
-

    - - -WHEN -INDEX

    - - - -WHEN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The when operator is used inside a rule to make the -execution of the rule depend on a boolean condition which is -evaluated at execution time. For the use see -rule. -

    -

    - - - -Syntax -INDEX

    -Syntax

    -
  • semicolon commandalias= (;)

    -

  • dollar commandalias= ($)

    -

  • percent commandalias= (%)

    -

  • dot operatoralias= (.)

    -

  • assign operatoralias= (: =)

    -

  • equalsign operatoralias= (=)

    -

  • replace operatoralias= (= >)

    -

  • plussign operatoralias= (+)

    -

  • minussign operatoralias= (-)

    -

  • asterisk operatoralias= (*)

    -

  • slash operatoralias= (/)

    -

  • power operatoralias= (* *)

    -

  • caret operatoralias= (^)

    -

  • geqsign operatoralias= (> =)

    -

  • greater operatoralias= (>)

    -

  • leqsign operatoralias= (< =)

    -

  • less operatoralias= (<)

    -

  • tilde operatoralias= (~)

    -

  • group commandalias= (< <)

    -

  • AND operator

    -

  • BEGIN command

    -

  • block command

    -

  • COMMENT command

    -

  • CONS operator

    -

  • END command

    -

  • EQUATION type

    -

  • FIRST operator

    -

  • FOR command

    -

  • FOREACH command

    -

  • GEQ operator

    -

  • GOTO command

    -

  • GREATERP operator

    -

  • IF command

    -

  • LIST operator

    -

  • OR operator

    -

  • PROCEDURE command

    -

  • REPEAT command

    -

  • REST operator

    -

  • RETURN command

    -

  • REVERSE operator

    -

  • RULE type

    -

  • Free Variable type

    -

  • Optional Free Variable type

    -

  • SECOND operator

    -

  • SET operator

    -

  • SETQ operator

    -

  • THIRD operator

    -

  • WHEN operator

    -

    •  - - -ARITHMETIC_OPERATIONS -INDEX

    - - - -ARITHMETIC\_OPERATIONS _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -This section considers operations defined in REDUCE that concern numbers, -or operators that can operate on numbers in addition, in most cases, to -more general expressions. -

    -

    - - - -ABS -INDEX

    - - - -ABS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The abs operator returns the absolute value of its argument. -

    -

    -

    -syntax:

    -abs(<expression>) -

    -

    -

    -<expression> can be any REDUCE scalar expression. -

    -

    -

    -examples:

    -

    
-abs(-a); 
-
-  ABS(A) 
-
-
-abs(-5); 
-
-  5 
-
-
-a := -10; 
-
-  A := -10 
-
-
-abs(a); 
-
-  10 
-
-
-abs(-a); 
-
-  10
-
-

    If the argument has had no numeric value assigned to it, such as a -n -identifier or polynomial, abs returns an expression involving -abs of its argument, doing as much simplification of the argument -as it can, such as dropping any preceding minus sign. -

    -

    -

    - - - -ADJPREC -INDEX

    - - - -ADJPREC _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When a real number is input, it is normally truncated to the - -precision in -effect at the time the number is read. If it is desired to keep the full -precision of all numbers input, the switch adjprec -(for <adjust precision>) can be turned on. While on, adjprec -will automatically increase the precision, when necessary, to match that -of any integer or real input, and a message printed to inform the user of -the precision increase. -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-1.23456789012345; 
-
-  1.23456789012 
-
-
-on adjprec; 
-
-1.23456789012345; 
-
-*** precision increased to 15 
-

    - - -ARG -INDEX

    - - - -ARG _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -If -complex and -rounded are on, and arg -evaluates to a complex number, arg returns the polar angle of -arg, measured in radians. Otherwise an expression in arg is -returned. -

    -

    -

    -examples:

    -

    
-arg(3+4i) 
-
-  ARG(3 + 4*I) 
-
-
-on rounded, complex; 
-
-ws; 
-
-  0.927295218002 
-
-
-arg a; 
-
-  ARG(A)
-
-

    - - -CEILING -INDEX

    - - - -CEILING _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -ceiling(<expression>) -

    -

    -

    -This operator returns the ceiling (i.e., the least integer greater than or -equal to its argument) if its argument has a numerical value. For -negative numbers, this is equivalent to -fix. For non-numeric -arguments, the value is an expression in the original operator. -

    -

    -

    -examples:

    -

    
-ceiling 3.4; 
-
-  4 
-
-
-fix 3.4; 
-
-  3 
-
-
-ceiling(-5.2); 
-
-  -5 
-
-
-fix(-5.2); 
-
-  -5 
-
-
-ceiling a; 
-
-  CEILING(A)
-
-

    - - -CHOOSE -INDEX

    - - - -CHOOSE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -choose(<m>,<m>) returns the number of ways of choosing -<m> objects from a collection of <n> distinct objects --- in other -words the binomial coefficient. If <m> and <n> are not positive -integers, or m >n, the expression is returned unchanged. -than or equal to -

    -examples:

    -

    
-choose(2,3); 
-
-  3 
-
-
-choose(3,2); 
-
-  CHOOSE(3,2) 
-
-
-choose(a,b); 
-
-  CHOOSE(A,B)
-
-

    -

    - - - -DEG2DMS -INDEX

    - - - -DEG2DMS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -deg2dms(<expression>) -

    -

    -

    -In -rounded mode, if <expression> is a real number, -the -operator deg2dms will interpret it as degrees, and convert it to a -list containing the equivalent degrees, minutes and seconds. In all other -cases, an expression in terms of the original operator is returned. -

    -

    -

    -examples:

    -

    
-deg2dms 60; 
-
-  DEG2DMS(60) 
-
-
-on rounded; 
-
-ws; 
-
-  {60,0,0} 
-
-
-deg2dms 42.4; 
-
-  {42,23,60.0} 
-
-
-deg2dms a; 
-
-  DEG2DMS(A)
-
-

    - - -DEG2RAD -INDEX

    - - - -DEG2RAD _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -deg2rad(<expression>) -

    -

    -

    -In -rounded mode, if <expression> is a real number, -the -operator deg2rad will interpret it as degrees, and convert it to -the equivalent radians. In all other cases, an expression in terms of the -original operator is returned. -

    -

    -

    -examples:

    -

    
-deg2rad 60; 
-
-  DEG2RAD(60) 
-
-
-on rounded; 
-
-ws; 
-
-  1.0471975512 
-
-
-deg2rad a; 
-
-  DEG2RAD(A)
-
-

    - - -DIFFERENCE -INDEX

    - - - -DIFFERENCE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The difference operator may be used as either an infix or prefix binary - -subtraction operator. It is identical to - as a binary operator. -

    -

    -

    -syntax:

    -difference(<expression>,<expression>) or -

    -

    -<expression> difference <expression> - {difference <expression>}* -

    -

    -

    -<expression> can be a number or any other valid REDUCE expression. Matrix - -expressions are allowed if they are of the same dimensions. -

    -

    -

    -examples:

    -

    
-
-difference(10,4); 
-
-  6 
-
-
-
-15 difference 5 difference 2; 
-
-  8 
-
-
-
-a difference b; 
-
-  A - B
-
-

    The difference operator is left associative, as shown in -the second -example above. -

    -

    -

    -

    - - - -DILOG -INDEX

    - - - -DILOG _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The dilog operator is known to the differentiation and integration -operators, but has numeric value attached only at dilog(0). Dilog is -defined by -

    -

    -dilog(x) = -int(log(x),x)/(x-1) -

    -

    -

    -examples:

    -

    
-df(dilog(x**2),x); 
-
-           2
-    2*LOG(X )*X
-  - ------------
-       2
-      X   - 1
-
-
-
-int(dilog(x),x); 
-
-  DILOG(X)*X - DILOG(X) + LOG(X)*X - X 
-
-
-
-dilog(0); 
-
-    2
-  PI
-  ----
-   6
-
-

    - - -DMS2DEG -INDEX

    - - - -DMS2DEG _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -dms2deg(<list>) -

    -

    -

    -In -rounded mode, if <list> is a list of three real -numbers, -the operator dms2deg will interpret the list as degrees, minutes -and seconds and convert it to the equivalent degrees. In all other cases, -an expression in terms of the original operator is returned. -

    -

    -

    -examples:

    -

    
-dms2deg {42,3,7}; 
-
-  DMS2DEG({42,3,7}) 
-
-
-on rounded; 
-
-ws; 
-
-  42.0519444444 
-
-
-dms2deg a; 
-
-  DMS2DEG(A)
-
-

    - - -DMS2RAD -INDEX

    - - - -DMS2RAD _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -dms2rad(<list>) -

    -

    -

    -In -rounded mode, if <list> is a list of three real -numbers, -the operator dms2rad will interpret the list as degrees, minutes -and seconds and convert it to the equivalent radians. In all other cases, -an expression in terms of the original operator is returned. -

    -

    -

    -examples:

    -

    
-dms2rad {42,3,7}; 
-
-  DMS2RAD({42,3,7}) 
-
-
-on rounded; 
-
-ws; 
-
-  0.733944887421 
-
-
-dms2rad a; 
-
-  DMS2RAD(A)
-
-

    - - -FACTORIAL -INDEX

    - - - -FACTORIAL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -factorial(<expression>) -

    -

    -

    -If the argument of factorial is a positive integer or zero, its -factorial is returned. Otherwise the result is expressed in terms of the -original operator. For more general operations, the -gamma operator -is available in the -Special Function Package. -

    -

    -

    -examples:

    -

    
-factorial 4; 
-
-  24 
-
-
-factorial 30 ; 
-
-  265252859812191058636308480000000 
-
-

    - - -FIX -INDEX

    - - - -FIX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -fix(<expression>) -

    -

    -

    -The operator fix returns the integer part of its argument, if that -argument has a numerical value. For positive numbers, this is equivalent -to -floor, and, for negative numbers, -ceiling. For -non-numeric arguments, the value is an expression in the original operator. -

    -

    -

    -examples:

    -

    
-fix 3.4; 
-
-  3 
-
-
-floor 3.4; 
-
-  3 
-
-
-ceiling 3.4; 
-
-  4 
-
-
-fix(-5.2); 
-
-  -5 
-
-
-floor(-5.2); 
-
-  -6 
-
-
-ceiling(-5.2); 
-
-  -5 
-
-
-fix(a); 
-
-  FIX(A)
-
-

    - - -FIXP -INDEX

    - - - -FIXP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The fixp logical operator returns true if its argument is an integer. -

    -syntax:

    -

    -

    -fixp(<expression>) or fixp <simple\_expression> -

    -

    -

    -<expression> can be any valid REDUCE expression, <simple\_expression -> -must be a single identifier or begin with a prefix operator. -

    -

    -

    -examples:

    -

    
-if fixp 1.5 then write "ok" else write "not";
-			 
-
-
-  not 
-
-
-if fixp(a) then write "ok" else write "not";
-			 
-
-
-  not 
-
-
-a := 15; 
-
-  A := 15 
-
-
-if fixp(a) then write "ok" else write "not";
-			 
-
-
-  ok
-
-

    Logical operators can only be used inside conditional expressions -such as -if...then or while...do. -

    -

    -

    - - - -FLOOR -INDEX

    - - - -FLOOR _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -floor(<expression>) -

    -

    -

    -This operator returns the floor (i.e., the greatest integer less than or -equal to its argument) if its argument has a numerical value. For -positive numbers, this is equivalent to -fix. For non-numeric -arguments, the value is an expression in the original operator. -

    -

    -

    -examples:

    -

    
-floor 3.4; 
-
-  3 
-
-
-fix 3.4; 
-
-  3 
-
-
-floor(-5.2); 
-
-  -6 
-
-
-fix(-5.2); 
-
-  -5 
-
-
-floor a; 
-
-  FLOOR(A)
-
-

    - - -EXPT -INDEX

    - - - -EXPT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The expt operator is both an infix and prefix binary exponentiation -operator. It is identical to ^ or **. -

    -syntax:

    -

    -

    -expt(<expression>,<expression>) - or <expression> expt <expression> -

    -

    -

    -

    -examples:

    -

    
-a expt b; 
-
-   B
-  A  
-
-
-expt(a,b); 
-
-   B
-  A  
-
-
-(x+y) expt 4; 
-
-   4      3        2  2        3    4
-  X  + 4*X *Y + 6*X *Y  + 4*X*Y  + Y
-
-

    Scalar expressions may be raised to fractional and floating-point -powers. -Square matrix expressions may be raised to positive powers, and also to -negative powers if non-singular. -

    -

    -exptis left associative. In other words, a expt b expt c is -equivalent to a expt (b*c), not a expt (b expt c), which -would be right associative. -

    -

    -

    - - - -GCD -INDEX

    - - - -GCD _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The gcd operator returns the greatest common divisor of two -polynomials. -

    -syntax:

    -

    -

    -gcd(<expression>,<expression>) -

    -

    -

    -<expression> must be a polynomial (or integer), otherwise an error -occurs. -

    -

    -

    -examples:

    -

    
-gcd(2*x**2 - 2*y**2,4*x + 4*y); 
-
-  2*(X + Y) 
-
-
-gcd(sin(x),x**2 + 1); 
-
-  1  
-
-
-gcd(765,68); 
-
-  17
-
-

    The operator gcd described here provides an explicit mean -s to find the -gcd of two expressions. The switch gcd described below simplifies -expressions by finding and canceling gcd's at every opportunity. When -the switch -ezgcd is also on, gcd's are figured using the EZ GCD -algorithm, which is usually faster. -

    -

    -

    - - - -LN -INDEX

    - - - -LN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -ln(<expression>) -

    -

    -

    -<expression> can be any valid scalar REDUCE expression. -

    -

    -The ln operator returns the natural logarithm of its argument. -However, unlike -log, there are no algebraic rules associated -with it; it will only evaluate when -rounded is on, and the -argument is a real number. -

    -

    -

    -examples:

    -

    
-ln(x); 
-
-  LN(X) 
-
-
-ln 4; 
-
-  LN(4) 
-
-
-ln(e); 
-
-  LN(E) 
-
-
-df(ln(x),x); 
-
-  DF(LN(X),X) 
-
-
-on rounded; 
-
-ln 4; 
-
-  1.38629436112 
-
-
-ln e; 
-
-  1
-
-

    Because of the restricted algebraic properties of ln, use -rs are -advised to use -log whenever possible. -

    -

    -

    - - - -LOG -INDEX

    - - - -LOG _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The log operator returns the natural logarithm of its argument. -

    -syntax:

    -

    -

    -log(<expression>) or log <expression> -

    -

    -

    -<expression> can be any valid scalar REDUCE expression. -

    -

    -

    -examples:

    -

    
-log(x); 
-
-  LOG(X) 
-
-
-log 4; 
-
-  LOG(4) 
-
-
-log(e); 
-
-  1 
-
-
-on rounded; 
-
-log 4; 
-
-  1.38629436112
-
-

    logreturns a numeric value only when -rounded is on. In that -case, use of a negative argument for log results in an error -message. No error is given on a negative argument when REDUCE is not in -that mode. -

    -

    -

    - - - -LOGB -INDEX

    - - - -LOGB _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -logb(<expression>,<integer>) -

    -

    -

    -<expression> can be any valid scalar REDUCE expression. -

    -

    -The logb operator returns the logarithm of its first argument using -the second argument as base. However, unlike -log, there are no -algebraic rules associated with it; it will only evaluate when - -rounded is on, and the first argument is a real number -. -

    -

    -

    -examples:

    -

    
-logb(x,2); 
-
-  LOGB(X,2) 
-
-
-logb(4,3); 
-
-  LOGB(4,3) 
-
-
-logb(2,2); 
-
-  LOGB(2,2) 
-
-
-df(logb(x,3),x); 
-
-  DF(LOGB(X,3),X) 
-
-
-on rounded; 
-
-logb(4,3); 
-
-  1.26185950714 
-
-
-logb(2,2); 
-
-  1
-
-

    - - -MAX -INDEX

    - - - -MAX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator max is an n-ary prefix operator, which returns the largest - -value in its arguments. -

    -syntax:

    -

    -

    -max(<expression>{,<expression>}*) -

    -

    -

    -

    -<expression> must evaluate to a number. max of an empty list -returns 0. -

    -

    -

    -examples:

    -

    
-max(4,6,10,-1); 
-
-  10 
-
-
-<<a := 23;b := 2*a;c := 4**2;max(a,b,c)>>;
-			 
-
-
-  46 
-
-
-max(-5,-10,-a); 
-
-  -5
-
-

    - - -MIN -INDEX

    - - - -MIN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator min is an n-ary prefix operator, which returns the -smallest value in its arguments. -

    -syntax:

    -

    -

    -min(<expression>{,<expression>}*) -

    -

    -

    -<expression> must evaluate to a number. min of an empty list -returns 0. -

    -examples:

    -

    
-min(-3,0,17,2); 
-
-  -3 
-
-
-<<a := 23;b := 2*a;c := 4**2;min(a,b,c)>>;
-			 
-
-
-  16 
-
-
-min(5,10,a); 
-
-  5
-
-

    -

    - - - -MINUS -INDEX

    - - - -MINUS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The minus operator is a unary minus, returning the negative of its -argument. It is equivalent to the unary -. -

    -syntax:

    -

    -

    -minus(<expression>) -

    -

    -

    -

    -<expression> may be any scalar REDUCE expression. -

    -

    -

    -examples:

    -

    
-minus(a); 
-
-  - A 
-
-
-minus(-1); 
-
-  1 
-
-
-minus((x+1)**4); 
-
-      4      3      2
-  - (X  + 4*X  + 6*X  + 4*X + 1)
-
-

    - - -NEXTPRIME -INDEX

    - - - -NEXTPRIME _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -nextprime(<expression>) -

    -

    -

    -If the argument of nextprime is an integer, the least prime greater -than that argument is returned. Otherwise, a type error results. -

    -

    -

    -examples:

    -

    
-nextprime 5001; 
-
-  5003  
-
-
-nextprime(10^30); 
-
-  1000000000000000000000000000057 
-
-
-nextprime a; 
-
-  ***** A invalid as integer
-
-

    - - -NOCONVERT -INDEX

    - - - -NOCONVERT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -Under normal circumstances when rounded is on, REDUCE converts the -number 1.0 to the integer 1. If this is not desired, the switch -noconvert can be turned on. -

    -examples:

    -

    
-on rounded; 
-
-1.0000000000001; 
-
-  1 
-
-
-on noconvert; 
-
-1.0000000000001; 
-
-  1.0 
-
-

    -

    - - - -NORM -INDEX

    - - - -NORM _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -norm(<expression>) -

    -

    -

    -If rounded is on, and the argument is a real number, <norm> -returns its absolute value. If complex is also on, <norm> -returns the square root of the sum of squares of the real and imaginary -parts of the argument. In all other cases, a result is returned in -terms of the original operator. -

    -

    -

    -examples:

    -

    
-norm (-2); 
-
-  NORM(-2) 
-
-
-on rounded;
-
-ws; 
-
-  2.0 
-
-
-norm(3+4i); 
-
-  NORM(4*I+3) 
-
-
-on complex;
-
-ws; 
-
-  5.0
-
-

    - - -PERM -INDEX

    - - - -PERM _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -perm(<expression1>,<expression2>) -

    -

    -

    -If <expression1> and <expression2> evaluate to positive integers, -perm returns the number of permutations possible in selecting -<expression1> objects from <expression2> objects. -In other cases, an expression in the original operator is returned. -

    -

    -

    -examples:

    -

    
-perm(1,1); 
-
-  1 
-
-
-perm(3,5); 
-
-  60 
-
-
-perm(-3,5); 
-
-  PERM(-3,5) 
-
-
-perm(a,b); 
-
-  PERM(A,B)
-
-

    - - -PLUS -INDEX

    - - - -PLUS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The plus operator is both an infix and prefix n-ary addition -operator. It exists because of the way in which REDUCE handles such -operators internally, and is not recommended for use in algebraic mode -programming. -plussign, which has the identical effect, should be -used instead. -

    -syntax:

    -

    -

    -plus(<expression>,<expression>{,<expression>} -*) or -

    -

    -<expression> plus <expression> {plus <expressio -n>}* -

    -

    -

    -<expression> can be any valid REDUCE expression, including matrix -expressions of the same dimensions. -

    -

    -

    -examples:

    -

    
-a plus b plus c plus d; 
-
-  A + B + C + D 
-
-
-4.5 plus 10; 
-
-  29
-  -- 
-  2
-
-
-
-plus(x**2,y**2); 
-
-   2    2
-  X  + Y
-
-

    - - -QUOTIENT -INDEX

    - - - -QUOTIENT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The quotient operator is both an infix and prefix binary operator that - -returns the quotient of its first argument divided by its second. It is -also a unary -reciprocal operator. It is identical to / and - - -slash. -

    -syntax:

    -

    -

    -quotient(<expression>,<expression>) or -<expression> quotient <expression> or -quotient(<expression>) or -quotient <expression> -

    -

    -

    -<expression> can be any valid REDUCE scalar expression. Matrix -expressions can also be used if the second expression is invertible and the -matrices are of the correct dimensions. -

    -examples:

    -

    
-quotient(a,x+1); 
-
-    A
-  ----- 
-  X + 1
-
-
-7 quotient 17; 
-
-  7
-  -- 
-  17
-
-
-on rounded; 
-
-4.5 quotient 2; 
-
-  2.25 
-
-
-quotient(x**2 + 3*x + 2,x+1); 
-
-  X + 2 
-
-
-matrix m,inverse; 
-
-m := mat((a,b),(c,d)); 
-
-  M(1,1) := A;
-  M(1,2) := B;
-  M(2,1) := C
-  M(2,2) := D
-
-
-
-inverse := quotient m; 
-
-                      D
-  INVERSE(1,1) := ----------
-                  A*D - B*C
-                        B
-  INVERSE(1,2) := - ----------
-                    A*D - B*C
-                        C
-  INVERSE(2,1) := - ----------
-                    A*D - B*C
-                      A
-  INVERSE(2,2) := ----------
-                  A*D - B*C
-
-

    -

    -The quotient operator is left associative: a quotient b quotient c - -is equivalent to (a quotient b) quotient c. -

    -

    -If a matrix argument to the unary quotient is not invertible, or if the - -second matrix argument to the binary quotient is not invertible, an error -message is given. -

    -

    -

    - - - -RAD2DEG -INDEX

    - - - -RAD2DEG _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -rad2deg(<expression>) -

    -

    -

    -In -rounded mode, if <expression> is a real number, -the -operator rad2deg will interpret it as radians, and convert it to -the equivalent degrees. In all other cases, an expression in terms of the -original operator is returned. -

    -

    -

    -examples:

    -

    
-rad2deg 1; 
-
-  RAD2DEG(1) 
-
-
-on rounded; 
-
-ws; 
-
-  57.2957795131 
-
-
-rad2deg a; 
-
-  RAD2DEG(A)
-
-

    - - -RAD2DMS -INDEX

    - - - -RAD2DMS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -rad2dms(<expression>) -

    -

    -

    -In -rounded mode, if <expression> is a real number, -the -operator rad2dms will interpret it as radians, and convert it to a -list containing the equivalent degrees, minutes and seconds. In all other -cases, an expression in terms of the original operator is returned. -

    -

    -

    -examples:

    -

    
-rad2dms 1; 
-
-  RAD2DMS(1) 
-
-
-on rounded; 
-
-ws; 
-
-  {57,17,44.8062470964} 
-
-
-rad2dms a; 
-
-  RAD2DMS(A)
-
-

    - - -RECIP -INDEX

    - - - -RECIP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -recip is the alphabetical name for the division operator / -or -slash used as a unary operator. The use of / -is preferred. -

    -

    -

    -examples:

    -

    
-recip a; 
-
-  1
-  - 
-  A
-
-
-recip 2; 
-
-  1
-  --
-  2
-
-

    - - -REMAINDER -INDEX

    - - - -REMAINDER _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The remainder operator returns the remainder after its first -argument is divided by its second argument. -

    -

    -

    -syntax:

    -remainder(<expression>,<expression>) -

    -

    -

    -<expression> can be any valid REDUCE polynomial, and is not limited -to numeric values. -

    -

    -

    -examples:

    -

    
-remainder(13,6); 
-
-  1 
-
-
-remainder(x**2 + 3*x + 2,x+1); 
-
-  0  
-
-
-remainder(x**3 + 12*x + 4,x**2 + 1); 
-
-
-  11*X + 4 
-
-
-remainder(sin(2*x),x*y); 
-
-  SIN(2*X)
-
-

    In the default case, remainders are calculated over the integers. -If you -need the remainder with respect to another domain, it must be declared -explicitly. -

    -

    -If the first argument to remainder contains a denominator not equal to - -1, an error occurs. -

    -

    -

    - - - -ROUND -INDEX

    - - - -ROUND _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -round(<expression>) -

    -

    -

    -If its argument has a numerical value, round rounds it to the -nearest integer. For non-numeric arguments, the value is an expression in -the original operator. -

    -

    -

    -examples:

    -

    
-round 3.4; 
-
-  3 
-
-
-round 3.5; 
-
-  4 
-
-
-round a; 
-
-  ROUND(A)
-
-

    - - -SETMOD -INDEX

    - - - -SETMOD _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The setmod command sets the modulus value for subsequent -modular -arithmetic. -

    -syntax:

    -

    -

    -setmod<integer> -

    -

    -

    -<integer> must be positive, and greater than 1. It need not be a prime -number. -

    -

    -

    -examples:

    -

    
-setmod 6; 
-
-  1 
-
-
-on modular; 
-
-16; 
-
-  4 
-
-
-x^2 + 5x + 7; 
-
-   2
-  X  + 5*X + 1 
-
-
-x/3; 
-
-  X
-  - 
-  3
-
-
-setmod 2; 
-
-  6 
-
-
-(x+1)^4; 
-
-   4
-  X  + 1 
-
-
-x/3; 
-
-  X
-
-

    setmodreturns the previous modulus, or 1 if none has been - set -before. setmod only has effect when -modular is on. -

    -

    -Modular operations are done only on numbers such as coefficients of -polynomials, not on the exponents. The modulus need not be prime. -Attempts to divide by a power of the modulus produces an error message, since th -e -operation is equivalent to dividing by 0. However, dividing by a factor -of a non-prime modulus does not produce an error message. -

    -

    -

    - - - -SIGN -INDEX

    - - - -SIGN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -sign<expression> -

    -

    -

    -signtries to evaluate the sign of its argument. If this -is possible sign returns one of 1, 0 or -1. Otherwise, the result -is the original form or a simplified variant. -

    -

    -

    -examples:

    -

    
-        sign(-5) 
-
-  -1
-
-
-        sign(-a^2*b) 
-
-  -SIGN(B)
-
-

    Even powers of formal expressions are assumed to be positive only -as long -as the switch -complex is off. -

    -

    -

    - - - -SQRT -INDEX

    - - - -SQRT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The sqrt operator returns the square root of its argument. -

    -syntax:

    -

    -

    -sqrt(<expression>) -

    -

    -

    -<expression> can be any REDUCE scalar expression. -

    -

    -

    -examples:

    -

    
-sqrt(16*a^3); 
-
-  4*SQRT(A)*A 
-
-
-sqrt(17); 
-
-  SQRT(17) 
-
-
-on rounded; 
-
-sqrt(17); 
-
-  4.12310562562 
-
-
-off rounded; 
-
-sqrt(a*b*c^5*d^3*27); 
-
-                                             2
-  3*SQRT(D)*SQRT(C)*SQRT(B)*SQRT(A)*SQRT(3)*C *D
-
-

    sqrtchecks its argument for squared factors and removes t -hem. -

    -

    -Numeric values for square roots that are not exact integers are given only -when -rounded is on. -

    -

    -Please note that sqrt(a**2) is given as a, which may be -incorrect if a eventually has a negative value. If you are -programming a calculation in which this is a concern, you can turn on the - -precise switch, which causes the absolute value of the - square root -to be returned. -

    -

    -

    - - - -TIMES -INDEX

    - - - -TIMES _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The times operator is an infix or prefix n-ary multiplication -operator. It is identical to *. -

    -syntax:

    -

    -

    -<expression> times <expression> {times <express -ion>}* -

    -

    -or times(<expression>,<expression> {,<expression>}*) - -

    -

    -

    -<expression> can be any valid REDUCE scalar or matrix expression. -Matrix expressions must be of the correct dimensions. Compatible scalar -and matrix expressions can be mixed. -

    -

    -

    -examples:

    -

    
-var1 times var2; 
-
-  VAR1*VAR2 
-
-
-times(6,5); 
-
-  30 
-
-
-matrix aa,bb; 
-
-aa := mat((1),(2),(x))$ 
-
-bb := mat((0,3,1))$ 
-
-aa times bb times 5; 
-
-  [0   15    5 ]
-  [            ]
-  [0   30   10 ]
-  [            ]
-  [0  15*X  5*X]
-
-

    - - -Arithmetic Operations -INDEX

    -Arithmetic Operations

    -
  • ARITHMETIC\_OPERATIONS introduction

    -

  • ABS operator

    -

  • ADJPREC switch

    -

  • ARG operator

    -

  • CEILING operator

    -

  • CHOOSE operator

    -

  • DEG2DMS operator

    -

  • DEG2RAD operator

    -

  • DIFFERENCE operator

    -

  • DILOG operator

    -

  • DMS2DEG operator

    -

  • DMS2RAD operator

    -

  • FACTORIAL operator

    -

  • FIX operator

    -

  • FIXP operator

    -

  • FLOOR operator

    -

  • EXPT operator

    -

  • GCD operator

    -

  • LN operator

    -

  • LOG operator

    -

  • LOGB operator

    -

  • MAX operator

    -

  • MIN operator

    -

  • MINUS operator

    -

  • NEXTPRIME operator

    -

  • NOCONVERT switch

    -

  • NORM operator

    -

  • PERM operator

    -

  • PLUS operator

    -

  • QUOTIENT operator

    -

  • RAD2DEG operator

    -

  • RAD2DMS operator

    -

  • RECIP operator

    -

  • REMAINDER operator

    -

  • ROUND operator

    -

  • SETMOD command

    -

  • SIGN operator

    -

  • SQRT operator

    -

  • TIMES operator

    -

    •  - - -boolean_value -INDEX

    - - - -BOOLEAN VALUE

    -

    - -There are no extra symbols for the truth values true -and false. Instead, -nil and the number zero -are interpreted as truth value false in algebraic -programs (see -false), while any different -value is considered as true (see -true). -

    -

    - - - -EQUAL -INDEX

    - - - -EQUAL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator equal is an infix binary comparison -operator. It is identical with =. It returns -true if its two -arguments are equal. -

    -

    -

    -syntax:

    -<expression> equal <expression> -

    -

    -

    -Equality is given between floating point numbers and integers that have -the same value. -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-a := 4; 
-
-  A := 4 
-
-
-b := 4.0; 
-
-  B := 4.0 
-
-
-if a equal b then write "true" else write "false";
-			 
-
-
-  true 
-
-
-if a equal 5 then write "true" else write "false";
-			 
-
-
-  false 
-
-
-if a equal sqrt(16) then write "true" else write "false";
-			 
-
-
-  true
-
-

    Comparison operators can only be used as conditions in conditional - commands -such as if...then and repeat...until. -<equal> can also be used as a prefix operator. However, this use -is not encouraged. -

    -

    -

    - - - -EVENP -INDEX

    - - - -EVENP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The evenp logical operator returns -true if its argument is an -even integer, and -nil if its argument is an odd integer. An error -message is returned if its argument is not an integer. -

    -

    -

    -syntax:

    -evenp(<integer>) or evenp <integer> -

    -

    -

    -<integer> must evaluate to an integer. -

    -

    -

    -examples:

    -

    
-aa := 1782; 
-
-  AA := 1782 
-
-
-if evenp aa then yes else no; 
-
-  YES 
-
-
-if evenp(-3) then yes else no; 
-
-  NO 
-
-

    Although you would not ordinarily enter an expression such as the -last -example above, note that the negative term must be enclosed in parentheses -to be correctly parsed. The evenp operator can only be used in -conditional statements such as if...then...else -or while...do. -

    -

    -

    - - - -false -INDEX

    - - - -FALSE

    -

    - -The symbol -nil and the number zero are considered -as -boolean value false if used in a place where -a boolean value is required. Most builtin operators return - -nil as false value. Algebraic programs use better zero -. -Note that nil is not printed when returned as result to -a top level evaluation. -

    -

    - - - -FREEOF -INDEX

    - - - -FREEOF _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The freeof logical operator returns - -true if its first argument does -not contain its second argument anywhere in its structure. -

    -syntax:

    -

    -

    -freeof(<expression>,<kernel>) or -<expression> freeof <kernel> -

    -

    -

    -<expression> can be any valid scalar REDUCE expression, <kernel> mus -t -be a kernel expression (see kernel). -

    -

    -

    -examples:

    -

    
-a := x + sin(y)**2 + log sin z;
-			 
-
-
-                           2
-  A := LOG(SIN(Z)) + SIN(Y)   + X 
-
-
-if freeof(a,sin(y)) then write "free" else write "not free";
-			 
-
-
-  not free 
-
-
-if freeof(a,sin(x)) then write "free" else write "not free";
-			 
-
-
-  free 
-
-
-if a freeof sin z then write "free" else write "not free";
-			 
-
-
-  not free
-
-

    Logical operators can only be used in conditional expressions such - as -

    -

    -if...then or while...do. -

    -

    -

    - - - -LEQ -INDEX

    - - - -LEQ _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The leq operator is a binary infix or prefix logical operator. It -returns -true if its first argument is less than or equal to it -s second -argument. As an infix operator it is identical with <=. -

    -syntax:

    -

    -

    -leq(<expression>,<expression>) or <expression> -leq <expression> -

    -

    -

    -

    -<expression> can be any valid REDUCE expression that evaluates to a -number. -

    -

    -

    -examples:

    -

    
-a := 15; 
-
-  A := 15 
-
-
-if leq(a,25) then write "yes" else write "no";
-			 
-
-
-  yes 
-
-
-if leq(a,15) then write "yes" else write "no";
-			 
-
-
-  yes 
-
-
-if leq(a,5) then write "yes" else write "no";
-			 
-
-
-  no
-
-

    Logical operators can only be used in conditional statements such -as -

    -

    -if...then...else or while...do. -

    -

    -

    - - - -LESSP -INDEX

    - - - -LESSP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The lessp operator is a binary infix or prefix logical operator. It -returns -true if its first argument is strictly less than its s -econd -argument. As an infix operator it is identical with <. -

    -syntax:

    -

    -

    -lessp(<expression>,<expression>) -or <expression> lessp <expression> -

    -

    -

    -

    -<expression> can be any valid REDUCE expression that evaluates to a -number. -

    -

    -

    -examples:

    -

    
-a := 15; 
-
-  A := 15 
-
-
-if lessp(a,25) then write "yes" else write "no";
-			 
-
-
-  yes 
-
-
-if lessp(a,15) then write "yes" else write "no";
-			 
-
-
-  no 
-
-
-if lessp(a,5) then write "yes" else write "no";
-			 
-
-
-  no
-
-

    Logical operators can only be used in conditional statements such -as -

    -

    -if...then...else or while...do. -

    -

    -

    - - - -MEMBER -INDEX

    - - - -MEMBER _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -<expression> member <list> -

    -

    -

    -memberis an infix binary comparison operator that evaluates to - -true if <expression> is -equal to a member of -the -list <list>. -

    -

    -

    -examples:

    -

    
-if a member {a,b} then 1 else 0; 
-
-  1 
-
-
-if 1 member(1,2,3) then a else b; 
-
-  a 
-
-
-if 1 member(1.0,2) then a else b; 
-
-  b
-
-

    Logical operators can only be used in conditional statements such -as -

    -

    -if...then...else or while...do. -<member> can also be used as a prefix operator. However, this use -is not encouraged. Finally, -equal (=) is used for the test -within the list, so expressions must be of the same type to match. -

    -

    -

    - - - -NEQ -INDEX

    - - - -NEQ _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The operator neq is an infix binary comparison -operator. It returns -true if its two -arguments are not -equal. -

    -

    -

    -syntax:

    -<expression> neq <expression> -

    -

    -

    -An inequality is satisfied between floating point numbers and integers -that have the same value. -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-a := 4; 
-
-  A := 4 
-
-
-b := 4.0; 
-
-  B := 4.0 
-
-
-if a neq b then write "true" else write "false";
-			 
-
-
-  false 
-
-
-if a neq 5 then write "true" else write "false";
-			 
-
-
-  true
-
-

    Comparison operators can only be used as conditions in conditional - commands -such as if...then and repeat...until. -<neq> can also be used as a prefix operator. However, this use -is not encouraged. -

    -

    -

    - - - -NOT -INDEX

    - - - -NOT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The not operator returns -true if its argument evaluates to - -nil, and nil if its argument is true -. -

    -syntax:

    -

    -

    -not(<logical expression>) -

    -

    -

    -

    -examples:

    -

    
-if not numberp(a) then write "indeterminate" else write a;
-			 
-
-
-  indeterminate; 
-
-
-a := 10; 
-
-  A := 10 
-
-
-if not numberp(a) then write "indeterminate" else write a;
-			 
-
-
-  10 
-
-
-if not(numberp(a) and a < 0) then write "positive number";
-			 
-
-
-  positive number
-
-

    Logical operators can only be used in conditional statements such -as -

    -

    -if...then...else or while...do. -

    -

    -

    - - - -NUMBERP -INDEX

    - - - -NUMBERP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The numberp operator returns -true if its argument is a number, -and -nil otherwise. -

    -syntax:

    -

    -

    -numberp(<expression>) or numberp <expression> -

    -

    -

    -<expression> can be any REDUCE scalar expression. -

    -

    -

    -examples:

    -

    
-cc := 15.3; 
-
-  CC := 15.3 
-
-
-if numberp(cc) then write "number" else write "nonnumber"; 
-
-
-  number 
-
-
-if numberp(cb) then write "number" else write "nonnumber"; 
-
-
-  nonnumber
-
-

    Logical operators can only be used in conditional expressions, suc -h as -

    -

    -if...then...else and while...do. -

    -

    -

    - - - -ORDP -INDEX

    - - - -ORDP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The ordp logical operator returns -true if its first argument is -ordered ahead of its second argument in canonical internal ordering, or is -identical to it. -

    -syntax:

    -

    -

    -ordp(<expression1>,<expression2>) -

    -

    -

    -

    -<expression1> and <expression2> can be any valid REDUCE scalar -expression. -

    -

    -

    -examples:

    -

    
-if ordp(x**2 + 1,x**3 + 3) then write "yes" else write "no";
-			 
-
-
-  no 
-
-
-if ordp(101,100) then write "yes" else write "no";
-			 
-
-
-  yes 
-
-
-if ordp(x,x) then write "yes" else write "no";
-			 
-
-
-  yes
-
-

    Logical operators can only be used in conditional expressions, suc -h as -

    -

    -if...then...else and while...do. -

    -

    -

    - - - -PRIMEP -INDEX

    - - - -PRIMEP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -primep(<expression>) or primep <simple\_expression -> -

    -

    -

    -If <expression> evaluates to a integer, primep returns - -true

    -

    -if <expression> is a prime number (i.e., a number other than 0 and -plus or minus 1 which is only exactly divisible by itself or a unit) -and -nil otherwise. -If <expression> does not have an integer value, a type error occurs. -

    -

    -

    -examples:

    -

    
-if primep 3 then write "yes" else write "no"; 
-
-
-  YES 
-
-
-if primep a then 1; 
-
-  ***** A invalid as integer
-
-

    - - -TRUE -INDEX

    - - - -TRUE

    -

    - -

    -

    -Any value of the boolean part of a logical expression which is neither - -nil nor 0 is considered as true. Mos -t -builtin test and compare functions return -t for true -and -nil for false. -

    -

    -

    -examples:

    -

    
-if member(3,{1,2,3}) then 1 else -1;
-
-
-  1
-
-
-if floor(1.7) then 1 else -1; 
-
-  1 
-
-
-if floor(0.7) then 1 else -1; 
-
-  -1
-
-

    - - -Boolean Operators -INDEX

    -Boolean Operators

    -
  • boolean value concept

    -

  • EQUAL operator

    -

  • EVENP operator

    -

  • false concept

    -

  • FREEOF operator

    -

  • LEQ operator

    -

  • LESSP operator

    -

  • MEMBER operator

    -

  • NEQ operator

    -

  • NOT operator

    -

  • NUMBERP operator

    -

  • ORDP operator

    -

  • PRIMEP operator

    -

  • TRUE concept

    -

    •  - - -BYE -INDEX

    - - - -BYE _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The bye command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are at -the top level, the bye command exits REDUCE. quit is a -synonym for bye. -

    -

    - - - -CONT -INDEX

    - - - -CONT _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The command cont returns control to an interactive file after a - -pause command that has been answered with n. - -

    -

    -

    -examples:

    -

    Suppose you are in the middle of an interactive file. -

     
-
- 
-
-  factorize(x**2 + 17*x + 60); 
-
-
- 
-
-  {{X + 12,1},{X + 5,1}} 
-
-
-   pause; 
-
-  Cont? (Y or N) 
-
-
-n 
-
-saveas results; 
-
-factor1 := first results; 
-
-  FACTOR1 := {X + 12,1} 
-
-
-factor2 := second results; 
-
-  FACTOR2 := {X + 5,1} 
-
-
-cont;

    the file resumes

    
-
-

    -

    -

    -A -pause allows you to enter your own REDUCE commands, ch -ange -switch values, inquire about results, or other such activities. When you -wish to resume operation of the interactive file, use cont. -

    -

    -

    -

    - - - -DISPLAY -INDEX

    - - - -DISPLAY _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -When given a numeric argument <n>, display prints the <n> -most recent input statements, identified by prompt numbers. If an empty -pair of parentheses is given, or if <n> is greater than the current -number of statements, all the input statements since the beginning of -the session are printed. -

    -

    -

    -syntax:

    -display(<n>) or display() -

    -

    -

    -<n> should be a positive integer. However, if it is a real number, the -truncated integer value is used, and if a non-numeric argument is used, all -the input statements are printed. -

    -

    -The statements are displayed in upper case, with lines split at semicolons or -dollar signs, as they are in editing. If long files have been input during -the session, the display command is slow to format these for -printing. -

    -

    -

    - - - -LOAD_PACKAGE -INDEX

    - - - -LOAD\_PACKAGE _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The load_package command is used to load REDUCE packages, such as -gentran that are not automatically loaded by the system. -

    -syntax:

    -

    -

    -load_package "<package\_name>" -

    -

    -

    -A package is only loaded once; subsequent calls of load_package -for the same package name are ignored. -

    -

    - - - -PAUSE -INDEX

    - - - -PAUSE _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The pause command, given in an interactive file, stops operation and -asks if you want to continue or not. -

    -

    -

    -examples:

    -

    An interactive file is running, and at some point you -see the -question

     
-
-				   Cont? (Y or N) 
-

    If you type

     
-
-ykey{Return}
-

    the file continues to run until the next pause or the end.

    - 
-

    If you type 

     
-
-nkey{Return} 
-

    you will get a numbered REDUCE prompt, and be allowed to -enter and execute any REDUCE statements. If you later wish to continue with -the file, type

     
-
-cont; 
-

    and the file resumes.

    -

    -

    -To use pause in your own interactive files, type -

    -

    -pause;in the file wherever you want it. -

    -

    -pausedoes not allow you to continue without typing either y -or n. Its use is to slow down scrolling of interactive files, or to -let you change parameters or switch settings for the calculations. -

    -

    -If you have stopped an interactive file at a pause, and do not wish to - -resume the file, type end;. This does not end the REDUCE session, but -stops input from the file. A second end; ends the REDUCE session. -However, if you have pauses from more than one file stacked up, an end; - -brings you back to the top level, not the file directly above. -

    -

    -A pause typed from the terminal has no effect. -

    -

    -

    - - - -QUIT -INDEX

    - - - -QUIT _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The quit command ends the REDUCE session, returning control to the -program (e.g., the operating system) that called REDUCE. When you are at -the top level, the quit command exits REDUCE. -bye is a -synonym for quit. -

    -

    - - - -RECLAIM -INDEX

    - - - -RECLAIM _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -REDUCE's memory is in a storage structure called a heap. As REDUCE -statements execute, chunks of memory are used up. When these chunks are no -longer needed, they remain idle. When the memory is almost full, -the system executes a garbage collection, reclaiming space that is no -longer needed, and putting all the free space at one end. Depending on -the size of the image REDUCE is using, -garbage collection needs to be done more or less often. A -larger image means fewer but longer garbage collections. -Regardless of memory size, -if you ask REDUCE to do something ridiculous, like factorial(2000), it -may -garbage collect many times. -

    -

    -

    - - - -REDERR -INDEX

    - - - -REDERR _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The rederr command allows you to print an error message from inside -a -procedure or a -block statement. -The calculation is gracefully terminated. -

    -syntax:

    -

    -

    -rederr<message> -

    -

    -

    -<message> is an error message, usually inside double quotation marks -(a -string). -

    -

    -

    -examples:

    -

    
-procedure fac(n);
-   if not (fixp(n) and n>=0)
-     then  rederr "Choose nonneg. integer only"
-    else for i := 0:n-1 product i+1;
- 
-
-  fac 
-
-
-fac a; 
-
-  	   ***** Choose nonneg. integer only 
-
-
-fac 5; 
-
-  120
-
-

    The above procedure finds the factorial of its argument. -If n is not a positive integer or 0, an error message is returned. -

    -

    -If your procedure is executed in a file, the usual error message is -printed, followed by Cont? (Y or N), just as any other error does from - -a file. Although the procedure is gracefully terminated, any switch settings or - -variable assignments you made before the error occurred are not undone. If you -need to clean up such items before exiting, use a group statement, with the -rederr command as its last statement. -

    -

    -

    - - - -RETRY -INDEX

    - - - -RETRY _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The retry command allows you to retry the latest statement that resulte -d -in an error message. -

    -

    -

    -examples:

    -

    
-matrix a; 
-
-det a; 
-
-  ***** Matrix A not set 
-
-
-a := mat((1,2),(3,4)); 
-
-  A(1,1) := 1
-  A(1,2) := 2
-  A(2,1) := 3
-  A(2,2) := 4
-
-
-retry; 
-
-  -2
-
-

    retryremembers only the most recent statement that result -ed in an -error message. It allows you to stop and fix something obvious, then -continue on your way without retyping the original command. -

    -

    -

    - - - -SAVEAS -INDEX

    - - - -SAVEAS _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The saveas command saves the current workspace under the name of its -argument. -

    -syntax:

    -

    -

    -saveas<identifier> -

    -

    -

    -<identifier> can be any valid REDUCE identifier. -

    -

    -

    -examples:

    -

    (The numbered prompts are shown below, unlike in most -examples)

    
-
-1: solve(x^2-3);
-
-  {x=sqrt(3),x= - sqrt(3)}
-
-
-2: saveas rts(0)$
-
-3: rts(0);
-
-  {x=sqrt(3),x= - sqrt(3)}
-
-

    -

    -

    -saveasworks only for the current workspace, the last algebraic -expression produced by REDUCE. This allows you to save a result that you -did not assign to an identifier when you originally typed the input. -For access to previous output use -ws. -

    -

    -

    - - - -SHOWTIME -INDEX

    - - - -SHOWTIME _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The showtime command prints the elapsed system time since the last -call of this command or since the beginning of the session, if it has not -been called before. -

    -

    -

    -examples:

    -

    
-showtime; 
-
-  Time: 1020 ms 
-
-
-factorize(x^4 - 8x^4 + 8x^2 - 136x - 153);
-			 
-
-
-          2
-  {X - 9,X  + 17,X + 1} 
-
-
-showtime; 
-
-  Time: 920 ms
-
-

    The time printed is either the elapsed cpu time or the elapsed wal -l clock -time, depending on your system. showtime allows you to see the -system time resources REDUCE uses in its calculations. Your time readings -will of course vary from this example according to the system you use. -

    -

    -

    - - - -WRITE -INDEX

    - - - -WRITE _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The write command explicitly writes its arguments to the output device - -(terminal or file). -

    -syntax:

    -

    -

    -write<item>{,<item>}* -

    -

    -

    -<item> can be an expression, an assignment or a -string -enclosed in double quotation marks ("). -

    -

    -

    -examples:

    -

    
-write a, sin x, "this is a string"; 
-
-
-  ASIN(X)this is a string 
-
-
-write a," ",sin x," this is a string"; 
-
-
-  A SIN(X) this is a string 
-
-
-if not numberp(a) then write "the symbol ",a;
-							
-
-
-  the symbol A 
-
-
-array m(10); 
-
-for i := 1:5 do write m(i) := 2*i; 
-
-
-  M(1) := 2
-  M(2) := 4
-  M(3) := 6
-  M(4) := 8
-  M(5) := 10
-
-
-m(4); 
-
-  8
-
-

    The items specified by a single write statement print on -a single line -unless they are too long. A printed line is always ended with a carriage -return, so the next item printed starts a new line. -

    -

    -When an assignment statement is printed, the assignment is also made. This -allows you to get feedback on filling slots in an array with a -for - statement, as shown in the last example above. -

    -

    -

    - - - -General Commands -INDEX

    -General Commands

    -
  • BYE command

    -

  • CONT command

    -

  • DISPLAY command

    -

  • LOAD\_PACKAGE command

    -

  • PAUSE command

    -

  • QUIT command

    -

  • RECLAIM operator

    -

  • REDERR command

    -

  • RETRY command

    -

  • SAVEAS command

    -

  • SHOWTIME command

    -

  • WRITE command

    -

    •  - - -APPEND -INDEX

    - - - -APPEND _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The append operator constructs a new -list -from the elements of its two arguments (which must be lists). -

    -

    -

    -syntax:

    -append(<list>,<list>) -

    -

    -

    -<list> must be a list, though it may be the empty list ({}). -Any arguments beyond the first two are ignored. -

    -

    -

    -examples:

    -

    
-alist := {1,2,{a,b}}; 
-
-  ALIST := {1,2,{A,B}} 
-
-
-blist := {3,4,5,sin(y)}; 
-
-  BLIST := {3,4,5,SIN(Y)} 
-
-
-append(alist,blist); 
-
-  {1,2,{A,B},3,4,5,SIN(Y)} 
-
-
-append(alist,{}); 
-
-  {1,2,{A,B}} 
-
-
-append(list z,blist); 
-
-  {Z,3,4,5,SIN(Y)}
-
-

    The new list consists of the elements of the second list appended -to the -elements of the first list. You can append new elements to the -beginning or end of an existing list by putting the new element in a -list (use curly braces or the operator list). This is -particularly helpful in an iterative loop. -

    -

    -

    - - - -ARBINT -INDEX

    - - - -ARBINT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator arbint is used to express arbitrary integer parts -of an expression, e.g. in the result of -solve when - -allbranch is on. -

    -examples:

    -

    
-
-solve(log(sin(x+3)),x); 
-
-  {X=2*ARBINT(1)*PI - ASIN(1) - 3,
-   X=2*ARBINT(1)*PI + ASIN(1) + PI - 3}
-
-

    -

    - - - -ARBCOMPLEX -INDEX

    - - - -ARBCOMPLEX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator arbcomplex is used to express arbitrary scalar parts -of an expression, e.g. in the result of -solve when -the solution is parametric in one of the variable. -

    -examples:

    -

    
-
-solve({x+3=y-2z,y-3x=0},{x,y,z}); 
-
-
-     2*ARBCOMPLEX(1) + 3
-  {X=-------------------,
-              2
-      3*ARBCOMPLEX(1) + 3
-    Y=-------------------,
-               2
-    Z=ARBCOMPLEX(1)}
-
-

    -

    - - - -ARGLENGTH -INDEX

    - - - -ARGLENGTH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator arglength returns the number of arguments of the top-level - -operator in its argument. -

    -

    -

    -syntax:

    -arglength(<expression>) -

    -

    -

    -<expression> can be any valid REDUCE algebraic expression. -

    -

    -

    -examples:

    -

    
-arglength(a + b + c + d); 
-
-  4 
-
-
-arglength(a/b/c); 
-
-  2 
-
-
-arglength(log(sin(df(r**3*x,x)))); 
-
-
-  1
-
-

    In the first example, + is an n-ary operator, so the numb -er of terms -is returned. In the second example, since / is a binary operator, the -argument is actually (a/b)/c, so there are two terms at the top level. In -the last example, no matter how deeply the operators are nested, there is -still only one argument at the top level. -

    -

    -

    - - - -COEFF -INDEX

    - - - -COEFF _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The coeff operator returns the coefficients of the powers of the -specified variable in the given expression, in a -list. -

    -

    -

    -syntax:

    -coeff(<expression>,<variable>) -

    -

    -

    -<expression> is expected to be a polynomial expression, not a rational -expression. Rational expressions are accepted when the switch - -ratarg is on. <variable> must be a kernel. The r -esults are -returned in a list. -

    -

    -

    -examples:

    -

    
-coeff((x+y)**3,x); 
-
-    3     2
-  {Y  ,3*Y  ,3*Y,1} 
-
-
-coeff((x+2)**4 + sin(x),x); 
-
-  {SIN(X) + 16,32,24,8,1} 
-
-
-high_pow; 
-
-  4 
-
-
-low_pow; 
-
-  0 
-
-
-ab := x**9 + sin(x)*x**7 + sqrt(y); 
- 
-
-
-                          7     9
-  AB := SQRT(Y) + SIN(X)*X   + X
-
-
-coeff(ab,x); 
-
-  {SQRT(Y),0,0,0,0,0,0,SIN(X),0,1}
-
-

    The variables -high_pow and -low_pow are set to the -highest and lowest powers of the variable, respectively, appearing in the -expression. -

    -

    -The coefficients are put into a list, with the coefficient of the lowest -(constant) term first. You can use the usual list access methods -(first, second, third, rest, length -, and -part) to extract them. If a power does not appear in the -expression, the corresponding element of the list is zero. Terms involving -functions of the specified variable but not including powers of it (for -example in the expression x**4 + 3*x**2 + tan(x)) are placed in the -constant term. -

    -

    -Since the coeff command deals with the expanded form of the expression, - -you may get unexpected results when -exp is off, or when - -factor or -ifactor are on. -

    -

    -If you want only a specific coefficient rather than all of them, use the - -coeffn operator. -

    -

    -

    -

    - - - -COEFFN -INDEX

    - - - -COEFFN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The coeffn operator takes three arguments: an expression, a kernel, and - -a non-negative integer. It returns the coefficient of the kernel to that -integer power, appearing in the expression. -

    -

    -

    -syntax:

    -coeffn(<expression>,<kernel>,<integer>) -

    -

    -

    -<expression> must be a polynomial, unless -ratarg is on which -allows rational expressions. <kernel> must be a kernel, and -<integer> must be a non-negative integer. -

    -

    -

    -examples:

    -

    
-
-ff := x**7 + sin(y)*x**5 + y**4 + x + 7; 
-
-
-                5     7         4
-  FF := SIN(Y)*X   + X   + X + Y   + 7 
-
-
-coeffn(ff,x,5); 
-
-  SIN(Y) 
-
-
-coeffn(ff,z,3); 
-
-  0 
-
-
-coeffn(ff,y,0); 
-
-          5     7
-  SIN(Y)*X   + X   + X + 7 
-
-
-
-rr := 1/y**2+y**3+sin(y); 
-
-                2     5
-        SIN(Y)*Y   + Y   + 1
-  RR := -------------------- 
-                  2
-                 Y
-
-
-on ratarg; 
-
-
-coeffn(rr,y,-2); 
-
-  ***** -2 invalid as COEFFN index 
-
-
-
-coeffn(rr,y,5); 
-
-  1
-  ---
-   2
-  Y
-
-

    If the given power of the kernel does not appear in the expression -, -coeffn returns 0. Negative powers are never detected, even if -they appear in the expression and -ratarg are on. coeffn -with an integer argument of 0 returns any terms in the expression that -do not contain the given kernel. -

    -

    -

    - - - -CONJ -INDEX

    - - - -CONJ _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -conj(<expression>) or conj <simple\_expression> -

    -

    -

    -This operator returns the complex conjugate of an expression, if that -argument has an numerical value. A non-numerical argument is returned as -an expression in the operators -repart and -impart. -

    -

    -

    -examples:

    -

    
-conj(1+i); 
-
-  1-I 
-
-
-conj(a+i*b); 
-
-  REPART(A) - REPART(B)*I - IMPART(A)*I - IMPART(B)
-
-

    - - -CONTINUED_FRACTION -INDEX

    - - - -CONTINUED_FRACTION _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -continued_fraction(<num>) -or continued_fraction( <num>,<size>) -

    -

    -

    -This operator approximates the real number <num> -( -rational number, -rounded number) -into a continued fraction. The result is a list of two elements: the -first one is the rational value of the approximation, the second one -is the list of terms of the continued fraction which represents the -same value according to the definition t0 +1/(t1 + 1/(t2 + ...)). -Precision: the second optional parameter <size> is an upper bound -for the absolute value of the result denominator. If omitted, the -approximation is performed up to the current system precision. -

    -

    -

    -examples:

    -

    
-continued_fraction pi;
- 
-
-   1146408
-  {-------,{3,7,15,1,292,1,1,1,2,1}} 
-   364913
-
-
-continued_fraction(pi,100);
- 
-
-   22
-  {--,{3,7}} 
-   7
-
-

    - - -DECOMPOSE -INDEX

    - - - -DECOMPOSE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The decompose operator takes a multivariate polynomial as argument, -and returns an expression and a -list of - -equations from which the -original polynomial can be found by composition. -

    -

    -

    -syntax:

    -decompose(<expression>) or decompose - <simple\_expression> -

    -

    -

    -

    -examples:

    -

    
-decompose(x^8-88*x^7+2924*x^6-43912*x^5+263431*x^4-
-          218900*x^3+65690*x^2-7700*x+234)
-
- 
-
-   2                  2            2
-  U  + 35*U + 234, U=V  + 10*V, V=X  - 22*X 
-
-
-     decompose(u^2+v^2+2u*v+1) 
-
-   2
-  W   + 1, W=U + V
-
-

    Unlike factorization, this decomposition is not unique. Further -details can be found in V.S. Alagar, M.Tanh, <Fast Polynomial -Decomposition>, Proc. EUROCAL 1985, pp 150-153 (Springer) and J. von zur -Gathen, <Functional> -<Decomposition of Polynomials: the Tame Case>, J. -Symbolic Computation (1990) 9, 281-299. -

    -

    -

    - - - -DEG -INDEX

    - - - -DEG _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator deg returns the highest degree of its variable argument -found in its expression argument. -

    -

    -

    -syntax:

    -deg(<expression>,<kernel>) -

    -

    -

    -<expression> is expected to be a polynomial expression, not a rational -expression. Rational expressions are accepted when the switch - -ratarg is on. <variable> must be a -kernel. The -results are returned in a list. -

    -

    -

    -examples:

    -

    
-
-deg((x+y)**5,x); 
-
-  5 
-
-
-
-deg((a+b)*(c+2*d)**2,d); 
-
-  2 
-
-
-
-deg(x**2 + cos(y),sin(x)); 
-
-
-deg((x**2 + sin(x))**5,sin(x)); 
-
-  5
-
-

    - - -DEN -INDEX

    - - - -DEN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The den operator returns the denominator of its argument. -

    -

    -

    -syntax:

    -den(<expression>) -

    -

    -

    -<expression> is ordinarily a rational expression, but may be any valid -scalar REDUCE expression. -

    -

    -

    -examples:

    -

    
-
-a := x**3 + 3*x**2 + 12*x; 
-
-           2
-  A := X*(X   + 3*X + 12) 
-
-
-
-b := 4*x*y + x*sin(x); 
-
-  B := X*(SIN(X) + 4*Y) 
-
-
-
-den(a/b); 
-
-  SIN(X) + 4*Y 
-
-
-
-den(aa/4 + bb/5); 
-
-  20 
-
-
-
-den(100/6); 
-
-  3 
-
-
-
-den(sin(x)); 
-
-  1
-
-

    denreturns the denominator of the expression after it has - been -simplified by REDUCE. As seen in the examples, this includes putting -sums of rational expressions over a common denominator, and reducing -common factors where possible. If the expression does not have any -other denominator, 1 is returned. -

    -

    -Switch settings, such as -mcd or -rational, have an -effect on the denominator of an expression. -

    -

    -

    - - - -DF -INDEX

    - - - -DF _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The df operator finds partial derivatives with respect to one or -more variables. -

    -

    -

    -syntax:

    -df(<expression>,<var> - [,<number>] - {,<var> [ ,<number>] } ) -

    -

    -

    -<expression> can be any valid REDUCE algebraic expression. <var> -must be a -kernel, and is the differentiation variable. -<number> must be a non-negative integer. -

    -

    -

    -examples:

    -

    
-
-df(x**2,x); 
-
-  2*X 
-
-
-
-df(x**2*y + sin(y),y); 
-
-            2
-  COS(Y) + X  
-
-
-
-df((x+y)**10,z); 
-
-  0 
-
-
-
-
-df(1/x**2,x,2); 
-
-  6
-  ---
-   4
-  X
-
-
-
-df(x**4*y + sin(y),y,x,3); 
-
-  24*X 
-
-
-
-for all x let df(tan(x),x) = sec(x)**2; 
-
-
-df(tan(3*x),x); 
-
-            2
-  3*SEC(3*X)
-
-

    An error message results if a non-kernel is entered as a different -iation -operator. If the optional number is omitted, it is assumed to be 1. -See the declaration -depend to establish dependencies for implicit -differentiation. -

    -

    -You can define your own differentiation rules, expanding REDUCE's -capabilities, using the -let command as shown in the last example -above. Note that once you add your own rule for differentiating a -function, it supersedes REDUCE's normal handling of that function for the -duration of the REDUCE session. If you clear the rule -( -clearrules), you don't get back -to the previous rule. -

    -

    -

    - - - -EXPAND_CASES -INDEX

    - - - -EXPAND\_CASES _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -When a -root_of form in a result of -solve -has been converted to a -one_of form, expand_cases -can be used to convert this into form corresponding to the -normal explicit results of -solve. See -root_of. -

    -

    - - - -EXPREAD -INDEX

    - - - -EXPREAD _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -expread() -

    -

    -

    -expreadreads one well-formed expression from the current input -buffer and returns its value. -

    -

    -

    -examples:

    -

    
-expread(); a+b; 
-
-  A + B
-
-

    - - -FACTORIZE -INDEX

    - - - -FACTORIZE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The factorize operator factors a given expression into a list of -{factor,power} pairs. -

    -syntax:

    -

    -

    -factorize(<expression>) -

    -

    -

    -<expression> should be a polynomial, otherwise an error will result. -

    -

    -

    -examples:

    -

    
-
-fff := factorize(x^3 - y^3); 
-
-         2          2
-  		  {{X  + X*Y + Y ,1},{X - Y,1}} 
-
-
-fac1 := first fff; 
-
-             2          2
-  FAC1 := {{X  + X*Y + Y ,1} 
-
-
-factorize(x^15 - 1); 
-
-       8    7    6    5    4
-   {{ X  - X  + X  - X  + X  - X + 1,1},
-     4    3    2
-   {X  + X  + X  + X + 1,1},
-     2
-   {X  + X + 1,1},
-   {X - 1,1}}
-
-
-lastone := part(ws,length ws); 
-
-  	LASTONE := {X - 1,1} 
-
-
-setmod 2; 
-
-  1 
-
-
-on modular; 
-
-factorize(x^15 - 1); 
-
-     4    3    2
-  {{X  + X  + X  + X + 1,1},
-     4    3
-   {X  + X  + 1,1},
-     4
-   {X  + X + 1,1},
-      2
-   { X  + X + 1,1},
-   {X + 1,1}}
-
-

    The factorize command returns the factor,power pairs as a - -list. -You can therefore use the usual list access methods ( -first, - -second, -third, -rest, -length and - -part) to extract these pairs. -

    -

    -If the <expression> given to factorize is an integer, it will be - -factored into its prime components. To factor any integer factor of a -non-numerical expression, the switch -ifactor should be turned on. -Its default is off. -ifactor has effect only when factoring is -explicitly done by factorize, not when factoring is automatically -done with the -factor switch. If full factorization is not -needed the switch -limitedfactors allows you to reduce the -computing time of calls to factorize. -

    -

    -Factoring can be done in a modular domain by calling factorize when - -modular is on. You can set the modulus with the -setmod -command. The last example above shows factoring modulo 2. -

    -

    -For general comments on factoring, see comments under the switch - -factor. -

    -

    -

    - - - -HYPOT -INDEX

    - - - -HYPOT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -hypot(<expression>,<expression>) -

    -

    -

    -If rounded is on, and the two arguments evaluate to numbers, this -operator returns the square root of the sums of the squares of the -arguments in a manner that avoids intermediate overflow. In other cases, -an expression in the original operator is returned. -

    -

    -

    -examples:

    -

    
-hypot(3,4); 
-
-  HYPOT(3,4) 
-
-
-on rounded; 
-
-ws; 
-
-  5.0 
-
-
-hypot(a,b); 
-
-  HYPOT(A,B)
-
-

    - - -IMPART -INDEX

    - - - -IMPART _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -impart(<expression>) or impart <simple\_expression -> -

    -

    -

    -This operator returns the imaginary part of an expression, if that -argument has an numerical value. A non-numerical argument is returned as -an expression in the operators -repart and impart. -

    -examples:

    -

    
-impart(1+i); 
-
-  1 
-
-
-impart(a+i*b); 
-
-  REPART(B) + IMPART(A)
-
-

    -

    - - - -INT -INDEX

    - - - -INT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The int operator performs analytic integration on a variety of -functions. -

    -

    -

    -syntax:

    -int(<expression>,<kernel>) -

    -

    -

    -<expression> can be any scalar expression. involving polynomials, log -functions, exponential functions, or tangent or arctangent expressions. -int attempts expressions involving error functions, dilogarithms -and other trigonometric expressions. Integrals involving algebraic -extensions (such as square roots) may not succeed. <kernel> must be a -REDUCE -kernel. -

    -

    -

    -examples:

    -

    
-int(x**3 + 3,x); 
-
-      3
-  X*(X  + 12)
-  ----------- 
-       4
-
-
-
-int(sin(x)*exp(2*x),x);
- 
-
-     2*X
-    E   *(COS(X) - 2*SIN(X))
-  - ------------------------ 
-               5
-
-
-int(1/(x^2-2),x);
- 
-
-  SQRT(2)*(LOG( - SQRT(2) + X) - LOG(SQRT(2) + X))
-  ------------------------------------------------ 
-                         4
-
-
-int(sin(x)/(4 + cos(x)**2),x);
- 
-
-         COS(X)
-    ATAN(------)
-           2
-  - ------------ 
-         2
-
-
-
-int(1/sqrt(x^2-x),x); 
-
-      SQRT(X)*SQRT(X - 1)
-  INT(-------------------,X)
-              2
-             X -X
-
-

    Note that REDUCE couldn't handle the last integral with its defaul -t -integrator, since the integrand involves a square root. However, -the integral can be found using the -algint package. -Alternatively, you could add a rule using the -let statement -to evaluate this integral. -

    -

    -The arbitrary constant of integration is not shown. Definite integrals can -be found by evaluating the result at the limits of integration (use - -rounded) and subtracting the lower from the higher. Ev -aluation can -be easily done by the -sub operator. -

    -

    -When int cannot find an integral it returns an expression -involving formal int expressions unless the switch - -failhard has been set. If not all of the expression -can be integrated, the switch -nolnr controls whether a partially -integrated result should be returned or not. -

    -

    -

    -

    - - - -INTERPOL -INDEX

    - - - -INTERPOL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -interpolgenerates an interpolation polynomial. -

    -syntax:

    -

    -

    -interpol(<values>,<variable>,<points>) -

    -

    -

    -<values> and <points> are -lists of equal length and -<variable> is an algebraic expression (preferably a -kernel). -The interpolation polynomial is generated in the given variable of degree -length(<values>)-1. The unique polynomial f is defined by the -property that for corresponding elements v of <values> and -p of <points> the relation f(p)=v holds. -

    -

    -

    -examples:

    -

    
-f := for i:=1:4 collect(i**3-1); 
-
-  F := 0,7,26,63 
-
-
-p := {1,2,3,4}; 
-
-  P := 1,2,3,4 
-
-
-interpol(f,x,p); 
-
-   3
-  X  - 1
-
-

    The Aitken-Neville interpolation algorithm is used which guarantee -s a -stable result even with rounded numbers and an ill-conditioned problem. -

    -

    -

    - - - -LCOF -INDEX

    - - - -LCOF _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The lcof operator returns the leading coefficient of a given expression - -with respect to a given variable. -

    -syntax:

    -

    -

    -lcof(<expression>,<kernel>) -

    -

    -

    -<expression> is ordinarily a polynomial. If -ratarg is on, -a rational expression may also be used, otherwise an error results. -<kernel> must be a -kernel. -

    -

    -

    -examples:

    -

    
-lcof((x+2*y)**5,y); 
-
-  32 
-
-
-lcof((x + y*sin(x))**2 + cos(x)*sin(x)**2,sin(x));
-			 
-
-
-        2
-  COS(X)  + Y 
-
-
-lcof(x**2 + 3*x + 17,y); 
-
-   2
-  X  + 3*X + 17
-
-

    If the kernel does not appear in the expression, lcof ret -urns the -expression. -

    -

    -

    - - - -LENGTH -INDEX

    - - - -LENGTH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The length operator returns the number of items in a -list, the -number of -terms in an expression, or the dimensions of an array or matrix. -

    -syntax:

    -

    -

    -length(<expr>) or length <expr> -

    -

    -

    -<expr> can be a list structure, an array, a matrix, or a scalar expression -. -

    -

    -

    -examples:

    -

    
-alist := {a,b,{ww,xx,yy,zz}}; 
-
-  ALIST := {A,B,{WW,XX,YY,ZZ}} 
-
-
-length alist; 
-
-  3  
-
-
-length third alist; 
-
-  4  
-
-
-dlist := {d}; 
-
-  DLIST := {D} 
-
-
-length rest dlist; 
-
-  0  
-
-
-matrix mmm(4,5); 
-
-length mmm; 
-
-  {4,5} 
-
-
-array aaa(5,3,2); 
-
-length aaa; 
-
-  {6,4,3} 
-
-
-eex := (x+3)**2/(x-y); 
-
-          2
-         X  + 6*X + 9
-  EEX := ------------ 
-            X - Y
-
-
-length eex; 
-
-  5
-
-

    An item in a list that is itself a list only counts as one item. A -n error -message will be printed if length is called on a matrix which has -not had its dimensions set. The length of an array includes the -zeroth element of each dimension, showing the full number of elements -allocated. (Declaring an array A with n elements -allocates A(0),A(1),...,A(n).) The -length of an expression is the total number of additive terms -appearing in the numerator and denominator of the expression. Note that -subtraction of a term is represented internally as addition of a negative -term. -

    -

    -

    - - - -LHS -INDEX

    - - - -LHS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The lhs operator returns the left-hand side of an -equation, -such as those -returned in a list by -solve. -

    -syntax:

    -

    -

    -lhs(<equation>) or lhs <equation> -

    -

    -

    -

    -<equation> must be an equation of the form -

    -

    -left-hand side=right-hand side. -

    -

    -

    -examples:

    -

    
-polly := (x+3)*(x^4+2x+1); 
-
-            5      4      2
-  POLLY := X  + 3*X  + 2*X  + 7*X + 3 
-
-
-pollyroots := solve(polly,x); 
-
-  POLLYROOTS := {X=ROOT F(X3 - X2 + X + 1,X ,
-                       O                   )
-                 X=-1,
-                 X=-3}
-
-
-variable := lhs first pollyroots; 
-
-  VARIABLE := X
-
-

    - - -LIMIT -INDEX

    - - - -LIMIT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -LIMITS is a fast limit package for REDUCE for functions which are -continuous except for computable poles and singularities, based on -some earlier work by Ian Cohen and John P. Fitch. The Truncated -Power Series package is used for non-critical points, at which -the value of the function is the constant term in the expansion -around that point. l'Hopital's rule is used in critical cases, -with preprocessing of 1-1 forms and reformatting of product forms -in order to apply l'Hopital's rule. A limited amount of bounded -arithmetic is also employed where applicable. -

    -

    -

    -syntax:

    -limit(<expr>,<var>,<limpoint>) or -

    -

    -limit!+(<expr>,<var>,<limpoint>) or -

    -

    -limit!-(<expr>,<var>,<limpoint>) -

    -

    -

    -where <expr> is an expression depending of the variable <var> -(a -kernel) and <limpoint> is the limit point. -If the limit depends upon the direction of approach to the <limpoint>, -the operators limit!+ and limit!- may be used. -

    -

    -

    -examples:

    -

    
-limit(x*cot(x),x,0);
-
-  0
-
-
-limit((2x+5)/(3x-2),x,infinity);
-
-  2
-  --
-  3
-
-

    - - -LPOWER -INDEX

    - - - -LPOWER _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The lpower operator returns the leading power of an expression with -respect to a kernel. 1 is returned if the expression does not depend on -the kernel. -

    -syntax:

    -

    -

    -lpower(<expression>,<kernel>) -

    -

    -

    -<expression> is ordinarily a polynomial. If -ratarg is on, -a rational expression may also be used, otherwise an error results. -<kernel> must be a -kernel. -

    -

    -

    -examples:

    -

    
-lpower((x+2*y)**6,y); 
-
-   6
-  Y  
-
-
-lpower((x + cos(x))**8 + df(x**2,x),cos(x));
-			 
-
-
-        8
-  COS(X)  
-
-
-lpower(x**3 + 3*x,y); 
-
-  1
-
-

    - - -LTERM -INDEX

    - - - -LTERM _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The lterm operator returns the leading term of an expression with -respect to a kernel. The expression is returned if it does not depend on -the kernel. -

    -syntax:

    -

    -

    -lterm(<expression>,<kernel>) -

    -

    -

    -<expression> is ordinarily a polynomial. If -ratarg is on, -a rational expression may also be used, otherwise an error results. -<kernel> must be a -kernel. -

    -

    -

    -examples:

    -

    
-lterm((x+2*y)**6,y); 
-
-      6
-  64*Y  
-
-
-lterm((x + cos(x))**8 + df(x**2,x),cos(x));
-			 
-
-
-        8
-  COS(X)  
-
-
-lterm(x**3 + 3*x,y); 
-
-   3
-  X  + 3X
-
-

    - - -MAINVAR -INDEX

    - - - -MAINVAR _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The mainvar operator returns the main variable (in the system's -internal representation) of its argument. -

    -syntax:

    -

    -

    -mainvar(<expression>) -

    -

    -

    -

    -<expression> is usually a polynomial, but may be any valid REDUCE -scalar expression. In the case of a rational function, the main variable -of the numerator is returned. The main variable returned is a - -kernel. -

    -

    -

    -examples:

    -

    
-test := (a + b + c)**2; 
-
-           2                    2            2
-  TEST := A  + 2*A*B + 2*A*C + B  + 2*B*C + C  
-
-
-mainvar(test); 
-
-  A 
-
-
-korder c,b,a; 
-
-mainvar(test); 
-
-  C 
-
-
-mainvar(2*cos(x)**2); 
-
-  COS(X) 
-
-
-mainvar(17); 
-
-  0
-
-

    The main variable is the first variable in the canonical ordering -of -kernels. Generally, alphabetically ordered functions come first, then -alphabetically ordered identifiers (variables). Numbers come last, and as -far as mainvar is concerned belong in the family 0. The -canonical ordering can be changed by the declaration -korder, as -shown above. -

    -

    -

    - - - -MAP -INDEX

    - - - -MAP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The map operator applies a uniform evaluation pattern -to all members of a composite structure: a -matrix, -a -list or the arguments of an -operator expression. -The evaluation pattern can be a -unary procedure, an operator, or an algebraic expression with -one free variable. -

    -syntax:

    -

    -

    -map(<function>,<object>) -

    -

    -

    -<object> is a list, a matrix or an operator expression. -

    -

    -<function> is -the name of an operator for a single argument: the operator -is evaluated once with each element of <object> as its single argument, -

    -

    -or an algebraic expression with exactly one -free variable, that is -a variable preceded by the tilde symbol: the expression - is evaluated for each element of <object> where the element is - substituted for the free variable, -

    -

    -or a replacement -rule of the form -

    -syntax:

    -

    -

    -var=> rep -

    -

    -

    -where <var> is a variable (a <kernel> without subscript) - and <rep> is an expression which contains <var>. - Here rep is evaluated for each element of <object> where - the element is substituted for var. var may be - optionally preceded by a tilde. -

    -

    -The rule form for <function> is needed when more than -one free variable occurs. -

    -

    -

    -examples:

    -

    
-map(abs,{1,-2,a,-a}); 
-
-  1,2,abs(a),abs(a) 
-
-
-map(int(~w,x), mat((x^2,x^5),(x^4,x^5))); 
-
-
-          [  3     6 ]
-          [ x     x  ]
-          [----  ----]
-          [ 3     6  ]
-          [          ]
-          [  5     6 ]
-          [ x     x  ]
-          [----  ----]
-  	[ 5     6  ]
-
-
-map(~w*6, x^2/3 = y^3/2 -1); 
-
-     2     3
-  2*x =3*(y -2)
-
-

    You can use map in nested expressions. It is not allowed -to -apply map for a non-composed object, e.g. an identifier or a number. -

    -

    -

    - - - -MKID -INDEX

    - - - -MKID _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The mkid command constructs an identifier, given a stem and an identifi -er -or an integer. -

    -syntax:

    -

    -

    -mkid(<stem>,<leaf>) -

    -

    -

    -<stem> can be any valid REDUCE identifier that does not include escaped -special characters. <leaf> may be an integer, including one given by a -local variable in a -for loop, or any other legal group of -characters. -

    -

    -

    -examples:

    -

    
-mkid(x,3); 
-
-  X3 
-
-
-factorize(x^15 - 1); 
-
-  {X - 1,
-    2
-   X  + X + 1,
-    4    3    2
-   X  + X  + X  + X + 1,
-    8    7    5    4    3
-   X  - X  + X  - X  + X   - X + 1}
-
-
-
-for i := 1:length ws do write set(mkid(f,i),part(ws,i));
-	 
-
-
-   8    7    5    4    3
-  X  - X  + X  - X  + X  - X + 1
-   4    3    2
-  X  + X  + X  + X + 1
-   2
-  X  + X + 1
-  X - 1
-
-

    You can use mkid to construct identifiers from inside pro -cedures. This -allows you to handle an unknown number of factors, or deal with variable -amounts of data. It is particularly helpful to attach identifiers to the -answers returned by factorize and solve. -

    -

    -

    - - - -NPRIMITIVE -INDEX

    - - - -NPRIMITIVE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -nprimitive(<expression>) or nprimitive - <simple\_expression> -

    -

    -

    -This operator returns the numerically-primitive part of any scalar -expression. In other words, any overall integer factors in the expression -are removed. -

    -

    -

    -examples:

    -

    
-nprimitive((2x+2y)^2); 
-
-   2            2
-  X  + 2*X*Y + Y  
-
-
-nprimitive(3*a*b*c); 
-
-  3*A*B*C
-
-

    - - -NUM -INDEX

    - - - -NUM _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The num operator returns the numerator of its argument. -

    -syntax:

    -

    -

    -num(<expression>) or num <simple\_expression> -

    -

    -

    -<expression> can be any valid REDUCE scalar expression. -

    -

    -

    -examples:

    -

    
-num(100/6); 
-
-  50 
-
-
-num(a/5 + b/6); 
-
-  6*A + 5*B 
-
-
-num(sin(x)); 
-
-  SIN(X)
-
-

    numreturns the numerator of the expression after it has b -een simplified -by REDUCE. As seen in the examples, this includes putting sums of rational -expressions over a common denominator, and reducing common factors where -possible. If the expression is not a rational expression, it is returned -unchanged. -

    -

    -

    - - - -ODESOLVE -INDEX

    - - - -ODESOLVE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The odesolve package is a solver for ordinary differential -equations. At the present time it has still limited capabilities: -

    -

    -1. it can handle only a single scalar equation presented as an - algebraic expression or equation, and -

    -

    -2. it can solve only first-order equations of simple types, linear - equations with constant coefficients and Euler equations. -

    -

    -These solvable types are exactly those for which Lie symmetry -techniques give no useful information. -

    -

    -

    -syntax:

    -odesolve(<expr>,<var1>,<var2>) -

    -

    -

    -

    -<expr> is a single scalar expression such that <expr>=0 -is the ordinary differential equation (ODE for short) to be solved, or -is an equivalent -equation. -

    -

    -<var1> is the name of the dependent variable, -<var2> is the name of the independent variable. -

    -

    -A differential in <expr> is expressed using the -df -operator. Note that in most cases you must declare explicitly -<var1> to depend of <var2> using a -depend -declaration -- otherwise the derivative might be evaluated to -zero on input to odesolve. -

    -

    -The returned value is a list containing the equation giving the general -solution of the ODE (for simultaneous equations this will be a -list of equations eventually). It will contain occurrences of -the operator arbconst for the arbitrary constants in the general -solution. The arguments of arbconst should be new. -A counter !!arbconst is used to arrange this. -

    -

    -

    -examples:

    -

    
-depend y,x;
-
-% A first-order linear equation, with an initial condition
-
-ode:=df(y,x) + y * sin x/cos x - 1/cos x$
-
-odesolve(ode,y,x); 
-
-  {y=arbconst(1)*cos(x) + sin(x)}
-
-

    - - -ONE_OF -INDEX

    - - - -ONE\_OF _ _ _ _ _ _ _ _ _ _ _ _ type

    -

    - -The operator one_of is used to represent an indefinite choice -of one element from a finite set of objects. -

    -examples:

    -

    
-x=one_of{1,2,5}
-

    this equation encodes that x can take one of the values -1,2 or 5

    
-

    -REDUCE generates a one_of form in cases when an implicit -root_of expression could be converted to an explicit solution set. -A one_of form can be converted to a solve solution using - -expand_cases. See -root_of. -

    -

    - - - -PART -INDEX

    - - - -PART _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator part permits the extraction of various parts or -operators of expressions and -lists. -

    -syntax:

    -

    -

    -part(<expression,integer>{,<integer>}*) -

    -

    -

    -<expression> can be any valid REDUCE expression or a list, -integer may be an expression that evaluates to a positive or negative -integer or 0. A positive integer <n> picks up the n th term, -counting from the first term toward the end. A negative integer n -picks up the n th term, counting from the back toward the front. The -integer 0 picks up the operator (which is LIST when the expression -is a -list). -

    -

    -

    -examples:

    -

    
-part((x + y)**5,4); 
-
-      2  3
-  10*X *Y  
-
-
-part((x + y)**5,4,2); 
-
-   2
-  X  
-
-
-part((x + y)**5,4,2,1); 
-
-  X 
-
-
-part((x + y)**5,0); 
-
-  PLUS 
-
-
-part((x + y)**5,-5); 
-
-        4
-  5*X *Y  
-
-
-part((x + y)**5,4) := sin(x); 
-
-   5      4         3  2                 4    5
-  X  + 5*X *Y + 10*X *Y  + SIN(X) + 5*X*Y  + Y  
-
-
-alist := {x,y,{aa,bb,cc},x**2*sqrt(y)}; 
-
-
-                                        2
-  			 ALIST := {X,Y,{AA,BB,CC},SQRT(Y)*X } 
-
-
-part(alist,3,2); 
-
-  BB 
-
-
-part(alist,4,0); 
-
-  TIMES
-
-

    Additional integer arguments after the first one examine the -terms recursively, as shown above. In the third line, the fourth term -is picked from the original polynomial, 10x^2y^3, -then the second term from that, x^2, and finally the first -component, x. If an integer's absolute value is too large for -the appropriate expression, a message is given. -

    -

    -partworks on the form of the expression as printed, or as it would -have been printed at that point of the calculation, bearing in mind the -current switch settings. It is important to realize that the switch settings -change the operation of part. -pri must be on when -part is used. -

    -

    -When part is used on a polynomial expression that has minus signs, the - -+ is always returned as the top-level operator. The minus is found -as a unary operator attached to the negative term. -

    -

    -partcan also be used to change the relevant part of the expression or -list as shown in the sixth example line. The part operator returns the - -changed expression, though original expression is not changed. You can -also use part to change the operator. -

    -

    -

    - - - -PF -INDEX

    - - - -PF _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -pf(<expression>,<variable>) -

    -

    -

    -pftransforms <expression> into a -list of partial fraction -s -with respect to the main variable, <variable>. pf does a -complete partial fraction decomposition, and as the algorithms used are -fairly unsophisticated (factorization and the extended Euclidean -algorithm), the code may be unacceptably slow in complicated cases. -

    -examples:

    -

    
-pf(2/((x+1)^2*(x+2)),x); 
-
-      2    -2        2
-  	{-----,-----,------------} 
-    X + 2 X + 1  2
-                X  + 2*X + 1
-
-
-off exp; 
-
-pf(2/((x+1)^2*(x+2)),x);
- 
-
-     2    - 2     2
-  {-----,-----,--------} 
-   X + 2 X + 1        2
-               (X + 1)
-
-
-for each j in ws sum j; 
-
-         2
-  ----------------
-                2
-  ( + 2)*(X + 1)
-
-

    -

    -If you want the denominators in factored form, turn -exp off, as -shown in the second example above. As shown in the final example, the - -for each construct can be used to recombine t -he terms. -Alternatively, one can use the operations on lists to extract any desired -term. -

    -

    -

    - - - -PROD -INDEX

    - - - -PROD _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator prod returns -the indefinite or definite product of a given expression. -

    -

    -

    -syntax:

    -prod(<expr>,<k>[,<lolim> [,<uplim> ]]) -

    -

    -

    -

    -where <expr> is the expression to be multiplied, <k> is the -control variable (a -kernel), and <lolim> and <uplim> -uplim are the optional lower and upper limits. If <uplim> is -not supplied the upper limit is taken as <k>. The -Gosper algorithm is used. If there is no closed form solution, -the operator returns the input unchanged. -

    -

    -

    -examples:

    -

    
-prod(k/(k-2),k);
-
-  k*( - k + 1)
-
-

    - - -REDUCT -INDEX

    - - - -REDUCT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The reduct operator returns the remainder of its expression after the -leading term with respect to the kernel in the second argument is removed. -

    -syntax:

    -

    -

    -reduct(<expression>,<kernel>) -

    -

    -

    -<expression> is ordinarily a polynomial. If -ratarg is on, -a rational expression may also be used, otherwise an error results. -<kernel> must be a -kernel. -

    -

    -

    -examples:

    -

    
-reduct((x+y)**3,x); 
-
-        2            2
-  Y*(3*X  + 3*X*Y + Y ) 
-
-
-reduct(x + sin(x)**3,sin(x)); 
-
-  X 
-
-
-reduct(x + sin(x)**3,y); 
-
-  0
-
-

    If the expression does not contain the kernel, reduct ret -urns 0. -

    -

    -

    - - - -REPART -INDEX

    - - - -REPART _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -repart(<expression>) or repart <simple\_expression -> -

    -

    -

    -This operator returns the real part of an expression, if that argument has an -numerical value. A non-numerical argument is returned as an expression in -the operators repart and -impart. -

    -examples:

    -

    
-repart(1+i); 
-
-  1 
-
-
-repart(a+i*b); 
-
-  REPART(A) - IMPART(B)
-
-

    -

    - - - -RESULTANT -INDEX

    - - - -RESULTANT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The resultant operator computes the resultant of two polynomials with -respect to a given variable. If the resultant is 0, the polynomials have -a root in common. -

    -syntax:

    -

    -

    -resultant(<expression>,<expression>,<kernel>) -

    -

    -

    -<expression> must be a polynomial containing <kernel> ; -<kernel> must be a -kernel. -

    -

    -

    -examples:

    -

    
-resultant(x**2 + 2*x + 1,x+1,x); 
-
-  0 
-
-
-resultant(x**2 + 2*x + 1,x-3,x); 
-
-  16 
-
-
-resultant(z**3 + z**2 + 5*z + 5,
-          z**4 - 6*z**3 + 16*z**2 - 30*z + 55,
-          z);
- 
-
-  0 
-
-
-resultant(x**3*y + 4*x*y + 10,y**2 + 6*y + 4,y); 
-
-
-   6       5        4        3        2
-  Y  + 18*Y  + 120*Y  + 360*Y  + 480*Y  + 288*Y + 64
-
-

    The resultant is the determinant of the Sylvester matrix, formed f -rom the -coefficients of the two polynomials in the following way: -

    -

    -Given two polynomials: -

    -

    -

    
-    n       n-1 
- a x  + a1 x     + ... + an
-
-

    and -

    -

    -

    
-    m       m-1 
- b x  + b1 x     + ... + bm
-
-

    form the (m+n)x(m+n-1) Sylvester matrix by the following means: -

    -

    -

    
-   0.......0 a  a1 .......... an
-   0....0 a  a1 .......... an  0
-       .    .   .   .  
-   a0 a1 .......... an 0.......0
-   0.......0 b  b1 .......... bm
-   0....0 b  b1 .......... bm  0
-       .    .   .   .  
-   b  b1 .......... bm 0.......0  
-
-

    If the determinant of this matrix is 0, the two polynomials have a - common -root. Finding the resultant of large expressions is time-consuming, due -to the time needed to find a large determinant. -

    -

    -The sign conventions resultant uses are those given in the article, -``Computing in Algebraic Extensions,'' by R. Loos, appearing in -<Computer Algebra--Symbolic and Algebraic Computation>, 2nd ed., -edited by B. Buchberger, G.E. Collins and R. Loos, and published by -Springer-Verlag, 1983. -These are: -

    -

    -

    
-  resultant(p(x),q(x),x) = (-1)^{deg p(x)*deg q(x)} * resultant(q(x),p(x),x),
-  resultant(a,p(x),x)    = a^{deg p(x)},
-  resultant(a,b,x)       = 1
-

    where p(x) and q(x) are polynomials which have x as a variable, an -d -a and b are free of x. -

    -

    -Error messages are given if resultant is given a non-polynomial -expression, or a non-kernel variable. -

    -

    -

    - - - -RHS -INDEX

    - - - -RHS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The rhs operator returns the right-hand side of an -equation, -such as those returned in a -list by -solve. -

    -syntax:

    -

    -

    -rhs(<equation>) or rhs <equation> -

    -

    -

    -<equation> must be an equation of the form left-hand side = right-hand -side. -

    -

    -

    -examples:

    -

    
-roots := solve(x**2 + 6*x*y + 5x + 3y**2,x); 
-
-
-                              2
-                     SQRT(24*Y  + 60*Y + 25) + 6*Y + 5
-      ROOTS := {X= - ---------------------------------,
-                                     2
-                             2
-                    SQRT(24*Y  + 60*Y + 25) - 6*Y - 5
-                 X= ---------------------------------}
-                                    2
-
-
-root1 := rhs first roots; 
-
-                      2
-             SQRT(24*Y  + 60*Y + 25) + 6*Y + 5
-  ROOT1 := - --------------------------------- 
-                             2
-
-
-root2 := rhs second roots; 
-
-                    2
-           SQRT(24*Y  + 60*Y + 25) - 6*Y - 5
-  ROOT2 := ----------------------------------
-                           2
-
-

    An error message is given if rhs is applied to something -other than an -equation. -

    -

    -

    - - - -ROOT_OF -INDEX

    - - - -ROOT\_OF _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -When the operator -solve is unable to find an explicit solution -or if that solution would be too complicated, the result is presented -as formal root expression using the internal operator root_of -and a new local variable. An expression with a top level root_of -is implicitly a list with an unknown number of elements since we -can't always know how many solutions an equation has. If a -substitution is made into such an expression, closed form solutions -can emerge. If this occurs, the root_of construct is -replaced by an operator -one_of. At this point it is -of course possible to transform the result if the original solve -operator expression into a standard solve solution. To -effect this, the operator -expand_cases can be used. -

    -

    -

    -examples:

    -

    
-solve(a*x^7-x^2+1,x);
-
-                 7     2
-  {x=root_of(a*x_  - x_  + 1,x_)}
-
-
-sub(a=0,ws);
-
-  {x=one_of(1,-1)}
-
-
-expand_cases ws;
-
-  x=1,x=-1
-
-

    The components of root_of and one_of expressions - can be -processed as usual with operators -arglength and -part. -A higher power of a root_of expression with a polynomial -as first argument is simplified by using the polynomial as a side relation. -

    -

    - - - -SELECT -INDEX

    - - - -SELECT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The select operator extracts from a list -or from the arguments of an n--ary operator elements corresponding -to a boolean predicate. The predicate pattern can be a -unary procedure, an operator or an algebraic expression with -one -free variable. -

    -syntax:

    -

    -

    -select(<function>,<object>) -

    -

    -

    -<object> is a -list. -

    -

    -<function> is -the name of an operator for a single argument: the operator - is evaluated once with each element of <object> as its single argument, -

    -

    -or an algebraic expression with exactly one -free variable, that is -a variable preceded by the tilde symbol: the expression - is evaluated for each element of <object> where the element is - substituted for the free variable, -

    -

    -or a replacement -rule of the form -

    -syntax:

    -

    -

    -var=> rep -

    -

    -

    -where <var> is a variable (a <kernel> without subscript) - and <rep> is an expression which contains <var>. - Here rep is evaluated for each element of <object> where - the element is substituted for var. var may be - optionally preceded by a tilde. -

    -

    -The rule form for <function> is needed when more than -one free variable occurs. The evaluation result of <function> is -interpreted as -boolean value corresponding to the conventions of -REDUCE. The result value is built with the leading operator of the -input expression. -

    -examples:

    -

    
-  select( ~w>0 , {1,-1,2,-3,3}) 
-
-  {1,2,3} 
-
-
-  q:=(part((x+y)^5,0):=list)
-
-  select(evenp deg(~w,y),q);
-
-    5      3   2       4
-  {x  ,10*x  *y  ,5*x*y  }
-
-
-  select(evenp deg(~w,x),2x^2+3x^3+4x^4);
-
-
-    2   4
-  2x +4x
-
-

    -

    - - - -SHOWRULES -INDEX

    - - - -SHOWRULES _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -showrules(<expression>) or - showrules <simple\_expression> -

    -

    -

    -showrulesreturns in -rule-list form any - -operator rules associated with its argument. -

    -

    -

    -examples:

    -

    
-showrules log; 
-
-  {LOG(E) => 1,
-   LOG(1) => 0,
-        ~X
-   LOG(E   ) => ~X,
-                     1
-   DF(LOG(~X),~X) => --}
-                     ~X
-
-

    Such rules can then be manipulated further as with any -list. For -example -rhs first ws; has the value 1. -

    -

    -An operator may have properties that cannot be displayed in such a form, -such as the fact it is an -odd function, or has a definition defined -as a procedure. -

    -

    -

    - - - -SOLVE -INDEX

    - - - -SOLVE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The solve operator solves a single algebraic -equation or a -system of simultaneous equations. -

    -

    -

    -syntax:

    -solve(<expression> [ , <kernel>]) or -

    -

    -solve({<expression>,...} [ ,{ <kernel> ,...}] ) -

    -

    -

    -

    -If the number of equations equals the number of distinct kernels, the -optional kernel argument(s) may be omitted. <expression> is either a -scalar expression or an -equation. -When more than one expression is given, -the -list of expressions is surrounded by curly braces. -The optional list -of -kernels follows, also in curly braces. -

    -

    -

    -examples:

    -

    
-sss := solve(x^2 + 7); 
-
-  Unknown: X
-  SSS := {X= - SQRT(7)*I,
-          X=SQRT(7)*I}
-
-
-rhs first sss; 
-
-  - SQRT(7)*I 
-
-
-solve(sin(x^2*y),y); 
-
-     2*ARBINT(1)*PI
-  {Y=---------------
-            2
-           X
-     PI*(2*ARBINT(1) + 1)
-   Y=--------------------}
-               2
-              X
-
-
-off allbranch; 
-
-solve(sin(x**2*y),y); 
-
-  {Y=0} 
-
-
-solve({3x + 5y = -4,2*x + y = -10},{x,y});
- 
-
-
-         22   46
-  {{X= - --,Y=--}} 
-         7    7
-
-
-solve({x + a*y + z,2x + 5},{x,y});
- 
-
-
-         5      2*Z - 5
-  {{X= - -,Y= - -------}} 
-         2        2*A
-
-
-ab := (x+2)^2*(x^6 + 17x + 1);
- 
-
-         8      7      6       3       2
-  AB := X  + 4*X  + 4*X  + 17*X  + 69*X  + 72*X + 4 
-
-
-www := solve(ab,x); 
-
-  {X=ROOT F(X6 + 17*X + 1),X=-2} 
-         O            
-
-
-root_multiplicities; 
-
-  {1,2}
-
-

    Results of the solve operator are returned as -equations -in a -list. -You can use the usual list access methods ( -first, - -second, -third, -rest and -part) to -extract the desired equation, and then use the operators -rhs and - -lhs to access the right-hand or left-hand expression o -f the -equation. When solve is unable to solve an equation, it returns the -unsolved part as the argument of root_of, with the variable renamed -to avoid confusion, as shown in the last example above. -

    -

    -For one equation, solve uses square-free factorization, roots of -unity, and the known inverses of the -log, -sin, - -cos, -acos, -asin, and -exponentiation operators. The quadratic, cubic and quartic formulas are -used if necessary, but these are applied only when the switch - -fullroots is set on; otherwise or when no closed form -is available -the result is returned as - -root_of expression. The switch -trigform -determines which type of cubic and quartic formula is used. -The multiplicity of each solution is given in a list as -the system variable -root_multiplicities. For systems of -simultaneous linear equations, matrix inversion is used. For nonlinear -systems, the Groebner basis method is used. -

    -

    -Linear equation system solving is influenced by the switch -cramer. -

    -

    -Singular systems can be solved when the switch -solvesingular is -on, which is the default setting. An empty list is returned the system of -equations is inconsistent. For a linear inconsistent system with parameters -the variable -requirements constraints -conditions for the system to become consistent. -

    -

    -For a solvable linear and polynomial system with parameters -the variable -assumptions -contains a list side relations for the parameters: the solution is -valid only as long as none of these expressions is zero. -

    -

    -If the switch -varopt is on (default), the system rearranges the -variable sequence for minimal computation time. Without varopt -the user supplied variable sequence is maintained. -

    -

    -If the solution has free variables (dimension of the solution is greater -than zero), these are represented by -arbcomplex expressions -as long as the switch -arbvars is on (default). Without -arbvars no explicit equations are generated for free variables. -

    -

    -

    -

    -related:

    -

    - _ _ _ -allbranchswitch -

    - _ _ _ -arbvars switch -

    - _ _ _ -assumptions variable -

    - _ _ _ -fullroots switch -

    - _ _ _ -requirements variable -

    - _ _ _ -roots operator -

    - _ _ _ -root_of operator -

    - _ _ _ -trigform switch -

    - _ _ _ -varopt switch -

    -

    -

    - - - -SORT -INDEX

    - - - -SORT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The sort operator sorts the elements of a list according to -an arbitrary comparison operator. -

    -syntax:

    -

    -

    -sort(<lst>,<comp>) -

    -

    -

    -<lst> is a -list of algebraic expressions. -<comp> is a comparison operator which defines a partial -ordering among the members of <lst>. <comp> may be -one of the builtin comparison operators like -<( -lessp), <=( -leq) -etc., or <comp> may be the name of a comparison procedure. -Such a procedure has two arguments, and it returns - -true if the first argument -ranges before the second one, and 0 or -nil otherwise. -The result of sort is a new list which contains the -elements of <lst> in a sequence corresponding to <comp>. -

    -examples:

    -

    
- procedure ce(a,b);
-
-   if evenp a and not evenp b then 1 else 0;
-
-for i:=1:10 collect random(50)$
-
-sort(ws,>=); 
-
-  {41,38,33,30,28,25,20,17,8,5}
-
-
-sort(ws,<); 
-
-  {5,8,17,20,25,28,30,33,38,41}
-
-
-sort(ws,ce); 
-
-  {8,20,28,30,38,5,17,25,33,41}
-
-
-  procedure cd(a,b);
-
-  if deg(a,x)>deg(b,x) then 1 else
-
-  if deg(a,x)<deg(b,x) then 0 else
-
-  if deg(a,y)>deg(b,y) then 1 else 0;
-
-sort({x^2,y^2,x*y},cd);
-
-    2      2
-  {x ,x*y,y }
-
-

    -

    - - - -STRUCTR -INDEX

    - - - -STRUCTR _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The structr operator breaks its argument expression into named -subexpressions. -

    -

    -

    -syntax:

    -structr(<expression> [,<identifier>[,<identifier> ... -]]) -

    -

    -

    -<expression> may be any valid REDUCE scalar expression. -<identifier> may be any valid REDUCE identifier. The first -identifier -is the stem for subexpression names, the second is the name to be assigned -to the structured expression. -

    -

    -

    -examples:

    -

    
-structr(sqrt(x**2 + 2*x) + sin(x**2*z)); 
-
-
-  ANS1 + ANS2
-      where
-                       2
-          ANS2 := SIN(X *Z)
-                             1/2
-          ANS1 := ((X + 2)*X)
-
-
-ans3; 
-
-  ANS3 
-
-
-on fort; 
-
-structr((x+1)**5 + tan(x*y*z),var,aa); 
-
-
-  VAR1=TAN(X*Y*Z)
-  AA=VAR1+X**5+5.*X**4+10.*X**3+10.X**2+5.*X+1
-
-

    The second argument to structr is optional. If it is not -given, the -default stem ANS is used by REDUCE to construct names for the -subexpression. The names are only for display purposes: REDUCE does not -store the names and their values unless the switch -savestructr is -on. -

    -

    -If a third argument is given, the structured expression as a whole is named by -this argument, when -fort is on. The expression is not stored -under this -name. You can send these structured Fortran expressions to a file with the -out command. -

    -

    -

    - - - -SUB -INDEX

    - - - -SUB _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The sub operator substitutes a new expression for a kernel in an -expression. -

    -syntax:

    -

    -

    -sub(<kernel>=<expression> - {,<kernel>=<expression>}*, - <expression>) or -

    -

    -sub({<kernel>=<expression>*, - <kernel>=expression},<expression>) -

    -

    -

    -<kernel> must be a -kernel, <expression> can be any REDUCE -scalar expression. -

    -

    -

    -examples:

    -

    
-sub(x=3,y=4,(x+y)**3); 
-
-  343 
-
-
-x; 
-
-  X 
-
-
-sub({cos=sin,sin=cos},cos a+sin b) 
-
-
-  COS(B) + SIN(A)
-
-

    Note in the second example that operators can be replaced using th -e -sub operator. -

    -

    -

    - - - -SUM -INDEX

    - - - -SUM _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator sum returns -the indefinite or definite summation of a given expression. -

    -

    -

    -syntax:

    -sum(<expr>,<k>[,<lolim> [,<uplim> ]]) -

    -

    -

    -

    -where <expr> is the expression to be added, <k> is the -control variable (a -kernel), and <lolim> and <uplim> -are the optional lower and upper limits. If <uplim> is -not supplied the upper limit is taken as <k>. The Gosper -algorithm is used. If there is no closed form solution, the operator -returns the input unchanged. -

    -

    -

    -examples:

    -

    
-sum(4n**3,n); 
-
-   2    2
-  n  *(n   + 2*n + 1)
-
-
-sum(2a+2k*r,k,0,n-1);
-
-  n*(2*a + n*r - r)
-
-

    - - -WS -INDEX

    - - - -WS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The ws operator alone returns the last result; ws with a -number argument returns the results of the REDUCE statement executed after -that numbered prompt. -

    -syntax:

    -

    -

    -wsor ws(<number>) -

    -

    -

    -<number> must be an integer between 1 and the current REDUCE prompt number -. -

    -

    -

    -examples:

    -

    (In the following examples, unlike most others, the nu -mbered -prompt is shown.)

     
-
-1: df(sin y,y); 
-
-  COS(Y) 
-
-
-2: ws^2; 
-
-        2
-  COS(Y)  
-
-
-3: df(ws 1,y); 
-
-  -SIN(Y)
-
-

    -

    -

    -wsand ws(<number>) can be used anywher -e the -expression they stand for can be used. Calling a number for which no -result was produced, such as a switch setting, will give an error message. -

    -

    -The current workspace always contains the results of the last REDUCE -command that produced an expression, even if several input statements -that do not produce expressions have intervened. For example, if you do -a differentiation, producing a result expression, then change several -switches, the operator ws; returns the results of the differentiation. - -The current workspace (ws) can also be used inside files, though the -numbered workspace contains only the in command that input the file. -

    -

    -There are three history lists kept in your REDUCE session. The first -stores raw input, suitable for the statement editor. The second stores -parsed input, ready to execute and accessible by -input. The -third stores results, when they are produced by statements, which are -accessible by the ws< n> operator. If your session is very -long, storage space begins to fill up with these expressions, so it is a -good idea to end the session once in a while, saving needed expressions to -files with the -saveas and -out commands. -

    -

    -An error message is given if a reference number has not yet been used. -

    -

    -

    - - - -Algebraic Operators -INDEX

    -Algebraic Operators

    -
  • APPEND operator

    -

  • ARBINT operator

    -

  • ARBCOMPLEX operator

    -

  • ARGLENGTH operator

    -

  • COEFF operator

    -

  • COEFFN operator

    -

  • CONJ operator

    -

  • CONTINUED_FRACTION operator

    -

  • DECOMPOSE operator

    -

  • DEG operator

    -

  • DEN operator

    -

  • DF operator

    -

  • EXPAND\_CASES operator

    -

  • EXPREAD operator

    -

  • FACTORIZE operator

    -

  • HYPOT operator

    -

  • IMPART operator

    -

  • INT operator

    -

  • INTERPOL operator

    -

  • LCOF operator

    -

  • LENGTH operator

    -

  • LHS operator

    -

  • LIMIT operator

    -

  • LPOWER operator

    -

  • LTERM operator

    -

  • MAINVAR operator

    -

  • MAP operator

    -

  • MKID command

    -

  • NPRIMITIVE operator

    -

  • NUM operator

    -

  • ODESOLVE operator

    -

  • ONE\_OF type

    -

  • PART operator

    -

  • PF operator

    -

  • PROD operator

    -

  • REDUCT operator

    -

  • REPART operator

    -

  • RESULTANT operator

    -

  • RHS operator

    -

  • ROOT\_OF operator

    -

  • SELECT operator

    -

  • SHOWRULES operator

    -

  • SOLVE operator

    -

  • SORT operator

    -

  • STRUCTR operator

    -

  • SUB operator

    -

  • SUM operator

    -

  • WS operator

    -

    •  - - -ALGEBRAIC -INDEX

    - - - -ALGEBRAIC _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The algebraic command changes REDUCE's mode of operation to -algebraic. When algebraic is used as an operator (with an -argument inside parentheses) that argument is evaluated in algebraic -mode, but REDUCE's mode is not changed. -

    -

    -

    -examples:

    -

    
-algebraic; 
-
-symbolic; 
-
-  NIL 
-
-
-algebraic(x**2); 
-
-   2
-  X  
-
-
-x**2; 
-
-    ***** The symbol X has no value.
-
-

    REDUCE's symbolic mode does not know about most algebraic commands -. -Error messages in this mode may also depend on the particular Lisp -used for the REDUCE implementation. -

    -

    -

    - - - -ANTISYMMETRIC -INDEX

    - - - -ANTISYMMETRIC _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -When an operator is declared antisymmetric, its arguments are -reordered to conform to the internal ordering of the system. If an odd -number of argument interchanges are required to do this ordering, -the sign of the expression is changed. -

    -

    -

    -syntax:

    -antisymmetric<identifier>{,<identifier>}* -

    -

    -

    -<identifier> is an identifier that has been declared as an operator. -

    -

    -

    -examples:

    -

    
-operator m,n; 
-
-antisymmetric m,n; 
-
-m(x,n(1,2)); 
-
-  - M( - N(2,1),X) 
-
-
-operator p; 
-
-antisymmetric p; 
-
-p(a,b,c); 
-
-  P(A,B,C) 
-
-
-p(b,a,c); 
-
-  - P(A,B,C)
-
-

    If <identifier> has not been declared an operator, the flag - -antisymmetric is still attached to it. When <identifier> is -subsequently used as an operator, the message Declare <identifier -> - operator? (Y or N) is printed. If the user replies y, the -antisymmetric property of the operator is used. -

    -

    -Note in the first example, identifiers are customarily ordered -alphabetically, while numbers are ordered from largest to smallest. -The operators may have any desired number of arguments (less than 128). -

    -

    -

    - - - -ARRAY -INDEX

    - - - -ARRAY _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The array declaration declares a list of identifiers to be of type -array, and sets all their entries to 0. -

    -syntax:

    -

    -

    -array<identifier>(<dimensions>) - {,<identifier>(<dimensions>)}* -

    -

    -

    -<identifier> may be any valid REDUCE identifier. If the identifier -was already an array, a warning message is given that the array has been -redefined. <dimensions> are of form - <integer>{,<integer>}*. -

    -

    -

    -examples:

    -

    
-array a(2,5),b(3,3,3),c(200); 
-
-array a(3,5); 
-
-  *** ARRAY A REDEFINED 
-
-
-a(3,4); 
-
-  0 
-
-
-length a; 
-
-  {4,6}
-
-

    Arrays are always global, even if defined inside a procedure or bl -ock -statement. Their status as an array remains until the variable is -reset by -clear. Arrays may not have the same names as operators -, -procedures or scalar variables. -

    -

    -Array elements are referred to by the usual notation: a(i,j) -returns the jth element of the ith row. The -assignment operator -:= is used to put values into the array. Arrays as a whole -cannot be subject to assignment by -let or := ; the -assignment operator := is only valid for individual elements. -

    -

    -When you use -let on an array element, the contents of that -element become the argument to let. Thus, if the element -contains a number or some other expression that is not a valid argument -for this command, you get an error message. If the element contains an -identifier, the identifier has the substitution rule attached to it -globally. The same behavior occurs with -clear. If the array -element contains an identifier or simple_expression, it is cleared. Do -<not> use clear to try to set an array element to 0. Because -of the side effects of either let or clear, it is unwise -to apply either of these to array elements. -

    -

    -Array indices always start with 0, so that the declaration array a(5) -sets aside 6 units of space, indexed from 0 through 5, and initializes -them to 0. The -length command returns a list of the true number of -elements in each dimension. -

    -

    -

    - - - -CLEAR -INDEX

    - - - -CLEAR _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The clear command is used to remove assignments or remove substitution - -rules from any expression. -

    -

    -

    -syntax:

    -clear<identifier>{,<identifier>}+ or -

    -

    -<let-type statement> clear <identifier> -

    -

    -

    -<identifier> can be any scalar, -matrix, -or -array variable or - -procedure name. <let-type statement> can be any -general -or specific -let statement (see below in Comments). -

    -

    -

    -examples:

    -

    
-array a(2,3); 
-
-a(2,2) := 15; 
-
-  A(2,2) := 15 
-
-
-clear a; 
-
-a(2,2); 
-
-  Declare A operator? (Y or N) 
-
-
-let x = y + z; 
-
-sin(x); 
-
-  SIN(Y + Z) 
-
-
-clear x; 
-
-sin(x); 
-
-  SIN(X) 
-
-
-let x**5 = 7; 
-
-clear x; 
-
-x**5; 
-
-  7 
-
-
-clear x**5; 
-
-x**5; 
-
-   5
-  X
-
-

    Although it is not a good idea, operators of the same name but tak -ing -different numbers of arguments can be defined. Using a clear statement - -on any of these operators clears every one with the same name, even if the -number of arguments is different. -

    -

    -The clear command is used to ``forget" matrices, arrays, operators - -and scalar variables, returning their identifiers to the pristine state -to be used for other purposes. When clear is applied to array -elements, the contents of the array element becomes the argument for -clear. Thus, you get an error message if the element contains a -number, or some other expression that is not a legal argument to -clear. If the element contains an identifier, it is cleared. -When clear is applied to matrix elements, an error message is returned -if the element evaluates to a number, otherwise there is no effect. Do - not try to use clear to set array or matrix elements to 0. -You will not be pleased with the results. -

    -

    -If you are trying to clear power or product substitution rules made with -either -let or -forall...let, you must -reproduce the rule, exactly as you typed it with the same arguments, up to -but not including the equal sign, using the word clear instead of -the word let. This is shown in the last example. Any other type of -let or forall...let substitution can be cleared -with just the variable or operator name. -match behaves the same as - -let in this situation. There is a more complicated exa -mple under - -forall. -

    -

    -

    -

    - - - -CLEARRULES -INDEX

    - - - -CLEARRULES _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -

    -syntax:

    -clearrules<list>{,<list>}+ -

    -

    -

    -The operator clearrules is used to remove previously defined - -rule lists from the system. <list> can be an exp -licit rule -list, or evaluate to a rule list. -

    -

    -

    -examples:

    -

    
-trig1 := {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2,
-          cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2,
-          sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2,
-          cos(~x)^2       => (1+cos(2*x))/2,
-          sin(~x)^2       => (1-cos(2*x))/2}$ 
-
-let trig1;
-cos(a)*cos(b); 
-
-  COS(A - B) + COS(A + B)
-  ----------------------- 
-             2
-
-
-clearrules trig1;
-cos(a)*cos(b); 
-
-  COS(A)*COS(B)
-
-

    - - -DEFINE -INDEX

    - - - -DEFINE _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The command define allows you to supply a new name for an identifier -or replace it by any valid REDUCE expression. -

    -

    -

    -syntax:

    -define<identifier>=<substitution> - {,<identifier>=<substitution>}* -

    -

    -

    -<identifier> is any valid REDUCE identifier, <substitution> can be a - -number, an identifier, an operator, a reserved word, or an expression. -

    -

    -

    -examples:

    -

    
-
-define is= :=, xx=y+z; 
-
-
-a is 10; 
-
-  A := 10 
-
-
-
-xx**2; 
-
-   2             2
-  Y   + 2*Y*Z + Z  
-
-
-
-xx := 10; 
-
-  Y + Z := 10
-
-

    The renaming is done at the input level, and therefore takes prece -dence -over any other replacement or substitution declared for the same identifier. -It remains in effect until the end of the REDUCE session. Be careful with -it, since you cannot easily undo it without ending the session. -

    -

    -

    - - - -DEPEND -INDEX

    - - - -DEPEND _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -dependdeclares that its first argument depends on the rest of its -arguments. -

    -

    -

    -syntax:

    -depend<kernel>{,<kernel>}+ -

    -

    -

    -<kernel> must be a legal variable name or a prefix operator (see - -kernel). -

    -

    -

    -examples:

    -

    
-
-depend y,x; 
-
-
-df(y**2,x); 
-
-  2*DF(Y,X)*Y 
-
-
-
-depend z,cos(x),y; 
-
-
-df(sin(z),cos(x)); 
-
-  COS(Z)*DF(Z,COS(X)) 
-
-
-
-df(z**2,x); 
-
-  2*DF(Z,X)*Z 
-
-
-
-nodepend z,y; 
-
-
-df(z**2,x); 
-
-  2*DF(Z,X)*Z 
-
-
-
-cc := df(y**2,x); 
-
-  CC := 2*DF(Y,X)*Y 
-
-
-
-y := tan x; 
-
-  Y := TAN(X); 
-
-
-
-cc; 
-
-                  2
-  2*TAN(X)*(TAN(X)   + 1)
-
-

    Dependencies can be removed by using the declaration -nodepend. -The differentiation operator uses this information, as shown in the -examples above. Linear operators also use knowledge of dependencies -(see -linear). Note that dependencies can be nested: Having - -declared y to depend on x, and z -to depend on y, we -see that the chain rule was applied to the derivative of a function of -z with respect to x. If the explicit function of the -dependency is later entered into the system, terms with DF(Y,X), -for example, are expanded when they are displayed again, as shown in the -last example. The boolean operator -freeof allows you to -check the dependency between two algebraic objects. -

    -

    -

    - - - -EVEN -INDEX

    - - - -EVEN _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -syntax:

    -

    -

    -even<identifier>{,<identifier>}* -

    -

    -

    -This declaration is used to declare an operator even in its first -argument. Expressions involving an operator declared in this manner are -transformed if the first argument contains a minus sign. Any other -arguments are not affected. -

    -examples:

    -

    
-        even f; 
-
-        f(-a) 
-
-  F(A) 
-
-
-        f(-a,-b) 
-
-  F(A,-B)
-
-

    -

    - - - -FACTOR_declaration -INDEX

    - - - -FACTOR _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -When a kernel is declared by factor, all terms involving fixed -powers of that kernel are printed as a product of the fixed powers and -the rest of the terms. -

    -syntax:

    -

    -

    -factor<kernel> {,<kernel>}* -

    -

    -

    -<kernel> must be a -kernel or a -list of -kernels. -

    -

    -

    -examples:

    -

    
-a := (x + y + z)**2; 
-
-        2                    2            2
-  A := X  + 2*X*Y + 2*X*Z + Y  + 2*Y*Z + Z  
-
-
-factor y; 
-
-a; 
-
-   2                  2            2
-  Y  + 2*Y*(X + Z) + X  + 2*X*Z + Z  
-
-
-factor sin(x); 
-
-c := df(sin(x)**4*x**2*z,x); 
-
-               4               3         2
-  C := 2*SIN(X) *X*Z + 4*SIN(X) *COS(X)*X *Z 
-
-
-remfac sin(x); 
-
-c; 
-
-          3
-  2*SIN(X) *X*Z*(2*COS(X)*X + SIN(X))
-
-

    Use the factor declaration to display variables of intere -st so that -you can see their powers more clearly, as shown in the example. Remove -this special treatment with the declaration -remfac. The -factor declaration is only effective when the switch -pri -is on. -

    -

    -The factor declaration is not a factoring command; to factor -expressions use the -factor switch or the -factorize command. -

    -

    -The factor declaration is helpful in such cases as Taylor polynomials -where the explicit powers of the variable are expected at the top level, not -buried in various factored forms. -

    -

    -Note that factor does not affect the order of its arguments. You -should also use -order if this is important. -

    -

    -

    - - - -FORALL -INDEX

    - - - -FORALL _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The forall or (preferably) for all command is used as a -modifier for -let statements, indicating the universal applicability - -of the rule, with possible qualifications. -

    -syntax:

    -

    -

    -for all<identifier>{,<identifier>}* let -<let statement> -

    -

    -or -

    -

    -for all<identifier>{,<identifier>}* - such that <condition> let <let statement> -

    -

    -

    -<identifier> may be any valid REDUCE identifier, <let statement> -can be an operator, a product or power, or a group or block statement. -<condition> must be a logical or comparison operator returning true or -false. -

    -

    -

    -examples:

    -

    
-for all x let f(x) = sin(x**2);
- 
-
-  Declare F operator ? (Y or N) 
-
-
-y 
-
-f(a); 
-
-       2
-  SIN(A ) 
-
-
-operator pos; 
-
-for all x such that x>=0 let pos(x) = sqrt(x + 1); 
-
-pos(5); 
-
-  SQRT(6) 
-
-
-pos(-5); 
-
-  POS(-5) 
-
-
-clear pos; 
-
-pos(5); 
-
-  Declare POS operator ? (Y or N) 
-
-
-for all a such that numberp a let x**a = 1; 
-
-x**4; 
-
-  1 
-
-
-clear x**a; 
-
-  *** X**A not found 
-
-
-for all a  clear x**a; 
-
-x**4; 
-
-  1 
-
-
-for all a such that numberp a clear x**a; 
-
-x**4; 
-
-   4
-  X
-
-

    Substitution rules defined by for all or for -all...such that commands that involve products or powers are -cleared by reproducing the command, with exactly the same variable names -used, up to but not including the equal sign, with -clear -replacing let, as shown in the last example. Other substitutions -involving variables or operator names can be cleared with just the name, -like any other variable. -

    -

    -The -match command can also be used in product and power su -bstitutions. -The syntax of its use and clearing is exactly like let. A match - -substitution only replaces the term if it is exactly like the pattern, for -example match x**5 = 1 replaces only terms of x**5 and not -terms of higher powers. -

    -

    -It is easier to declare your potential operator before defining the -for all rule, since the system will ask you to declare it an -operator anyway. Names of declared arrays or matrices or scalar -variables are invalid as operator names, to avoid ambiguity. Either -for all...let statements or procedures are often used to defin -e -operators. One difference is that procedures implement ``call by value" -meaning that assignments involving their formal parameters do not change -the calling variables that replace them. If you use assignment statements -on the formal parameters in a for all...let statement, the -effects are seen in the calling variables. Be careful not to redefine a -system operator unless you mean it: the statement for all x let -sin(x)=0; has exactly that effect, and the usual definition for sin(x) has - -been lost for the remainder of the REDUCE session.

    -

    - -

    -

    - - - -INFIX -INDEX

    - - - -INFIX _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -infixdeclares identifiers to be infix operators. -

    -syntax:

    -

    -

    -infix<identifier>{,<identifier>}* -

    -

    -

    -<identifier> can be any valid REDUCE identifier, which has not already -been declared an operator, array or matrix, and is not reserved by the -system. -

    -

    -

    -examples:

    -

    
-infix aa; 
-
-for all x,y let aa(x,y) = cos(x)*cos(y) - sin(x)*sin(y); 
-
-x aa y; 
-
-  COS(X)*COS(Y) - SIN(X)*SIN(Y) 
-
-
-pi/3 aa pi/2; 
-
-    SQRT(3)
-  - ------- 
-       2
-
-
-aa(pi,pi); 
-
-  1
-
-

    A -let statement must be used to attach functionality to - -the operator. Note that the operator is defined in prefix form in -the let statement. -After its definition, the operator may be used in either prefix or infix -mode. The above operator aa finds the cosine of the sum -of two angles by the formula -

    -

    -cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y). -

    -

    -Precedence may be attached to infix operators with the - -precedence declaration. -

    -

    -User-defined infix operators may be used in prefix form. If they are used -in infix form, a space must be left on each side of the operator to avoid -ambiguity. Infix operators are always binary. -

    -

    -

    - - - -INTEGER -INDEX

    - - - -INTEGER _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The integer declaration must be made immediately after a - -begin (or other variable declaration such as -real -and -scalar) and declares local integer variables. They are - -initialized to 0. -

    -syntax:

    -

    -

    -integer<identifier>{,<identifier>}* -

    -

    -

    -<identifier> may be any valid REDUCE identifier, except -t or nil. -

    -

    -Integer variables remain local, and do not share values with variables of -the same name outside the -begin...end block. When the -block is finished, the variables are removed. You may use the words - -real or -scalar in the place of integer. -integer does not indicate typechecking by the -current REDUCE; it is only for your own information. Declaration -statements must immediately follow the begin, without a semicolon -between begin and the first variable declaration. -

    -

    -Any variables used inside begin...end blocks that were not -declared scalar, real or integer are global, and any - -change made to them inside the block affects their global value. Any - -array or -matrix declared inside a block is always global. -

    -

    -

    - - - -KORDER -INDEX

    - - - -KORDER _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -The korder declaration changes the internal canonical ordering of -kernels. -

    -syntax:

    -

    -

    -korder<kernel>{,<kernel>}* -

    -

    -

    -<kernel> must be a REDUCE -kernel or a -list of -kernels. -

    -

    -The declaration korder changes the internal ordering, but not the print - -ordering, so the effects cannot be seen on output. However, in some -calculations, the order of the variables can have significant effects on the -time and space demands of a calculation. If you are doing a demanding -calculation with several kernels, you can experiment with changing the -canonical ordering to improve behavior. -

    -

    -The first kernel in the argument list is given the highest priority, the -second gets the next highest, and so on. Kernels not named in a -korder ordering otherwise. A new korder declaration replaces -the previous one. To return to canonical ordering, use the command -korder nil. -

    -

    -To change the print ordering, use the declaration -order. -

    -

    -

    - - - -LET -INDEX

    - - - -LET _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The let command defines general or specific substitution rules. -

    -syntax:

    -

    -

    -let<identifier> = <expression>{,<identifier> - -= <expression>}* -

    -

    -

    -<identifier> can be any valid REDUCE identifier except an array, and in -some cases can be an expression; <expression> can be any valid REDUCE -expression. -

    -

    -

    -examples:

    -

    
-let a = sin(x); 
-
-b := a; 
-
-  B := SIN X; 
-
-
-let c = a; 
-
-exp(a); 
-
-   SIN(X)
-  E       
-
-
-a := x**2; 
-
-        2
-  A := X  
-
-
-exp(a); 
-
-    2
-   X
-  E   
-
-
-exp(b); 
-
-   SIN(X)
-  E       
-
-
-exp(c); 
-
-    2
-   X
-  E   
-
-
-let m + n = p; 
-
-(m + n)**5; 
-
-   5
-  P  
-
-
-operator h; 
-
-let h(u,v) = u - v; 
-
-h(u,v); 
-
-  U - V 
-
-
-h(x,y); 
-
-  H(X,Y) 
-
-
-array q(10); 
-
-let q(1) = 15; 
-
-  ***** Substitution for 0 not allowed
-
-

    The let command is also used to activate a rule sets -. -

    -syntax:

    -

    -

    -let<list>{,<list>}+ -

    -

    -

    -<list> can be an explicit -rule list, or evaluate -to a rule list. -

    -

    -

    -examples:

    -

    
-trig1 := {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2,
-          cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2,
-          sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2,
-          cos(~x)^2       => (1+cos(2*x))/2,
-          sin(~x)^2       => (1-cos(2*x))/2}$ 
-
-let trig1;
-cos(a)*cos(b); 
-
-  COS(A - B) + COS(A + B)
-  ------------------------
-             2
-
-

    A let command returns no value, though the substitution r -ule is -entered. Assignment rules made by -assign and let -rules are at the -same level, and cancel each other. There is a difference in their -operation, however, as shown in the first example: a let assignment -tracks the changes in what it is assigned to, while a := assignment -is fixed at the value it originally had. -

    -

    -The use of expressions as left-hand sides of let statements is a -little complicated. The rules of operation are: -

    -

    - _ _ _ (i) -Expressions of the form A*B = C do not change A, B or C, but set A*B to C. -

    -

    - _ _ _ (ii) -Expressions of the form A+B = C substitute C - B for A, but do not change -B or C. -

    -

    - _ _ _ (iii) -Expressions of the form A-B = C substitute B + C for A, but do not change -B or C. -

    -

    - _ _ _ (iv) -Expressions of the form A/B = C substitute B*C for A, but do not change B or -C. -

    -

    - _ _ _ (v) -Expressions of the form A**N = C substitute C for A**N in every expression of -a power of A to N or greater. An asymptotic command such as A**N = 0 sets -all terms involving A to powers greater than or equal to N to 0. Finite -fields may be generated by requiring modular arithmetic (the -modular -switch) and defining the primitive polynomial via a let statement. -

    -

    -

    -letsubstitutions involving expressions are cleared by using -the -clear command with exactly the same expression. -

    -

    -Note when a simple let statement is used to assign functionality to an - -operator, it is valid only for the exact identifiers used. For the use of the -let command to attach more general functionality to an operator, -see -forall. -

    -

    -Arrays as a whole cannot be arguments to let statements, but -matrices as a whole can be legal arguments, provided both arguments are -matrices. However, it is important to note that the two matrices are then -linked. Any change to an element of one matrix changes the corresponding -value in the other. Unless you want this behavior, you should not use -let for matrices. The assignment operator -assign can be used -for non-tracking assignments, avoiding the side effects. Matrices are -redimensioned as needed in let statements. -

    -

    -When array or matrix elements are used as the left-hand side of let -statements, the contents of that element is used as the argument. When the -contents is a number or some other expression that is not a valid left-hand -side for let, you get an error message. If the contents is an -identifier or simple expression, the let rule is globally attached -to that identifier, and is in effect not only inside the array or matrix, -but everywhere. Because of such unwanted side effects, you should not -use let with array or matrix elements. The assignment operator -:= can be used to put values into array or matrix elements without -the side effects. -

    -

    -Local variables declared inside begin...end blocks cannot -be used as the left-hand side of let statements. However, - -begin...end blocks themselves can be used as -the -right-hand side of let statements. The construction: -

    -syntax:

    -

    -

    -for all<vars> - let<operator>(<vars>)=<block> -

    -

    -

    -is an alternative to the -

    -syntax:

    -

    -

    -procedure<name>(<vars>);<block> -

    -

    -

    -construction. One important difference between the two constructions is that -the <vars> as formal parameters to a procedure have their global values -protected against change by the procedure, while the <vars> of a -let statement are changed globally by its actions. -

    -

    -Be careful in using a construction such as let x = x + 1 except inside - -a controlled loop statement. The process of resubstitution continues until -a stack overflow message is given. -

    -

    -The let statement may be used to make global changes to variables from - -inside procedures. If x is a formal parameter to a procedure, the -command let x = ... makes the change to the calling variable. -For example, if a procedure was defined by -

    
-        procedure f(x,y);
-        let x = 15;
-

    -

    -and the procedure was called as -

    
-        f(a,b);
-

    -

    -awould have its value changed to 15. Be careful when using let - -statements inside procedures to avoid unwanted side effects. -

    -

    -It is also important to be careful when replacing let statements with -other let statements. The overlapping of these substitutions can be -unpredictable. Ordinarily the latest-entered rule is the first to be applied. -Sometimes the previous rule is superseded completely; other times it stays -around as a special case. The order of entering a set of related let -expressions is very important to their eventual behavior. The best -approach is to assume that the rules will be applied in an arbitrary order. -

    -

    -

    - - - -LINEAR -INDEX

    - - - -LINEAR _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -An operator can be declared linear in its first argument over powers of -its second argument by the declaration linear. -

    -syntax:

    -

    -

    -linear<operator>{,<operator>}* -

    -

    -

    -<operator> must have been declared to be an operator. Be careful not -to use a system operator name, because this command may change its definition. -The operator being declared must have at least two arguments, and the -second one must be a -kernel. -

    -

    -

    -examples:

    -

    
-operator f; 
-
-linear f; 
-
-f(0,x); 
-
-  0 
-
-
-f(-y,x); 
-
-  - F(1,X)*Y 
-
-
-f(y+z,x); 
-
-  F(1,X)*(Y + Z) 
-
-
-f(y*z,x); 
-
-  F(1,X)*Y*Z 
-
-
-depend z,x; 
-
-f(y*z,x); 
-
-  F(Z,X)*Y 
-
-
-f(y/z,x); 
-
-    1
-  F(-,X)*Y 
-    Z
-
-
-depend y,x; 
-
-f(y/z,x); 
-
-    Y
-  F(-,X) 
-    Z
-
-
-nodepend z,x; 
-
-f(y/z,x); 
-
-  F(Y,X)
-  ------ 
-    Z
-
-
-f(2*e**sin(x),x); 
-
-       SIN(X)
-  2*F(E      ,X)
-
-

    Even when the operator has not had its functionality attached, it -exhibits -linear properties as shown in the examples. Notice the difference when -dependencies are added. Dependencies are also in effect when the operator's -first argument contains its second, as in the last line above. -

    -

    -For a fully-developed example of the use of linear operators, refer to the -article in the <Journal of Computational Physics>, Vol. 14 (1974), pp. -301-317, ``Analytic Computation of Some Integrals in Fourth Order Quantum -Electrodynamics," by J.A. Fox and A.C. Hearn. The article includes the -complete listing of REDUCE procedures used for this work. -

    -

    -

    - - - -LINELENGTH -INDEX

    - - - -LINELENGTH _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -The linelength declaration sets the length of the output line. Default - -is 80. -

    -syntax:

    -

    -

    -linelength<expression> -

    -

    -

    -To change the linelength, -<expression> must evaluate to a positive integer less than 128 -(although this varies from system to system), and should not be less than -20 or so for proper operation. -

    -

    -linelengthreturns the previous linelength. If you want the current -linelength value, but not change it, say linelength nil. -

    -

    -

    - - - -LISP -INDEX

    - - - -LISP _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The lisp command changes REDUCE's mode of operation to symbolic. When -lisp is followed by an expression, that expression is evaluated in -symbolic mode, but REDUCE's mode is not changed. This command is -equivalent to -symbolic. -

    -

    -

    -examples:

    -

    
-lisp; 
-
-  NIL 
-
-
-car '(a b c d e); 
-
-  A  
-
-
-algebraic; 
-
-c := (lisp car '(first second))**2; 
-
- 
-
-            2
-  C := FIRST
-
-

    - - -LISTARGP -INDEX

    - - - -LISTARGP _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -

    -syntax:

    -listargp<operator>{,<operator>}* -

    -

    -

    -If an operator other than those specifically defined for lists is given a -single argument that is a -list, then the result of this -operation will be a list in which that operator is applied to each element -of the list. -This process can be inhibited for a specific operator, or list of operators, -by using the declaration listargp. -

    -

    -

    -examples:

    -

    
-log {a,b,c}; 
-
-  LOG(A),LOG(B),LOG(C) 
-
-
-listargp log; 
-
-log {a,b,c}; 
-
-  LOG(A,B,C)
-
-

    It is possible to inhibit such distribution globally by turning on - the -switch -listargs. In addition, if an operator has more than on -e -argument, no such distribution occurs, so listargp has no effect. -

    -

    -

    - - - -NODEPEND -INDEX

    - - - -NODEPEND _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -The nodepend declaration removes the dependency declared with - -depend. -

    -syntax:

    -

    -

    -nodepend<dep-kernel>{,<kernel>}+ -

    -

    -

    -

    -<dep-kernel> must be a kernel that has had a dependency declared upon the - -one or more other kernels that are its other arguments. -

    -

    -

    -examples:

    -

    
-depend y,x,z; 
-
-df(sin y,x); 
-
-  COS(Y)*DF(Y,X) 
-
-
-df(sin y,x,z); 
-
-  COS(Y)*DF(Y,X,Z) - DF(Y,X)*DF(Y,Z)*SIN(Y) 
-
-
-nodepend y,z; 
-
-df(sin y,x); 
-
-  COS(Y)*DF(Y,X) 
-
-
-df(sin y,x,z); 
-
-  0
-
-

    A warning message is printed if the dependency had not been declar -ed by -depend. -

    -

    -

    - - - -MATCH -INDEX

    - - - -MATCH _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The match command is similar to the -let command, except -that it matches only explicit powers in substitution. -

    -syntax:

    -

    -

    -match<expr> = <expression>{,<expr> - =<expression>}* -

    -

    -

    -<expr> is generally a term involving powers, and is limited by -the rules for the -let command. <expression> may be -any valid REDUCE scalar expression. -

    -

    -

    -examples:

    -

    
-match c**2*a**2 = d;
-(a+c)**4; 
-
-   4       3          3    4
-  A   + 4*A *C + 4*A*C  + C  + 6*D 
-
-
-match a+b = c; 
-
-a + 2*b; 
-
-  B + C 
-
-
-(a + b + c)**2; 
-
-   2     2               2
-  A   - B   + 2*B*C + 3*C  
-
-
-clear a+b; 
-
-(a + b + c)**2; 
-
-    2                    2            2
-  A   + 2*A*B + 2*A*C + B  + 2*B*C + C  
-
-
-let p*r = s; 
-
-match p*q = ss; 
-
-(a + p*r)**2; 
-
-   2            2
-  A  + 2*A*S + S  
-
-
-(a + p*q)**2; 
-
-   2              2  2
-  A   + 2*A*SS + P *Q
-
-

    Note in the last example that a + b has been explicitly m -atched -after the squaring was done, replacing each single power of a by -c - b. This kind of substitution, although following the rules, is -confusing and could lead to unrecognizable results. It is better to use -match with explicit powers or products only. match should -not be used inside procedures for the same reasons that let should -not be. -

    -

    -Unlike -let substitutions, match substitutions are ex -ecuted -after all other operations are complete. The last example shows the -difference. match commands can be cleared by using -clear, -with exactly the expression that the original match took. -match commands can also be done more generally with for all -or -forall...such that commands. -

    -

    -

    - - - -NONCOM -INDEX

    - - - -NONCOM _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -noncomdeclares that already-declared operators are noncommutative -under multiplication. -

    -syntax:

    -

    -

    -noncom<operator>{,<operator>}* -

    -

    -

    -<operator> must have been declared an -operator, or a warning -message is given. -

    -

    -

    -examples:

    -

    
-operator f,h; 
-
-noncom f; 
-
-f(a)*f(b) - f(b)*f(a); 
-
-  F(A)*F(B) - F(B)*F(A) 
-
-
-h(a)*h(b) - h(b)*h(a); 
-
-  0 
-
-
-operator comm; 
-
-for all x,y such that x neq y and ordp(x,y)
-        let f(x)*f(y) = f(y)*f(x) + comm(x,y);
-
-
-f(1)*f(2); 
-
-  F(1)*F(2) 
-
-
-f(2)*f(1); 
-
-  COMM(2,1) + F(1)*F(2)
-
-

    The last example introduces the commutator of f(x) and f(y) -for all x and y. The equality check is to prevent an infinite loop. The -operator f can have other functionality attached to it if desired, or it -can remain an indeterminate operator. -

    -

    -

    - - - -NONZERO -INDEX

    - - - -NONZERO _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -

    -syntax:

    -nonzero<identifier>{,<identifier>}* -

    -

    -

    -If an -operator f is declared -odd, then f(0) -is replaced by zero unless f is also declared non zero by the -declaration nonzero. -

    -examples:

    -

    
-        odd f; 
-
-        f(0) 
-
-  0 
-
-
-        nonzero f; 
-
-        f(0) 
-
-  F(0)
-
-

    -

    - - - -ODD -INDEX

    - - - -ODD _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -

    -syntax:

    -odd<identifier>{,<identifier>}* -

    -

    -

    -This declaration is used to declare an operator odd in its first -argument. Expressions involving an operator declared in this manner are -transformed if the first argument contains a minus sign. Any other -arguments are not affected. -

    -examples:

    -

    
-        odd f; 
-
-        f(-a) 
-
-  -F(A) 
-
-
-        f(-a,-b) 
-
-  -F(A,-B) 
-
-
-        f(a,-b) 
-
-  F(A,-B)
-
-

    -

    -If say f is declared odd, then f(0) is replaced by zero -unless f is also declared non zero by the declaration - -nonzero. -

    -

    -

    - - - -OFF -INDEX

    - - - -OFF _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The off command is used to turn switches off. -

    -syntax:

    -

    -

    -off<switch>{,<switch>}* -

    -

    -

    -<switch> can be any switch name. There is no problem if the -switch is already off. If the switch name is mistyped, an error message is -given. -

    -

    - - - -ON -INDEX

    - - - -ON _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The on command is used to turn switches on. -

    -syntax:

    -

    -

    -on<switch>{,<switch>}* -

    -

    -

    -<switch> can be any switch name. There is no problem if the -switch is already on. If the switch name is mistyped, an error message is -given. -

    -

    - - - -OPERATOR -INDEX

    - - - -OPERATOR _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -Use the operator declaration to declare your own operators. -

    -syntax:

    -

    -

    -operator<identifier>{,<identifier>}* -

    -

    -

    -<identifier> can be any valid REDUCE identifier, which is not the name -of a -matrix, -array, scalar variable or previously-defined -operator. -

    -

    -

    -examples:

    -

    
-operator dis,fac; 
-
-let dis(~x,~y) = sqrt(x^2 + y^2); 
-
-dis(1,2); 
-
-  SQRT(5) 
-
-
-dis(a,10); 
-
-        2
-  SQRT(A  + 100) 
-
-
-on rounded; 
-
-dis(1.5,7.2); 
-
-  7.35459040329
-
-
-let fac(~n) = if n=0 then 1
-               else if not(fixp n and n>0)
-                then rederr "choose non-negative integer"
-               else for i := 1:n product i;
- 
-
-fac(5); 
-
-  120 
-
-
-fac(-2); 
-
-  ***** choose non-negative integer
-
-

    The first operator is the Euclidean distance metric, the distance -of point -(x,y) from the origin. The second operator is the factorial. -

    -

    -Operators can have various properties assigned to them; they can be -declared -infix, -linear, -symmetric, - -antisymmetric, or -noncommutative. -The default operator is prefix, nonlinear, and commutative. -Precedence can also be assigned to operators using the declaration - -precedence. -

    -

    -Functionality is assigned to an operator by a -let statement or -a -forall...let statement, -(or possibly by a procedure with the name -of the operator). Be careful not to redefine a system operator by -accident. REDUCE permits you to redefine system operators, giving you a -warning message that the operator was already defined. This flexibility -allows you to add mathematical rules that do what you want them to do, but -can produce odd or erroneous behavior if you are not careful. -

    -

    -You can declare operators from inside -procedures, as long as they -are not local variables. Operators defined inside procedures are global. -A formal parameter may be declared as an operator, and has the effect of -declaring the calling variable as the operator. -

    -

    -

    - - - -ORDER -INDEX

    - - - -ORDER _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -The order declaration changes the order of precedence of kernels for -display purposes only. -

    -syntax:

    -

    -

    -order<identifier>{,<identifier>}* -

    -

    -

    -<kernel> must be a valid -kernel or -operator name -complete with argument or a -list of such objects. -

    -

    -

    -examples:

    -

    
-x + y + z + cos(a); 
-
-  COS(A) + X + Y + Z 
-
-
-order z,y,x,cos(a); 
-
-x + y + z + cos(a); 
-
-  Z + Y + X + COS(A) 
-
-
-(x + y)**2; 
-
-   2            2
-  Y  + 2*Y*X + X  
-
-
-order nil; 
-
-(z + cos(z))**2; 
-
-        2                 2
-  COS(Z)  + 2*COS(Z)*Z + Z
-
-

    orderaffects the printing order of the identifiers only; -internal -order is unchanged. Change internal order of evaluation with the -declaration -korder. You can use order to feature variable -s -or functions you are particularly interested in. -

    -

    -Declarations made with order are cumulative: kernels in new order -declarations are ordered behind those in previous declarations, and -previous declarations retain their relative order. Of course, specific -kernels named in new declarations are removed from previous ones and given -the new priority. Return to the standard canonical printing order with the -statement order nil. -

    -

    -The print order specified by order commands is not in effect if the -switch -pri is off. -

    -

    -

    - - - -PRECEDENCE -INDEX

    - - - -PRECEDENCE _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -The precedence declaration attaches a precedence to an infix operator. - -

    -syntax:

    -

    -

    -precedence<operator>,<known\_operator> -

    -

    -

    -<operator> should have been declared an operator but may be a REDUCE -identifier that is not already an operator, array, or matrix. -<known\_operator> must be a system infix operator or have had its -precedence already declared. -

    -

    -

    -examples:

    -

    
-operator f,h; 
-
-precedence f,+; 
-
-precedence h,*; 
-
-a + f(1,2)*c; 
-
-  (1 F 2)*C + A 
-
-
-a + h(1,2)*c; 
-
-  1 H 2*C + A 
-
-
-a*1 f 2*c; 
-
-  A F 2*C 
-
-
-a*1 h 2*c; 
-
-  1 H 2*A*C
-
-

    The operator whose precedence is being declared is inserted into t -he infix -operator precedence list at the next higher place than <known\_operator>. - -

    -

    -Attaching a precedence to an operator has the side effect of declaring the -operator to be infix. If the identifier argument for precedence has -not been declared to be an operator, an attempt to use it causes an error -message. After declaring it to be an operator, it becomes an infix operator -with the precedence previously given. Infix operators may be used in prefix -form; if they are used in infix form, a space must be left on each side of -the operator to avoid ambiguity. Declared infix operators are always binary. -

    -

    -To see the infix operator precedence list, enter symbolic mode and type -preclis!*;. The lowest precedence operator is listed first. -

    -

    -All prefix operators have precedence higher than infix operators. -

    -

    -

    - - - -PRECISION -INDEX

    - - - -PRECISION _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -The precision declaration sets the number of decimal places used when - -rounded is on. Default is system dependent, and normal -ly about 12. -

    -syntax:

    -

    -

    -precision(<integer>) or precision <integer> -

    -

    -

    -<integer> must be a positive integer. When <integer> is 0, the -current precision is displayed, but not changed. There is no upper limit, -but precision of greater than several hundred causes unpleasantly slow -operation on numeric calculations. -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-7/9; 
-
-  0.777777777778 
-
-
-precision 20; 
-
-  20 
-
-
-7/9; 
-
-  0.77777777777777777778 
-
-
-sin(pi/4); 
-
-  0.7071067811865475244
-
-

    Trailing zeroes are dropped, so sometimes fewer than 20 decimal pl -aces are -printed as in the last example. Turn on the switch -fullprec if -you want to print all significant digits. The -rounded mode -carries calculations to two more places than given by precision, and -rounds off. -

    -

    -

    - - - -PRINT_PRECISION -INDEX

    - - - -PRINT\_PRECISION _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -

    -syntax:

    -print_precision(<integer>) - or print_precision <integer> -

    -

    -

    -In -rounded mode, numbers are normally printed to the spec -ified -precision. If the user wishes to print such numbers with less precision, -the printing precision can be set by the declaration print_precision. -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-1/3; 
-
-  0.333333333333 
-
-
-print_precision 5; 
-
-1/3 
-
-  0.33333
-
-

    - - -REAL -INDEX

    - - - -REAL _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The real declaration must be made immediately after a - -begin (or other variable declaration such as -integer -and -scalar) and declares local integer variables. They are - -initialized to zero. -

    -syntax:

    -

    -

    -real<identifier>{,<identifier>}* -

    -

    -

    -<identifier> may be any valid REDUCE identifier, except -t or nil. -

    -

    -Real variables remain local, and do not share values with variables of the -same name outside the -begin...end block. When the -block is finished, the variables are removed. You may use the words - -integer or -scalar in the place of real. -real does not indicate typechecking by the current REDUCE; it is -only for your own information. Declaration statements must immediately -follow the begin, without a semicolon between begin and the -first variable declaration. -

    -

    -Any variables used inside a begin...end -block -that were not declared scalar, real or integer are -global, and any change made to them inside the block affects their global -value. Any -array or -matrix declared inside a block is always -global. -

    -

    -

    - - - -REMFAC -INDEX

    - - - -REMFAC _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -The remfac declaration removes the special factoring treatment of its -arguments that was declared with -factor. -

    -syntax:

    -

    -

    -remfac<kernel>{,<kernel>}+ -

    -

    -

    -<kernel> must be a -kernel or -operator name that -was declared as special with the -factor declaration. -

    -

    - - - -SCALAR -INDEX

    - - - -SCALAR _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The scalar declaration must be made immediately after a - -begin (or other variable declaration such as -integer -and -real) and declares local scalar variables. They are -initialized to 0. -

    -syntax:

    -

    -

    -scalar<identifier>{,<identifier>}* -

    -

    -

    -<identifier> may be any valid REDUCE identifier, except t or -nil. -

    -

    -Scalar variables remain local, and do not share values with variables of -the same name outside the -begin...end -block. -When the block is finished, the variables are removed. You may use the -words -real or -integer in the place of scalar. -real and integer do not indicate typechecking by the current -REDUCE; they are only for your own information. Declaration statements -must immediately follow the begin, without a semicolon between -begin and the first variable declaration. -

    -

    -Any variables used inside begin...end blocks that were not -declared scalar, real or integer are global, and any - -change made to them inside the block affects their global value. Arrays -declared inside a block are always global. -

    -

    -

    - - - -SCIENTIFIC_NOTATION -INDEX

    - - - -SCIENTIFIC\_NOTATION _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -

    -syntax:

    -scientific_notation(<m>) or -scientific_notation({<m>,<n>}) -

    -

    -

    -<m> and <n> are positive integers. -scientific_notation controls the output format of floating point -numbers. At the default settings, any number with five or less digits -before the decimal point is printed in a fixed-point notation, e.g., -12345.6. Numbers with more than five digits are printed in scientific -notation, e.g., 1.234567E+5. Similarly, by default, any number with -eleven or more zeros after the decimal point is printed in scientific -notation. -

    -

    -When scientific_notation is called with the numerical argument - m a number with more than m digits before the decimal point, -or m or more zeros after the decimal point, is printed in scientific -notation. When scientific_notation is called with a list -{<m>,<n>}, a number with more than m digits before the -decimal point, or n or more zeros after the decimal point is -printed in scientific notation. -

    -

    -

    -examples:

    -

    
-
-on rounded;
-
-
-12345.6;
-
-  12345.6
-
-
-
-123456.5;
-
-  1.234565e+5
-
-
-
-0.00000000000000012;
-
-  1.2e-16
-
-
-
-scientific_notation 20;
-
-  5,11
-
-
-
-5: 123456.7;
-
-  123456.7
-
-
-
-0.00000000000000012;
-
-  0.00000000000000012
-
-

    - - -SHARE -INDEX

    - - - -SHARE _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The share declaration allows access to its arguments by both -algebraic and symbolic modes. -

    -syntax:

    -

    -

    -share<identifier>{,<identifier>}* -

    -

    -

    -<identifier> can be any valid REDUCE identifier. -

    -

    -Programming in -symbolic as well as algebraic mode allows -you a wider range -of techniques than just algebraic mode alone. Expressions do not cross the -boundary since they have different representations, unless the share -declaration is used. For more information on using symbolic mode, see -the <REDUCE User's Manual>, and the <Standard Lisp Report>. -

    -

    -You should be aware that a previously-declared array is destroyed by the -share declaration. Scalar variables retain their values. You can -share a declared -matrix that has not yet -been dimensioned so that it can be -used by both modes. Values that are later put into the matrix are -accessible from symbolic mode too, but not by the usual matrix reference -mechanism. In symbolic mode, a matrix is stored as a list whose first -element is -MAT, and whose next elements are the rows of the matri -x -stored as lists of the individual elements. Access in symbolic mode is by -the operators -first, -second, -third and - -rest. -

    -

    -

    - - - -SYMBOLIC -INDEX

    - - - -SYMBOLIC _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The symbolic command changes REDUCE's mode of operation to symbolic. -When symbolic is followed by an expression, that expression is -evaluated in symbolic mode, but REDUCE's mode is not changed. It is -equivalent to the -lisp command. -

    -

    -

    -examples:

    -

    
-symbolic; 
-
-  NIL 
-
-
-cdr '(a b c); 
-
-  (B C) 
-
-
-algebraic; 
-
-x + symbolic car '(y z); 
-
-  X + Y
-
-

    - - -SYMMETRIC -INDEX

    - - - -SYMMETRIC _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -When an operator is declared symmetric, its arguments are reordered -to conform to the internal ordering of the system. -

    -syntax:

    -

    -

    -symmetric<identifier>{,<identifier>}* -

    -

    -

    -<identifier> is an identifier that has been declared an operator. -

    -

    -

    -examples:

    -

    
-operator m,n; 
-
-symmetric m,n; 
-
-m(y,a,sin(x)); 
-
-  M(SIN(X),A,Y) 
-
-
-n(z,m(b,a,q)); 
-
-  N(M(A,B,Q),Z)
-
-

    If <identifier> has not been declared to be an operator, the - flag -symmetric is still attached to it. When <identifier> is -subsequently used as an operator, the message Declare<identifier> - - operator ? (Y or N) is printed. If the user replies y, the -symmetric property of the operator is used. -

    -

    -

    - - - -TR -INDEX

    - - - -TR _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -The tr declaration is used to trace system or user-written procedures. - -It is only useful to those with a good knowledge of both Lisp and the -internal formats used by REDUCE. -

    -

    -

    -syntax:

    -tr<name>{,<name>}* -

    -

    -

    -<name> is the name of a REDUCE system procedure or one of your own -procedures. -

    -

    -

    -examples:

    -

    The system procedure prepsq is traced, - which prepares REDUCE standard -forms for printing by converting them to a Lisp prefix form.

     
-
-tr prepsq; 
-
-  (PREPSQ) 
-
-
-x**2 + y; 
-
-  PREPSQ entry:
-    Arg 1: (((((X . 2) . 1) ((Y . 1) . 1)) . 1)
-  PREPSQ return value = (PLUS (EXPT X 2) Y)
-  PREPSQ entry:
-    Arg 1: (1 . 1)
-  PREPSQ return value = 1
-   2
-  X  + Y
-
-
-untr prepsq; 
-
-  (PREPSQ)
-
-

    -

    -

    -This example is for a PSL-based system; the above format will vary if -other Lisp systems are used. -

    -

    -When a procedure is traced, the first lines show entry to the procedure and -the arguments it is given. The value returned by the procedure is printed -upon exit. If you are tracing several procedures, with a call to one of -them inside the other, the inner trace will be indented showing procedure -nesting. There are no trace options. However, the format of the trace -depends on the underlying Lisp system used. The trace can be removed with -the command -untr. Note that trace, below, is a matrix -operator, while tr does procedure tracing. -

    -

    -

    - - - -UNTR -INDEX

    - - - -UNTR _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -

    -

    -The untr declaration is used to remove a trace from system or -user-written procedures declared with -tr. It is only useful to -those with a good knowledge of both Lisp and the internal formats used by -REDUCE. -

    -

    -

    -syntax:

    -untr<name>{,<name>}* -

    -

    -

    -<name> is the name of a REDUCE system procedure or one of your own -procedures that has previously been the argument of a tr -declaration. -

    -

    - - - -VARNAME -INDEX

    - - - -VARNAME _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The declaration varname instructs REDUCE to use its argument as the -default Fortran (when -fort is on) or -structr identifier -and identifier stem, rather than using ANS. -

    -syntax:

    -

    -

    -varname<identifier> -

    -

    -

    -<identifier> can be any combination of one or more alphanumeric -characters. Try to avoid REDUCE reserved words. -

    -

    -

    -examples:

    -

    
-varname ident; 
-
-  IDENT 
-
-
-on fort; 
-
-x**2 + 1; 
-
-  IDENT=X**2+1. 
-
-
-off fort,exp; 
-
-structr(((x+y)**2 + z)**3); 
-
-        3
-  IDENT2
-      where
-                         2
-         IDENT2 := IDENT1  + Z
-  IDENT1 := X + Y
-
-

    -expwas turned off so that -structr could show the -structure. If exp had been on, the expression would have been -expanded into a polynomial. -

    -

    -

    - - - -WEIGHT -INDEX

    - - - -WEIGHT _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The weight command is used to attach weights to kernels for asymptotic - -constraints. -

    -syntax:

    -

    -

    -weight<kernel> =<number> -

    -

    -

    -<kernel> must be a REDUCE -kernel, <number> must be a -positive integer, not 0. -

    -

    -

    -examples:

    -

    
-a := (x+y)**4; 
-
-        4      3        2  2        3    4
-  A := X  + 4*X *Y + 6*X *Y  + 4*X*Y  + Y  
-
-
-weight x=2,y=3; 
-
-wtlevel 8; 
-
-a; 
-
-   4
-  X  
-
-
-wtlevel 10; 
-
-a; 
-
-   2     2             2
-  X *(6*Y  + 4*X*Y  + X ) 
-
-
-int(x**2,x); 
-
-  ***** X invalid as KERNEL
-
-

    Weights and -wtlevel are used for asymptotic constraints, where -higher-order terms are considered insignificant. -

    -

    -Weights are originally equivalent to 0 until set by a weight -command. To remove a weight from a kernel, use the -clear -command. Weights once assigned cannot be changed without clearing the -identifier. Once a weight is assigned to a kernel, it is no longer a -kernel and cannot be used in any REDUCE commands or operators that require -kernels, until the weight is cleared. Note that terms are ordered by -greatest weight. -

    -

    -The weight level of the system is set by -wtlevel, initially at -2. Since no kernels have weights, no effect from wtlevel can be -seen. Once you assign weights to kernels, you must set wtlevel -correctly for the desired operation. When weighted variables appear in a -term, their weights are summed for the total weight of the term (powers of -variables multiply their weights). When a term exceeds the weight level -of the system, it is discarded from the result expression. -

    -

    -

    - - - -WHERE -INDEX

    - - - -WHERE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The where operator provides an infix notation for one-time -substitutions for kernels in expressions. -

    -syntax:

    -

    -

    -<expression> where <kernel> - =<expression> - {,<kernel> =<expression>}* -

    -

    -

    -<expression> can be any REDUCE scalar expression, <kernel> must -be a -kernel. Alternatively a -rule or a rule list -can be a member of the right-hand part of a where expression. -

    -

    -

    -examples:

    -

    
-x**2 + 17*x*y + 4*y**2 where x=1,y=2; 
-
-
-  51 
-
-
-for i := 1:5 collect x**i*q where q= for j := 1:i product j;
- 
-
-
-        2    3     4      5
-  {X,2*X ,6*X ,24*X ,120*X } 
-
-
-x**2 + y + z where z=y**3,y=3; 
-
-   2    3
-  X  + Y  + 3
-
-

    Substitution inside a where expression has no effect upon - the values -of the kernels outside the expression. The where operator has the -lowest precedence of all the infix operators, which are lower than prefix -operators, so that the substitutions apply to the entire expression -preceding the where operator. However, where is applied -before command keywords such as then, repeat, or do. - -

    -

    -A -rule or a rule set in the right-hand part of -the -where expression act as if the rules were activated by -let -immediately before the evaluation of the expression and deactivated -by -clearrules immediately afterwards. -

    -

    -wheregives you a natural notation for auxiliary variables in -expressions. As the second example shows, the substitute expression can be -a command to be evaluated. The substitute assignments are made in -parallel, rather than sequentially, as the last example shows. The -expression resulting from the first round of substitutions is not -reexamined to see if any further such substitutions can be made. -where can also be used to define auxiliary variables in - -procedure definitions. -

    -

    -

    - - - -WHILE -INDEX

    - - - -WHILE _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The while command causes a statement to be repeatedly executed until a - -given condition is true. If the condition is initially false, the statement -is not executed at all. -

    -syntax:

    -

    -

    -while<condition> do <statement> -

    -

    -

    -<condition> is given by a logical operator, <statement> must be a -single REDUCE statement, or a -group (<<...>>) or - -begin...end -block. -

    -

    -

    -examples:

    -

    
-a := 10; 
-
-  A := 10 
-
-
-while a <= 12 do <<write a; a := a + 1>>;
- 
-
-
-  10 
-
-
-                                          11 
-
-                                          12 
-
-while a < 5 do <<write a; a := a + 1>>;
- 
-
-
-      nothing is printed
-
-

    - - -WTLEVEL -INDEX

    - - - -WTLEVEL _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -In conjunction with -weight, wtlevel is used to implement -asymptotic constraints. Its default value is 2. -

    -syntax:

    -

    -

    -wtlevel<expression> -

    -

    -

    -To change the weight level, <expression> must evaluate to a positive -integer that is the greatest weight term to be retained in expressions -involving kernels with weight assignments. wtlevel returns the -new weight level. If you want the current weight level, but not -change it, say wtlevel nil. -

    -

    -

    -examples:

    -

    
-(x+y)**4;          
- 
-
-   4      3        2  2        3    4
-  X  + 4*X *Y + 6*X *Y  + 4*X*Y  + Y  
-
-
-weight x=2,y=3; 
-
-wtlevel 8; 
-
-(x+y)**4; 
-
-   4
-  X  
-
-
-wtlevel 10; 
-
-(x+y)**4; 
-
-   2     2            2
-  X *(6*Y  + 4*X*Y + X ) 
-
-
-int(x**2,x); 
-
-  ***** X invalid as KERNEL
-
-

    wtlevelis used in conjunction with the command -weight to -enable asymptotic constraints. Weight of a term is computed by multiplying -the weights of each variable in it by the power to which it has been -raised, and adding the resulting weights for each variable. If the weight -of the term is greater than wtlevel, the term is dropped from the -expression, and not used in any further computation involving the -expression. -

    -

    -Once a weight has been attached to a -kernel, it is no longer -recognized by the system as a kernel, though still a variable. It cannot -be used in REDUCE commands and operators that need kernels. The weight -attachment can be undone with a -clear command. wtlevel can -be changed as desired. -

    -

    -

    - - - -Declarations -INDEX

    -Declarations

    -
  • ALGEBRAIC command

    -

  • ANTISYMMETRIC declaration

    -

  • ARRAY declaration

    -

  • CLEAR command

    -

  • CLEARRULES command

    -

  • DEFINE command

    -

  • DEPEND declaration

    -

  • EVEN declaration

    -

  • FACTOR declaration

    -

  • FORALL command

    -

  • INFIX declaration

    -

  • INTEGER declaration

    -

  • KORDER declaration

    -

  • LET command

    -

  • LINEAR declaration

    -

  • LINELENGTH declaration

    -

  • LISP command

    -

  • LISTARGP declaration

    -

  • NODEPEND declaration

    -

  • MATCH command

    -

  • NONCOM declaration

    -

  • NONZERO declaration

    -

  • ODD declaration

    -

  • OFF command

    -

  • ON command

    -

  • OPERATOR declaration

    -

  • ORDER declaration

    -

  • PRECEDENCE declaration

    -

  • PRECISION declaration

    -

  • PRINT\_PRECISION declaration

    -

  • REAL declaration

    -

  • REMFAC declaration

    -

  • SCALAR declaration

    -

  • SCIENTIFIC\_NOTATION declaration

    -

  • SHARE declaration

    -

  • SYMBOLIC command

    -

  • SYMMETRIC declaration

    -

  • TR declaration

    -

  • UNTR declaration

    -

  • VARNAME declaration

    -

  • WEIGHT command

    -

  • WHERE operator

    -

  • WHILE command

    -

  • WTLEVEL command

    -

    •  - - -IN -INDEX

    - - - -IN _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The in command takes a list of file names and inputs each file into -the system. -

    -syntax:

    -

    -

    -in<filename>{,<filename>}* -

    -

    -

    -<filename> must be in the current directory, or be a valid pathname. -If the file name is not an identifier, double quote marks (") are - -needed around the file name. -

    -

    -A message is given if the file cannot be found, or has a mistake -in it. -

    -

    -Ending the command with a semicolon causes the file to be echoed to the -screen; ending it with a dollar sign does not echo the file. If you want -some but not all of a file echoed, turn the switch -echo on or off -in the file. -

    -

    -An efficient way to develop procedures in REDUCE is to write them into a file -using a system editor of your choice, and then input the -files into an active REDUCE session. REDUCE reparses the procedure as -it takes information from the file, overwriting the previous procedure -definition. When it accepts the procedure, it echoes its name to the screen. -Data can also be input to the system from files. -

    -

    -Files to be read in should always end in -end; to avoid -end-of-file problems. Note that this is an additional end; to any -ending procedures in the file. -

    -

    -

    - - - -INPUT -INDEX

    - - - -INPUT _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The input command returns the input expression to the REDUCE numbered -prompt that is its argument. -

    -syntax:

    -

    -

    -input(<number>) or input <number> -

    -

    -

    -

    -<number> must be between 1 and the current REDUCE prompt number. -

    -

    -An expression brought back by input can be reexecuted with new -values or switch settings, or used as an argument in another expression. -The command -ws brings back the results of a numbered REDUCE -statement. Two lists contain every input and every output statement since -the beginning of the session. If your session is very long, storage space -begins to fill up with these expressions, so it is a good idea to end the -session once in a while, saving needed expressions to files with the - -saveas and -out commands. -

    -

    -Switch settings and -let statements can also be reexecuted by using -input. -

    -

    -An error message is given if a number is called for that has not yet been used. - -

    -

    -

    - - - -OUT -INDEX

    - - - -OUT _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The out command directs output to the filename that is its argument, -until another out changes the output file, or -shut closes it. -

    -syntax:

    -

    -

    -out<filename> or out "<pathname> " - or out t -

    -

    -

    -<filename> must be in the current directory, or be a valid complete -file description for your system. If the file name is not -in the current directory, quote marks are needed around the file name. -If the file already exists, a message is printed allowing you to decide -whether to supersede the contents of the file with new material. -

    -

    -To restore output to the terminal, type out t, or -shut the -file. When you use out t, the file remains available, and if you -open it again (with another out), new material is appended rather -than overwriting. -

    -

    -To write a file using out that can be input at a later time, the -switch -nat must be turned off, so that the standard linear fo -rm -is saved that can be read in by -in. If nat is on, exponents -are printed on the line above the expression, which causes trouble -when REDUCE tries to read the file. -

    -

    -There is a slight complication if you are using the out command from -inside a file to create another file. The -echo switch is normally -off at the top-level and on while reading files (so you can see what is -being read in). If you create a file using out at the top-level, -the result lines are printed into the file as you want them. But if you -create such a file from inside a file, the echo switch is on, and -every line is echoed, first as you typed it, then as REDUCE parsed it, and -then once more for the file. Therefore, when you create a file from -a file, you need to turn echo off explicitly before the out -command, and turn it back on when you shut the created file, so your -executing file echoes as it should. This behavior also means that as you -watch the file execute, you cannot see the lines that are being put into -the out file. As soon as you turn echo on, you can see -output again. -

    -

    -

    - - - -SHUT -INDEX

    - - - -SHUT _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The shut command closes output files. -

    -syntax:

    -

    -

    -shut<filename>{,<filename>}* -

    -

    -

    -<filename> must have been a file opened by -out. -

    -

    -A file that has been opened by -out must be shut before it is -brought in by -in. Files that have been opened by out should - -always be shut before the end of the REDUCE session, to avoid either -loss of information or the printing of extraneous information into the file. -In most systems, terminating a session by -bye closes all open -output files. -

    -

    -

    - - - -Input and Output -INDEX

    -Input and Output

    -
  • IN command

    -

  • INPUT command

    -

  • OUT command

    -

  • SHUT command

    -

    •  - - -ACOS -INDEX

    - - - -ACOS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The acos operator returns the arccosine of its argument. -

    -

    -

    -syntax:

    -acos(<expression>) or acos <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-acos(ab); 
-
-  ACOS(AB) 
-
-
-acos 15; 
-
-  ACOS(15) 
-
-
-df(acos(x*y),x); 
-
-           2  2
-  SQRT( - X *Y  + 1)*Y
-  -------------------- 
-        2  2
-       X *Y  - 1
-
-
-on rounded; 
-
-res := acos(sqrt(2)/2); 
-
-  RES := 0.785398163397 
-
-
-res-pi/4; 
-
-  0
-
-

    An explicit numeric value is not given unless the switch -rounded is -on and the argument has an absolute numeric value less than or equal to 1. -

    -

    -

    - - - -ACOSH -INDEX

    - - - -ACOSH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -acoshrepresents the hyperbolic arccosine of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of -acosh is known to the system. Numerical values may also be found by -turning on the switch -rounded. -

    -

    -

    -syntax:

    -acosh(<expression>) or acosh <simple\_expression> - -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-acosh a; 
-
-  ACOSH(A) 
-
-
-acosh(0); 
-
-  ACOSH(0) 
-
-
-df(acosh(a**2),a); 
-
-          4
-  2*SQRT(A  - 1)*A
-  ---------------- 
-        4
-       A  - 1
-
-
-int(acosh(x),x); 
-
-  INT(ACOSH(X),X)
-
-

    You may attach functionality by defining acosh to be the -inverse of -cosh. This is done by the commands -

    
-        put('cosh,'inverse,'acosh);
-        put('acosh,'inverse,'cosh);
-

    -

    -You can write a procedure to attach integrals or other -functions to acosh. You may wish to add a check to see that its -argument is properly restricted. -

    -

    -

    - - - -ACOT -INDEX

    - - - -ACOT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -acotrepresents the arccotangent of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of -acot is known to the system. Numerical values may also be found by -turning on the switch -rounded. -

    -

    -

    -syntax:

    -acot(<expression>) or acot <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -You can add functionality yourself with let and procedures. -

    -

    - - - -ACOTH -INDEX

    - - - -ACOTH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -acothrepresents the inverse hyperbolic cotangent of its argument. -It takes an arbitrary scalar expression as its argument. The derivative -of acoth is known to the system. Numerical values may also be found -by turning on the switch -rounded. -

    -

    -

    -syntax:

    -acoth(<expression>) or acoth <simple\_expression> - -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, -matrix or vector expression. <simple\_expression> must be a single -identifier or begin with a prefix operator name. You can add -functionality yourself with let and procedures. -

    -

    - - - -ACSC -INDEX

    - - - -ACSC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The acsc operator returns the arccosecant of its argument. -

    -

    -

    -syntax:

    -acsc(<expression>) or acsc <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-acsc(ab); 
-
-  ACSC(AB) 
-
-
-acsc 15; 
-
-  ACSC(15) 
-
-
-df(acsc(x*y),x); 
-
-         2  2
-  -SQRT(X *Y  - 1)
-  ---------------- 
-       2  2
-   X*(X *Y  - 1)
-
-
-on rounded; 
-
-res := acsc(2/sqrt(3)); 
-
-  RES := 1.0471975512 
-
-
-res-pi/3; 
-
-  0
-
-

    An explicit numeric value is not given unless the switch round -ed is -on and the argument has an absolute numeric value less than or equal to 1. -

    -

    -

    - - - -ACSCH -INDEX

    - - - -ACSCH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The acsch operator returns the hyperbolic arccosecant of its argument. - -

    -

    -

    -syntax:

    -acsch(<expression>) or acsch <simple\_expression> - -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-acsch(ab); 
-
-  ACSCH(AB) 
-
-
-acsch 15; 
-
-  ACSCH(15) 
-
-
-df(acsch(x*y),x); 
-
-         2  2
-  -SQRT(X *Y  + 1)
-  ---------------- 
-       2  2
-   X*(X *Y  + 1)
-
-
-on rounded; 
-
-res := acsch(3); 
-
-  RES := 0.327450150237
-
-

    An explicit numeric value is not given unless the switch round -ed is -on and the argument has an absolute numeric value less than or equal to 1. -

    -

    -

    - - - -ASEC -INDEX

    - - - -ASEC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The asec operator returns the arccosecant of its argument. -

    -

    -

    -syntax:

    -asec(<expression>) or asec <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-asec(ab); 
-
-  ASEC(AB) 
-
-
-asec 15; 
-
-  ASEC(15) 
-
-
-df(asec(x*y),x); 
-
-        2  2
-  SQRT(X *Y  - 1)
-  --------------- 
-       2  2
-   X*(X *Y  - 1)
-
-
-on rounded; 
-
-res := asec sqrt(2); 
-
-  RES := 0.785398163397 
-
-
-res-pi/4; 
-
-  0
-
-

    An explicit numeric value is not given unless the switch round -ed is -on and the argument has an absolute numeric value greater or equal to 1. -

    -

    -

    - - - -ASECH -INDEX

    - - - -ASECH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -asechrepresents the hyperbolic arccosecant of its argument. It takes -an arbitrary scalar expression as its argument. The derivative of -asech is known to the system. Numerical values may also be found by -turning on the switch -rounded. -

    -

    -

    -syntax:

    -asech(<expression>) or asech <simple\_expression> - -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-asech a; 
-
-  ASECH(A) 
-
-
-asech(1); 
-
-  0 
-
-
-df(acosh(a**2),a); 
-
-            4
-  2*SQRT(- A  + 1)
-  ---------------- 
-         4
-     A*(A  - 1)
-
-
-int(asech(x),x); 
-
-  INT(ASECH(X),X)
-
-

    You may attach functionality by defining asech to be the -inverse of -sech. This is done by the commands -

    
-        put('sech,'inverse,'asech);
-        put('asech,'inverse,'sech);
-

    -

    -You can write a procedure to attach integrals or other -functions to asech. You may wish to add a check to see that its -argument is properly restricted. -

    -

    -

    - - - -ASIN -INDEX

    - - - -ASIN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The asin operator returns the arcsine of its argument. -

    -

    -

    -syntax:

    -asin(<expression>) or asin <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-asin(givenangle); 
-
-  ASIN(GIVENANGLE) 
-
-
-asin(5); 
-
-  ASIN(5) 
-
-
-df(asin(2*x),x); 
-
-                 2
-    2*SQRT( - 4*X  + 1))
-  - -------------------- 
-             2
-          4*X  - 1
-
-
-on rounded; 
-
-asin .5; 
-
-  0.523598775598 
-
-
-asin(sqrt(3)); 
-
-  ASIN(1.73205080757) 
-
-
-asin(sqrt(3)/2); 
-
-  1.04719755120 
-
-

    A numeric value is not returned by asin unless the switch - -rounded is on and its argument has an absolute value less than or -equal to 1. -

    -

    -

    - - - -ASINH -INDEX

    - - - -ASINH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The asinh operator returns the hyperbolic arcsine of its argument. -The derivative of asinh and some simple transformations are known -to the system. -

    -

    -

    -syntax:

    -asinh(<expression>) or asinh <simple\_expression> - -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-asinh d; 
-
-  ASINH(D) 
-
-
-asinh(1); 
-
-  ASINH(1) 
-
-
-df(asinh(2*x),x); 
-
-            2
-  2*SQRT(4*X  + 1))
-  ----------------- 
-         2
-      4*X  + 1
-
-

    You may attach further functionality by defining asinh to - be the -inverse of sinh. This is done by the commands -

    
-        put('sinh,'inverse,'asinh);
-        put('asinh,'inverse,'sinh);
-

    -

    -A numeric value is not returned by asinh unless the switch -rounded is on and its argument evaluates to a number. -

    -

    -

    - - - -ATAN -INDEX

    - - - -ATAN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The atan operator returns the arctangent of its argument. -

    -

    -

    -syntax:

    -atan(<expression>) or atan <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-atan(middle); 
-
-  ATAN(MIDDLE) 
-
-
-on rounded; 
-
-atan 45; 
-
-  1.54857776147 
-
-
-off rounded; 
-
-int(atan(x),x); 
-
-                     2
-  2*ATAN(X)*X - LOG(X  + 1)
-  ------------------------- 
-              2
-
-
-df(atan(y**2),y); 
-
-   2*Y
-  -------
-   4
-  Y  + 1
-
-

    A numeric value is not returned by atan unless the switch - - -rounded is on and its argument evaluates to a number. - -

    -

    -

    - - - -ATANH -INDEX

    - - - -ATANH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The atanh operator returns the hyperbolic arctangent of its argument. -The derivative of asinh and some simple transformations are known -to the system. -

    -

    -

    -syntax:

    -atanh(<expression>) or atanh <simple\_expression> - -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-atanh aa; 
-
-  ATANH(AA) 
-
-
-atanh(1); 
-
-  ATANH(1) 
-
-
-df(atanh(x*y),y); 
-
-     - X
-  ----------
-   2  2
-  X *Y  - 1
-
-

    A numeric value is not returned by asinh unless the switc -h -rounded is on and its argument evaluates to a number. -You may attach additional functionality by defining atanh to be the -inverse of tanh. This is done by the commands -

    -

    -

    
-        put('tanh,'inverse,'atanh);
-        put('atanh,'inverse,'tanh);
-

    -

    - - - -ATAN2 -INDEX

    - - - -ATAN2 _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -atan2(<expression>,<expression>) -

    -

    -

    -<expression> is any valid scalar REDUCE expression. In - -rounded mode, if a numerical value exists, atan2 - returns -the principal value of the arc tangent of the second argument divided by -the first in the range [-pi,+pi] radians, using the signs of both -arguments to determine the quadrant of the return value. An expression in -terms of atan2 is returned in other cases. -

    -

    -

    -examples:

    -

    
-atan2(3,2); 
-
-  ATAN2(3,2); 
-
-
-on rounded; 
-
-atan2(3,2); 
-
-  0.982793723247 
-
-
-atan2(a,b); 
-
-  ATAN2(A,B); 
-
-
-atan2(1,0); 
-
-  1.57079632679
-
-

    atan2returns a numeric value only if -rounded is on. Then -atan2 is calculated to the current degree of floating point precision. - -

    -

    -

    -

    - - - -COS -INDEX

    - - - -COS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The cos operator returns the cosine of its argument. -

    -

    -

    -syntax:

    -cos(<expression>) or cos <simple\_expression> -

    -

    -

    -<expression> is any valid scalar REDUCE expression, -<simple\_expression> is a single identifier or begins with a prefix -operator name. -

    -

    -

    -examples:

    -

    
-
-
-cos abc; 
-
-  COS(ABC) 
-
-
-
-cos(pi); 
-
-  -1 
-
-
-
-cos 4; 
-
-  COS(4) 
-
-
-
-on rounded; 
-
-
-cos(4); 
-
-  - 0.653643620864 
-
-
-
-cos log 5; 
-
-  - 0.0386319699339
-
-

    cosreturns a numeric value only if -rounded is on. Then the -cosine is calculated to the current degree of floating point precision. -

    -

    -

    -

    - - - -COSH -INDEX

    - - - -COSH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The cosh operator returns the hyperbolic cosine of its argument. -The derivative of cosh and some simple transformations are known -to the system. -

    -

    -

    -syntax:

    -cosh(<expression>) or cosh <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-
-cosh b; 
-
-  COSH(B) 
-
-
-
-cosh(0); 
-
-  1 
-
-
-
-df(cosh(x*y),x); 
-
-  SINH(X*Y)*Y 
-
-
-
-int(cosh(x),x); 
-
-  SINH(X)
-
-

    You may attach further functionality by defining its inverse (see - - -acosh). -A numeric value is not returned by cosh unless the switch - -rounded is on and its argument evaluates to a number. - -

    -

    -

    -

    - - - -COT -INDEX

    - - - -COT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -cot represents the cotangent of its argument. It takes an arbitrary -scalar expression as its argument. The derivative of acot and some -simple properties are known to the system. -

    -

    -

    -syntax:

    -cot(<expression>) or cot <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression. <simple\_expression -> -must be a single identifier or begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-cot(a)*tan(a); 
-
-  COT(A)*TAN(A)) 
-
-
-cot(1); 
-
-  COT(1) 
-
-
-df(cot(2*x),x); 
-
-               2
-  - 2*(COT(2*X)   + 1)
-
-

    Numerical values of expressions involving cot may be foun -d by -turning on the switch -rounded. -

    -

    -

    -

    - - - -COTH -INDEX

    - - - -COTH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The coth operator returns the hyperbolic cotangent of its argument. -The derivative of coth and some simple transformations are known -to the system. -

    -

    -

    -syntax:

    -coth(<expression>) or coth <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression. <simple\_expression -> -must be a single identifier or begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-df(coth(x*y),x); 
-
-                2
-  - Y*(COTH(X*Y)   - 1) 
-
-
-
-coth acoth z; 
-
-  Z
-
-

    You can write -let statements and procedures to add further -functionality to coth if you wish. Numerical values of expressions -involving coth may also be found by turning on the switch - -rounded. -

    -

    -

    -

    - - - -CSC -INDEX

    - - - -CSC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The csc operator returns the cosecant of its argument. -The derivative of csc and some simple transformations are known -to the system. -

    -

    -

    -syntax:

    -csc(<expression>) or csc <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression. <simple\_expression -> -must be a single identifier or begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-
-csc(q)*sin(q); 
-
-  CSC(Q)*SIN(Q) 
-
-
-
-df(csc(x*y),x); 
-
-  -COT(X*Y)*CSC(X*Y)*Y
-
-

    You can write -let statements and procedures to add further -functionality to csc if you wish. Numerical values of expressions -involving csc may also be found by turning on the switch - -rounded. -

    -

    -

    -

    - - - -CSCH -INDEX

    - - - -CSCH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The cosh operator returns the hyperbolic cosecant of its argument. -The derivative of csch and some simple transformations are known -to the system. -

    -

    -

    -syntax:

    -csch(<expression>) or csch <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-
-csch b; 
-
-  CSCH(B) 
-
-
-
-csch(0); 
-
-  0 
-
-
-
-df(csch(x*y),x); 
-
-  - COTH(X*Y)*CSCH(X*Y)*Y 
-
-
-
-int(csch(x),x); 
-
-  INT(CSCH(X),X)
-
-

    A numeric value is not returned by csch unless the switch - - -rounded is on and its argument evaluates to a number. - -

    -

    -

    - - - -ERF -INDEX

    - - - -ERF _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The erf operator represents the error function, defined by -

    -

    -erf(x) = (2/sqrt(pi))*int(e^(-x^2),x) -

    -

    -A limited number of its properties are known to the system, including the -fact that it is an odd function. Its derivative is known, and from this, -some integrals may be computed. However, a complete integration procedure -for this operator is not currently included. -

    -

    -

    -examples:

    -

    
-erf(0); 
-
-  0 
-
-
-erf(-a); 
-
-  - ERF(A) 
-
-
-df(erf(x**2),x); 
-
-  4*SQRT(PI)*X
-  ------------ 
-       4
-      X
-     E  *PI
-
-
-
-int(erf(x),x); 
-
-    2
-   X
-  E  *ERF(X)*PI*X + SQRT(PI)
-  ---------------------------
-              2
-             X
-            E  *PI
-
-

    - - -EXP -INDEX

    - - - -EXP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The exp operator returns e raised to the power of its argument -. -

    -syntax:

    -

    -

    -exp(<expression>) or exp <simple\_expression> -

    -

    -

    -<expression> can be any valid REDUCE scalar expression. -<simple\_expression> must be a single identifier or begin with a -prefix operator. -

    -

    -

    -examples:

    -

    
-exp(sin(x)); 
-
-   SIN X
-  E      
-
-
-exp(11); 
-
-   11
-  E   
-
-
-on rounded; 
-
-exp sin(pi/3); 
-
-  2.37744267524
-
-

    Numeric values are returned only when rounded is on. -The single letter e with the exponential operator ^ or -** may be substituted for exp without change of function. -

    -

    -

    - - - -SEC -INDEX

    - - - -SEC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The sec operator returns the secant of its argument. -

    -

    -

    -syntax:

    -sec(<expression>) or sec <simple\_expression> -

    -

    -

    -<expression> is any valid scalar REDUCE expression, -<simple\_expression> is a single identifier or begins with a prefix -operator name. -

    -

    -

    -examples:

    -

    
-
-
-sec abc; 
-
-  SEC(ABC) 
-
-
-
-sec(pi); 
-
-  -1 
-
-
-
-sec 4; 
-
-  SEC(4) 
-
-
-
-on rounded; 
-
-
-sec(4); 
-
-  - 1.52988565647 
-
-
-
-sec log 5; 
-
-  - 25.8852966005
-
-

    secreturns a numeric value only if -rounded is on. Then the -secant is calculated to the current degree of floating point precision. -

    -

    -

    - - - -SECH -INDEX

    - - - -SECH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The sech operator returns the hyperbolic secant of its argument. -

    -

    -

    -syntax:

    -sech(<expression>) or sech <simple\_expression> -

    -

    -

    -<expression> is any valid scalar REDUCE expression, -<simple\_expression> is a single identifier or begins with a prefix -operator name. -

    -

    -

    -examples:

    -

    
-sech abc; 
-
-  SECH(ABC) 
-
-
-
-sech(0); 
-
-  1 
-
-
-
-sech 4; 
-
-  SECH(4) 
-
-
-
-on rounded; 
-
-
-sech(4); 
-
-  0.0366189934737 
-
-
-
-sech log 5; 
-
-  0.384615384615
-
-

    sechreturns a numeric value only if -rounded is on. Then the -expression is calculated to the current degree of floating point precision. -

    -

    -

    - - - -SIN -INDEX

    - - - -SIN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The sin operator returns the sine of its argument. -

    -syntax:

    -

    -

    -sin(<expression>) or sin <simple\_expression> -

    -

    -

    -<expression> is any valid scalar REDUCE expression, -<simple\_expression> is a single identifier or begins with a prefix -operator name. -

    -

    -

    -examples:

    -

    
-sin aa; 
-
-  SIN(AA) 
-
-
-sin(pi/2); 
-
-  1 
-
-
-on rounded; 
-
-sin 3; 
-
-  0.14112000806 
-
-
-sin(pi/2); 
-
-  1.0
-
-

    sinreturns a numeric value only if rounded is on -. -Then the sine is calculated to the current degree of floating point precision. -The argument in this case is assumed to be in radians. -

    -

    -

    - - - -SINH -INDEX

    - - - -SINH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The sinh operator returns the hyperbolic sine of its argument. -The derivative of sinh and some simple transformations are known -to the system. -

    -

    -

    -syntax:

    -sinh(<expression>) or sinh <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-
-sinh b; 
-
-  SINH(B) 
-
-
-
-sinh(0); 
-
-  0 
-
-
-df(sinh(x**2),x); 
-
-          2
-  2*COSH(X )*X 
-
-
-int(sinh(4*x),x); 
-
-  COSH(4*X)
-  --------- 
-      4
-
-
-on rounded; 
-
-sinh 4; 
-
-  27.2899171971
-
-

    You may attach further functionality by defining its inverse (see - - -asinh). -A numeric value is not returned by sinh unless the switch - -rounded is on and its argument evaluates to a number. - -

    -

    -

    - - - -TAN -INDEX

    - - - -TAN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The tan operator returns the tangent of its argument. -

    -syntax:

    -

    -

    -tan(<expression>) or tan <simple\_expression> -

    -

    -

    -

    -<expression> is any valid scalar REDUCE expression, -<simple\_expression> is a single identifier or begins with a prefix -operator name. -

    -

    -

    -examples:

    -

    
-tan a; 
-
-  TAN(A) 
-
-
-tan(pi/5); 
-
-      PI
-  TAN(--) 
-      5
-
-
-on rounded;
-tan(pi/5); 
-
-  0.726542528005
-
-

    tanreturns a numeric value only if rounded is on -. Then the -tangent is calculated to the current degree of floating point accuracy. -

    -

    -When -rounded is on, -no check is made to see if the argument of tan is a multiple of -pi/2, for which the tangent goes to positive or negative infinity. -(Of course, since REDUCE uses a fixed-point representation of pi/2, -it produces a large but not infinite number.) You need to make a check for -multiples of pi/2 in any program you use that might possibly ask -for the tangent of such a quantity. -

    -

    -

    - - - -TANH -INDEX

    - - - -TANH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The tanh operator returns the hyperbolic tangent of its argument. -The derivative of tanh and some simple transformations are known -to the system. -

    -

    -

    -syntax:

    -tanh(<expression>) or tanh <simple\_expression> -

    -

    -

    -<expression> may be any scalar REDUCE expression, not an array, matrix or - -vector expression. <simple\_expression> must be a single identifier or -begin with a prefix operator name. -

    -

    -

    -examples:

    -

    
-tanh b; 
-
-  TANH(B) 
-
-
-tanh(0); 
-
-  0 
-
-
-df(tanh(x*y),x); 
-
-                 2
-  Y*( - TANH(X*Y)  + 1) 
-
-
-int(tanh(x),x); 
-
-       2*X
-  LOG(E    + 1) - X 
-
-
-on rounded; tanh 2; 
-
-  0.964027580076
-
-

    You may attach further functionality by defining its inverse (see - - -atanh). -A numeric value is not returned by tanh unless the switch - -rounded is on and its argument evaluates to a number. - -

    -

    -

    - - - -Elementary Functions -INDEX

    -Elementary Functions

    -
  • ACOS operator

    -

  • ACOSH operator

    -

  • ACOT operator

    -

  • ACOTH operator

    -

  • ACSC operator

    -

  • ACSCH operator

    -

  • ASEC operator

    -

  • ASECH operator

    -

  • ASIN operator

    -

  • ASINH operator

    -

  • ATAN operator

    -

  • ATANH operator

    -

  • ATAN2 operator

    -

  • COS operator

    -

  • COSH operator

    -

  • COT operator

    -

  • COTH operator

    -

  • CSC operator

    -

  • CSCH operator

    -

  • ERF operator

    -

  • EXP operator

    -

  • SEC operator

    -

  • SECH operator

    -

  • SIN operator

    -

  • SINH operator

    -

  • TAN operator

    -

  • TANH operator

    -

    •  - - -SWITCHES -INDEX

    - - - -SWITCHES _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -Switches are set on or off using the commands -on or - -off, respectively. -The default setting of the switches described in this section is - -off unless stated otherwise. -

    -

    - - - -ALGINT -INDEX

    - - - -ALGINT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the algint switch is on, the algebraic integration module (which -must be loaded from the REDUCE library) is used for integration. -

    -

    -Loading algint from the library automatically turns on the -algint switch. An error message will be given if algint is -turned on when the algint has not been loaded from the library. -

    -

    -

    - - - -ALLBRANCH -INDEX

    - - - -ALLBRANCH _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -When allbranch is on, the operator -solve selects all -branches of solutions. -When allbranch is off, it selects only the principal -branches. Default is on. -

    -

    -

    -examples:

    -

    
-
-solve(log(sin(x+3)),x); 
-
-  {X=2*ARBINT(1)*PI - ASIN(1) - 3,
-   X=2*ARBINT(1)*PI + ASIN(1) + PI - 3}
-
-
-off allbranch; 
-
-solve(log(sin(x+3)),x); 
-
-  X=ASIN(1) - 3
-
-

    -arbint(1) indicates an arbitrary integer, which is giv -en a -unique identifier by REDUCE, showing that there are infinitely many -solutions of this type. When allbranch is off, the single -canonical solution is given. -

    -

    -

    - - - -ALLFAC -INDEX

    - - - -ALLFAC _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -The allfac switch, when on, causes REDUCE to factor out automatically -common products in the output of expressions. Default is on. -

    -

    -

    -examples:

    -

    
-x + x*y**3 + x**2*cos(z); 
-
-                 3
-  X*(COS(Z)*X + Y   + 1) 
-
-
-off allfac; 
-
-x + x*y**3 + x**2*cos(z); 
-
-          2      3
-  COS(Z)*X  + X*Y   + X
-
-

    The allfac switch has no effect when pri is off. - Although the -switch setting stays as it was, printing behavior is as if it were off. -

    -

    -

    - - - -ARBVARS -INDEX

    - - - -ARBVARS _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When arbvars is on, the solutions of singular or underdetermined -systems of equations are presented in terms of arbitrary complex variables -(see -arbcomplex). Otherwise, the solution is parametrized i -n -terms of some of the input variables. Default is on. -

    -

    -

    -examples:

    -

    
-solve({2x + y,4x + 2y},{x,y}); 
-
-         arbcomplex(1)
-  {{x= - -------------,y=arbcomplex(1)}} 
-               2
-
-
-solve({sqrt(x)+ y**3-1},{x,y});				
-
-
-                            6       3
-  		   {{y=arbcomplex(2),x=y   - 2*y   + 1}} 
-
-
-off arbvars; 
-
-solve({2x + y,4x + 2y},{x,y}); 
-
-         y
-  {{x= - -}} 
-         2
-
-
-solve({sqrt(x)+ y**3-1},{x,y});				
-
-
-            6       3
-  		   {{x=y   - 2*y   + 1}} 
-
-

    With arbvars off, the return value {{}} means th -at the -equations given to -solve imply no relation among the input -variables. -

    -

    -

    - - - -BALANCED_MOD -INDEX

    - - - -BALANCED\_MOD _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    - -modularnumbers are normally produced in the range [0,. -..<n>), -where -<n> is the current modulus. With balanced_mod on, the range -[-<n>/2,<n>/2], or more precisely -[-floor((<n>-1)/2), ceiling((<n>-1)/2)], is used instead. -

    -

    -

    -examples:

    -

    
-setmod 7; 
-
-  1 
-
-
-on modular; 
-
-4; 
-
-  4 
-
-
-on balanced_mod; 
-
-4; 
-
-  -3
-
-

    - - -BFSPACE -INDEX

    - - - -BFSPACE _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -Floating point numbers are normally printed in a compact notation (either -fixed point or in scientific notation if -SCIENTIFIC_NOTATION -has been used). In some (but not all) cases, it helps comprehensibility -if spaces are inserted in the number at regular intervals. The switch -bfspace, if on, will cause a blank to be inserted in the number after -every five characters. -

    -examples:

    -

    
-on rounded; 
-
-1.2345678; 
-
-  1.2345678 
-
-
-on bfspace; 
-
-1.2345678; 
-
-  1.234 5678
-
-

    -

    -bfspaceis normally off. -

    -

    -

    - - - -COMBINEEXPT -INDEX

    - - - -COMBINEEXPT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -REDUCE is in general poor at surd simplification. However, when the -switch combineexpt is on, the system attempts to combine -exponentials whenever possible. -

    -

    -

    -examples:

    -

    
-3^(1/2)*3^(1/3)*3^(1/6); 
-
-           1/3  1/6
-  SQRT(3)*3   *3    
-
-
-on combineexpt; 
-
-ws; 
-
-  1
-
-

    - - -COMBINELOGS -INDEX

    - - - -COMBINELOGS _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -In many cases it is desirable to expand product arguments of logarithms, -or collect a sum of logarithms into a single logarithm. Since these are -inverse operations, it is not possible to provide rules for doing both at -the same time and preserve the REDUCE concept of idempotent evaluation. -As an alternative, REDUCE provides two switches -expandlogs and -combinelogs to carry out these operations. -

    -examples:

    -

    
-on expandlogs; 
-
-log(x*y); 
-
-  LOG(X) + LOG(Y) 
-
-
-on combinelogs; 
-
-ws; 
-
-  LOG(X*Y)
-
-

    -

    -At the present time, it is possible to have both switches on at once, -which could lead to infinite recursion. However, an expression is -switched from one form to the other in this case. Users should not rely -on this behavior, since it may change in the next release. -

    -

    -

    - - - -COMP -INDEX

    - - - -COMP _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When comp is on, any succeeding function definitions are compiled -into a faster-running form. Default is off. -

    -

    -

    -examples:

    -

    The following procedure finds Fibonacci numbers recurs -ively. -Create a new file ``refib" in your current directory with the following -lines in it:

     
-
-procedure refib(n);
-   if fixp n and n >= 0 then
-     if n <= 1 then 1
-       else refib(n-1) + refib(n-2)
-    else rederr "nonnegative integer only";
-
-end;
-
-

    Now load REDUCE and run the following:

    
-
-on time; 
-
-  Time: 100 ms 
-
-
-
-in "refib"$ 
-
-  Time: 0 ms 
-
-
-
- 
-
-  REFIB 
-
-
-
- 
-
-  Time: 260 ms 
-
-
-
- 
-
-  Time: 20 ms 
-
-
-
-refib(80); 
-
-  37889062373143906 
-
-
-
- 
-
-  Time: 14840 ms 
-
-
-
-on comp; 
-
-  Time: 80 ms 
-
-
-
-in "refib"$ 
-
-  Time: 20 ms 
-
-
-
- 
-
-  REFIB 
-
-
-
- 
-
-  Time: 640 ms 
-
-
-
-refib(80); 
-
-  37889062373143906 
-
-
-
- 
-
-  Time: 10940 ms
-
-

    -

    -

    -Note that the compiled procedure runs faster. Your time messages will -differ depending upon which system you have. Compiled functions remain so -for the duration of the REDUCE session, and are then lost. They must be -recompiled if wanted in another session. With the switch -time on -as shown above, the CPU time used in executing the command is returned in -milliseconds. Be careful not to leave comp on unless you want it, -as it makes the processing of procedures much slower. -

    -

    -

    -

    - - - -COMPLEX -INDEX

    - - - -COMPLEX _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the complex switch is on, full complex arithmetic is used in -simplification, function evaluation, and factorization. Default is off. - -

    -

    -

    -examples:

    -

    
-
-factorize(a**2 + b**2); 
-
-     2     2
-  {{A   + B ,1}} 
-
-
-on complex; 
-
-
-factorize(a**2 + b**2); 
-
-  {{A + I*B,1},{A - I*B,1}} 
-
-
-
-(x**2 + y**2)/(x + i*y); 
-
-  X - I*Y 
-
-
-
-on rounded; 
-
-      *** Domain mode COMPLEX changed to COMPLEX_FLOAT 
-
-
-
-sqrt(-17); 
-
-  4.12310562562*I 
-
-
-
-log(7*i); 
-
-  1.94591014906 + 1.57079632679*I
-
-

    Complex floating-point can be done by turning on -rounded in -addition to complex. With complex off however, REDUCE knows -that i is the square root of -1 but will not -carry out more complicated complex operations. If you want complex -denominators cleared by multiplication by their conjugates, turn on the -switch -rationalize. -

    -

    -

    - - - -CREF -INDEX

    - - - -CREF _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -The switch cref invokes the CREF cross-reference program that -processes a set of procedure definitions to produce a summary of their -entry points, undefined procedures, non-local variables and so on. The -program will also check that procedures are called with a consistent -number of arguments, and print a diagnostic message otherwise. -

    -

    -The output is alphabetized on the first seven characters of each function -name. -

    -

    -To invoke the cross-reference program, cref is first turned on. -This causes the program to load and the cross-referencing process to -begin. After all the required definitions are loaded, turning cref -off will cause a cross-reference listing to be produced. -

    -

    -Algebraic procedures in REDUCE are treated as if they were symbolic, so -that algebraic constructs will actually appear as calls to symbolic -functions, such as aeval. -

    -

    -

    - - - -CRAMER -INDEX

    - - - -CRAMER _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the cramer switch is on, -matrix inversion -and linear equation -solving (operator -solve) is done by Cramer's rule, through exterior -multiplication. Default is off. -

    -

    -

    -examples:

    -

    
-on time; 
-
-  Time: 80 ms 
-
-
-off output; 
-
-  Time: 100 ms 
-
-
-mm := mat((a,b,c,d,f),(a,a,c,f,b),(b,c,a,c,d), (c,c,a,b,f),
-          (d,a,d,e,f));
- 
-
-  Time: 300 ms 
-
-
-inverse := 1/mm; 
-
-  Time: 18460 ms 
-
-
-on cramer; 
-
-  Time: 80 ms 
-
-
-cramersinv := 1/mm; 
-
-  Time: 9260 ms
-
-

    Your time readings will vary depending on the REDUCE version you u -se. -After you invert the matrix, turn on -output and ask for one of -the elements of the inverse matrix, such as cramersinv(3,2), so that -you can see the size of the expressions produced. -

    -

    -Inversion of matrices and the solution of linear equations with dense -symbolic entries in many variables is generally considerably faster with -cramer on. However, inversion of numeric-valued matrices is -slower. Consider the matrices you're inverting before deciding whether to -turn cramer on or off. A substantial portion of the time in matrix -inversion is given to formatting the results for printing. To save this -time, turn output off, as shown in this example or terminate the -expression with a dollar sign instead of a semicolon. The results are -still available to you in the workspace associated with your prompt -number, or you can assign them to an identifier for further use. -

    -

    -

    - - - -DEFN -INDEX

    - - - -DEFN _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the switch defn is on, the Standard Lisp equivalent of the -input statement or procedure is printed, but not evaluated. Default is -off. -

    -

    -

    -examples:

    -

    
-
-on defn; 
-
-
-17/3; 
-
-  (AEVAL (LIST 'QUOTIENT 17 3)) 
-
-
-
-df(sin(x),x,2);          
- 
-
-  (AEVAL (LIST 'DF (LIST 'SIN 'X) 'X 2)) 
-
-
-procedure coshval(a);
-   begin scalar g;
-      g := (exp(a) + exp(-a))/2;
-      return g
-   end;
- 
-
-  (AEVAL 
-    (PROGN 
-      (FLAG '(COSHVAL) 'OPFN) 
-      (DE COSHVAL (A) 
-        (PROG (G) 
-          (SETQ G 
-            (AEVAL 
-               (LIST 
-                  'QUOTIENT 
-                  (LIST 
-                     'PLUS 
-                     (LIST 'EXP A) 
-                     (LIST 'EXP (LIST 'MINUS A))) 
-                  2))) 
-         (RETURN G)))) ) 
-
-
-
-coshval(1); 
-
-  (AEVAL (LIST 'COSHVAL 1)) 
-
-
-
-off defn; 
-
-
-coshval(1); 
-
-  Declare COSHVAL operator? (Y or N) 
-
-
-
-n 
-
-procedure coshval(a);
-   begin scalar g;
-      g := (exp(a) + exp(-a))/2;
-      return g
-   end;
- 
-
-  COSHVAL 
-
-
-
-on rounded; 
-
-
-coshval(1); 
-
-  1.54308063482
-
-

    The above function coshval finds the hyperbolic cosine (c -osh) of its -argument. When defn is on, you can see the Standard Lisp equivalent -of the function, but it is not entered into the system as shown by the -message Declare COSHVAL operator?. It must be reentered with -defn off to be recognized. This procedure is used as an example; a -more efficient procedure would eliminate the unnecessary local variable -with -

    
-      procedure coshval(a);
-         (exp(a) + exp(-a))/2;
-

    -

    -

    -

    - - - -DEMO -INDEX

    - - - -DEMO _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -The demo switch is used for interactive files, causing the system -to pause after each command in the file until you type a Return. -Default is off. -

    -

    -The switch demo has no effect on top level interactive -statements. Use it when you want to slow down operations in a file so -you can see what is happening. -

    -

    -You can either include the on demo command in the file, or enter -it from the top level before bringing in any file. Unlike the - -pause command, on demo does not permit you to - interrupt -the file for questions of your own. -

    -

    -

    -

    - - - -DFPRINT -INDEX

    - - - -DFPRINT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When dfprint is on, expressions in the differentiation operator - -df are printed in a more ``natural'' notation, with th -e -differentiation variables appearing as subscripts. In addition, if the -switch -noarg is on (the default), the arguments of the -differentiated operator are suppressed. -

    -

    -

    -examples:

    -

    
-operator f; 
-
-df(f x,x); 
-
-  DF(F(X),X); 
-
-
-on dfprint; 
-
-ws; 
-
-  F  
-   X
-
-
-df(f(x,y),x,y); 
-
-  F  
-   Y
-
-
-off noarg; 
-
-ws; 
-
-  F(X,Y)
-        X
-
-

    - - -DIV -INDEX

    - - - -DIV _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When div is on, the system divides any simple factors found in -the denominator of an expression into the numerator. Default is off. -

    -

    -

    -examples:

    -

    
-
-on div; 
-
-
-a := x**2/y**2; 
-
-        2   -2
-  A := X  *Y   
-
-
-
-b := a/(3*z); 
-
-       1  2   -2    -1
-  B := -*X  *Y    *Z   
-       3
-
-
-
-off div; 
-
-
-a; 
-
-   2
-  X
-  ---
-   2
-  Y
-
-
-
-b; 
-
-     2
-    X
-  -------                                       
-     2
-  3*Y  *Z
-
-

    The div switch only has effect when the -pri switch is on. -When pri is off, regardless of the setting of div, the -printing behavior is as if div were off. -

    -

    -

    - - - -ECHO -INDEX

    - - - -ECHO _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -The echo switch is normally off for top-level entry, and on when files - -are brought in. If echo is turned on at the top level, your input -statements are echoed to the screen (thus appearing twice). Default -off (but note default on for files). -

    -

    -If you want to display certain portions of a file and not others, use the -commands off echo and on echo inside the file. If you want -no display of the file, use the input command -

    -

    - in filename$ -

    -

    -rather than using the semicolon delimiter. -

    -

    -Be careful when you use commands within a file to generate another file. -Since echo is on for files, the output file echoes input statements -(unlike its behavior from the top level). You should explicitly turn off -echo when writing output, and turn it back on when you're done. -

    -

    -

    - - - -ERRCONT -INDEX

    - - - -ERRCONT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the errcont switch is on, error conditions do not stop file -execution. Error messages will be printed whether errcont is on or -off. -

    -

    -Default is off. -

    -

    -The following describes what happens when an error occurs in a file under -each setting of errcont and int: -

    -

    -Both off: Message is printed and parsing continues, but no further -statements are executed; no commands from keyboard accepted except bye or -end; -

    -

    -errcontoff, int on: Message is printed, and you are asked -if you wish to continue. (This is the default behavior); -

    -

    -errconton, int off: Message is printed, and file continues -to execute without pause; -

    -

    -Both on: Message is printed, and file continues to execute without pause. -

    -

    -

    -

    - - - -EVALLHSEQP -INDEX

    - - - -EVALLHSEQP _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -Under normal circumstances, the right-hand-side of an -equation -is evaluated but not the left-hand-side. This also applies to any -substitutions made by the -sub operator. If both sides are to be -evaluated, the switch evallhseqp should be turned on. -

    -

    - - - -EXP_switch -INDEX

    - - - -EXP _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the exp switch is on, powers and products of expressions are -expanded. Default is on. -

    -

    -

    -examples:

    -

    
-(x+1)**3; 
-
-   3      2
-  X  + 3*X  + 3*X + 1 
-
-
-(a + b*i)*(c + d*i); 
-
-  A*C + A*D*I + B*C*I - B*D 
-
-
-off exp; 
-
-(x+1)**3; 
-
-         3
-  (X + 1)  
-
-
-(a + b*i)*(c + d*i); 
-
-  (A + B*I)*(C + D*I) 
-
-
-length((x+1)**2/(y+1)); 
-
-  2
-
-

    Note that REDUCE knows that i^2 = -1. -When exp is off, equivalent expressions may not simplify to the same -form, although zero expressions still simplify to zero. Several operators -that expect a polynomial argument behave differently when exp is -off, such as -length. Be cautious about leaving exp off. -

    -

    -

    - - - -EXPANDLOGS -INDEX

    - - - -EXPANDLOGS _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -In many cases it is desirable to expand product arguments of logarithms, -or collect a sum of logarithms into a single logarithm. Since these are -inverse operations, it is not possible to provide rules for doing both at -the same time and preserve the REDUCE concept of idempotent evaluation. -As an alternative, REDUCE provides two switches expandlogs and - -combinelogs to carry out these operations. Both are of -f by default. -

    -examples:

    -

    
-on expandlogs; 
-
-log(x*y); 
-
-  LOG(X) + LOG(Y) 
-
-
-on combinelogs; 
-
-ws; 
-
-  LOG(X*Y)
-
-

    -

    -At the present time, it is possible to have both switches on at once, -which could lead to infinite recursion. However, an expression is -switched from one form to the other in this case. Users should not rely -on this behavior, since it may change in the next release. -

    -

    -

    - - - -EZGCD -INDEX

    - - - -EZGCD _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When ezgcd and -gcd are on, greatest common divisors are -computed using the EZ GCD algorithm that uses modular arithmetic (and is -usually faster). Default is off. -

    -

    -As a side effect of the gcd calculation, the expressions involved are -factored, though not the heavy-duty factoring of -factorize. The -EZ GCD algorithm was introduced in a paper by J. Moses and D.Y.Y. Yun in -<Proceedings of the ACM>, 1973, pp. 159-166. -

    -

    -Note that the -gcd switch must also be on for ezgcd to have - -effect. -

    -

    -

    - - - -FACTOR -INDEX

    - - - -FACTOR _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the factor switch is on, input expressions and results are -automatically factored. -

    -

    -

    -examples:

    -

    
-
-on factor; 
-
-
-aa := 3*x**3*a + 6*x**2*y*a + 3*x**3*b + 6*x**2*y*b
-
-+ x*y*a + 2*y**2*a + x*y*b + 2*y**2*b;
-			 
-
-
-                    2
-  AA := (A + B)*(3*X  + Y)*(X + 2*Y) 
-
-
-off factor; 
-
-aa; 
-
-       3        2                  2         3        2
-  3*A*X  + 6*A*X *Y + A*X*Y + 2*A*Y   + 3*B*X  + 6*B*X *Y
-
-
-+ B*X*Y + 2*B*Y^{2} 
-
-on factor; 
-
-ab := x**2 - 2; 
-
-         2
-  AB := X  - 2
-
-

    REDUCE factors univariate and multivariate polynomials with -integer coefficients, finding any factors that also have integer coefficients. -The factoring is done by reducing multivariate problems to univariate -ones with symbolic coefficients, and then solving the univariate ones modulo -small primes. The results of these calculations are merged to -determine the factors of the original polynomial. The factorizer normally -selects evaluation points and primes using a random number generator. -Thus, the detailed factoring behavior may be different each time any -particular problem is tackled. -

    -

    -When the factor switch is turned on, the -exp switch is -turned off, and when the factor switch is turned off, the - -exp switch is turned on, whether it was on previously -or not. -

    -

    -When the switch -trfac is on, informative messages are generated at -each call to the factorizer. The -trallfac switch causes the -production of a more verbose trace message. It takes precedence over -trfac if they are both on. -

    -

    -To factor a polynomial explicitly and store the results, use the operator - -factorize. -

    -

    -

    - - - -FAILHARD -INDEX

    - - - -FAILHARD _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the failhard switch is on, the integration operator -int -terminates with an error message if the integral cannot be done in closed -terms. -Default is off. -

    -

    -Use the failhard switch when you are dealing with complicated integrals - -and want to know immediately if REDUCE was unable to handle them. The -integration operator sometimes returns a formal integration form that is -more complicated than the original expression, when it is unable to -complete the integration. -

    -

    -

    - - - -FORT -INDEX

    - - - -FORT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When fort is on, output is given Fortran-compatible syntax. Default -is off. -

    -

    -

    -examples:

    -

    
-on fort; 
-
-df(sin(7*x + y),x); 
-
-  ANS=7.*COS(7*X+Y) 
-
-
-on rounded; 
-
-b := log(sin(pi/5 + n*pi)); 
-
-  	       B=LOG(SIN(3.14159265359*N+0.628318530718))
-
-

    REDUCE results can be written to a file (using -out) and used as data -by Fortran programs when fort is in effect. fort knows about -correct statement length, continuation characters, defining a symbol when -it is first used, and other Fortran details. -

    -

    -The -GENTRAN package offers many more possibilities than th -e -fort switch. It produces Fortran (or C or Ratfor) code from REDUCE -procedures or structured specifications, including facilities for producing -double precision output. -

    -

    -

    - - - -FORTUPPER -INDEX

    - - - -FORTUPPER _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When fortupper is on, any Fortran-style output appears in upper case. -Default is off. -

    -

    -

    -examples:

    -

    
-on fort; 
-
-df(sin(7*x + y),x); 
-
-  ans=7.*cos(7*x+y) 
-
-
-on fortupper; 
-
-df(sin(7*x + y),x); 
-
-  ANS=7.*COS(7*X+Y) 
-
-

    - - -FULLPREC -INDEX

    - - - -FULLPREC _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -Trailing zeroes of rounded numbers to the full system precision are -normally not printed. If this information is needed, for example to get a -more understandable indication of the accuracy of certain data, the switch -fullprec can be turned on. -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-1/2; 
-
-  0.5 
-
-
-on fullprec; 
-
-ws; 
-
-  0.500000000000
-
-

    This is just an output options which neither influences -the accuracy of the computation nor does it give additional -information about the precision of the results. -See also -scientific_notation. -

    -

    -

    - - - -FULLROOTS -INDEX

    - - - -FULLROOTS _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -Since roots of cubic and quartic polynomials can often be very -messy, a switch fullroots controls the production -of results in closed form. -solve will apply the -formulas for explicit forms for degrees 3 and 4 only if -fullroots is on. Otherwise the result forms -are built using -root_of. Default is off. -

    -

    - - - -GC -INDEX

    - - - -GC _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -With the gc switch, you can turn the garbage collection messages on -or off. The form of the message depends on the particular Lisp used for -the REDUCE implementation. -

    -

    -See -reclaim for an explanation of garbage collection. REDU -CE does -garbage collection when needed even if you have turned the notices off. -

    -

    -

    - - - -GCD_switch -INDEX

    - - - -GCD _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When gcd is on, common factors in numerators and denominators of -expressions are canceled. Default is off. -

    -

    -

    -examples:

    -

    
-
-(2*(f*h)**2 - f**2*g*h - (f*g)**2 - f*h**3 + f*h*g**2
-   - h**4 + g*h**3)/(f**2*h - f**2*g - f*h**2 + 2*f*g*h
-   - f*g**2 - g*h**2 + g**2*h);
- 
-
-   2  2    2          2  2      2        3      3    4
-  F *G  + F *G*H - 2*F *H  - F*G *H + F*H  - G*H  + H
-  ---------------------------------------------------- 
-    2      2        2                2    2        2
-   F *G - F *H + F*G  - 2*F*G*H + F*H  - G *H + G*H
-
-
-on gcd; 
-
-ws; 
-
-                 2
-  F*G + 2*F*H + H
-  ---------------- 
-       F + G
-
-
-e2 := a*c + a*d + b*c + b*d; 
-
-  E2 := A*C + A*D + B*C + B*D 
-
-
-off exp; 
-
-e2; 
-
-  (A + B)*(C + D)
-
-

    Even with gcd off, a check is automatically made for comm -on variable -and numerical products in the numerators and denominators of expression, -and the appropriate cancellations made. Thus the example demonstrating the -use of gcd is somewhat complicated. Note when -exp is off, -gcd has the side effect of factoring the expression. -

    -

    -

    - - - -HORNER -INDEX

    - - - -HORNER _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the horner switch is on, polynomial expressions are printed -in Horner's form for faster and safer numerical evaluation. Default -is off. The leading variable of the expression is selected as -Horner variable. To select the Horner variable explicitly use the - -korder declaration. -

    -

    -

    -examples:

    -

    
-on horner;
-
-(13p-4q)^3;
-
-           3            2
-  ( - 64)*q   + p*(624*q   + p*(( - 2028)*q + p*2197))
-
-
-korder q;
-
-ws;
-
-        3                  2
-  2197*p   + q*(( - 2028)*p   + q*(624*p + q*(-64)))
-
-

    - - -IFACTOR -INDEX

    - - - -IFACTOR _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the ifactor switch is on, any integer terms appearing as a result - -of the -factorize command are factored themselves into primes. - Default -is off. If the argument of factorize is an integer, -ifactor has no effect, since the integer is always factored. -

    -

    -

    -examples:

    -

    
-factorize(4*x**2 + 28*x + 48); 
-
-  {{4,1},{X + 4,1},{X + 3,1}} 
-
-
-factorize(22587); 
-
-  {{3,1},{7529,1}} 
-
-
-on ifactor; 
-
-factorize(4*x**2 + 28*x + 48); 
-
-  {{2,2},{X + 4,1},{X + 3,1}} 
-
-
-factorize(22587); 
-
-  {{3,1},{7529,1}} 
-
-

    Constant terms that appear within nonconstant -polynomial factors are not factored. -

    -

    -The ifactor switch affects only factoring done specifically -with -factorize, not on factoring done automatically when th -e - -factor switch is on. -

    -

    -

    - - - -INT_switch -INDEX

    - - - -INT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -The int switch specifies an interactive mode of operation. Default -on. -

    -

    -There is no reason to turn int off during interactive calculations, -since there are no benefits to be gained. If you do have int off -while inputting a file, and REDUCE finds an error, it prints the message -``Continuing with parsing only." In this state, REDUCE accepts only - -end; or -bye; from the keyboard; -everything else is ignored, even the command on int. -

    -

    -

    - - - -INTSTR -INDEX

    - - - -INTSTR _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -If intstr (for ``internal structure'') is on, arguments of an -operator are printed in a more structured form. -

    -

    -

    -examples:

    -

    
-operator f; 
-
-f(2x+2y); 
-
-  F(2*X + 2*Y) 
-
-
-on intstr; 
-
-ws; 
-
-  F(2*(X + Y))
-
-

    - - -LCM -INDEX

    - - - -LCM _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -The lcm switch instructs REDUCE to compute the least common multiple -of denominators whenever rational expressions occur. Default is on. -

    -

    -

    -examples:

    -

    
-off lcm; 
-
-z := 1/(x**2 - y**2) + 1/(x-y)**2;  
-			 
-
-
-              2*X*(X - Y)
-  Z := ------------------------- 
-        4      3          3    4
-       X  - 2*X *Y + 2*X*Y  - Y
-
-
-on lcm; 
-
-z; 
-
-         2*X*(X - Y)
-  ------------------------- 
-   4      3          3    4
-  X  - 2*X *Y + 2*X*Y  - Y
-
-
-zz := 1/(x**2 - y**2) + 1/(x-y)**2;
-			 
-
-
-                 2*X
-  ZZ := --------------------- 
-         3    2        2    3
-        X  - X *Y - X*Y  + Y
-
-
-on gcd; 
-
-z; 
-
-           2*X
-  ----------------------
-   3    2        2    3
-  X  - X *Y - X*Y  + Y
-
-

    Note that lcm has effect only when rational expressions a -re first -combined. It does not examine existing structures for simplifications on -display. That is shown above when z is entered with -lcm off. It remains unsimplified even after lcm is turned -back on. However, a new variable containing the same expression is -simplified on entry. The switch -gcd does examine existing -structures, as shown in the last example line above. -

    -

    -Full greatest common divisor calculations become expensive if work with -large rational expressions is required. A considerable savings of time -can be had if a full gcd check is made only when denominators are combined, -and only a partial check for numerators. This is the effect of the lcm - -switch. -

    -

    -

    - - - -LESSSPACE -INDEX

    - - - -LESSSPACE _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -You can turn on the switch lessspace if you want fewer -blank lines in your output. -

    -

    - - - -LIMITEDFACTORS -INDEX

    - - - -LIMITEDFACTORS _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -To get limited factorization in cases where it is too expensive to use -full multivariate polynomial factorization, the switch -limitedfactors can be turned on. In that case, only ``inexpensive'' -factoring operations, such as square-free factorization, will be used -when -factorize is called. -

    -

    -

    -examples:

    -

    
-a := (y-x)^2*(y^3+2x*y+5)*(y^2-3x*y+7)$ 
-
-factorize a; 
-
-              2
-  {- 3*X*Y + Y  + 7,1}
-            3
-  {2*X*Y + Y  + 5,1},
-  {X - Y,2}}
-
-
-on limitedfactors; 
-
-factorize a; 
-
-        2  2        4        3          5      3      2
-  {- 6*X *Y  - 3*X*Y  + 2*X*Y  - X*Y + Y  + 7*Y  + 5*Y  + 35,1},
-  {X - Y,2}}
-
-

    - - -LIST_switch -INDEX

    - - - -LIST _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -The list switch causes REDUCE to print each term in any sum on -separate lines. -

    -

    -

    -examples:

    -

    
-x**2*(y**2 + 2*y) + x*(y**2 + z)/(2*a);
-			 
-
-
-            2              2
-  X*(2*A*X*Y  + 4*A*X*Y + Y  +Z)
-  ------------------------------ 
-               2*A
-
-
-on list; 
-
-ws; 
-
-             2
-  (X*(2*A*X*Y
-    + 4*A*X*Y
-       2
-    + Y
-    + Z))/(2*A)
-
-

    - - -LISTARGS -INDEX

    - - - -LISTARGS _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -If an operator other than those specifically defined for lists is given a -single argument that is a list, then the result of this operation will be -a list in which that operator is applied to each element of the list. -This process can be inhibited globally by turning on the switch -listargs. -

    -

    -

    -examples:

    -

    
-log {a,b,c}; 
-
-  LOG(A),LOG(B),LOG(C) 
-
-
-on listargs; 
-
-log {a,b,c}; 
-
-  LOG(A,B,C)
-
-

    It is possible to inhibit such distribution for a specific operato -r by -using the declaration -listargp. In addition, if an operator has -more than one argument, no such distribution occurs, so listargs -has no effect. -

    -

    -

    - - - -MCD -INDEX

    - - - -MCD _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When mcd is on, sums and differences of rational expressions are put -on a common denominator. Default is on. -

    -

    -

    -examples:

    -

    
-a/(x+1) + b/5; 
-
-  5*A + B*X + B
-  ------------- 
-    5*(X + 1)
-
-
-off mcd; 
-
-a/(x+1) + b/5; 
-
-         -1
-  (X + 1)  *A + 1/5*B 
-
-
-1/6 + 1/7; 
-
-  13/42
-
-

    Even with mcd off, rational expressions involving only nu -mbers are still -put over a common denominator. -

    -

    -Turning mcd off is useful when explicit negative powers are needed, -or if no greatest common divisor calculations are desired, or when -differentiating complicated rational expressions. Results when mcd -is off are no longer in canonical form, and expressions equivalent to zero -may not simplify to 0. Some operations, such as factoring cannot be done -while mcd is off. This option should therefore be used with some -caution. Turning mcd off is most valuable in intermediate parts of -a complicated calculation, and should be turned back on for the last stage. -

    -

    -

    - - - -MODULAR -INDEX

    - - - -MODULAR _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When modular is on, polynomial coefficients are reduced by the -modulus set by -setmod. If no modulus has been set, modular -has no effect. -

    -

    -

    -examples:

    -

    
-setmod 2; 
-
-  1 
-
-
-on modular; 
-
-(x+y)**2; 
-
-   2    2
-  X  + Y  
-
-
-145*x**2 + 20*x**3 + 17 + 15*x*y;
-			 
-
-
-   2
-  X  + X*Y + 1
-
-

    Modular operations are only conducted on the coefficients, not the - -exponents. The modulus is not restricted to being prime. When the modulus -is prime, division by a number not relatively prime to the modulus results -in a <Zero divisor> error message. When the modulus is a composite -number, division by a power of the modulus results in an error message, but -division by an integer which is a factor of the modulus does not. -The representation of modular number can be influenced by - -balanced_mod. -

    -

    -

    - - - -MSG -INDEX

    - - - -MSG _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When msg is off, the printing of warning messages is suppressed. Error - -messages are still printed. -

    -

    -Warning messages include those about redimensioning an -array -or declaring an -operator where one is expected. -

    -

    -

    - - - -MULTIPLICITIES -INDEX

    - - - -MULTIPLICITIES _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When -solve is applied to a set of equations with multiple r -oots, -solution multiplicities are normally stored in the global variable - -root_multiplicities rather than the solution list. If -you want -the multiplicities explicitly displayed, the switch multiplicities -should be turned on. In this case, root_multiplicities has no value. -

    -

    -

    -examples:

    -

    
-solve(x^2=2x-1,x); 
-
-  X=1 
-
-
-root_multiplicities; 
-
-  2 
-
-
-on multiplicities; 
-
-solve(x^2=2x-1,x); 
-
-  X=1,X=1 
-
-
-root_multiplicities; 
-
-

    - - -NAT -INDEX

    - - - -NAT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When nat is on, output is printed to the screen in natural form, with -raised exponents. nat should be turned off when outputting expressions - -to a file for future input. Default is on. -

    -

    -

    -examples:

    -

    
-(x + y)**3; 
-
-   3      2          2    3
-  X  + 3*X *Y + 3*X*Y  + Y  
-
-
-off nat; 
-
-(x + y)**3; 
-
-  X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3$ 
-
-
-on fort; 
-
-(x + y)**3; 
-
-  ANS=X**3+3.*X**2*Y+3.*X*Y**2+Y**3
-
-

    With nat off, a dollar sign is printed at the end of each - expression. -An output file written with nat off is ready to be read into REDUCE -using the command -in. -

    -

    -

    - - - -NERO -INDEX

    - - - -NERO _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When nero is on, zero assignments (such as matrix elements) are not -printed. -

    -

    -

    -examples:

    -

    
-matrix a;
-a := mat((1,0),(0,1)); 
-
-  A(1,1) := 1
-  A(1,2) := 0
-  A(2,1) := 0
-  A(2,2) := 1
-
-
-on nero; 
-
-a; 
-
-  MAT(1,1) := 1
-  MAT(2,2) := 1
-
-
-a(1,2);

    nothing is printed.

    
-
-
-b := 0;

    nothing is printed.

    
-
-
-off nero; 
-
-b := 0; 
-
-  B := 0
-
-

    -

    -

    -nerois often used when dealing with large sparse matrices, to avoid -being overloaded with zero assignments. -

    -

    -

    - - - -NOARG -INDEX

    - - - -NOARG _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When -dfprint is on, expressions in the differentiation oper -ator - -df are printed in a more ``natural'' notation, with th -e -differentiation variables appearing as subscripts. When noarg -is on (the default), the arguments of the differentiated operator are also -suppressed. -

    -

    -

    -examples:

    -

    
-operator f; 
-
-df(f x,x); 
-
-  DF(F(X),X); 
-
-
-on dfprint; 
-
-ws; 
-
-  F  
-   X
-
-
-off noarg; 
-
-ws; 
-
-  F(X)
-      X
-
-

    - - -NOLNR -INDEX

    - - - -NOLNR _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When nolnr is on, the linear properties of the integration operator - -int are suppressed if the integral cannot be found in -closed terms. -

    -

    -REDUCE uses the linear properties of integration to attempt to break down -an integral into manageable pieces. If an integral cannot be found in -closed terms, these pieces are returned. When the nolnr switch is off, - -as many of the pieces as possible are integrated. When it is on, if any piece -fails, the rest of them remain unevaluated. -

    -

    -

    - - - -NOSPLIT -INDEX

    - - - -NOSPLIT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -Under normal circumstances, the printing routines try to break an expression -across lines at a natural point. This is a fairly expensive process. If -you are not overly concerned about where the end-of-line breaks come, you -can speed up the printing of expressions by turning off the switch -nosplit. This switch is normally on. -

    -

    - - - -NUMVAL -INDEX

    - - - -NUMVAL _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -With -rounded on, elementary functions with numerical argume -nts -will return a numerical answer where appropriate. If you wish to inhibit -this evaluation, numval should be turned off. It is normally on. -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-cos 3.4; 
-
-  - 0.966798192579 
-
-
-off numval; 
-
-cos 3.4; 
-
-  COS(3.4)
-
-

    - - -OUTPUT -INDEX

    - - - -OUTPUT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When output is off, no output is printed from any REDUCE calculation. -The calculations have their usual effects other than printing. Default is -on. -

    -

    -Turn output off if you do not wish to see output when executing -large files, or to save the time REDUCE spends formatting large expressions -for display. Results are still available with -ws, or in their -assigned variables. -

    -

    -

    - - - -OVERVIEW -INDEX

    - - - -OVERVIEW _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When overview is on, the amount of detail reported by the factorizer -switches -trfac and -trallfac is reduced. -

    -

    - - - -PERIOD -INDEX

    - - - -PERIOD _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When period is on, periods are added after integers in -Fortran-compatible output (when -fort is on). There is no effect -when fort is off. Default is on. -

    -

    - - - -PRECISE -INDEX

    - - - -PRECISE _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the precise switch is on, simplification of roots of even -powers returns absolute values, a more precise answer mathematically. -Default is on. -

    -

    -

    -examples:

    -

    
-sqrt(x**2); 
-
-  X 
-
-
-(x**2)**(1/4); 
-
-  SQRT(X) 
-
-
-on precise; 
-
-sqrt(x**2); 
-
-  ABS(X) 
-
-
-(x**2)**(1/4); 
-
-  SQRT(ABS(X))
-
-

    In many types of mathematical work, simplification of powers and s -urds can -proceed by the fastest means of simplifying the exponents arithmetically. -When it is important to you that the positive root be returned, turn -precise on. One situation where this is important is when graphing -square-root expressions such as sqrt(x^2+y^2) to -avoid a spike caused by REDUCE simplifying -sqrt(y^2) to y when x is -zero. -

    -

    -

    - - - -PRET -INDEX

    - - - -PRET _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When pret is on, input is printed in standard REDUCE format and then -evaluated. -

    -

    -

    -examples:

    -

    
-on pret; 
-
- (x+1)^3; 
-
-   (x + 1)**3;
-   3      2
-  X  + 3*X  + 3*X + 1
-
-
-
-procedure fac(n);
-   if not (fixp(n) and n>=0)
-     then rederr "Choose nonneg. integer only"
-    else for i := 0:n-1 product i+1;
- 
-
-  procedure fac n;
-     if not (fixp n and n>=0)
-       then rederr "Choose nonneg. integer only"
-      else for i := 0:n - 1 product i + 1;
-  FAC
-
-
-
-fac 5; 
-
-  fac 5;
-  120
-
-

    Note that all input is converted to lower case except strings (whi -ch keep -the same case) all operators with a single argument have had the -parentheses removed, and all infix operators have had a space added on each -side. In addition, syntactical constructs like -if...then...else are printed in a standard format. -

    -

    -

    - - - -PRI -INDEX

    - - - -PRI _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When pri is on, the declarations -order and -factor can -be used, and the switches -allfac, -div, -rat, -and -revpri take effect when they are on. Default is on -. -

    -

    -Printing of expressions is faster with pri off. The expressions are -then returned in one standard form, without any of the display options that -can be used to feature or display various parts of the expression. You can -also gain insight into REDUCE's representation of expressions with -pri off. -

    -

    -

    - - - -RAISE -INDEX

    - - - -RAISE _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When raise is on, lower case letters are automatically converted to -upper case on input. raise is normally on. -

    -

    -This conversion affects the internal representation of the letter, and is -independent of the case with which a letter is printed, which is normally -lower case. -

    -

    -

    - - - -RAT -INDEX

    - - - -RAT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the rat switch is on, and kernels have been selected to display -with the -factor declaration, the denominator is printed with ea -ch -term rather than one common denominator at the end of an expression. -

    -

    -

    -examples:

    -

    
-(x+1)/x + x**2/sin y;        
- 
-
-                       3
-  SIN(Y)*X + SIN(Y) + X
-  ---------------------- factor x; 
-         SIN(Y)*X
-
-
-(x+1)/x + x**2/sin y;        
- 
-
-   3
-  X  + X*SIN(Y) + SIN(Y)
-  ---------------------- on rat;  
-         X*SIN(Y)
-
-
-(x+1)/x + x**2/sin y;       
- 
-
-     2
-    X           -1
-  ------ + 1 + X
-  SIN(Y)
-
-

    The rat switch only has effect when the -pri switch is on. -When pri is off, regardless of the setting of rat, the -printing behavior is as if rat were off. rat only has -effect upon the display of expressions, not their internal form. -

    -

    -

    - - - -RATARG -INDEX

    - - - -RATARG _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When ratarg is on, rational expressions can be given to operators -such as -coeff and -lterm that normally require -polynomials in one of their arguments. When ratarg is off, rational -expressions cause an error message. -

    -

    -

    -examples:

    -

    
-aa := x/y**2 + 1/x + y/x**2; 
- 
-
-         3      2    3
-        X  + X*Y  + Y
-  AA := -------------- 
-             2  2
-            X *Y
-
-
-coeff(aa,x); 
-
-         3      2    3
-        X  + X*Y  + Y
-  ***** -------------- invalid as POLYNOMIAL
-             2  2
-            X *Y
-
-
-on ratarg; 
-
-coeff(aa,x);                
- 
-
-   Y  1      1
-  {--,--,0,-----}
-    2  2    2  2
-   X  X    X *Y
-
-

    - - -RATIONAL -INDEX

    - - - -RATIONAL _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When rational is on, polynomial expressions with rational coefficients - -are produced. -

    -

    -

    -examples:

    -

    
-x/2 + 3*y/4; 
-
-  2*X + 3*Y
-  --------- 
-      4
-
-
-(x**2 + 5*x + 17)/2; 
-
-   2
-  X  + 5*X + 17
-  ------------- 
-        2
-
-
-on rational; 
-
-x/2 + 3y/4; 
-
-  1      3
-  -*(X + -*Y) 
-  2      2
-
-
-(x**2 + 5*x + 17)/2; 
-
-  1   2
-  -*(X  + 5*X + 17)
-  2
-
-

    By using rational, polynomial expressions with rational -coefficients can be used in some commands that expect polynomials. With -rational off, such a polynomial becomes a rational expression, with -denominator the least common multiple of the denominators of the rational -number coefficients.

    -

    - -

    -

    - - - -RATIONALIZE -INDEX

    - - - -RATIONALIZE _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the rationalize switch is on, denominators of rational expressions - -that contain complex numbers or root expressions are simplified by -multiplication by their conjugates. -

    -

    -

    -examples:

    -

    
-qq := (1+sqrt(3))/(sqrt(3)-7); 
-
-        SQRT(3) + 1
-  QQ := ----------- 
-        SQRT(3) - 7
-
-
-on rationalize; 
-
-qq; 
-
-  - 4*SQRT(3) - 5
-  --------------- 
-        23
-
-
-2/(4 + 6**(1/3)); 
-
-   2/3      1/3
-  6    - 4*6    + 16
-  ------------------ 
-          35
-
-
-(i-1)/(i+3); 
-
-  2*I - 1
-  ------- 
-     5
-
-
-off rationalize; 
-
-(i-1)/(i+3); 
-
-  I - 1
-  ------
-  I + 3
-
-

    - - -RATPRI -INDEX

    - - - -RATPRI _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the ratpri switch is on, rational expressions and fractions are -printed as two lines separated by a fraction bar, rather than in a linear -style. Default is on. -

    -

    -

    -examples:

    -

    
-3/17; 
-
-  3
-  -- 
-  17
-
-
-2/b + 3/y; 
-
-  3*B + 2*Y
-  --------- 
-     B*Y
-
-
-off ratpri; 
-
-3/17; 
-
-  3/17 
-
-
-2/b + 3/y; 
-
-  (3*B + 2*Y)/(B*Y)
-
-

    - - -REVPRI -INDEX

    - - - -REVPRI _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When the revpri switch is on, terms are printed in reverse order from -the normal printing order. -

    -

    -

    -examples:

    -

    
-x**5 + x**2 + 18 + sqrt(y); 
-
-             5    2
-  SQRT(Y) + X  + X  + 18 
-
-
-a + b + c + w; 
-
-  A + B + C + W 
-
-
-on revpri; 
-
-x**5 + x**2 + 18 + sqrt(y); 
-
-        2    5
-  17 + X  + X  + SQRT(Y) 
-
-
-a + b + c + w; 
-
-  W + C + B + A
-
-

    Turn revpri on when you want to display a polynomial in a -scending -rather than descending order. -

    -

    -

    - - - -RLISP88 -INDEX

    - - - -RLISP88 _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -Rlisp '88 is a superset of the Rlisp that has been traditionally used for -the support of REDUCE. It is fully documented in the book Marti, J.B., -``RLISP '88: An Evolutionary Approach to Program Design and Reuse'', -World Scientific, Singapore (1993). It supports different looping -constructs from the traditional Rlisp, and treats ``-'' as a letter unless -separated by spaces. Turning on the switch rlisp88 converts to -Rlisp '88 parsing conventions in symbolic mode, and enables the use of -Rlisp '88 extensions. Turning off the switch reverts to the traditional -Rlisp and the previous mode ( ( -symbolic or -algebraic) -in force before rlisp88 was turned on. -

    -

    - - - -ROUNDALL -INDEX

    - - - -ROUNDALL _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -In -rounded mode, rational numbers are normally converted -to a -floating point representation. If roundall is off, this conversion -does not occur. roundall is normally on. -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-1/2; 
-
-  0.5 
-
-
-off roundall; 
-

    - - -ROUNDBF -INDEX

    - - - -ROUNDBF _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -When -rounded is on, the normal defaults cause underflows to - be -converted to zero. If you really want the small number that results in -such cases, roundbf can be turned on. -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-exp(-100000.1^2); 
-
-  0 
-
-
-on roundbf; 
-
-exp(-100000.1^2); 
-
-  1.18441281937E-4342953505
-
-

    If a polynomial is input in -rounded mode at the default -precision into any -roots function, and it is not possible to -represent any of the coefficients of the polynomial precisely in the -system floating point representation, the switch roundbf will be -automatically turned on. All rounded computation will use the internal -bigfloat representation until the user subsequently turns roundbf -off. (A message is output to indicate that this condition is in effect.) -

    -

    -

    - - - -ROUNDED -INDEX

    - - - -ROUNDED _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When rounded is on, floating-point arithmetic is enabled, with -precision initially at a system default value, which is usually 12 digits. -The precise number can be found by the command -precision(0). -

    -examples:

    -

    
-pi; 
-
-  PI 
-
-
-35/217; 
-
-  5
-  -- 
-  31
-
-
-on rounded; 
-
-pi; 
-
-  3.14159265359 
-
-
-35/217; 
-
-  0.161 
-
-
-sqrt(3); 
-
-  1.73205080756
-
-

    -

    -If more than the default number of decimal places are required, use the - -precision command to set the required number. -

    -

    -

    - - - -SAVESTRUCTR -INDEX

    - - - -SAVESTRUCTR _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When savestructr is on, results of the -structr command are -returned as a list whose first element is the representation for the -expression and the remaining elements are equations showing the -relationships of the generated variables. -

    -

    -

    -examples:

    -

    
-off exp; 
-
-structr((x+y)^3 + sin(x)^2); 
-
-  ANS3
-     where
-                    3       2
-        ANS3 := ANS1  + ANS2
-        ANS2 := SIN(X)
-        ANS1 := X + Y
-
-
-ans3; 
-
-  ANS3 
-
-
-on savestructr; 
-
-structr((x+y)^{3} + sin(x)^{2}); 
-
-                3       2
-  ANS3,ANS3=ANS1  + ANS2 ,ANS2=SIN(X),ANS1=X + Y 
-
-
-ans3 where rest ws; 
-
-         3         2
-  (X + Y)  + SIN(X)
-
-

    In normal operation, -structr is only a display command. With -savestructr on, you can access the various parts of the expression -produced by structr. -

    -

    -The generic system names use the stem ANS. You can change this to your - -own stem by the command -varname. REDUCE adds integers to this stem -to make unique identifiers. -

    -

    -

    - - - -SOLVESINGULAR -INDEX

    - - - -SOLVESINGULAR _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When solvesingular is on, singular or underdetermined systems of -linear equations are solved, using arbitrary real, complex or integer -variables in the answer. Default is on. -

    -

    -

    -examples:

    -

    
-solve({2x + y,4x + 2y},{x,y}); 
-
-         ARBCOMPLEX(1)
-  {{X= - -------------,Y=ARBCOMPLEX(1)}} 
-               2
-
-
-solve({7x + 15y - z,x - y - z},{x,y,z}); 
-
-
-      8*ARBCOMPLEX(3)
-  {{X=----------------
-            11
-         3*ARBCOMPLEX(3)
-    Y= - ----------------
-               11
-    Z=ARBCOMPLEX(3)}}
-
-
-off solvesingular; 
-
-solve({2x + y,4x + 2y},{x,y}); 
-
-  ***** SOLVE given singular equations 
-
-
-solve({7x + 15y - z,x - y - z},{x,y,z}); 
-
-
-  ***** SOLVE given singular equations
-
-

    The integer following the identifier -arbcomplex above is assigned by -the system, and serves to identify the variable uniquely. It has no other -significance. -

    -

    -

    - - - -TIME -INDEX

    - - - -TIME _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When time is on, the system time used in executing each REDUCE -statement is printed after the answer is printed. -

    -

    -

    -examples:

    -

    
-on time; 
-
-  Time: 4940 ms 
-
-
-df(sin(x**2 + y),y); 
-
-            2
-  COS(X  + Y )
-  Time: 180 ms
-
-
-solve(x**2 - 6*y,x); 
-
-  {X= - SQRT(Y)*SQRT(6),
-   X=SQRT(Y)*SQRT(6)}
-  Time: 320 ms
-
-

    When time is first turned on, the time since the beginnin -g of the -REDUCE session is printed. After that, the time used in computation, -(usually in milliseconds, though this is system dependent) is printed after -the results of each command. Idle time or time spent typing in commands is -not counted. If time is turned off, the first reading after it is -turned on again gives the time elapsed since it was turned off. The time -printed is CPU or wall clock time, depending on the system. -

    -

    -

    - - - -TRALLFAC -INDEX

    - - - -TRALLFAC _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When trallfac is on, a more detailed trace of factorizer calls is -generated. -

    -

    -The trallfac switch takes precedence over -trfac if they are -both on. trfac gives a factorization trace with less detail in it. -When the -factor switch is on also, all input polynomials are se -nt to -the factorizer automatically and trace information is generated. The - -out command saves the results of the factoring, but no -t the trace. -

    -

    -

    - - - -TRFAC -INDEX

    - - - -TRFAC _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When trfac is on, a narrative trace of any calls to the factorizer is -generated. Default is off. -

    -

    -When the switch -factor is on, and trfac is on, every input -polynomial is sent to the factorizer, and a trace generated. With -factor off, only polynomials that are explicitly factored with the -command -factorize generate trace information. -

    -

    -The -out command saves the results of the factoring, but no -t -the trace. The -trallfac switch gives trace information to a -greater level of detail. -

    -

    -

    - - - -TRIGFORM -INDEX

    - - - -TRIGFORM _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When -fullroots is on, -solve will compute the -roots of a cubic or quartic polynomial is closed form. When -trigform is on, the roots will be expressed by trigonometric -forms. Otherwise nested surds are used. Default is on. -

    -

    - - - -TRINT -INDEX

    - - - -TRINT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When trint is on, a narrative tracing various steps in the -integration process is produced. -

    -

    -The -out command saves the results of the integration, but -not the -trace. -

    -

    -

    - - - -TRNONLNR -INDEX

    - - - -TRNONLNR _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When trnonlnr is on, a narrative tracing various steps in -the process for solving non-linear equations is produced. -

    -

    -trnonlnrcan only be used after the solve package has been loaded -(e.g., by an explicit call of -load_package). The -out -command saves the results of the equation solving, but not the trace. -

    -

    -

    - - - -VAROPT -INDEX

    - - - -VAROPT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -When varopt is on, the sequence of variables is optimized by - -solve with respect to execution speed. Otherwise, the -sequence -given in the call to -solve is preserved. Default is on. -

    -

    -In combination with the switch -arbvars, varopt can be used -to control variable elimination. -

    -

    -

    -examples:

    -

    
-off arbvars; 
-
-solve({x+2z,x-3y},{x,y,z});				
-
-           x      x
-  		   {{y=-,z= - -}} 
-           3      2
-
-
-solve({x*y=1,z=x},{x,y,z});				
-
-               1
-  		   {{z=x,y=-}} 
-               x
-
-
-off varopt; 
-
-solve({x+2z,x-3y},{x,y,z});				
-
-                       2*z
-  		   {{x= - 2*z,y= - ---}} 
-                        3
-
-
-solve({x*y=1,z=x},{x,y,z});				
-
-           1
-  		   {{y=-,x=z}} 
-           z
-
-

    - - -General Switches -INDEX

    -General Switches

    -
  • SWITCHES introduction

    -

  • ALGINT switch

    -

  • ALLBRANCH switch

    -

  • ALLFAC switch

    -

  • ARBVARS switch

    -

  • BALANCED\_MOD switch

    -

  • BFSPACE switch

    -

  • COMBINEEXPT switch

    -

  • COMBINELOGS switch

    -

  • COMP switch

    -

  • COMPLEX switch

    -

  • CREF switch

    -

  • CRAMER switch

    -

  • DEFN switch

    -

  • DEMO switch

    -

  • DFPRINT switch

    -

  • DIV switch

    -

  • ECHO switch

    -

  • ERRCONT switch

    -

  • EVALLHSEQP switch

    -

  • EXP switch

    -

  • EXPANDLOGS switch

    -

  • EZGCD switch

    -

  • FACTOR switch

    -

  • FAILHARD switch

    -

  • FORT switch

    -

  • FORTUPPER switch

    -

  • FULLPREC switch

    -

  • FULLROOTS switch

    -

  • GC switch

    -

  • GCD switch

    -

  • HORNER switch

    -

  • IFACTOR switch

    -

  • INT switch

    -

  • INTSTR switch

    -

  • LCM switch

    -

  • LESSSPACE switch

    -

  • LIMITEDFACTORS switch

    -

  • LIST switch

    -

  • LISTARGS switch

    -

  • MCD switch

    -

  • MODULAR switch

    -

  • MSG switch

    -

  • MULTIPLICITIES switch

    -

  • NAT switch

    -

  • NERO switch

    -

  • NOARG switch

    -

  • NOLNR switch

    -

  • NOSPLIT switch

    -

  • NUMVAL switch

    -

  • OUTPUT switch

    -

  • OVERVIEW switch

    -

  • PERIOD switch

    -

  • PRECISE switch

    -

  • PRET switch

    -

  • PRI switch

    -

  • RAISE switch

    -

  • RAT switch

    -

  • RATARG switch

    -

  • RATIONAL switch

    -

  • RATIONALIZE switch

    -

  • RATPRI switch

    -

  • REVPRI switch

    -

  • RLISP88 switch

    -

  • ROUNDALL switch

    -

  • ROUNDBF switch

    -

  • ROUNDED switch

    -

  • SAVESTRUCTR switch

    -

  • SOLVESINGULAR switch

    -

  • TIME switch

    -

  • TRALLFAC switch

    -

  • TRFAC switch

    -

  • TRIGFORM switch

    -

  • TRINT switch

    -

  • TRNONLNR switch

    -

  • VAROPT switch

    -

    •  - - -COFACTOR -INDEX

    - - - -COFACTOR _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator cofactor returns the cofactor of the element in row -<row> and column <column> of a -matrix. Errors occur -if <row> or <column> do not evaluate to integer expressions or if -the matrix is not square. -

    -

    -

    -syntax:

    -cofactor(<matrix\_expression>,<row>,<column>) -

    -

    -

    -

    -examples:

    -

    
-cofactor(mat((a,b,c),(d,e,f),(p,q,r)),2,2); 
-
-
-  A*R - C*P 
-
-
-cofactor(mat((a,b,c),(d,e,f)),1,1); 
-
-
-  ***** non-square matrix
-
-

    - - -DET -INDEX

    - - - -DET _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The det operator returns the determinant of its -(square -matrix) argument. -

    -

    -

    -syntax:

    -det(<expression>) or det <expression> -

    -

    -

    -<expression> must evaluate to a square matrix. -

    -

    -

    -examples:

    -

    
-
-matrix m,n; 
-
-
-m := mat((a,b),(c,d)); 
-
-  M(1,1) := A
-  M(1,2) := B
-  M(2,1) := C
-  M(2,2) := D
-                                 
-
-
-det m; 
-
-  A*D - B*C 
-
-
-n := mat((1,2),(1,2)); 
-
-  N(1,1) := 1
-  N(1,2) := 2
-  N(2,1) := 1
-  N(2,2) := 2
-                                 
-
-
-
-det(n); 
-
-  0 
-
-
-
-det(5); 
-
-  5
-
-

    Given a numerical argument, det returns the number. Howev -er, given a -variable name that has not been declared of type matrix, or a non-square -matrix, det returns an error message. -

    -

    -

    - - - -MAT -INDEX

    - - - -MAT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The mat operator is used to represent a two-dimensional - -matrix. -

    -syntax:

    -

    -

    -mat((<expr>{,<expr>}*) {(<expr>{,<expr ->}*)}*) -

    -

    -

    -<expr> may be any valid REDUCE scalar expression. -

    -

    -

    -examples:

    -

    
-mat((1,2),(3,4)); 
-
-  MAT(1,1) := 1
-  MAT(2,3) := 2
-  MAT(2,1) := 3
-  MAT(2,2) := 4
-
-
-mat(2,1); 
-
-  ***** Matrix mismatch
-  Cont? (Y or N) 
-
-
-matrix qt; 
-
-qt := ws; 
-
-  QT(1,1) := 1
-  QT(1,2) := 2
-  QT(2,1) := 3
-  QT(2,2) := 4 
-
-
-matrix a,b; 
-
-a := mat((x),(y),(z)); 
-
-  A(1,1) := X
-  A(2,1) := Y
-  A(3,1) := Z 
-
-
-b := mat((sin x,cos x,1)); 
-
-  B(1,1) := SIN(X)
-  B(1,2) := COS(X)
-  B(1,3) := 1
-
-

    Matrices need not have a size declared (unlike arrays). mat - -redimensions a matrix variable as needed. It is necessary, of course, -that all rows be the same length. An anonymous matrix, as shown in the -first example, must be named before it can be referenced (note error -message). When using mat to fill a 1 x n -matrix, the row of values must be inside a second set of parentheses, to -eliminate ambiguity. -

    -

    -

    - - - -MATEIGEN -INDEX

    - - - -MATEIGEN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The mateigen operator calculates the eigenvalue equation and the -corresponding eigenvectors of a -matrix. -

    -syntax:

    -

    -

    -mateigen(<matrix-id>,<tag-id>) -

    -

    -

    -<matrix-id> must be a declared matrix of values, and <tag-id> must b -e -a legal REDUCE identifier. -

    -

    -

    -examples:

    -

    
-aa := mat((2,5),(1,0))$ 
-
-mateigen(aa,alpha); 
-
-         2
-  {{ALPHA  - 2*ALPHA - 5,
-    1,
-                5*ARBCOMPLEX(1)
-    MAT(1,1) := ---------------,
-                   ALPHA - 2
-    MAT(2,1) := ARBCOMPLEX(1)
-    }}
-
-
-charpoly := first first ws; 
-
-                   2
-  CHARPOLY := ALPHA  - 2*ALPHA - 5 
-
-
-bb := mat((1,0,1),(1,1,0),(0,0,1))$ 
-
-mateigen(bb,lamb); 
-
-  {{LAMB - 1,3,
-    [      0      ]
-    [ARBCOMPLEX(2)]
-    [      0      ]
-    }}
-
-

    The mateigen operator returns a list of lists of three -elements. The first element is a square free factor of the characteristic -polynomial; the second element is its multiplicity; and the third element -is the corresponding eigenvector. If the characteristic polynomial can be -completely factored, the product of the first elements of all the sublists -will produce the minimal polynomial. You can access the various parts of -the answer with the usual list access operators. -

    -

    -If the matrix is degenerate, more than one eigenvector can be produced for -the same eigenvalue, as shown by more than one arbitrary variable in the -eigenvector. The identification numbers of the arbitrary complex variables -shown in the examples above may not be the same as yours. Note that since -lambda is a reserved word in REDUCE, you cannot use it as a -tag-id for this operator. -

    -

    -

    - - - -MATRIX -INDEX

    - - - -MATRIX _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -Identifiers are declared to be of type matrix. -

    -syntax:

    -

    -

    -matrix<identifier> _ _ _ option (<index>,<index>) -

    -

    -{,<identifier> _ _ _ option - (<index>,<index>)}* -

    -

    -

    -<identifier> must not be an already-defined operator or array or -the name of a scalar variable. Dimensions are optional, and if used appear -inside parentheses. <index> must be a positive integer. -

    -

    -

    -examples:

    -

    
-matrix a,b(1,4),c(4,4); 
-
-b(1,1); 
-
-  0 
-
-
-a(1,1); 
-
-  ***** Matrix A not set 
-
-
-a := mat((x0,y0),(x1,y1)); 
-
-  A(1,1) := X0
-  A(1,2) := Y0
-  A(2,1) := X0
-  A(2,2) := X1
-
-
-length a; 
-
-  {2,2} 
-
-
-b := a**2; 
-
-              2
-  B(1,1) := X0  + X1*Y0
-  B(1,2) := Y0*(X0 + Y1)
-  B(2,1) := X1*(X0 + Y1)
-                      2
-  B(2,2) := X1*Y0 + Y1
-
-

    When a matrix variable has not been dimensioned, matrix elements c -annot be -referenced until the matrix is set by the -mat operator. When a -matrix is dimensioned in its declaration, matrix elements are set to 0. -Matrix elements cannot stand for themselves. When you use -let on -a matrix element, there is no effect unless the element contains a -constant, in which case an error message is returned. The same behavior -occurs with -clear. Do <not> use -clear to try to -set a matrix element to 0. -let statements can be applied to -matrices as a whole, if the right-hand side of the expression is a matrix -expression, and the left-hand side identifier has been declared to be a matrix. - -

    -

    -Arithmetical operators apply to matrices of the correct dimensions. The -operators + and - can be used with matrices of the same -dimensions. The operator * can be used to multiply -m x n matrices by n x p -matrices. Matrix multiplication is non-commutative. Scalars can also be -multiplied with matrices, with the result that each element of the matrix -is multiplied by the scalar. The operator / applied to two -matrices computes the first matrix multiplied by the inverse of the -second, if the inverse exists, and produces an error message otherwise. -Matrices can be divided by scalars, which results in dividing each element -of the matrix. Scalars can also be divided by matrices when the matrices -are invertible, and the result is the multiplication of the scalar by the -inverse of the matrix. Matrix inverses can by found by 1/A or -/A, where A is a matrix. Square matrices can be raised to -positive integer powers, and also to negative integer powers if they are -nonsingular. -

    -

    -When a matrix variable is assigned to the results of a calculation, the -matrix is redimensioned if necessary. -

    -

    -

    - - - -NULLSPACE -INDEX

    - - - -NULLSPACE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -nullspace(<matrix\_expression>) -

    -

    -

    -<nullspace> calculates for its -matrix argument, -a, a list of -linear independent vectors (a basis) whose linear combinations satisfy the -equation a x = 0. The basis is provided in a form such that as many -upper components as possible are isolated. -

    -

    -

    -examples:

    -

    
-nullspace mat((1,2,3,4),(5,6,7,8)); 
-
-
-         {
-           [ 1  ]
-           [    ]
-           [ 0  ]
-           [    ]
-           [ - 3]
-           [    ]
-           [ 2  ]
-           ,
-           [ 0  ]
-           [    ]
-           [ 1  ]
-           [    ]
-           [ - 2]
-           [    ]
-           [ 1  ]
-          }
-
-

    Note that with b := nullspace a, the expression lengt -h b is -the nullity/ of A, and that second length a - length b -calculates the rank/ of A. The rank of a matrix expression can -also be found more directly by the -rank operator. -

    -

    -In addition to the REDUCE matrix form, nullspace accepts as input a -matrix given as a -list of lists, that is interpreted as a row matrix. If - -that form of input is chosen, the vectors in the result will be -represented by lists as well. This additional input syntax facilitates -the use of nullspace in applications different from classical linear -algebra. -

    -

    -

    - - - -RANK -INDEX

    - - - -RANK _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -rank(<matrix\_expression>) -

    -

    -

    -rankcalculates the rank of its matrix argument. -

    -

    -

    -examples:

    -

    
-rank mat((a,b,c),(d,e,f)); 
-
-  2
-
-

    The argument to rank can also be a -list of lists, interpreted -either as a row matrix or a set of equations. If that form of input is -chosen, the vectors in the result will be represented by lists as well. -This additional input syntax facilitates the use of rank in -applications different from classical linear algebra. -

    -

    -

    - - - -TP -INDEX

    - - - -TP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The tp operator returns the transpose of its -matrix - argument. -

    -syntax:

    -

    -

    -tp<identifier> or tp(<identifier>) -

    -

    -

    -<identifier> must be a matrix, which either has had its dimensions set -in its declaration, or has had values put into it by mat. -

    -

    -

    -examples:

    -

    
-matrix m,n; 
-
-m := mat((1,2,3),(4,5,6))$ 
-
-n := tp m; 
-
-  N(1,1) := 1
-  N(1,2) := 4
-  N(2,1) := 2
-  N(2,2) := 5
-  N(3,1) := 3
-  N(3,2) := 6
-
-

    In an assignment statement involving tp, the matrix ident -ifier on the -left-hand side is redimensioned to the correct size for the transpose. -

    -

    -

    - - - -TRACE -INDEX

    - - - -TRACE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The trace operator finds the trace of its -matrix argument. -

    -syntax:

    -

    -

    -trace(<expression>) or trace <simple\_expression> - -

    -

    -

    -<expression> or <simple\_expression> must evaluate to a square -matrix. -

    -

    -

    -examples:

    -

    
-matrix a; 
-
-a := mat((x1,y1),(x2,y2))$ 
-
-trace a; 
-
-  X1 + Y2
-
-

    The trace is the sum of the entries along the diagonal of a square - matrix. -Given a non-matrix expression, or a non-square matrix, trace returns -an error message. -

    -

    -

    - - - -Matrix Operations -INDEX

    -Matrix Operations

    -
  • COFACTOR operator

    -

  • DET operator

    -

  • MAT operator

    -

  • MATEIGEN operator

    -

  • MATRIX declaration

    -

  • NULLSPACE operator

    -

  • RANK operator

    -

  • TP operator

    -

  • TRACE operator

    -

    •  - - -Groebner_bases -INDEX

    - - - -GROEBNER BASES _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -The GROEBNER package calculates Groebner bases using the - Buchberger algorithm and provides related algorithms -for arithmetic with ideal bases, such as ideal quotients, -Hilbert polynomials ( Hollmann algorithm), -basis conversion ( - Faugere-Gianni-Lazard-Mora algorithm), independent -variable set ( Kredel-Weispfenning algorithm). -

    -

    -Some routines of the Groebner package are used by -solve - in -that context the package is loaded automatically. However, if you -want to use the package by explicit calls you must load it by -

    
-    load_package groebner;
-

    -

    -For the common parameter setting of most operators in this package -see -ideal parameters. -

    -

    - - - -Ideal_Parameters -INDEX

    - - - -IDEAL PARAMETERS

    -

    - -

    -

    -Most operators of the Groebner package compute expressions in a -polynomial ring which given as <R>[<var>,<var>,...] where -<R> is the current REDUCE coefficient domain. All algebraically -exact domains of REDUCE are supported. The package can operate over rings -and fields. The operation mode is distinguished automatically. In -general the ring mode is a bit faster than the field mode. The factoring -variant can be applied only over domains which allow you factoring of -multivariate polynomials. -

    -

    -The variable sequence <var> is either declared explicitly as argument -in form of a -list in -torder, or it is extracted -automatically from the expressions. In the second case the current REDUCE -system order is used (see -korder) for arranging the variables. -If some kernels should play the role of formal parameters (the ground -domain <R> then is the polynomial ring over these), the variable -sequences must be given explicitly. -

    -

    -All REDUCE -kernels can be used as variables. But please note, -that all variables are considered as independent. E.g. when using -sin(a) and cos(a) as variables, the basic relation -sin(a)^2+cos(a)^2-1=0 must be explicitly added to an equation set -because the Groebner operators don't include such knowledge automatically. -

    -

    -The terms (monomials) in polynomials are arranged according to the current - -term order. Note that the algebraic properties of the -computed -results only are valid as long as neither the ordering nor the variable -sequence changes. -

    -

    -The input expressions <exp> can be polynomials <p>, rational -functions <n>/<d> or equations <lh>=<rh> built from -polynomials or rational functions. Apart from the tracing -algorithms -groebnert and -preducet, where the equations -have a specific meaning, equations are converted to simple expressions by -taking the difference of the left-hand and right-hand sides -<lh>-<rh>=><p>. Rational functions are converted to -polynomials by converting the expression to a common denominator form -first, and then using the numerator only <n>=><p>. So eventual -zeros of the denominators are ignored. -

    -

    -A basis on input or output of an algorithm is coded as -list of -expressions {<exp>,<exp>,...} . -

    -

    - - - -Term_order -INDEX

    - - - -TERM ORDER _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -

    -

    -For all Groebner operations the polynomials are -represented in distributive form: a sum of terms (monomials). -The terms are ordered corresponding to the actual term order -which is set by the -torder operator, and to the -actual variable sequence which is either given as explicit -parameter or by the system -kernel order. -

    -

    - - - -torder -INDEX

    - - - -TORDER _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The operator torder sets the actual variable sequence and term order. -

    -

    -1. simple term order: -

    -syntax:

    -

    -

    -torder(<vl>, <m>) -

    -

    -

    -where <vl> is a -list of variables ( -kernels) and -<m> is the name of a simple -term order mode - -lex term order, -gradlex term order, - -revgradlex term order or another implemented parameter -less mode. -

    -

    -2. stepped term order: -

    -syntax:

    -

    -

    -torder(<vl>,<m>,<n>) -

    -

    -

    -

    -where <m> is the name of a two step term order, one of - -gradlexgradlex term order, -gradlexrevgradlex term order, - -lexgradlex term order or -lexrevgradlex term order, and -<n> is a positive integer. -

    -

    -3. weighted term order -

    -syntax:

    -

    -

    -torder(<vl>, weighted, <n>,<n>,...); -

    -

    -

    -where the <n> are positive integers, see -weighted term order. -

    -

    -4. matrix term order -

    -syntax:

    -

    -

    -torder(<vl>, matrix, <m>); -

    -

    -

    -where <m> is a matrix with integer elements, see - -torder_compile. -

    -

    -5. compiled term order -

    -syntax:

    -

    -

    -torder(<vl>, co); -

    -

    -

    -where <co> is the name of a routine generated by - -torder_compile. -

    -

    -tordersets the variable sequence and the term order mode. If the -an empty list is used as variable sequence, the automatic variable extraction -is activated. The defaults are the empty variable list an the - -lex term order. -The previous setting is returned as a list. -

    -

    -Alternatively to the above syntax the arguments of torder may be -collected in a -list and passed as one argument to -torder. -

    -

    - - - -torder_compile -INDEX

    - - - -TORDER_COMPILE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -A matrix can be converted into -a compilable LISP program for faster execution by using -

    -syntax:

    -

    -

    -torder_compile(<name>,<mat>) -

    -

    -

    -where <name> is an identifier for the new term order and <mat> -is an integer matrix to be used as -matrix term order. Afterwards -the term order can be activated by using <name> in a -torder -expression. The resulting program is compiled if the switch -comp -is on, or if the torder_compile expression is part of a compiled -module. -

    -

    - - - -lex_term_order -INDEX

    - - - -LEX TERM ORDER

    -

    - -

    -

    -The terms are ordered lexicographically: two terms t1 t2 -are compared for their degrees -along the fixed variable sequence: t1 is higher than t2 -if the first different degree is higher in t1. -This order has the elimination property -for groebner basis calculations. -If the ideal has a univariate polynomial in the last -variable the groebner basis will contain -such polynomial. Lex is best -suited for solving of polynomial equation systems. -

    -

    - - - -gradlex_term_order -INDEX

    - - - -GRADLEX TERM ORDER

    -

    - -

    -

    -The terms are ordered first with their total -degree, and if the total degree is identical -the comparison is -lex term order. -With groebner basis calculations this term order -produces polynomials of lowest degree. -

    -

    - - - -revgradlex_term_order -INDEX

    - - - -REVGRADLEX TERM ORDER

    -

    - -

    -

    -The terms are ordered first with their total -degree (degree sum), and if the total degree is identical -the comparison is the inverse of -lex term order. -With -groebner and -groebnerf -calculations this term order -is similar to -gradlex term order; it is known -as most efficient ordering with respect to computing time. -

    -

    - - - -gradlexgradlex_term_order -INDEX

    - - - -GRADLEXGRADLEX TERM ORDER

    -

    - -

    -

    -The terms are separated into two groups where the -second parameter of the -torder call determines -the length of the first group. For a comparison first -the total degrees of both variable groups are compared. -If both are equal - -gradlex term order comparison is applied to the first - -group, and if that does not decide -gradlex term order -is applied for the second group. This order has the elimination -property for the variable groups. It can be used e.g. for -separating variables from parameters. -

    -

    - - - -gradlexrevgradlex_term_order -INDEX

    - - - -GRADLEXREVGRADLEX TERM ORDER

    -

    - -

    -

    -Similar to -gradlexgradlex term order, but using - -revgradlex term order for the second group. -

    -

    - - - -lexgradlex_term_order -INDEX

    - - - -LEXGRADLEX TERM ORDER

    -

    - -

    -

    -Similar to -gradlexgradlex term order, but using - -lex term order for the first group. -

    -

    - - - -lexrevgradlex_term_order -INDEX

    - - - -LEXREVGRADLEX TERM ORDER

    -

    - -

    -

    -Similar to -gradlexgradlex term order, but using - -lex term order for the first group - -revgradlex term order for the second group. -

    -

    - - - -weighted_term_order -INDEX

    - - - -WEIGHTED TERM ORDER

    -

    - -

    -

    -establishes a graduated ordering -similar to -gradlex term order, where the exponents first are -multiplied by the given weights. If there are less weight values than -variables, the weight list is extended by ones. If the weighted degree -comparison is not decidable, the - -lex term order is used. -

    -

    - - - -graded_term_order -INDEX

    - - - -GRADED TERM ORDER

    -

    - -

    -

    -establishes a cascaded term ordering: first a graduated ordering -similar to -gradlex term order is used, where the exponents first -are -multiplied by the given weights. If there are less weight values than -variables, the weight list is extended by ones. If the weighted degree -comparison is not decidable, the term ordering described in the following -parameters of the -torder command is used. -

    -

    - - - -matrix_term_order -INDEX

    - - - -MATRIX TERM ORDER

    -

    - -

    -

    -Any arbitrary term order mode can be installed by a matrix with -integer elements where the row length corresponds to the variable -number. The matrix must have at least as many rows as columns. -It must have full rank, and the top nonzero element of each column -must be positive. -

    -

    -The matrix term order mode -defines a term order where the exponent vectors of the monomials are -first multiplied by the matrix and the resulting vectors are compared -lexicographically. -

    -

    -If the switch -comp is on, the matrix is converted into -a compiled LISP program for faster execution. A matrix can also be -compiled explicitly, see -torder_compile. -

    -

    - - - -Term order -INDEX

    -Term order

    -
  • Term order introduction

    -

  • torder operator

    -

  • torder_compile operator

    -

  • lex term order concept

    -

  • gradlex term order concept

    -

  • revgradlex term order concept

    -

  • gradlexgradlex term order concept

    -

  • gradlexrevgradlex term order concept

    -

  • lexgradlex term order concept

    -

  • lexrevgradlex term order concept

    -

  • weighted term order concept

    -

  • graded term order concept

    -

  • matrix term order concept

    -

    •  - - -gvars -INDEX

    - - - -GVARS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -syntax:

    -

    -

    -gvars({<exp>,<exp>,... }) -

    -

    -

    -

    -where <exp> are expressions or -equations. -

    -

    -gvarsextracts from the expressions the -kernels -which can -play the role of variables for a -groebner or -groebnerf -calculation. -

    -

    - - - -groebner -INDEX

    - - - -GROEBNER _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -groebner({exp, ...}) -

    -

    -

    -

    -where {exp, ... } is a list of -expressions or equations. -

    -

    -The operator groebner implements the Buchberger algorithm -for computing Groebner bases for a given set of -expressions with respect to the given set of variables in the order -given. As a side effect, the sequence of variables is stored as a REDUCE list -in the shared variable -gvarslast - this is important in cases -where the algorithm rearranges the variable sequence because -groebopt -is on. -

    -

    -

    -examples:

    -

    
-   groebner({x**2+y**2-1,x-y}) 
-
-  {X - Y,2*Y**2 -1}
-
-

    -related:

    -

    - _ _ _ -groebnerfoperator -

    - _ _ _ -gvarslast variable -

    - _ _ _ -groebopt switch -

    - _ _ _ -groebprereduce switch -

    - _ _ _ -groebfullreduction switch -

    - _ _ _ -gltbasis switch -

    - _ _ _ -gltb variable -

    - _ _ _ -glterms variable -

    - _ _ _ -groebstat switch -

    - _ _ _ -trgroeb switch -

    - _ _ _ -trgroebs switch -

    - _ _ _ -groebprot switch -

    - _ _ _ -groebprotfile variable -

    - _ _ _ -groebnert operator -

    -

    -

    - - - -groebner_walk -INDEX

    - - - -GROEBNER\_WALK _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The operator groebner_walk computes a lex basis -from a given graded (or weighted) one. -

    -syntax:

    -

    -

    -groebner_walk(<g>) -

    -

    -

    -where <g> is a graded basis (or weighted basis -with a weight vector with one repeated element) of the polynomial ideal. -Groebner_walk computes a sequence of monomial bases, each -time lifting the full system to a complete basis. Groebner_walk -should be called only in cases, where a normal kex computation -would take too much computer time. -

    -

    -The operator -torder has to be called before in order to -define the variable sequence and the term order mode of <g>. -

    -

    -The variable -gvarslast is not set. -

    -

    -Do not call groebner_walk with on -groebopt. -

    -

    -Groebner_walkincludes some overhead (such as e. g. -computation with division). On the other hand, sometimes -groebner_walk is faster than a direct lex computation. -

    -

    - - - -groebopt -INDEX

    - - - -GROEBOPT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -If groebopt is set ON, the sequence of variables is optimized -with respect to execution speed of groebner calculations; -note that the final list of variables is available in -gvarslast. -By default groebopt is off, conserving the original variable -sequence. -

    -

    -An explicitly declared dependency using the -depend -declaration supersedes the variable optimization. -

    -examples:

    -

    -

    -guarantees that a will be placed in front of x and y. -

    -

    - - - -gvarslast -INDEX

    - - - -GVARSLAST _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -After a -groebner or -groebnerf calculation -the actual variable sequence is stored in the variable -gvarslast. If -groebopt is on -gvarslast shows the variable sequence after reordering. -

    -

    - - - -groebprereduce -INDEX

    - - - -GROEBPREREDUCE _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -If groebprereduce set ON, -groebner -and -groebnerf try to simplify the -input expressions: if the head term of an input expression is a -multiple of the head term of another expression, it can be reduced; -these reductions are done cyclicly as long as possible in order to -shorten the main part of the algorithm. -

    -

    -By default groebprereduce is off. -

    -

    - - - -groebfullreduction -INDEX

    - - - -GROEBFULLREDUCTION _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -If groebfullreduction set off, the polynomial reduction steps during - -groebner and -groebnerf are limited to the pure head -term reduction; subsequent terms are reduced otherwise. -

    -

    -By default groebfullreduction is on. -

    -

    - - - -gltbasis -INDEX

    - - - -GLTBASIS _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -If gltbasis set on, the leading terms of the result basis -of a -groebner or -groebnerf calculation are -extracted. They are collected as a basis of monomials, which is -available as value of the global variable -gltb. -

    -

    - - - -gltb -INDEX

    - - - -GLTB _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -See -gltbasis -

    -

    - - - -glterms -INDEX

    - - - -GLTERMS _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -If the expressions in a -groebner or -groebnerf -call contain parameters (symbols -which are not member of the variable list), the share variable -glterms is set to a list of expression which during the -calculation were assumed to be nonzero. The calculated bases -are valid only under the assumption that all these expressions do -not vanish. -

    -

    - - - -groebstat -INDEX

    - - - -GROEBSTAT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -if groebstat is on, a summary of the - -groebner or -groebnerf computation is printed -at the end -including the computing time, the number of intermediate -H polynomials and the counters for the criteria hits. -

    -

    - - - -trgroeb -INDEX

    - - - -TRGROEB _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -if trgroeb is on, intermediate H polynomials are -printed during a -groebner -or -groebnerf calculation. -

    -

    - - - -trgroebs -INDEX

    - - - -TRGROEBS _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -if trgroebs is on, intermediate H and S polynomials are -printed during a -groebner or -groebnerf calculation. -

    -

    - - - -gzerodim_ -INDEX

    - - - -GZERODIM? _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -syntax:

    -

    -

    -gzerodim!?(<basis>) -

    -

    -

    -

    -where <bas> is a Groebner basis in the current - -term order with the actual setting -(see -ideal parameters). -

    -

    -gzerodim!?tests whether the ideal spanned by the given basis -has dimension zero. If yes, the number of zeros is returned, - -nil otherwise. -

    -

    - - - -gdimension -INDEX

    - - - -GDIMENSION _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -gdimension(<bas>) -

    -

    -

    -

    -where <bas> is a -groebner basis in the current -term order (see -ideal parameters). -gdimension computes the dimension of the ideal -spanned by the given basis and returns the dimension as an integer -number. The Kredel-Weispfenning algorithm is used: the dimension -is the length of the longest independent variable set, -see -gindependent_sets -

    -

    - - - -gindependent_sets -INDEX

    - - - -GINDEPENDENT\_SETS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -gindependent_sets(<bas>) -

    -

    -

    -

    -where <bas> is a -groebner basis in any term order -(which must be the current term order) with the specified -variables (see -ideal parameters). -

    -

    -Gindependent_setscomputes the maximal -left independent variable sets of the ideal, that are -the variable sets which play the role of free parameters in the -current ideal basis. Each set is a list which is a subset of the -variable list. The result is a list of these sets. For an -ideal with dimension zero the list is empty. -The Kredel-Weispfenning algorithm is used. -

    -

    - - - -dd_groebner -INDEX

    - - - -DD_GROEBNER _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -For a homogeneous system of polynomials under - -graded term order, -gradlex term order, - -revgradlex term order

    -

    -or -weighted term order -a Groebner Base can be computed with limiting the grade -of the intermediate S polynomials: -

    -syntax:

    -

    -

    -dd_groebner(<d1>,<d2>,<plist>) -

    -

    -

    -where <d1> is a non negative integer and <d2> is an integer -or ``infinity". A pair of polynomials is considered -only if the grade of the lcm of their head terms is between -<d1> and <d2>. -For the term orders graded or weighted the (first) weight -vector is used for the grade computation. Otherwise the total -degree of a term is used. -

    -

    - - - -glexconvert -INDEX

    - - - -GLEXCONVERT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -glexconvert(<bas>[,<vars>][,MAXDEG=<mx>] -[,NEWVARS=<nv>]) -

    -

    -

    -

    -where <bas> is a -groebner basis -in the current term order, <mx> (optional) is a positive -integer and <nvl> (optional) is a list of variables -(see -ideal parameters). -

    -

    -The operator glexconvert converts the basis -of a zero-dimensional ideal (finite number -of isolated solutions) from arbitrary ordering into a basis under - -lex term order. -

    -

    -The parameter <newvars> defines the new variable sequence. -If omitted, the -original variable sequence is used. If only a subset of variables is -specified here, the partial ideal basis is evaluated. -

    -

    -If <newvars> is a list with one element, the minimal - univariate polynomial is computed. -

    -

    -<maxdeg> is an upper limit for the degrees. The algorithm stops with -an error message, if this limit is reached. -

    -

    -A warning occurs, if the ideal is not zero dimensional. -

    -

    -During the call the term order of the input basis must -be active. -

    -

    -

    - - - -greduce -INDEX

    - - - -GREDUCE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -syntax:

    -

    -

    -greduce(exp, {exp1, exp2, ... , expm}) -

    -

    -

    -

    -where exp is an expression, and {exp1, exp2, ... , expm} is -a list of expressions or equations. -

    -

    -greduceis functionally equivalent with a call to - -groebner and then a call to -preduce. -

    -

    - - - -preduce -INDEX

    - - - -PREDUCE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -syntax:

    -

    -

    -preduce(<p>, {<exp>, ... }) -

    -

    -

    -

    -where <p> is an expression, and {<exp>, ... } is -a list of expressions or equations. -

    -

    -Preducecomputes the remainder of exp -modulo the given set of polynomials resp. equations. -This result is unique (canonical) only if the given set -is a groebner basis under the current -term order -

    -

    -see also: -preducet operator. -

    -

    - - - -idealquotient -INDEX

    - - - -IDEALQUOTIENT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -syntax:

    -

    -

    -idealquotient({<exp>, ...}, <d>) -

    -

    -

    -

    -where {<exp>,...} is a list of -expressions or equations, <d> is a single expression or equation. -

    -

    -Idealquotientcomputes the ideal quotient: -ideal spanned by the expressions {<exp>,...} -divided by the single polynomial/expression <f>. The result -is the -groebner basis of the quotient ideal. -

    -

    - - - -hilbertpolynomial -INDEX

    - - - -HILBERTPOLYNOMIAL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -hilbertpolynomial(<bas>) -

    -

    -

    -

    -where <bas> is a -groebner basis in the -current -term order. -

    -

    -The degree of the Hilbert polynomial is the -dimension of the ideal spanned by the basis. For an -ideal of dimension zero the Hilbert polynomial is a -constant which is the number of common zeros of the -ideal (including eventual multiplicities). -The Hollmann algorithm is used. -

    -

    - - - -saturation -INDEX

    - - - -SATURATION _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -syntax:

    -

    -

    -saturation({<exp>, ...}, <p>) -

    -

    -

    -

    -where {<exp>,...} is a list of -expressions or equations, <p> is a single polynomial. -

    -

    -Saturationcomputes the quotient of the polynomial <p> -and a power (with unknown but finite exponent) of the ideal built from -{<exp>, ...}. The result is the computed quotient. Saturation -calls -idealquotient several times until the result does not -change -any more. -

    -

    - - - -Basic Groebner operators -INDEX

    -Basic Groebner operators

    -
  • gvars operator

    -

  • groebner operator

    -

  • groebner\_walk operator

    -

  • groebopt switch

    -

  • gvarslast variable

    -

  • groebprereduce switch

    -

  • groebfullreduction switch

    -

  • gltbasis switch

    -

  • gltb variable

    -

  • glterms variable

    -

  • groebstat switch

    -

  • trgroeb switch

    -

  • trgroebs switch

    -

  • gzerodim? operator

    -

  • gdimension operator

    -

  • gindependent\_sets operator

    -

  • dd_groebner operator

    -

  • glexconvert operator

    -

  • greduce operator

    -

  • preduce operator

    -

  • idealquotient operator

    -

  • hilbertpolynomial operator

    -

  • saturation operator

    -

    •  - - -groebnerf -INDEX

    - - - -GROEBNERF _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -syntax:

    -

    -

    -groebnerf({<exp>, ...}[,{},{<nz>, ... }]); -

    -

    -

    -

    -where {<exp>, ... } is a list of expressions or -equations, and {<nz>,... } is -an optional list of polynomials to be considered as non zero -for this calculation. An empty list must be passed as second argument -if the non-zero list is specified. -

    -

    -groebnerftries to separate polynomials into individual factors and -to branch the computation in a recursive manner (factorization tree). -The result is a list of partial Groebner bases. -Multiplicities (one factor with a higher power, the same partial basis -twice) are deleted as early as possible in order to speed up the -calculation. -

    -

    -The third parameter of groebnerf declares some polynomials -nonzero. If any of these is found in a branch of the calculation -the branch is canceled. -

    -

    -

    -example:

    -

    
-groebnerf({ 3*x**2*y+2*x*y+y+9*x**2+5*x = 3,  
-            2*x**3*y-x*y-y+6*x**3-2*x**2-3*x = -3, 
-            x**3*y+x**2*y+3*x**3+2*x**2 }, {y,x});
-
-       {{Y - 3,X},
-
-                      2
-    {2*Y + 2*X - 1,2*X  - 5*X - 5}}
-

    -related:

    -

    - _ _ _ -groebresmaxvariable -

    - _ _ _ -groebmonfac variable -

    - _ _ _ -groebrestriction variable -

    - _ _ _ -groebner operator -

    - _ _ _ -gvarslast variable -

    - _ _ _ -groebopt switch -

    - _ _ _ -groebprereduce switch -

    - _ _ _ -groebfullreduction switch -

    - _ _ _ -gltbasis switch -

    - _ _ _ -gltb variable -

    - _ _ _ -glterms variable -

    - _ _ _ -groebstat switch -

    - _ _ _ -trgroeb switch -

    - _ _ _ -trgroebs switch -

    - _ _ _ -groebnert operator -

    -

    -

    - - - -groebmonfac -INDEX

    - - - -GROEBMONFAC _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -The variable groebmonfac is connected to -the handling of monomial factors. A monomial factor is a product -of variable powers as a factor, e.g. x**2*y in x**3*y - -2*x**2*y**2. A monomial factor represents a solution of the type - x = 0 or y = 0 with a certain multiplicity. With - -groebnerf the multiplicity of monomial factors is lowe -red -to the value of the shared variable groebmonfac -which by default is 1 (= monomial factors remain present, but their -multiplicity is brought down). With -groebmonfac:= 0 -the monomial factors are suppressed completely. -

    -

    - - - -groebresmax -INDEX

    - - - -GROEBRESMAX _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -The variable groebresmax -controls during -groebnerf calculations -the number of partial results. Its default value is 300. If -more partial results are calculated, the calculation is -terminated. -

    -

    - - - -groebrestriction -INDEX

    - - - -GROEBRESTRICTION _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -During -groebnerf calculations -irrelevant branches can be excluded -by setting the variable groebrestriction. The -following restrictions are implemented: -

    -syntax:

    -

    -

    -groebrestriction:= nonnegative -

    -

    -groebrestriction:= positive -

    -

    -groebrestriction:= zeropoint -

    -

    -

    -With nonnegative branches are excluded where one -polynomial has no nonnegative real zeros; with positive -the restriction is sharpened to positive zeros only. -The restriction zeropoint excludes all branches -which do not have the origin (0,0,...0) in their solution -set. -

    -

    - - - -Factorizing Groebner bases -INDEX

    -Factorizing Groebner bases

    -
  • groebnerf operator

    -

  • groebmonfac variable

    -

  • groebresmax variable

    -

  • groebrestriction variable

    -

    •  - - -groebprot -INDEX

    - - - -GROEBPROT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -If groebprot is ON the computation steps during - -preduce, -greduce and -groebner -are collected in a list which is assigned to the variable - -groebprotfile. -

    -

    - - - -groebprotfile -INDEX

    - - - -GROEBPROTFILE _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -See -groebprot switch. -

    -

    - - - -groebnert -INDEX

    - - - -GROEBNERT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -syntax:

    -

    -

    -groebnert({<v>=<exp>,...}) -

    -

    -

    -

    -where <v> are -kernels (simple or indexed variables), -<exp> are polynomials. -

    -

    -groebnertis functionally equivalent to a -groebner -call for {<exp>,...}, but the result is a set of -equations where the left-hand sides are the basis elements while -the right-hand sides are the same values expressed as combinations -of the input formulas, expressed in terms of the names <v> -

    -example:

    -

    
-    groebnert({p1=2*x**2+4*y**2-100,p2=2*x-y+1});
-
-   GB1 := {2*X - Y + 1=P2,
-
-           2
-        9*Y  - 2*Y - 199= - 2*X*P2 - Y*P2 + 2*P1 + P2}
-

    -

    - - - -preducet -INDEX

    - - - -PREDUCET _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -syntax:

    -

    -

    -preduce(<p>,{<v>=<exp>...}) -

    -

    -

    -where <p> is an expression, <v> are kernels -(simple or indexed variables), -exp are polynomials. -

    -

    -preducetcomputes the remainder of <p> modulo {<exp>,...} -similar to -preduce, but the result is an equation -which expresses the remainder as combination of the polynomials. -

    -example:

    -

    
-                             
-   GB2 := {G1=2*X - Y + 1,G2=9*Y**2  - 2*Y - 199}
-   preducet(q=x**2,gb2);
-
- - 16*Y + 208= - 18*X*G1 - 9*Y*G1 + 36*Q + 9*G1 - G2
-

    -

    - - - -Tracing Groebner bases -INDEX

    -Tracing Groebner bases

    -
  • groebprot switch

    -

  • groebprotfile variable

    -

  • groebnert operator

    -

  • preducet operator

    -

    •  - - -Module -INDEX

    - - - -MODULE

    -

    - -Given a polynomial ring, e.g. R=Z[x,y,...] and an integer n>1. -The vectors with n elements of R form a free MODULE under -elementwise addition and multiplication with elements of R. -

    -

    -For a submodule given by a finite basis a Groebner basis -can be computed, and the facilities of the GROEBNER package -are available except the operators -groebnerf -and groesolve. The vectors are encoded using auxiliary -variables which represent the unit vectors in the module. -These are declared in the share variable -gmodule. -

    -

    - - - -gmodule -INDEX

    - - - -GMODULE _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -The vectors of a free -module over a polynomial ring R -are encoded as linear combinations with unit vectors of -M which are represented by auxiliary variables. These -must be collected in the variable gmodule before -any call to an operator of the Groebner package. -

    -

    -

    
-   torder({x,y,v1,v2,v3})$
-   gmodule := {v1,v2,v3}$
-   g:=groebner({x^2*v1 + y*v2,x*y*v1 - v3,2y*v1 + y*v3});
-

    compute the Groebner basis of the submodule -

    -

    -

    
-      ([x^2,y,0],[xy,0,-1],[0,2y,y])
-

    The members of the list gmodule are automatically -appended to the end of the variable list, if they are not -yet members there. They take part in the actual term ordering. -

    -

    - - - -Groebner Bases for Modules -INDEX

    -Groebner Bases for Modules

    -
  • Module concept

    -

  • gmodule variable

    -

    •  - - -gsort -INDEX

    - - - -GSORT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -gsort(<p>) -

    -

    -

    -where <p> is a polynomial or a list of polynomials. -

    -

    -The polynomials are reordered and sorted corresponding to -the current -term order. -

    -examples:

    -

    
-
-  torder lex;
-  
-  gsort(x**2+2x*y+y**2,{y,x}); 
-
-  y**2+2y*x+x**2
-
-

    -

    - - - -gsplit -INDEX

    - - - -GSPLIT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -gsplit(<p>[,<vars>]); -

    -

    -

    -where <p> is a polynomial or a list of polynomials. -

    -

    -The polynomial is reordered corresponding to the -the current -term order and then -separated into leading term and reductum. Result is -a list with the leading term as first and the reductum -as second element. -

    -examples:

    -

    
-
-  torder lex;
-  
-  gsplit(x**2+2x*y+y**2,{y,x}); 
-
-  {y**2,2y*x+x**2}
-
-

    -

    - - - -gspoly -INDEX

    - - - -GSPOLY _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -gspoly(<p1>,<p2>); -

    -

    -

    -

    -where <p1> and <p2> are polynomials. -

    -

    -The subtraction polynomial of p1 and p2 is computed -corresponding to the method of the Buchberger algorithm for -computing groebner bases: p1 and p2 are multiplied -with terms such that when subtracting them the leading terms -cancel each other. -

    -

    - - - -Computing with distributive polynomials -INDEX

    -Computing with distributive polynomials

    -
  • gsort operator

    -

  • gsplit operator

    -

  • gspoly operator

    -

    •  - - -Groebner package -INDEX

    -Groebner package

    -
  • Groebner bases introduction

    -

  • Ideal Parameters concept

    -

  • Term order

    -

  • Basic Groebner operators

    -

  • Factorizing Groebner bases

    -

  • Tracing Groebner bases

    -

  • Groebner Bases for Modules

    -

  • Computing with distributive polynomials

    -

    •  - - -HEPHYS -INDEX

    - - - -HEPHYS _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -The High-energy Physics package is historic for REDUCE, since REDUCE -originated as a program to aid in computations with Dirac expressions. -The commutation algebra of the gamma matrices is independent of their -representation, and is a natural subject for symbolic mathematics. Dirac -theory is applied to beta decay and the computation of -cross-sections and scattering. The high-energy physics operators are -available in the REDUCE main program, rather than as a module which must -be loaded. -

    -

    - - - -HE_dot -INDEX

    - - - -. _ _ _ HE-DOT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The . operator is used to denote the scalar product of two Lorentz -four-vectors. -

    -syntax:

    -

    -

    -<vector> . <vector> -

    -

    -

    -<vector> must be an identifier declared to be of type vector to h -ave -the scalar product definition. When applied to arguments that are not -vectors, the -cons operator is used, -whose symbol is also ``dot.'' -

    -

    -

    -examples:

    -

    
-vector aa,bb,cc; 
-
-let aa.bb = 0; 
-
-aa.bb; 
-
-  0 
-
-
-aa.cc; 
-
-  AA.CC 
-
-
-q := aa.cc; 
-
-  Q := AA.CC 
-
-
-q; 
-
-  AA.CC
-
-

    Since vectors are special high-energy physics entities that do not - contain -values, the . product will not return a true scalar product. You can -assign a scalar identifier to the result of a . operation, or assign a . -operation to have the value of the scalar you supply, as shown above. Note -that the result of a . operation is a scalar, not a vector. -

    -

    -The metric tensor g(u,v) can be represented by u.v. If contraction -over the indices is required, u and v should be declared to -be of type -index. -

    -

    -The dot operator has the highest precedence of the infix operators, so -expressions involving . and other operators have the scalar product -evaluated first before other operations are done. -

    -

    -

    - - - -EPS -INDEX

    - - - -EPS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The eps operator denotes the completely antisymmetric tensor of -order 4 and its contraction with Lorentz four-vectors, as used in -high-energy physics calculations. -

    -syntax:

    -

    -

    -eps(<vector-expr>,<vector-expr>,<vector-expr>, -<vector-expr>) -

    -

    -

    -<vector-expr> must be a valid vector expression, and may be an index. -

    -

    -

    -examples:

    -

    
-vector g0,g1,g2,g3; 
-
-eps(g1,g0,g2,g3); 
-
-  - EPS(G0,G1,G2,G3); 
-
-
-eps(g1,g2,g0,g3); 
-
-  EPS(G0,G1,G2,G3); 
-
-
-eps(g1,g2,g3,g1); 
-
-  0
-
-

    Vector identifiers are ordered alphabetically by REDUCE. When an o -dd number -of transpositions is required to restore the canonical order to the four -arguments of eps, the term is ordered and carries a minus sign. When an - -even number of transpositions is required, the term is returned ordered and -positive. When one of the arguments is repeated, the value 0 is returned. -A contraction of the form -eps(_i j mu nu p_mu q_nu) -is represented by eps(i,j,p,q) when i and j have been - -declared to be of type -index. -

    -

    -

    - - - -G -INDEX

    - - - -G _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -g is an n-ary operator used to denote a product of gamma matrices -contracted with Lorentz four-vectors, in high-energy physics. -

    -syntax:

    -

    -

    -g(<identifier>,<vector-expr> -{,<vector-expr>}*) -

    -

    -

    -<identifier> is a scalar identifier representing a fermion line -identifier, <vector-expr> can be any valid vector expression, -representing a vector or a gamma matrix. -

    -

    -

    -examples:

    -

    
-vector aa,bb,cc; 
-
-vector a; 
-
-g(line1,aa,bb); 
-
-  AA.BB 
-
-
-g(line2,aa,a); 
-
-  0 
-
-
-g(id,aa,bb,cc); 
-
-  0 
-
-
-g(li1,aa,bb) + k; 
-
-  AA.BB + K 
-
-
-let aa.bb = m*k; 
-
-g(ln1,aa)*g(ln1,bb); 
-
-  K*M 
-
-
-g(ln1,aa)*g(ln2,bb); 
-
-  0
-
-

    The vector A is reserved in arguments of g to de -note the -special gamma matrix gamma_5. It must be declared to -be a vector before you use it. -

    -

    -Gamma matrix expressions are associated with fermion lines in a Feynman -diagram. If more than one line occurs in an expression, the gamma -matrices involved are separate (operating in independent spin space), as -shown in the last two example lines above. A product of gamma matrices -associated with a single line can be entered either as a single g -command with several vector arguments, or as products of separate g -commands each with a single argument. -

    -

    -While the product of vectors is not defined, the product, sum and -difference of several gamma expressions are defined, as is the product of -a gamma expression with a scalar. If an expression involving gamma -matrices includes a scalar, the scalar is treated as if it were the -product of itself with a unit 4 x 4 matrix. -

    -

    -Dirac expressions are evaluated by computing the trace of the expression -using the commutation algebra of gamma matrices. The algorithms used are -described in articles by J. S. R. Chisholm in <Il Nuovo Cimento X,> Vol. -30, p. 426, 1963, and J. Kahane, <Journal of Mathematical Physics>, -Vol. 9, p. 1732, 1968. The trace is then divided by 4 to distinguish -between the trace of a scalar and the trace of an expression that is the -product of a scalar with a unit 4 x 4 matrix. -

    -

    -Trace calculations may be prevented over any line identifier by declaring it -to be -nospur. If it is later desired to evaluate these trace -s, -the declaration can be undone with the -spur declaration. -

    -

    -The notation of Bjorken and Drell, <Relativistic Quantum Mechanics,> -1964, is assumed in all operations involving gamma matrices. For an -example of the use of g in a calculation, see the <REDUCE -User's Manual>. -

    -

    -

    - - - -INDEX -INDEX

    - - - -INDEX _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The declaration index flags a four-vector as an index for subsequent -high-energy physics calculations. -

    -syntax:

    -

    -

    -index<vector-id>{,<vector-id>}* -

    -

    -

    -<vector-id> must have been declared of type vector. -

    -

    -

    -examples:

    -

    
-vector aa,bb,cc; 
-
-index uu; 
-
-let aa.bb = 0; 
-
-(aa.uu)*(bb.uu); 
-
-  0 
-
-
-(aa.uu)*(cc.uu); 
-
-  AA.CC
-
-

    Index variables are used to represent contraction over components -of -vectors when scalar products are taken by the . operator, as well as -indicating contraction for the -eps operator or metric tensor. -

    -

    -The special status of a vector as an index can be revoked with the -declaration -remind. The object remains a vector, however. -

    -

    -

    - - - -MASS -INDEX

    - - - -MASS _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The mass command associates a scalar variable as a mass with -the corresponding vector variable, in high-energy physics calculations. -

    -syntax:

    -

    -

    -mass<vector-var>=<scalar-var> -{,<vector-var>=<scalar-var>}* -

    -

    -

    -<vector-var> can be a declared vector variable; mass will declare - -it to be of type vector if it is not. This may override an existing -matrix variable by that name. <scalar-var> must be a scalar variable. -

    -

    -

    -examples:

    -

    
-vector bb,cc; 
-
-mass cc=m; 
-
-mshell cc; 
-
-cc.cc; 
-
-   2
-  M
-
-

    Once a mass has been attached to a vector with a mass dec -laration, -the -mshell declaration puts the associated particle ``on t -he mass -shell.'' Subsequent scalar (.) products of the vector with itself will be -replaced by the square of the mass expression. -

    -

    -

    - - - -MSHELL -INDEX

    - - - -MSHELL _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The mshell command puts particles on the mass shell in high-energy -physics calculations. -

    -syntax:

    -

    -

    -mshell<vector-var>{,<vector-var>}* -

    -

    -

    -<vector-var> must have had a mass attached to it by a -mass -declaration. -

    -

    -

    -examples:

    -

    
-vector v1,v2; 
-
-mass v1=m,v2=q; 
-
-mshell v1; 
-
-v1.v1; 
-
-   2
-  M  
-
-
-v2.v2; 
-
-  V2.V2 
-
-
-mshell v2; 
-
-v1.v1*v2.v2; 
-
-   2  2
-  M *Q
-
-

    Even though a mass is attached to a vector variable representing a - -particle, the replacement does not take place until the mshell -declaration is given for that vector variable. -

    -

    -

    - - - -NOSPUR -INDEX

    - - - -NOSPUR _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The nospur declaration prevents the trace calculation over the given -line identifiers in high-energy physics calculations. -

    -syntax:

    -

    -

    -nospur<line-id>{,<line-id>}* -

    -

    -

    -<line-id> is a scalar identifier that will be used as a line identifier. -

    -

    -

    -examples:

    -

    
-vector a1,b1,c1; 
-
-g(line1,a1,b1)*g(line2,b1,c1); 
-
-  A1.B1*B1.C1 
-
-
-nospur line2; 
-
-g(line1,a1,b1)*g(line2,b1,c1); 
-
-  A1.B1*G(LINE2,B1,C1)
-
-

    Nospur declarations can be removed by making the declaration -spur. -

    -

    -

    - - - -REMIND -INDEX

    - - - -REMIND _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The remind declaration removes the special status of its arguments -as indices, which was set in the -index declaration, in -high-energy physics calculations. -

    -syntax:

    -

    -

    -remind<identifier>{,<identifier>}* -

    -

    -

    -<identifier> must have been declared to be of type -index. -

    -

    - - - -SPUR -INDEX

    - - - -SPUR _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The spur declaration removes the special exemption from trace -calculations that was declared by -nospur, in high-energy physics -calculations. -

    -syntax:

    -

    -

    -spur<line-id>{,<line-id>}* -

    -

    -

    -<line-id> must be a line-identifier that has previously been declared -nospur. -

    -

    - - - -VECDIM -INDEX

    - - - -VECDIM _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The command vecdim changes the vector dimension from 4 to an arbitrary - -integer or symbol. Used in high-energy physics calculations. -

    -syntax:

    -

    -

    -vecdim<dimension> -

    -

    -

    -<dimension> must be either an integer or a valid scalar identifier that -does not have a floating-point value. -

    -

    -The -eps operator and the gamma_5 -symbol (A) are not properly defined in anything except four -dimensions and will print an error message if you use them that way. The -other high-energy physics operators should work without problem. -

    -

    -

    - - - -VECTOR -INDEX

    - - - -VECTOR _ _ _ _ _ _ _ _ _ _ _ _ declaration

    -

    - -The vector declaration declares that its arguments are of type vect -or. -

    -syntax:

    -

    -

    -vector<identifier>{,<identifier>}* -

    -

    -

    -<identifier> must be a valid REDUCE identifier. It may have already been -used for a matrix, array, operator or scalar variable. After an identifier -has been declared to be a vector, it may not be used as a scalar variable. -

    -

    -Vectors are special entities for high-energy physics calculations. You -cannot put values into their coordinates; they do not have coordinates. -They are legal arguments for the high-energy physics operators - -eps, -g and . (dot). Vector variables are -used to represent gamma matrices and gamma matrices contracted with Lorentz -4-vectors, since there are no Dirac variables per se in the system. -Vectors do follow the usual vector rules for arithmetic operations: -+ and - operate upon two or more vectors, producing a -vector; * and / cannot be used between vectors; the -scalar product is represented by the . operator; and the product of a -scalar and vector expression is well defined, and is a vector. -

    -

    -You can represent components of vectors by including representations of unit -vectors in your system. For instance, letting E0 represent the unit -vector (1,0,0,0), the command -

    -

    -V1.E0 := 0;would set up the substitution of zero for the first componen -t of the vector -V1. -

    -

    -Identifiers that are declared by the index and mass declaratio -ns are -automatically declared to be vectors. -

    -

    -The following errors can occur in calculations using the high energy -physics package: -

    -

    -A represents only gamma5 in vector expressionsYou have tried to use A i -n some way other than gamma5 in a -high-energy physics expression. -

    -

    -

    -Gamma5 not allowed unless vecdim is 4You have used gamma_5 in a high-en -ergy physics -computation involving a vector dimension other than 4. -

    -

    -

    -<ID> has no mass -

    -

    -One of the arguments to -mshell has had no mass assigned to it, in -high-energy physics calculations. -

    -

    -

    -Missing arguments for G operatorA line symbol is missing in a gamma mat -rix expression in high-energy physics -calculations. -

    -

    -

    -Unmatched index<list> -

    -

    -The parser has found unmatched indices during the evaluation of a -gamma matrix expression in high-energy physics calculations. -

    -

    -

    -

    -

    - - - -High Energy Physics -INDEX

    -High Energy Physics

    -
  • HEPHYS introduction

    -

  • HE-dot operator

    -

  • EPS operator

    -

  • G operator

    -

  • INDEX declaration

    -

  • MASS command

    -

  • MSHELL command

    -

  • NOSPUR declaration

    -

  • REMIND declaration

    -

  • SPUR declaration

    -

  • VECDIM command

    -

  • VECTOR declaration

    -

    •  - - -Numeric_Package -INDEX

    - - - -NUMERIC PACKAGE _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -The numeric package supplies algorithms based on approximation -techniques of numerical mathematics. The algorithms use -the -rounded mode arithmetic of REDUCE, including -the variable precision feature which is exploited in some -algorithms in an adaptive manner in order to reach the -desired accuracy. -

    -

    - - - -Interval -INDEX

    - - - -INTERVAL _ _ _ _ _ _ _ _ _ _ _ _ type

    -

    - -Intervals are generally coded as lower bound and -upper bound connected by the operator .., usually -associated to a variable in an -equation. -

    -

    -

    -syntax:

    -<var> = (<low> .. <high>) -

    -

    -

    -where <var> is a -kernel and <low>, <high> are -numbers or expression which evaluate to numbers with <low><=<high ->. -

    -

    -

    -examples:

    -

    means that the variable x is taken in the range from 2 -.5 up to -3.5. -

    -

    - - - -numeric_accuracy -INDEX

    - - - -NUMERIC ACCURACY

    -

    - -The keyword parameters accuracy=a and iterations=i, -where aand i must be positive integer numbers, control the -iterative algorithms: the iteration is continued until -the local error is below 10**-a; if that is impossible -within i steps, the iteration is terminated with an -error message. The values reached so far are then returned -as the result. -

    -

    - - - -TRNUMERIC -INDEX

    - - - -TRNUMERIC _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -Normally the algorithms produce only a minimum of printed -output during their operation. In cases of an unsuccessful -or unexpected long operation a trace of the iteration can be -printed by setting trnumeric on. -

    -

    - - - -num_min -INDEX

    - - - -NUM_MIN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Fletcher Reeves version of the steepest descent -algorithms is used to find the minimum of a -function of one or more variables. The -function must have continuous partial derivatives with respect to all -variables. The starting point of the search can be -specified; if not, random values are taken instead. -The steepest descent algorithms in general find only local -minima. -

    -

    -

    -syntax:

    -num_min(<exp>, - <var>[=<val>] [,<var>[=<val>] ... - [,accuracy=<a>] [,iterations=<i>]) -

    -

    -or -

    -

    -num_min(exp, { - <var>[=<val>] [,<var>[=<val>] ...] } - [,accuracy=<a>] [,iterations=<i>]) -

    -

    -

    -where <exp> is a function expression, -<var> are the variables in <exp> and -<val> are the (optional) start values. -For <a> and <i> see -numeric accuracy. -

    -

    -Num_mintries to find the next local minimum along the descending -path starting at the given point. The result is a -list -with the minimum function value as first element followed by a list -of -equations, where the variables are equated to - the coordinates -of the result point. -

    -

    -

    -examples:

    -

    
-num_min(sin(x)+x/5, x)
-
-  {4.9489585606,{X=29.643767785}}
-
-
-num_min(sin(x)+x/5, x=0)
-
-  { - 1.3342267466,{X= - 1.7721582671}}
-
-

    - - -num_solve -INDEX

    - - - -NUM_SOLVE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -An adaptively damped Newton iteration is used to find -an approximative root of a function (function vector) or the -solution of an -equation (equation system). The expressions -must have continuous derivatives for all variables. -A starting point for the iteration can be given. If not given -random values are taken instead. When the number of -forms is not equal to the number of variables, the -Newton method cannot be applied. Then the minimum -of the sum of absolute squares is located instead. -

    -

    -With -complex on, solutions with imaginary parts can be -found, if either the expression(s) or the starting point -contain a nonzero imaginary part. -

    -

    -

    -syntax:

    -num_solve(<exp>, <var>[=<val>][,accuracy=<a>][, -iterations=<i>]) -

    -

    -or -

    -

    -num_solve({<exp>,...,<exp>}, <var>[=<val>],..., -<var>[=<val>] - [,accuracy=<a>][,iterations=<i>]) -

    -

    -or -

    -

    -num_solve({<exp>,...,<exp>}, {<var>[=<val>],... -,<var>[=<val>]} - [,accuracy=<a>][,iterations=<i>]) -

    -

    -

    -

    -where <exp> are function expressions, - <var> are the variables, - <val> are optional start values. -For <a> and <i> see -numeric accuracy. -

    -

    -num_solvetries to find a zero/solution of the expression(s). -Result is a list of equations, where the variables are -equated to the coordinates of the result point. -

    -

    -The Jacobian matrix is stored as side effect the shared -variable jacobian. -

    -

    -

    -examples:

    -

    
-num_solve({sin x=cos y, x + y = 1},{x=1,y=2});
-
-
-  {X= - 1.8561957251,Y=2.856195584}
-
-
-jacobian;
-
-      [COS(X)  SIN(Y)]
-      [              ]
-      [  1       1   ]
-
-

    - - -num_int -INDEX

    - - - -NUM_INT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -For the numerical evaluation of univariate integrals -over a finite interval the following strategy is used: -If -int finds a formal antiderivative - which is bounded in the integration interval, this - is evaluated and the end points and the difference - is returned. -Otherwise a -Chebyshev fit is computed, - starting with order 20, eventually up to order 80. - If that is recognized as sufficiently convergent - it is used for computing the integral by directly - integrating the coefficient sequence. -If none of these methods is successful, an - adaptive multilevel quadrature algorithm is used. -

    -

    -For multivariate integrals only the adaptive quadrature is used. -This algorithm tolerates isolated singularities. -The value iterations here limits the number of -local interval intersection levels. -<a> is a measure for the relative total discretization -error (comparison of order 1 and order 2 approximations). -

    -

    -

    -syntax:

    -num_int(<exp>,<var>=(<l> .. <u>) - [,<var>=(<l> .. <u>),...] - [,accuracy=<a>][,iterations=<i>]) -

    -

    -

    -where <exp> is the function to be integrated, -<var> are the integration variables, -<l> are the lower bounds, -<u> are the upper bounds. -

    -

    -Result is the value of the integral. -

    -

    -

    -examples:

    -

    
-num_int(sin x,x=(0 .. 3.1415926));
-
-  2.0000010334
-
-

    - - -num_odesolve -INDEX

    - - - -NUM_ODESOLVE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Runge-Kutta method of order 3 finds an approximate graph for -the solution of real ODE initial value problem. -

    -

    -

    -syntax:

    -num_odesolve(<exp>,<depvar>=<start>, - <indep>=(<from> .. <to>) - [,accuracy=<a>][,iterations=<i>]) -

    -

    -or -

    -

    -num_odesolve({<exp>,<exp>,...}, - { <depvar>=<start>,<depvar>=<start>,...} - <indep>=(<from> .. <to>) - [,accuracy=<a>][,iterations=<i>]) -

    -

    -

    -

    -where -<depvar> and <start> specify the dependent variable(s) -and the starting point value (vector), -<indep>, <from> and <to> specify the independent variable -and the integration interval (starting point and end point), -<exp> are equations or expressions which -contain the first derivative of the independent variable -with respect to the dependent variable. -

    -

    -The ODEs are converted to an explicit form, which then is -used for a Runge Kutta iteration over the given range. The -number of steps is controlled by the value of <i> -(default: 20). If the steps are too coarse to reach the desired -accuracy in the neighborhood of the starting point, the number is -increased automatically. -

    -

    -Result is a list of pairs, each representing a point of the -approximate solution of the ODE problem. -

    -

    -

    -examples:

    -

    
-depend(y,x);
-
-num_odesolve(df(y,x)=y,y=1,x=(0 .. 1), iterations=5);
-
-
-  ,{0.2,1.2214},{0.4,1.49181796},{0.6,1.8221064563},
-    {0.8,2.2255208258},{1.0,2.7182511366}}
-
-

    In most cases you must declare the dependency relation -between the variables explicitly using -depend; -otherwise the formal derivative might be converted to zero. -

    -

    -The operator -solve is used to convert the form into -an explicit ODE. If that process fails or if it has no unique result, -the evaluation is stopped with an error message. -

    -

    - - - -bounds -INDEX

    - - - -BOUNDS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -Upper and lower bounds of a real valued function over an - -interval or a rectangular multivariate domain are comp -uted -by the operator bounds. The algorithmic basis is the computation -with inequalities: starting from the interval(s) of the -variables, the bounds are propagated in the expression -using the rules for inequality computation. Some knowledge -about the behavior of special functions like ABS, SIN, COS, EXP, LOG, -fractional exponentials etc. is integrated and can be evaluated -if the operator bounds is called with rounded mode on -(otherwise only algebraic evaluation rules are available). -

    -

    -If bounds finds a singularity within an interval, the evaluation -is stopped with an error message indicating the problem part -of the expression. -

    -

    -

    -syntax:

    -bounds(<exp>,<var>=(<l> .. <u>) - [,<var>=(<l> .. <u>) ...]) -

    -

    -or -

    -

    -bounds(<exp>,{<var>=(<l> .. <u>) - [,<var>=(<l> .. <u>) ...]}) -

    -

    -

    -

    -where <exp> is the function to be investigated, -<var> are the variables of <exp>, -<l> and <u> specify the area as set of -intervals. -

    -

    -boundscomputes upper and lower bounds for the expression in the -given area. An -interval is returned. -

    -

    -

    -examples:

    -

    
-bounds(sin x,x=(1 .. 2));
-
-  -1 .. 1
-
-
-on rounded;
-
-bounds(sin x,x=(1 .. 2));
-
-  0.84147098481 .. 1
-
-
-bounds(x**2+x,x=(-0.5 .. 0.5));
-
-  - 0.25 .. 0.75
-
-

    - - -Chebyshev_fit -INDEX

    - - - -CHEBYSHEV FIT

    -

    - -

    -

    -The operator family Chebyshev_... implements approximation -and evaluation of functions by the Chebyshev method. -Let T(n,a,b,x) be the Chebyshev polynomial of order n -transformed to the interval (a,b). -Then a function f(x) can be -approximated in (a,b) by a series -

    -

    -

    
-  for i := 0:n sum c(i)*T(i,a,b,x)
-

    The operator chebyshev_fit computes this approximation an -d -returns a list, which has as first element the sum expressed -as a polynomial and as second element the sequence -of Chebyshev coefficients. -Chebyshev_df and Chebyshev_int transform a Chebyshev -coefficient list into the coefficients of the corresponding -derivative or integral respectively. For evaluating a Chebyshev -approximation at a given point in the basic interval the -operator Chebyshev_eval can be used. -Chebyshev_eval is based on a recurrence relation which is -in general more stable than a direct evaluation of the -complete polynomial. -

    -

    -

    -syntax:

    -chebyshev_fit(<fcn>,<var>=(<lo> .. <hi>),<n ->) -

    -

    -chebyshev_eval(<coeffs>,<var>=(<lo> .. <hi>), - <var>=<pt>) -

    -

    -chebyshev_df(<coeffs>,<var>=(<lo> .. <hi>)) -

    -

    -chebyshev_int(<coeffs>,<var>=(<lo> .. <hi>)) -

    -

    -

    -where <fcn> is an algebraic expression (the target function), -<var> is the variable of <fcn>, -<lo> and <hi> are -numerical real values which describe an -interval <lo> <<hi>, -the integer <n> is the approximation order (set to 20 if missing), -<pt> is a number in the interval and <coeffs> is -a series of Chebyshev coefficients. -

    -

    -

    -examples:

    -

    
-
-on rounded;
-
-
-w:=chebyshev_fit(sin x/x,x=(1 .. 3),5);
-
-
-                 3            2
-  w := {0.03824*x   - 0.2398*x   + 0.06514*x + 0.9778,
-        {0.8991,-0.4066,-0.005198,0.009464,-0.00009511}}                    
-
-
-chebyshev_eval(second w, x=(1 .. 3), x=2.1);
-
-
-  0.4111
-
-

    - - -num_fit -INDEX

    - - - -NUM_FIT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator num_fit finds for a set of -points the linear combination of a given set of -functions (function basis) which approximates the -points best under the objective of the least squares -criterion (minimum of the sum of the squares of the deviation). -The solution is found as zero of the -gradient vector of the sum of squared errors. -

    -

    -

    -syntax:

    -num_fit(<vals>,<basis>,<var>=<pts>) -

    -

    -

    -where <vals> is a list of numeric values, -<var> is a variable used for the approximation, -<pts> is a list of coordinate values which correspond to -<var>, -<basis> is a set of functions varying in var which is used - for the approximation. -

    -

    -The result is a list containing as first element the -function which approximates the given values, and as -second element a list of coefficients which were used -to build this function from the basis. -

    -

    -

    -examples:

    -

    
- 
-pts:=for i:=1 step 1 until 5 collect i$
-
-vals:=for i:=1 step 1 until 5 collect
-
-            for j:=1:i product j$
-
-num_fit(vals,{1,x,x**2},x=pts);
-
-                     2
-      {14.571428571*X   - 61.428571429*X + 54.6,{54.6,
-           - 61.428571429,14.571428571}}
-
-

    - - -Numeric Package -INDEX

    -Numeric Package

    -
  • Numeric Package introduction

    -

  • Interval type

    -

  • numeric accuracy concept

    -

  • TRNUMERIC switch

    -

  • num_min operator

    -

  • num_solve operator

    -

  • num_int operator

    -

  • num_odesolve operator

    -

  • bounds operator

    -

  • Chebyshev fit concept

    -

  • num_fit operator

    -

    •  - - -Roots_Package -INDEX

    - - - -ROOTS PACKAGE _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -

    -

    -The root finding package is designed so that it can -be used to find some or all of the roots of univariate -polynomials with real or complex coefficients, to the accuracy -specified by the user. -

    -

    -Not all operators of roots package are described here. For using -the operators -

    -

    -isolater(intervals isolating real roots) -

    -

    -rlrootno(number of real roots in an interval) -

    -

    -rootsat-prec(roots at system precision) -

    -

    -rootval(result in equation form) -

    -

    -firstroot(computing only one root) -

    -

    -getroot(selecting roots from a collection) -

    -

    -please consult the full documentation of the package. -

    -

    - - - -MKPOLY -INDEX

    - - - -MKPOLY _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -Given a roots list as returned by -roots, -the operator mkpoly constructs a -polynomial which has these numbers as roots. -

    -syntax:

    -

    -

    -mkpoly<rl> -

    -

    -

    -where <rl> is a -list with equations, which -all have the same -kernel on their left-hand sides -and numbers as right-hand sides. -

    -

    -

    -examples:

    -

    
-mkpoly{x=1,x=-2,x=i,x=-i};
-
-  x**4 + x**3 - x**2 + x - 2
-
-

    Note that this polynomial is unique only up to a numeric -factor. -

    -

    - - - -NEARESTROOT -INDEX

    - - - -NEARESTROOT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator nearestroot finds one root of a polynomial -with an iteration using a given starting point. -

    -

    -

    -syntax:

    -nearestroot(<p>,<pt>) -

    -

    -

    -where <p> is a univariate polynomial -and <pt> is a number. -

    -

    -

    -examples:

    -

    
-nearestroot(x^2+2,2);
-
-  {x=1.41421*i}
-
-

    The minimal accuracy of the result values is controlled by - -rootacc. -

    -

    - - - -REALROOTS -INDEX

    - - - -REALROOTS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator realroots finds that real roots of a polynomial -to an accuracy that is sufficient to separate them and which is -a minimum of 6 decimal places. -

    -

    -

    -syntax:

    -realroots(<p>) or -

    -

    -realroots(<p>,<from>,<to>) -

    -

    -

    -where <p> is a univariate polynomial. -The optional parameters <from> and <to> classify -an interval: if given, exactly the real roots in this -interval will be returned. <from> and <to> -can also take the values infinity or -infinity. -If omitted all real roots will be returned. -Result is a -list -of equations which represent the roots of the polynomial at the -given accuracy. -

    -

    -

    -examples:

    -

    
-realroots(x^5-2);
-
-  {x=1.1487}
-
-
-realroots(x^3-104*x^2+403*x-300,2,infinity);
-
-
-  {x=3.0,x=100.0}
-
-
-realroots(x^3-104*x^2+403*x-300,-infinity,2);
-
-
-  {x=1}
-
-

    The minimal accuracy of the result values is controlled by - -rootacc. -

    -

    - - - -ROOTACC -INDEX

    - - - -ROOTACC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator rootacc allows you to set the accuracy -up to which the roots package computes its results. -

    -syntax:

    -

    -

    -rootacc(<n>) -

    -

    -

    -Here <n> is an integer value. The internal accuracy of -the roots package is adjusted to a value of -max(6,n). The default value is 6. -

    -

    - - - -ROOTS -INDEX

    - - - -ROOTS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator roots -is the main top level function of the roots package. -It will find all roots, real and complex, of the polynomial p -to an accuracy that is sufficient to separate them and which is -a minimum of 6 decimal places. -

    -

    -

    -syntax:

    -roots(<p>) -

    -

    -

    -where <p> is a univariate polynomial. Result is a -list -of equations which represent the roots of the polynomial at the -given accuracy. In addition, roots stores -separate lists of real roots and complex roots in the global -variables -rootsreal and -rootscomplex. -

    -

    -

    -examples:

    -

    
-roots(x^5-2);
-
-  {x=-0.929316 + 0.675188*i,
-    x=-0.929316 - 0.675188*i,
-    x=0.354967 + 1.09248*i,
-    x=0.354967 - 1.09248*i, 
-    x=1.1487}
-
-

    The minimal accuracy of the result values is controlled by - -rootacc. -

    -

    - - - -ROOT_VAL -INDEX

    - - - -ROOT\_VAL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The operator root_val computes the roots of a -univariate polynomial at system precision -(or greater if required for root separation) and presents -its result as a list of numbers. -

    -syntax:

    -

    -

    -roots(<p>) -

    -

    -

    -where <p> is a univariate polynomial. -

    -

    -

    -examples:

    -

    
-root_val(x^5-2);
-
-  {-0.929316490603 + 0.6751879524*i,
-   -0.929316490603 - 0.6751879524*i,
-   0.354967313105 + 1.09247705578*i,
-   0.354967313105 - 1.09247705578*i,
-   1.148698355}
-
-

    - - -ROOTSCOMPLEX -INDEX

    - - - -ROOTSCOMPLEX _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -When the operator -roots is called the complex -roots are collected in the global variable rootscomplex -as -list. -

    -

    - - - -ROOTSREAL -INDEX

    - - - -ROOTSREAL _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -When the operator -roots is called the real -roots are collected in the global variable rootreal -as -list. -

    -

    - - - -Roots Package -INDEX

    -Roots Package

    -
  • Roots Package introduction

    -

  • MKPOLY operator

    -

  • NEARESTROOT operator

    -

  • REALROOTS operator

    -

  • ROOTACC operator

    -

  • ROOTS operator

    -

  • ROOT\_VAL operator

    -

  • ROOTSCOMPLEX variable

    -

  • ROOTSREAL variable

    -

    •  - - -Special_Function_Package -INDEX

    - - - -SPECIAL FUNCTION PACKAGE _ _ _ _ _ _ _ _ _ _ _ _ introduction -

    -

    - -The REDUCE Special Function Package supplies extended -algebraic and numeric support for a wide class of objects. -This package was released together with REDUCE 3.5 (October 1993) -for the first time, a major update is released with REDUCE 3.6. -

    -

    -The functions included in this package are in most cases (unless otherwise -stated) defined and named like in the book by Abramowitz and Stegun: -Handbook of Mathematical Functions, Dover Publications. -

    -

    -The aim is to collect as much information on the special functions -and simplification capabilities as possible, -i.e. algebraic simplifications and numeric (rounded mode) code, limits -of the functions together -with the definitions of the functions, which are in most cases a power -series, a (definite) integral and/or a differential equation. -

    -

    -What can be found: Some famous constants, a variety of Bessel functions, -special polynomials, -the Gamma function, the (Riemann) Zeta function, Elliptic Functions, Elliptic -Integrals, 3J symbols (Clebsch-Gordan coefficients) and integral functions. -

    -

    -What is missing: Mathieu functions, LerchPhi, etc.. -The information about the special functions which solve certain -differential equation is very limited. -In several cases numerical approximation is restricted to real -arguments or is missing completely. -

    -

    -The implementation of this package uses REDUCE rule sets to a large extent, -which guarantees a high 'readability' of the functions definitions in the -source file directory. It makes extensions to the special -functions code easy in most cases too. To look at these rules -it may be convenient to use the showrules operator e.g. -

    -

    - -showrulesBesseli; -

    -

    -. -

    -

    -Some evaluations are improved if the special function package is loaded, -e.g. some (infinite) sums and products leading to expressions including -special functions are known in this case. -

    -

    -Note: The special function package has to be loaded explicitly by calling -

    
-   load_package specfn;
-

    -

    -The functions -MeijerG and -hypergeometric require -additionally -

    
-   load_package specfn2;
-

    -

    - - - -Constants -INDEX

    - - - -CONSTANTS

    -

    - -

    -

    -There are a few constants known to the special function package, namely -

    -

    - _ _ _ Euler's constant (which can be computed as - -Psi(1)) and -

    - _ _ _ Khinchin's constant (which is defined in Khinchin's book - ``Continued Fractions'') and -

    - _ _ _ Golden_Ratio (which can be computed as (1 + sqrt 5)/2) and -

    - _ _ _ Catalan's constant (which is known as an infinite sum of recipro -cal -powers) -

    -

    -

    -examples:

    -

    
-on rounded;
-Euler_Gamma; 
-
-  0.577215664902 
-
-
-Khinchin; 
-
-  2.68545200107 
-
-
-Catalan 
-
-  0.915965594177 
-
-
-Golden_Ratio 
-
-  1.61803398875
-
-

    - - -BERNOULLI -INDEX

    - - - -BERNOULLI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The bernoulli operator returns the nth Bernoulli number. -

    -

    -

    -syntax:

    -Bernoulli(<integer>) -

    -

    -

    -

    -

    -examples:

    -

    
-bernoulli 20; 
-
-  - 174611 / 330 
-
-
-bernoulli 17; 
-
-  0
-
-

    All Bernoulli numbers with odd indices except for 1 are zero. -

    -

    -

    - - - -BERNOULLIP -INDEX

    - - - -BERNOULLIP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The BernoulliP operator returns the nth Bernoulli Polynomial -evaluated at x. -

    -

    -

    -syntax:

    -BernoulliP(<integer>,<expression>) -

    -

    -

    -

    -

    -examples:

    -

    
-BernoulliP(3,z); 
-
-        2
-  z*(2*z   - 3*z + 1)/2
-
-
-
-BernoulliP(10,3); 
-
-  338585 / 66
-
-

    The value of the nth Bernoulli Polynomial at 0 is the nth Bernoull -i number. -

    -

    -

    - - - -EULER -INDEX

    - - - -EULER _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The EULER operator returns the nth Euler number. -

    -

    -

    -syntax:

    -Euler(<integer>) -

    -

    -

    -

    -examples:

    -

    
-Euler 20; 
-
-  370371188237525 
-
-
-Euler 0; 
-
-  1
-
-

    The Euler numbers are evaluated by a recursive algorithm -which -makes it hard to compute Euler numbers above say 200. -

    -

    -Euler numbers appear in the coefficients of the power series -representation of 1/cos(z). -

    -

    -

    - - - -EULERP -INDEX

    - - - -EULERP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The EulerP operator returns the nth Euler Polynomial. -

    -

    -

    -syntax:

    -EulerP(<integer>,<expression>) -

    -

    -

    -

    -examples:

    -

    
-EulerP(2,xx); 
-
-  xx*(xx - 1) 
-
-
-EulerP(10,3); 
-
-  2046
-
-

    The Euler numbers are the values of the Euler Polynomials at 1/2 -multiplied by 2**n. -

    -

    -

    - - - -ZETA -INDEX

    - - - -ZETA _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Zeta operator returns Riemann's Zeta function, -

    -

    -Zeta (z) := sum(1/(k**z),k,1,infinity) -

    -

    -

    -syntax:

    -Zeta(<expression>) -

    -

    -

    -

    -examples:

    -

    
-Zeta(2); 
-
-    2
-  pi  / 6 
-
-
-on rounded; 
-
-Zeta 1.01; 
-
-  100.577943338
-
-

    Numerical computation for the Zeta function for arguments close to - 1 are -tedious, because the series is converging very slowly. In this case a formula -(e.g. found in Bender/Orzag: Advanced Mathematical Methods for -Scientists and Engineers, McGraw-Hill) is used. -

    -

    -No numerical approximation for complex arguments is done. -

    -

    -

    - - - -Bernoulli Euler Zeta -INDEX

    -Bernoulli Euler Zeta

    -
  • BERNOULLI operator

    -

  • BERNOULLIP operator

    -

  • EULER operator

    -

  • EULERP operator

    -

  • ZETA operator

    -

    •  - - -BESSELJ -INDEX

    - - - -BESSELJ _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The BesselJ operator returns the Bessel function of the first kind. -

    -

    -

    -syntax:

    -BesselJ(<order>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-BesselJ(1/2,pi); 
-
-  0 
-
-
-on rounded; 
-
-BesselJ(0,1); 
-
-  0.765197686558  
-
-

    - - -BESSELY -INDEX

    - - - -BESSELY _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The BesselY operator returns the Bessel function of the second kind. -

    -syntax:

    -

    -

    -BesselY(<order>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-BesselY (1/2,pi); 
-
-  - sqrt(2) / pi 
-
-
-on rounded; 
-
-BesselY (1,3); 
-
-  0.324674424792
-
-

    The operator BesselY is also called Weber's function. -

    -

    -

    - - - -HANKEL1 -INDEX

    - - - -HANKEL1 _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Hankel1 operator returns the Hankel function of the first kind. -

    -

    -

    -syntax:

    -Hankel1(<order>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-on complex; 
-
-Hankel1 (1/2,pi); 
-
-  - i * sqrt(2) / pi 
-
-
-Hankel1 (1,pi); 
-
-  besselj(1,pi) + i*bessely(1,pi)
-
-

    The operator Hankel1 is also called Bessel function of th -e third kind. -There is currently no numeric evaluation of Hankel functions. -

    -

    -

    - - - -HANKEL2 -INDEX

    - - - -HANKEL2 _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Hankel2 operator returns the Hankel function of the second kind. -

    -

    -

    -syntax:

    -Hankel2(<order>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-on complex; 
-
-Hankel2 (1/2,pi); 
-
-  - i * sqrt(2) / pi 
-
-
-Hankel2 (1,pi); 
-
-  besselj(1,pi) - i*bessely(1,pi)
-
-

    The operator Hankel2 is also called Bessel function of th -e third kind. -There is currently no numeric evaluation of Hankel functions. -

    -

    -

    - - - -BESSELI -INDEX

    - - - -BESSELI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The BesselI operator returns the modified Bessel function I. -

    -

    -

    -syntax:

    -BesselI(<order>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-on rounded; 
-
-Besseli (1,1); 
-
-  0.565159103992
-
-

    The knowledge about the operator BesselI is currently fai -rly limited. -

    -

    -

    - - - -BESSELK -INDEX

    - - - -BESSELK _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The BesselK operator returns the modified Bessel function K. -

    -

    -

    -syntax:

    -BesselK(<order>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-df(besselk(0,x),x); 
-
-  - besselk(1,x)
-
-

    There is currently no numeric support for the operator BesselK -. -

    -

    -

    - - - -StruveH -INDEX

    - - - -STRUVEH _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The StruveH operator returns Struve's H function. -

    -

    -

    -syntax:

    -StruveH(<order>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-struveh(-3/2,x); 
-
-  - besselj(3/2,x) / i
-
-

    - - -StruveL -INDEX

    - - - -STRUVEL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The StruveL operator returns the modified Struve L function . -

    -

    -

    -syntax:

    -StruveL(<order>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-struvel(-3/2,x); 
-
-  besseli(3/2,x)
-
-

    - - -KummerM -INDEX

    - - - -KUMMERM _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The KummerM operator returns Kummer's M function. -

    -

    -

    -syntax:

    -KummerM(<parameter>,<parameter>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-kummerm(1,1,x); 
-
-   x
-  e  
-
-
-on rounded; 
-
-kummerm(1,3,1.3); 
-
-  1.62046942914
-
-

    Kummer's M function is one of the Confluent Hypergeometric functio -ns. -For reference see the -hypergeometric operator. -

    -

    -

    - - - -KummerU -INDEX

    - - - -KUMMERU _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The KummerU operator returns Kummer's U function. -

    -

    -

    -syntax:

    -KummerU(<parameter>,<parameter>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-df(kummeru(1,1,x),x) 
-
-  - kummeru(2,2,x)
-
-

    Kummer's U function is one of the Confluent Hypergeometric functio -ns. -For reference see the -hypergeometric operator. -

    -

    -

    - - - -WhittakerW -INDEX

    - - - -WHITTAKERW _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The WhittakerW operator returns Whittaker's W function. -

    -

    -

    -syntax:

    -WhittakerW(<parameter>,<parameter>,<argument>) -

    -

    -

    -

    -examples:

    -

    
-WhittakerW(2,2,2); 
-
-                    1
-  4*sqrt(2)*kummeru(-,5,2)
-                    2
-  -------------------------
-             e
-
-

    Whittaker's W function is one of the Confluent Hypergeometric func -tions. -For reference see the -hypergeometric operator. -

    -

    -

    - - - -Bessel Functions -INDEX

    -Bessel Functions

    -
  • BESSELJ operator

    -

  • BESSELY operator

    -

  • HANKEL1 operator

    -

  • HANKEL2 operator

    -

  • BESSELI operator

    -

  • BESSELK operator

    -

  • StruveH operator

    -

  • StruveL operator

    -

  • KummerM operator

    -

  • KummerU operator

    -

  • WhittakerW operator

    -

    •  - - -Airy_Ai -INDEX

    - - - -AIRY_AI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Airy_Ai operator returns the Airy Ai function for a given argument. - -

    -

    -

    -syntax:

    -Airy_Ai(<argument>) -

    -

    -

    -

    -examples:

    -

    
-on complex;
-on rounded;
-Airy_Ai(0); 
-
-
-  0.355028053888          
-
-
-Airy_Ai(3.45 + 17.97i); 
-
-  - 5.5561528511e+9 - 8.80397899932e+9*i  
-
-

    - - -Airy_Bi -INDEX

    - - - -AIRY_BI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Airy_Bi operator returns the Airy Bi function for a given -argument. -

    -

    -

    -syntax:

    -Airy_Bi(<argument>) -

    -

    -

    -

    -examples:

    -

    
-Airy_Bi(0); 
-
-  0.614926627446          
-
-
-Airy_Bi(3.45 + 17.97i); 
-
-  8.80397899932e+9 - 5.5561528511e+9*i   
-
-

    - - -Airy_Aiprime -INDEX

    - - - -AIRY_AIPRIME _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Airy_Aiprime operator returns the Airy Aiprime function for a -given argument. -

    -

    -

    -syntax:

    -Airy_Aiprime(<argument>) -

    -

    -

    -

    -examples:

    -

    
-Airy_Aiprime(0); 
-
-  - 0.258819403793           
-
-
-Airy_Aiprime(3.45+17.97i);
-
-  - 3.83386421824e+19 + 2.16608828136e+19*i 
-
-

    - - -Airy_Biprime -INDEX

    - - - -AIRY_BIPRIME _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Airy_Biprime operator returns the Airy Biprime function for a -given argument. -

    -

    -

    -syntax:

    -Airy_Biprime(<argument>) -

    -

    -

    -

    -examples:

    -

    
-Airy_Biprime(0); 
-
-
-Airy_Biprime(3.45 + 17.97i); 
-
-  3.84251916792e+19 - 2.18006297399e+19*i
-
-

    - - -Airy Functions -INDEX

    -Airy Functions

    -
  • Airy_Ai operator

    -

  • Airy_Bi operator

    -

  • Airy_Aiprime operator

    -

  • Airy_Biprime operator

    -

    •  - - -JacobiSN -INDEX

    - - - -JACOBISN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobisn operator returns the Jacobi Elliptic function sn. -

    -

    -

    -syntax:

    -Jacobisn(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobisn(0.672, 0.36) 
-
-  0.609519691792 
-
-
-Jacobisn(1,0.9) 
-
-  0.770085724907881 
-
-

    - - -JacobiCN -INDEX

    - - - -JACOBICN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobicn operator returns the Jacobi Elliptic function cn. -

    -

    -

    -syntax:

    -Jacobicn(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobicn(7.2, 0.6) 
-
-  0.837288298482018  
-
-
-Jacobicn(0.11, 19) 
-
-  0.994403862690043 - 1.6219006985556e-16*i  
-
-

    - - -JacobiDN -INDEX

    - - - -JACOBIDN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobidn operator returns the Jacobi Elliptic function dn. -

    -

    -

    -syntax:

    -Jacobidn(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobidn(15, 0.683) 
-
-  0.640574162024592 
-
-
-Jacobidn(0,0) 
-
-  1 
-
-

    - - -JacobiCD -INDEX

    - - - -JACOBICD _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobicd operator returns the Jacobi Elliptic function cd. -

    -

    -

    -syntax:

    -Jacobicd(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobicd(1, 0.34) 
-
-  0.657683337805273 
-
-
-Jacobicd(0.8,0.8) 
-
-  0.925587311582301 
-
-

    - - -JacobiSD -INDEX

    - - - -JACOBISD _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobisd operator returns the Jacobi Elliptic function sd. -

    -

    -

    -syntax:

    -Jacobisd(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobisd(12, 0.4) 
-
-  0.357189729437272    
-
-
-Jacobisd(0.35,1) 
-
-  - 1.17713873203043  
-
-

    - - -JacobiND -INDEX

    - - - -JACOBIND _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobind operator returns the Jacobi Elliptic function nd. -

    -

    -

    -syntax:

    -Jacobind(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobind(0.2, 17) 
-
-  1.46553203037507 + 0.0000000000334032759313703*i 
-
-
-Jacobind(30, 0.001) 
-
-  1.00048958438  
-
-

    - - -JacobiDC -INDEX

    - - - -JACOBIDC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobidc operator returns the Jacobi Elliptic function dc. -

    -

    -

    -syntax:

    -Jacobidc(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobidc(0.003,1) 
-
-  1 
-
-
-Jacobidc(2, 0.75) 
-
-  6.43472885111  
-
-

    - - -JacobiNC -INDEX

    - - - -JACOBINC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobinc operator returns the Jacobi Elliptic function nc. -

    -

    -

    -syntax:

    -Jacobinc(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobinc(1,0) 
-
-  1.85081571768093 
-
-
-Jacobinc(56, 0.4387) 
-
-  39.304842663512  
-
-

    - - -JacobiSC -INDEX

    - - - -JACOBISC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobisc operator returns the Jacobi Elliptic function sc. -

    -

    -

    -syntax:

    -Jacobisc(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobisc(9, 0.88) 
-
-  - 1.16417697982095  
-
-
-Jacobisc(0.34, 7) 
-
-  0.305851938390775 - 9.8768100944891e-12*i 
-
-

    - - -JacobiNS -INDEX

    - - - -JACOBINS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobins operator returns the Jacobi Elliptic function ns. -

    -

    -

    -syntax:

    -Jacobins(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobins(3, 0.9) 
-
-  1.00945801599785 
-
-
-Jacobins(0.887, 15) 
-
-  0.683578280513975 - 0.85023411082469*i 
-
-

    - - -JacobiDS -INDEX

    - - - -JACOBIDS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobisn operator returns the Jacobi Elliptic function ds. -

    -

    -

    -syntax:

    -Jacobids(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobids(98,0.223) 
-
-  - 1.061253961477 
-
-
-Jacobids(0.36,0.6) 
-
-  2.76693172243692 
-
-

    - - -JacobiCS -INDEX

    - - - -JACOBICS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Jacobics operator returns the Jacobi Elliptic function cs. -

    -

    -

    -syntax:

    -Jacobics(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Jacobics(0, 0.767) 
-
-  infinity   
-
-
-Jacobics(1.43, 0) 
-
-  0.141734127352112 
-
-

    - - -JacobiAMPLITUDE -INDEX

    - - - -JACOBIAMPLITUDE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The JacobiAmplitude operator returns the amplitude of u. -

    -syntax:

    -

    -

    -JacobiAmplitude(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-JacobiAmplitude(7.239, 0.427) 
-
-  0.0520978301448978 
-
-
-JacobiAmplitude(0,0.1) 
-
-  0 
-
-

    Amplitude u = asin(Jacobisn(u,m)) -

    -

    -

    - - - -AGM_FUNCTION -INDEX

    - - - -AGM_FUNCTION _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The AGM_function operator returns a list of (N, AGM, - list of aNtoa0, list of bNtob0, list of cNtoc0) where a0, b0 and c0 -are the initial values; N is the index number of the last term -used to generate the AGM. AGM is the Arithmetic Geometric Mean. -

    -

    -

    -syntax:

    -AGM_function(<integer>,<integer>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-AGM_function(1,1,1) 
-
-  1,1,1,1,1,1,0,1  
-
-
-AGM_function(1, 0.1, 1.3) 
-
-  {6,
-   2.27985615996629, 
-   {2.27985615996629, 2.27985615996629,
-    2.2798561599706, 2.2798624278857, 
-    2.28742283656583, 2.55, 1},
-   {2.27985615996629, 2.27985615996629,
-    2.27985615996198, 2.2798498920555, 
-    2.27230201920557, 2.02484567313166, 4.1},
-   {0, 4.30803136219904e-12, 0.0000062679151007581,
-    0.00756040868012758, 0.262577163434171, - 1.55, 5.9}}
-
-

    The other Jacobi functions use this function with initial values -a0=1, b0=sqrt(1-m), c0=sqrt(m). -

    -

    -

    - - - -LANDENTRANS -INDEX

    - - - -LANDENTRANS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The landentrans operator generates the descending landen -transformation of the given imput values, returning a list of these -values; initial to final in each case. -

    -syntax:

    -

    -

    -landentrans(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-landentrans(0,0.1) 
-
-  {{0,0,0,0,0},{0.1,0.0025041751943776, 
-
-
- 
-
-  0.00000156772498954046,6.1444078 9914461e-13,0}}  
-
-

    The first list ascends in value, and the second descends in value. - -

    -

    -

    - - - -EllipticF -INDEX

    - - - -ELLIPTICF _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The EllipticF operator returns the Elliptic Integral of the -First Kind. -

    -syntax:

    -

    -

    -EllitpicF(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-EllipticF(0.3, 8.222) 
-
-  0.3 
-
-
-EllipticF(7.396, 0.1) 
-
-  7.58123216114307 
-
-

    The Complete Elliptic Integral of the First Kind can be found by -putting the first argument to pi/2 or by using EllipticK -and the second argument. -

    -

    -

    - - - -EllipticK -INDEX

    - - - -ELLIPTICK _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The EllipticK operator returns the Elliptic value K. -

    -

    -

    -syntax:

    -EllipticK(<integer>) -

    -

    -

    -

    -examples:

    -

    
-EllipticK(0.2) 
-
-  1.65962359861053   
-
-
-EllipticK(4.3) 
-
-  0.808442364282734 - 1.05562492399206*i  
-
-
-EllipticK(0.000481) 
-
-  1.57098526617635    
-
-

    The EllipticK function is the Complete Elliptic Integral -of -the First Kind. -

    -

    -

    - - - -EllipticKprime -INDEX

    - - - -ELLIPTICKPRIME _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The EllipticK' operator returns the Elliptic value K(m). -

    -

    -

    -syntax:

    -EllipticKprime(<integer>) -

    -

    -

    -

    -examples:

    -

    
-EllipticKprime(0.2) 
-
-  2.25720532682085 
-
-
-EllipticKprime(4.3) 
-
-  1.05562492399206 
-
-
-EllipticKprime(0.000481) 
-
-  5.206621921966   
-
-

    The EllipticKprime function is the Complete Elliptic Inte -gral of -the First Kind of (1-m). -

    -

    -

    - - - -EllipticE -INDEX

    - - - -ELLIPTICE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The EllipticE operator used with two arguments -returns the Elliptic Integral of the Second Kind. -

    -syntax:

    -

    -

    -EllipticE(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-EllipticE(1.2,0.22) 
-
-  1.15094019180949 
-
-
-EllipticE(0,4.35) 
-
-  0                
-
-
-EllipticE(9,0.00719) 
-
-  8.98312465929145  
-
-

    The Complete Elliptic Integral of the Second Kind can be obtained -by -using just the second argument, or by using pi/2 as the first argument. -

    -

    -

    -The EllipticE operator used with one argument -returns the Elliptic value E. -

    -syntax:

    -

    -

    -EllipticE(<integer>) -

    -

    -

    -

    -examples:

    -

    
-EllipticE(0.22) 
-
-  1.48046637439519  
-
-
-EllipticE(pi/2, 0.22) 
-
-  1.48046637439519  
-
-

    - - -EllipticTHETA -INDEX

    - - - -ELLIPTICTHETA _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The EllipticTheta operator returns one of the four Theta -functions. It cannot except any number other than 1,2,3 or 4 as -its first argument. -

    -

    -

    -syntax:

    -EllipticTheta(<integer>,<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-EllipticTheta(1, 1.4, 0.72) 
-
-  0.91634775373  
-
-
-EllipticTheta(2, 3.9, 6.1 ) 
-
-  -48.0202736969 + 20.9881034377 i 
-
-
-EllipticTheta(3, 0.67, 0.2) 
-
-  1.0083077448   
-
-
-EllipticTheta(4, 8, 0.75) 
-
-  0.894963369304 
-
-
-EllipticTheta(5, 1, 0.1) 
-
-  ***** In EllipticTheta(a,u,m); a = 1,2,3 or 4.   
-
-

    Theta functions are important because every one of the Jacobian -Elliptic functions can be expressed as the ratio of two theta functions. -

    -

    -

    - - - -JacobiZETA -INDEX

    - - - -JACOBIZETA _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The JacobiZeta operator returns the Jacobian function Zeta. -

    -

    -

    -syntax:

    -JacobiZeta(<expression>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-JacobiZeta(3.2, 0.8) 
-
-  - 0.254536403439 
-
-
-JacobiZeta(0.2, 1.6) 
-
-  0.171766095970451 - 0.0717028569800147*i  
-
-

    The Jacobian function Zeta is related to the Jacobian function The -ta. -But it is significantly different from Riemann's Zeta Function -Zeta. -

    -

    -

    - - - -Jacobi's Elliptic Functions and Elliptic Integrals -INDEX

    -Jacobi's Elliptic Functions and Elliptic Integrals

    -
  • JacobiSN operator

    -

  • JacobiCN operator

    -

  • JacobiDN operator

    -

  • JacobiCD operator

    -

  • JacobiSD operator

    -

  • JacobiND operator

    -

  • JacobiDC operator

    -

  • JacobiNC operator

    -

  • JacobiSC operator

    -

  • JacobiNS operator

    -

  • JacobiDS operator

    -

  • JacobiCS operator

    -

  • JacobiAMPLITUDE operator

    -

  • AGM_FUNCTION operator

    -

  • LANDENTRANS operator

    -

  • EllipticF operator

    -

  • EllipticK operator

    -

  • EllipticKprime operator

    -

  • EllipticE operator

    -

  • EllipticTHETA operator

    -

  • JacobiZETA operator

    -

    •  - - -POCHHAMMER -INDEX

    - - - -POCHHAMMER _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Pochhammer operator implements the Pochhammer notation -(shifted factorial). -

    -

    -

    -syntax:

    -Pochhammer(<expression>,<expression>) -

    -

    -

    -

    -examples:

    -

    
-pochhammer(17,4); 
-
-  116280 
-
-
-
-pochhammer(1/2,z); 
-
-    factorial(2*z)
-  --------------------
-    2*z
-  (2   *factorial(z))
-
-

    A number of complex rules for Pochhammer are inactive, be -cause they -cause a huge system load in algebraic mode. If one wants to use more rules -for the simplification of Pochhammer's notation, one can do: -

    -

    -let special!*pochhammer!*rules; -

    -

    -

    -

    - - - -GAMMA -INDEX

    - - - -GAMMA _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Gamma operator returns the Gamma function. -

    -

    -

    -syntax:

    -Gamma(<expression>) -

    -

    -

    -

    -

    -examples:

    -

    
-gamma(10); 
-
-  362880    
-
-
-gamma(1/2); 
-
-  sqrt(pi)
-
-

    - - -BETA -INDEX

    - - - -BETA _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Beta operator returns the Beta function defined by -

    -

    -Beta (z,w) := defint(t**(z-1)* (1 - t)**(w-1),t,0,1) . -

    -

    -

    -syntax:

    -Beta(<expression>,<expression>) -

    -

    -

    -

    -

    -examples:

    -

    
-Beta(2,2); 
-
-  1 / 6 
-
-
-Beta(x,y); 
-
-  gamma(x)*gamma(y) / gamma(x + y)
-
-

    The operator Beta is simplified towards the -GAMMA operator. -

    -

    -

    - - - -PSI -INDEX

    - - - -PSI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Psi operator returns the Psi (or DiGamma) function. -

    -

    -Psi(x) := df(Gamma(z),z)/ Gamma (z) -

    -

    -

    -syntax:

    -Gamma(<expression>) -

    -

    -

    -

    -

    -examples:

    -

    
-Psi(3); 
-
-  (2*log(2) + psi(1/2) + psi(1) + 3)/2 
-
-
-on rounded; 
-
-- Psi(1); 
-
-  0.577215664902
-
-

    Euler's constant can be found as - Psi(1). -

    -

    -

    - - - -POLYGAMMA -INDEX

    - - - -POLYGAMMA _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Polygamma operator returns the Polygamma function. -

    -

    -Polygamma(n,x) := df(Psi(z),z,n); -

    -

    -

    -syntax:

    -Polygamma(<integer>,<expression>) -

    -

    -

    -

    -examples:

    -

    
- Polygamma(1,2); 
-
-     2
-  (pi   - 6) / 6
-
-
-on rounded; 
-
-Polygamma(1,2.35); 
-
-  0.52849689109
-
-

    The Polygamma function is used for simplification of the -ZETA -function for some arguments. -

    -

    -

    - - - -Gamma and Related Functions -INDEX

    -Gamma and Related Functions

    -
  • POCHHAMMER operator

    -

  • GAMMA operator

    -

  • BETA operator

    -

  • PSI operator

    -

  • POLYGAMMA operator

    -

    •  - - -DILOG_extended -INDEX

    - - - -DILOG EXTENDED _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The package specfn supplies an extended support for the - -dilog operator which implements the dilogarithm fu -nction. -

    -

    -dilog(x) := - defint(log(t)/(t - 1),t,1,x); -

    -

    -

    -syntax:

    -Dilog(<order>,<expression>) -

    -

    -

    -

    -examples:

    -

    
-defint(log(t)/(t - 1),t,1,x); 
-
-  - dilog (x) 
-
-
-dilog 2; 
-
-      2
-  - pi  /12 
-
-
-
-on rounded; 
-
-Dilog 20; 
-
-  - 5.92783972438
-
-

    The operator Dilog is sometimes called Spence's Integral -for n = 2. -

    -

    -

    - - - -Lambert_W_function -INDEX

    - - - -LAMBERT\_W FUNCTION _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -Lambert's W function is the inverse of the function w * e**w. -It is used in the -solve package for equations containing -exponentials and logarithms. -

    -

    -

    -syntax:

    -Lambert_W(<z>) -

    -

    -

    -

    -examples:

    -

    
-Lambert_W(-1/e); 
-
-  -1 
-
-
-solve(w + log(w),w); 
-
-  w=lambert_w(1)
-
-
-on rounded; 
-
-Lambert_W(-0.05); 
-
-  - 0.0527059835515
-
-

    The current implementation will compute the principal branch in -rounded mode only. -

    -

    -

    - - - -Miscellaneous Functions -INDEX

    -Miscellaneous Functions

    -
  • DILOG extended operator

    -

  • Lambert\_W function operator

    -

    •  - - -ChebyshevT -INDEX

    - - - -CHEBYSHEVT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The ChebyshevT operator computes the nth Chebyshev T Polynomial (of the - -first kind). -

    -

    -

    -syntax:

    -ChebyshevT(<integer>,<expression>) -

    -

    -

    -

    -examples:

    -

    
-ChebyshevT(3,xx); 
-
-          2
-  xx*(4*xx   - 3) 
-
-
-
-ChebyshevT(3,4); 
-
-  244
-
-

    Chebyshev's T polynomials are computed using the recurrence relati -on: -

    -

    -ChebyshevT(n,x) := 2x*ChebyshevT(n-1,x) - ChebyshevT(n-2,x) with -

    -

    -ChebyshevT(0,x) := 0 and ChebyshevT(1,x) := x -

    -

    -

    - - - -ChebyshevU -INDEX

    - - - -CHEBYSHEVU _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The ChebyshevU operator returns the nth Chebyshev U Polynomial (of the - -second kind). -

    -

    -

    -syntax:

    -ChebyshevU(<integer>,<expression>) -

    -

    -

    -

    -examples:

    -

    
-ChebyshevU(3,xx); 
-
-          2
-  4*x*(2*x   - 1) 
-
-
-
-ChebyshevU(3,4); 
-
-  496
-
-

    Chebyshev's U polynomials are computed using the recurrence relati -on: -

    -

    -ChebyshevU(n,x) := 2x*ChebyshevU(n-1,x) - ChebyshevU(n-2,x) with -

    -

    -ChebyshevU(0,x) := 0 and ChebyshevU(1,x) := 2x -

    -

    -

    -

    - - - -HermiteP -INDEX

    - - - -HERMITEP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The HermiteP operator returns the nth Hermite Polynomial. -

    -

    -

    -syntax:

    -HermiteP(<integer>,<expression>) -

    -

    -

    -

    -examples:

    -

    
-HermiteP(3,xx); 
-
-            2
-  4*xx*(2*xx   - 3) 
-
-
-HermiteP(3,4); 
-
-  464
-
-

    Hermite polynomials are computed using the recurrence relation: -

    -

    -HermiteP(n,x) := 2x*HermiteP(n-1,x) - 2*(n-1)*HermiteP(n-2,x) with -

    -

    -HermiteP(0,x) := 1 and HermiteP(1,x) := 2x -

    -

    -

    -

    - - - -LaguerreP -INDEX

    - - - -LAGUERREP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The LaguerreP operator computes the nth Laguerre Polynomial. -The two argument call of LaguerreP is a (common) abbreviation of -LaguerreP(n,0,x). -

    -

    -

    -syntax:

    -LaguerreP(<integer>,<expression>) or -

    -

    -LaguerreP(<integer>,<expression>,<expression>) -

    -

    -

    -

    -examples:

    -

    
-LaguerreP(3,xx); 
-
-       3        2
-  (- xx   + 9*xx   - 18*xx + 6)/6
-
-
-
-LaguerreP(2,3,4); 
-
-  -2
-
-

    Laguerre polynomials are computed using the recurrence relation: -

    -

    -LaguerreP(n,a,x) := (2n+a-1-x)/n*LaguerreP(n-1,a,x) - - (n+a-1) * LaguerreP(n-2,a,x) with -

    -

    -LaguerreP(0,a,x) := 1 and LaguerreP(2,a,x) := -x+1+a -

    -

    -

    - - - -LegendreP -INDEX

    - - - -LEGENDREP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The binary LegendreP operator computes the nth Legendre -Polynomial which is -a special case of the nth Jacobi Polynomial with -

    -

    -LegendreP(n,x) := JacobiP(n,0,0,x) -

    -

    -The ternary form returns the associated Legendre Polynomial (see below). -

    -

    -

    -syntax:

    -LegendreP(<integer>,<expression>) or -

    -

    -LegendreP(<integer>,<expression>,<expression>) -

    -

    -

    -

    -examples:

    -

    
-LegendreP(3,xx); 
-
-          2
-  xx*(5*xx   - 3)
-  ----------------
-         2
-
-
-
-LegendreP(3,2,xx); 
-
-              2
-  15*xx*( - xx   + 1)
-
-

    The ternary form of the operator LegendreP is the associa -ted -Legendre Polynomial defined as -

    -

    -P(n,m,x) = (-1)**m * (1-x**2)**(m/2) * df(LegendreP(n,x),x,m) -

    -

    -

    - - - -JacobiP -INDEX

    - - - -JACOBIP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The JacobiP operator computes the nth Jacobi Polynomial. -

    -

    -

    -syntax:

    -JacobiP(<integer>,<expression>,<expression>, - <expression>) -

    -

    -

    -

    -examples:

    -

    
-JacobiP(3,4,5,xx); 
-
-          3         2
-  7*(65*xx   - 13*xx   - 13*xx + 1)
-  ----------------------------------
-                  8
-
-
-
-JacobiP(3,4,5,6); 
-
-  94465/8
-
-

    - - -GegenbauerP -INDEX

    - - - -GEGENBAUERP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The GegenbauerP operator computes Gegenbauer's (ultraspherical) -polynomials. -

    -

    -

    -syntax:

    -GegenbauerP(<integer>,<expression>,<expression>) -

    -

    -

    -

    -examples:

    -

    
-GegenbauerP(3,2,xx); 
-
-            2
-  4*xx*(8*xx   - 3)
-
-
-
-GegenbauerP(3,2,4); 
-
-  2000
-
-

    - - -SolidHarmonicY -INDEX

    - - - -SOLIDHARMONICY _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The SolidHarmonicY operator computes Solid harmonic (Laplace) -polynomials. -

    -

    -

    -syntax:

    -SolidHarmonicY(<integer>,<integer>, -<expression>,<expression>,<expression>,<expression>) -

    -

    -

    -

    -

    -examples:

    -

    
-
-SolidHarmonicY(3,-2,x,y,z,r2); 
-
-                           2    2
-  sqrt(105)*z*(-2*i*x*y + x  - y )
-  ---------------------------------
-         4*sqrt(pi)*sqrt(2)
-
-

    - - -SphericalHarmonicY -INDEX

    - - - -SPHERICALHARMONICY _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The SphericalHarmonicY operator computes Spherical harmonic (Laplace) -polynomials. These are special cases of the -solid harmonic polynomials, -SolidHarmonicY. -

    -

    -

    -syntax:

    -SphericalHarmonicY(<integer>,<integer>, -<expression>,<expression>) -

    -

    -

    -

    -

    -examples:

    -

    
-SphericalHarmonicY(3,2,theta,phi); 
-
-
-                                 2          2                               2
-  sqrt(105)*cos(theta)*sin(theta) *(cos(phi) +2*cos(phi)*sin(phi)*i-sin(phi) )
-  -----------------------------------------------------------------------------
-                               4*sqrt(pi)*sqrt(2)
-
-

    - - -Orthogonal Polynomials -INDEX

    -Orthogonal Polynomials

    -
  • ChebyshevT operator

    -

  • ChebyshevU operator

    -

  • HermiteP operator

    -

  • LaguerreP operator

    -

  • LegendreP operator

    -

  • JacobiP operator

    -

  • GegenbauerP operator

    -

  • SolidHarmonicY operator

    -

  • SphericalHarmonicY operator

    -

    •  - - -Si -INDEX

    - - - -SI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Si operator returns the Sine Integral function. -

    -

    -

    -syntax:

    -Si(<expression>) -

    -

    -

    -

    -examples:

    -

    
-limit(Si(x),x,infinity); 
-
-  pi / 2 
-
-
-on rounded; 
-
-Si(0.35); 
-
-  0.347626790989
-
-

    The numeric values for the operator Si are computed via t -he -power series representation, which limits the argument range. -

    -

    -

    - - - -Shi -INDEX

    - - - -SHI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Shi operator returns the hyperbolic Sine Integral function. -

    -

    -

    -syntax:

    -Shi(<expression>) -

    -

    -

    -

    -examples:

    -

    
-df(shi(x),x); 
-
-  sinh(x) / x 
-
-
-on rounded; 
-
-Shi(0.35); 
-
-  0.352390716351
-
-

    The numeric values for the operator Shi are computed via -the -power series representation, which limits the argument range. -

    -

    -

    - - - -s_i -INDEX

    - - - -S_I _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The s_i operator returns the Sine Integral function si. -

    -

    -

    -syntax:

    -s_i(<expression>) -

    -

    -

    -

    -examples:

    -

    
-s_i(xx); 
-
-  (2*Si(xx) - pi) / 2 
-
-
-df(s_i(x),x); 
-
-  sin(x) / x
-
-

    The operator name s_i is simplified towards -SI. -Since REDUCE is not case sensitive by default the name ``si'' can't be -used. -

    -

    -

    - - - -Ci -INDEX

    - - - -CI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Ci operator returns the Cosine Integral function. -

    -

    -

    -syntax:

    -Ci(<expression>) -

    -

    -

    -

    -examples:

    -

    
-defint(cos(t)/t,t,x,infinity); 
-
-  - ci (x) 
-
-
-on rounded; 
-
-Ci(0.35); 
-
-  - 0.50307556932
-
-

    The numeric values for the operator Ci are computed via t -he -power series representation, which limits the argument range. -

    -

    -

    - - - -Chi -INDEX

    - - - -CHI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Chi operator returns the Hyperbolic Cosine Integral function. -

    -

    -

    -syntax:

    -Chi(<expression>) -

    -

    -

    -

    -examples:

    -

    
-defint((cosh(t)-1)/t,t,0,x); 
-
-  - log(x) + psi(1) + chi(x)
-
-
-on rounded; 
-
-Chi(0.35); 
-
-  - 0.44182471827
-
-

    The numeric values for the operator Chi are computed via -the -power series representation, which limits the argument range. -

    -

    -

    - - - -ERF_extended -INDEX

    - - - -ERF EXTENDED _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The special function package supplies an extended support for the - -erf operator which implements the error function - -

    -

    -defint(e**(-x**2),x,0,infinity) * 2/sqrt(pi) -

    -

    -. -

    -

    -

    -syntax:

    -erf(<expression>) -

    -

    -

    -

    -examples:

    -

    
-erf(-x); 
-
-  - erf(x)
-
-
-on rounded; 
-
-erf(0.35); 
-
-  0.379382053562
-
-

    The numeric values for the operator erf are computed via -the -power series representation, which limits the argument range. -

    -

    -

    - - - -erfc -INDEX

    - - - -ERFC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The erfc operator returns the complementary Error function -

    -

    -1 - defint(e**(-x**2),x,0,infinity) * 2/sqrt(pi) -

    -

    -. -

    -

    -

    -syntax:

    -erfc(<expression>) -

    -

    -

    -

    -examples:

    -

    
-erfc(xx); 
-
-  - erf(xx) + 1
-
-

    The operator erfc is simplified towards the -erf operator. -

    -

    -

    - - - -Ei -INDEX

    - - - -EI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Ei operator returns the Exponential Integral function. -

    -

    -

    -syntax:

    -Ei(<expression>) -

    -

    -

    -

    -examples:

    -

    
-df(ei(x),x); 
-
-   x
-  e
-  ---
-  x
-
-
-on rounded; 
-
-Ei(0.35); 
-
-  - 0.0894340019184
-
-

    The numeric values for the operator Ei are computed via t -he -power series representation, which limits the argument range. -

    -

    -

    - - - -Fresnel_C -INDEX

    - - - -FRESNEL_C _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Fresnel_C operator represents Fresnel's Cosine function. -

    -

    -

    -syntax:

    -Fresnel_C(<expression>) -

    -

    -

    -

    -examples:

    -

    
-int(cos(t^2*pi/2),t,0,x); 
-
-  fresnel_c(x) 
-
-
-on rounded; 
-
-fresnel_c(2.1); 
-
-  0.581564135061
-
-

    The operator Fresnel_C has a limited numeric evaluation o -f -large values of its argument. -

    -

    -

    - - - -Fresnel_S -INDEX

    - - - -FRESNEL_S _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Fresnel_S operator represents Fresnel's Sine Integral function. -

    -

    -

    -syntax:

    -Fresnel_S(<expression>) -

    -

    -

    -

    -examples:

    -

    
-int(sin(t^2*pi/2),t,0,x); 
-
-  fresnel_s(x) 
-
-
-on rounded; 
-
-fresnel_s(2.1); 
-
-  0.374273359378
-
-

    The operator Fresnel_S has a limited numeric evaluation o -f -large values of its argument. -

    -

    -

    - - - -Integral Functions -INDEX

    -Integral Functions

    -
  • Si operator

    -

  • Shi operator

    -

  • s_i operator

    -

  • Ci operator

    -

  • Chi operator

    -

  • ERF extended operator

    -

  • erfc operator

    -

  • Ei operator

    -

  • Fresnel_C operator

    -

  • Fresnel_S operator

    -

    •  - - -BINOMIAL -INDEX

    - - - -BINOMIAL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Binomial operator returns the Binomial coefficient if both -parameter are integer and expressions involving the Gamma function otherwise. -

    -

    -

    -syntax:

    -Binomial(<integer>,<integer>) -

    -

    -

    -

    -

    -examples:

    -

    
-Binomial(49,6); 
-
-  13983816 
-
-
-
-Binomial(n,3); 
-
-   gamma(n + 1)
-  ---------------
-  6*gamma(n - 2)
-
-

    The operator Binomial evaluates the Binomial coefficients - from -the explicit form and therefore it is not the best algorithm if you -want to compute many binomial coefficients with big indices in which -case a recursive algorithm is preferable. -

    -

    -

    - - - -STIRLING1 -INDEX

    - - - -STIRLING1 _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Stirling1 operator returns the Stirling Numbers S(n,m) of the first - -kind, i.e. the number of permutations of n symbols which have exactly m cycles -(divided by (-1)**(n-m)). -

    -

    -

    -syntax:

    -Stirling1(<integer>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Stirling1 (17,4); 
-
-  -87077748875904 
-
-
-Stirling1 (n,n-1); 
-
-  -gamma(n+1)
-  -------------
-  2*gamma(n-1)
-
-

    The operator Stirling1 evaluates the Stirling numbers of -the -first kind by rulesets for special cases or by a computing the closed -form, which is a series involving the operators -BINOMIAL -and -STIRLING2. -

    -

    -

    - - - -STIRLING2 -INDEX

    - - - -STIRLING2 _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Stirling1 operator returns the Stirling Numbers S(n,m) of the -second kind, i.e. the number of ways of partitioning a set of n elements -into m non-empty subsets. -

    -

    -

    -syntax:

    -Stirling2(<integer>,<integer>) -

    -

    -

    -

    -examples:

    -

    
-Stirling2 (17,4); 
-
-  694337290 
-
-
-Stirling2 (n,n-1); 
-
-   gamma(n+1)
-  -------------
-  2*gamma(n-1)
-
-

    The operator Stirling2 evaluates the Stirling numbers of -the -second kind by rulesets for special cases or by a computing the closed -form. -

    -

    -

    - - - -Combinatorial Operators -INDEX

    -Combinatorial Operators

    -
  • BINOMIAL operator

    -

  • STIRLING1 operator

    -

  • STIRLING2 operator

    -

    •  - - -ThreejSymbol -INDEX

    - - - -THREEJSYMBOL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The ThreejSymbol operator implements the 3j symbol. -

    -

    -

    -syntax:

    -ThreejSymbol(<list of j1,m1>,<list of j2,m2>, -<list of j3,m3>) -

    -

    -

    -

    -

    -examples:

    -

    
-
-ThreejSymbol({j+1,m},{j+1,-m},{1,0}); 
-
-
-        j
-  ( - 1)  *(abs(j - m + 1) - abs(j + m + 1))
-  -------------------------------------------
-             3       2                    m
-   2*sqrt(2*j   + 9*j   + 13*j + 6)*( - 1)
-
-

    - - -Clebsch_Gordan -INDEX

    - - - -CLEBSCH_GORDAN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The Clebsch_Gordan operator implements the Clebsch_Gordan -coefficients. This is closely related to the -Threejsymbol. -

    -

    -

    -syntax:

    -Clebsch_Gordan(<list of j1,m1>,<list of j2,m2>, -<list of j3,m3>) -

    -

    -

    -

    -

    -examples:

    -

    
- Clebsch_Gordan({2,0},{2,0},{2,0}); 
-
-
-     -2
-  ---------
-  sqrt(14)
-
-

    - - -SixjSymbol -INDEX

    - - - -SIXJSYMBOL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The SixjSymbol operator implements the 6j symbol. -

    -syntax:

    -

    -

    -SixjSymbol(<list of j1,j2,j3>,<list of l1,l2,l3>) -

    -

    -

    -

    -

    -examples:

    -

    
-
-SixjSymbol({7,6,3},{2,4,6}); 
-
-       1
-  -------------
-  14*sqrt(858)
-
-

    The operator SixjSymbol uses the -ineq package in order -to find minima and maxima for the summation index. -

    -

    -

    - - - -3j and 6j symbols -INDEX

    -3j and 6j symbols

    -
  • ThreejSymbol operator

    -

  • Clebsch_Gordan operator

    -

  • SixjSymbol operator

    -

    •  - - -HYPERGEOMETRIC -INDEX

    - - - -HYPERGEOMETRIC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Hypergeometric operator provides simplifications for the -generalized hypergeometric functions. -The Hypergeometric operator is included in the package specfn2. -

    -

    -

    -syntax:

    -hypergeometric(<list of parameters>,<list of parameters>, - <argument>) -

    -

    -

    -

    -examples:

    -

    
-load specfn2;
-
-hypergeometric ({1/2,1},{3/2},-x^2); 
-
-
-  atan(x)
-  --------
-     x
-
-
-hypergeometric ({},{},z); 
-
-   z
-  e
-
-

    The special case where the length of the first list is equal to 2 -and -the length of the second list is equal to 1 is often called -``the hypergeometric function'' (notated as 2F1(a1,a2,b;x)). -

    -

    -

    - - - -MeijerG -INDEX

    - - - -MEIJERG _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The MeijerG operator provides simplifications for Meijer's G -function. The simplifications are performed towards polynomials, -elementary or -special functions or (generalized) -hypergeometric functions. -

    -

    -The MeijerG operator is included in the package specfn2. -

    -

    -

    -syntax:

    -MeijerG(<list of parameters>,<list of parameters>, - <argument>) -

    -

    -

    -The first element of the lists has to be the list containing the -first group (mostly called ``m'' and ``n'') of parameters. This passes -the four parameters of a Meijer's G function implicitly via the -length of the lists. -

    -

    -

    -examples:

    -

    
-load specfn2;
-
-MeijerG({{},1},{{0}},x); 
-
-  heaviside(-x+1)
-
-
-MeijerG({{}},{{1+1/4},1-1/4},(x^2)/4) * sqrt pi;
- 
-
-
-                  2
-  sqrt(2)*sin(x)*x
-  ------------------
-      4*sqrt(x)
-
-

    Many well-known functions can be written as G functions, -e.g. exponentials, logarithms, trigonometric functions, Bessel functions -and hypergeometric functions. -The formulae can be found e.g. in -

    -

    -A.P.Prudnikov, Yu.A.Brychkov, O.I.Marichev: -Integrals and Series, Volume 3: More special functions, -Gordon and Breach Science Publishers (1990). -

    -

    -

    - - - -Heaviside -INDEX

    - - - -HEAVISIDE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The Heaviside operator returns the Heaviside function. -

    -

    -Heaviside(~w) => if (w <0) then 0 else 1 -

    -

    -when numberp w; -

    -

    -

    -syntax:

    -Heaviside(<argument>) -

    -

    -

    -This operator is often included in the result of the simplification -of a generalized -hypergeometric function or a - -MeijerG function. -

    -

    -No simplification is done for this function. -

    -

    -

    - - - -erfi -INDEX

    - - - -ERFI _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -The erfi operator returns the error function of an imaginary argument. - -

    -

    -erfi(~x) => 2/sqrt(pi) * defint(e**(t**2),t,0,x); -

    -

    -

    -syntax:

    -erfi(<argument>) -

    -

    -

    -This operator is sometimes included in the result of the simplification -of a generalized -hypergeometric function or a - -MeijerG function. -

    -

    -No simplification is done for this function. -

    -

    -

    - - - -Miscellaneous -INDEX

    -Miscellaneous

    -
  • HYPERGEOMETRIC operator

    -

  • MeijerG operator

    -

  • Heaviside operator

    -

  • erfi operator

    -

    •  - - -Special Functions -INDEX

    -Special Functions

    -
  • Special Function Package introduction

    -

  • Constants concept

    -

  • Bernoulli Euler Zeta

    -

  • Bessel Functions

    -

  • Airy Functions

    -

  • Jacobi's Elliptic Functions and Elliptic Integrals -

    -

  • Gamma and Related Functions

    -

  • Miscellaneous Functions

    -

  • Orthogonal Polynomials

    -

  • Integral Functions

    -

  • Combinatorial Operators

    -

  • 3j and 6j symbols

    -

  • Miscellaneous

    -

    •  - - -TAYLOR_introduction -INDEX

    - - - -TAYLOR _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -This short note describes a package of REDUCE procedures that allow -Taylor expansion in one or more variables and efficient manipulation -of the resulting Taylor series. Capabilities include basic operations -(addition, subtraction, multiplication and division) and also -application of certain algebraic and transcendental functions. To a -certain extent, Laurent expansion can be performed as well. -

    -

    - - - -taylor -INDEX

    - - - -TAYLOR _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - - The taylor operator is used for expanding an expression into a - Taylor series. -

    -

    -

    -syntax:

    -taylor(<expression> - ,<var>, - <expression>,<number> -

    -

    -{,<var>, - <expression>,<number>}*) -

    -

    -

    -<expression> can be any valid REDUCE algebraic expression. - <var> must be a -kernel, and is the expansion - variable. The <expression> following it denotes the point - about which the expansion is to take place. <number> must be a - non-negative integer and denotes the maximum expansion order. If - more than one triple is specified taylor will expand its - first argument independently with respect to all the variables. - Note that once the expansion has been done it is not possible to - calculate higher orders. -

    -

    -Instead of a -kernel, <var> may also be a list of - kernels. In this case expansion will take place in a way so that - the sum/ of the degrees of the kernels does not exceed the - maximum expansion order. If the expansion point evaluates to the - special identifier infinity, taylor tries to expand in - a series in 1/<var>. -

    -

    -The expansion is performed variable per variable, i.e. in the - example above by first expanding - exp(x^2+y^2) - with respect to - x and then expanding every coefficient with respect to y. -

    -

    -

    -examples:

    -

    
-    taylor(e^(x^2+y^2),x,0,2,y,0,2); 
-
-
-       2    2    2  2      2  2
-  1 + Y  + X  + Y *X  + O(X ,Y )   
-
-
-    taylor(e^(x^2+y^2),{x,y},0,2); 
-
-
-       2    2       2  2
-  1 + Y  + X  + O({X ,Y })
-
-

    The following example shows the case of a non-analytical function. -

    
-
-    taylor(x*y/(x+y),x,0,2,y,0,2); 
-
-
-  ***** Not a unit in argument to QUOTTAYLOR 
-
-

    -

    -

    -Note that it is not generally possible to apply the standard - reduce operators to a Taylor kernel. For example, -part, - -coeff, or -coeffn cannot be used. Instead, the - expression at hand has to be converted to standard form first - using the -taylortostandard operator. -

    -

    -Differentiation of a Taylor expression is possible. If you - differentiate with respect to one of the Taylor variables the - order will decrease by one. -

    -

    -Substitution is a bit restricted: Taylor variables can only be - replaced by other kernels. There is one exception to this rule: - you can always substitute a Taylor variable by an expression that - evaluates to a constant. Note that REDUCE will not always be able - to determine that an expression is constant: an example is - sin(acos(4)). -

    -

    -Only simple taylor kernels can be integrated. More complicated - expressions that contain Taylor kernels as parts of themselves are - automatically converted into a standard representation by means of - the -taylortostandard operator. In this case a suitable - warning is printed. -

    -

    -

    -

    - - - -taylorautocombine -INDEX

    - - - -TAYLORAUTOCOMBINE _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - - If you set taylorautocombine to on, REDUCE - automatically combines Taylor expressions during the simplification - process. This is equivalent to applying -taylorcombine to - every expression that contains Taylor kernels. Default is - on. -

    -

    - - - -taylorautoexpand -INDEX

    - - - -TAYLORAUTOEXPAND _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - - taylorautoexpand makes Taylor expressions ``contagious'' in - the sense that -taylorcombine tries to Taylor expand all - non-Taylor subexpressions and to combine the result with the rest. - Default is off. -

    -

    - - - -taylorcombine -INDEX

    - - - -TAYLORCOMBINE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - - This operator tries to combine all Taylor kernels found in its - argument into one. Operations currently possible are: -

    -

    - _ _ _ Addition, subtraction, multiplication, and division. -

    - _ _ _ Roots, exponentials, and logarithms. -

    - _ _ _ Trigonometric and hyperbolic functions and their inverses. -

    -

    -

    -examples:

    -

    
-    hugo := taylor(exp(x),x,0,2); 
-
-                  1  2      3
-  HUGO := 1 + X + -*X  + O(X )
-                  2
-
-
-    taylorcombine log hugo; 
-
-         3
-  X + O(X )
-
-
-    taylorcombine(hugo + x); 
-
-           1  2      3
-  (1 + X + -*X  + O(X )) + X
-           2
-
-
-    on taylorautoexpand; 
-
-    taylorcombine(hugo + x); 
-
-            1  2      3
-  1 + 2*X + -*X  + O(X )  
-            2
-
-

    Application of unary operators like log and atan - - will nearly always succeed. For binary operations their arguments - have to be Taylor kernels with the same template. This means that - the expansion variable and the expansion point must match. - Expansion order is not so important, different order usually means - that one of them is truncated before doing the operation. -

    -

    -If -taylorkeeporiginal is set to on and if all - Taylor kernels in its argument have their original expressions - kept taylorcombine will also combine these and store the - result as the original expression of the resulting Taylor kernel. - There is also the switch -taylorautoexpand. -

    -

    -There are a few restrictions to avoid mathematically undefined - expressions: it is not possible to take the logarithm of a Taylor - kernel which has no terms (i.e. is zero), or to divide by such a - beast. There are some provisions made to detect singularities - during expansion: poles that arise because the denominator has - zeros at the expansion point are detected and properly treated, - i.e. the Taylor kernel will start with a negative power. (This - is accomplished by expanding numerator and denominator separately - and combining the results.) Essential singularities of the known - functions (see above) are handled correctly. -

    -

    -

    - - - -taylorkeeporiginal -INDEX

    - - - -TAYLORKEEPORIGINAL _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - - taylorkeeporiginal, if set to on, forces the - -taylor and all Taylor kernel manipulation operators to - - keep the original expression, i.e. the expression that was Taylor - expanded. All operations performed on the Taylor kernels are also - applied to this expression which can be recovered using the operator - -taylororiginal. Default is off. -

    -

    - - - -taylororiginal -INDEX

    - - - -TAYLORORIGINAL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - - Recovers the original expression (the one that was expanded) from - the Taylor kernel that is given as its argument. -

    -

    -

    -syntax:

    -taylororiginal(<expression>) or - taylororiginal <simple_expression> -

    -

    -

    -

    -examples:

    -

    
-    hugo := taylor(exp(x),x,0,2); 
-
-                  1  2      3
-  HUGO := 1 + X + -*X  + O(X )
-                  2
-
-
-    taylororiginal hugo; 
-
-  ***** Taylor kernel doesn't have an original part in TAYLORORIGINAL
-
-
-    on taylorkeeporiginal; 
-
-    hugo := taylor(exp(x),x,0,2); 
-
-                  1  2      3
-  HUGO := 1 + X + -*X  + O(X )
-                  2
-
-
-    taylororiginal hugo; 
-
-   X
-  E   
-
-

    An error is signalled if the argument is not a Taylor kernel or if - - the original expression was not kept, i.e. if - -taylorkeeporiginal was set off during expansi -on. -

    -

    -

    - - - -taylorprintorder -INDEX

    - - - -TAYLORPRINTORDER _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - - taylorprintorder, if set to on, causes the remainder - to be printed in big-O notation. Otherwise, three dots are printed. - Default is on. -

    -

    - - - -taylorprintterms -INDEX

    - - - -TAYLORPRINTTERMS _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - - Only a certain number of (non-zero) coefficients are printed. If - there are more, an expression of the form n terms is printed - to indicate how many non-zero terms have been suppressed. The - number of terms printed is given by the value of the shared - algebraic variable taylorprintterms. Allowed values are - integers and the special identifier all. The latter setting - specifies that all terms are to be printed. The default setting is - 5. -

    -

    -

    -examples:

    -

    
-    taylor(e^(x^2+y^2),x,0,4,y,0,4); 
-
-
-       2   1  4    2    2  2                              5  5
-  1 + Y  + -*Y  + X  + Y *X  +             (4 terms) + O(X ,Y )
-           2
-
-
-    taylorprintterms := all; 
-
-  TAYLORPRINTTERMS := ALL 
-
-
-    taylor(e^(x^2+y^2),x,0,4,y,0,4); 
-
-
-       2   1  4    2    2  2   1  4  2   1  4   1  2  4
-  1 + Y  + -*Y  + X  + Y *X  + -*Y *X  + -*X  + -*Y *X
-           2                   2         2      2
-     1  4  4      5  5
-   + -*Y *X  + O(X ,Y )
-     4
-      
-
-

    - - -taylorrevert -INDEX

    - - - -TAYLORREVERT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - - taylorrevert allows reversion of a Taylor series of a - function f, i.e., to compute the first terms of the expansion of the - inverse of f from the expansion of f. -

    -

    -

    -syntax:

    -taylorrevert(<expression>, - <var>,<var>) -

    -

    -

    -The first argument must evaluate to a Taylor kernel with the second - argument being one of its expansion variables. -

    -

    -

    -examples:

    -

    
-    taylor(u - u**2,u,0,5); 
-
-       2      6
-  U - U  + O(U ) 
-
-
-    taylorrevert (ws,u,x); 
-
-       2      3      4       5      6
-  X + X  + 2*X  + 5*X  + 14*X  + O(X )  
-
-

    - - -taylorseriesp -INDEX

    - - - -TAYLORSERIESP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - - This operator may be used to determine if its argument is a Taylor - kernel. -

    -

    -

    -syntax:

    -taylorseriesp(<expression>) or taylorseriesp - <simple_expression> -

    -

    -

    -

    -examples:

    -

    
-    hugo := taylor(exp(x),x,0,2); 
-
-                  1  2      3
-  HUGO := 1 + X + -*X  + O(X )
-                  2
-
-
-    if taylorseriesp hugo then OK;
-
-  OK 
-
-
-    if taylorseriesp(hugo + y) then OK else NO; 
-
-
-  NO  
-
-

    Note that this operator is subject to the same restrictions as, - e.g., ordp or numberp, i.e. it may only be used in - boolean expressions in if or let statements. -

    -

    -

    - - - -taylortemplate -INDEX

    - - - -TAYLORTEMPLATE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - - The template of a Taylor kernel, i.e. the list of all variables - with respect to which expansion took place together with expansion - point and order can be extracted using -

    -

    -

    -syntax:

    -taylortemplate(<expression>) or - taylortemplate <simple_expression> -

    -

    -

    -This returns a list of lists with the three elements - (VAR,VAR0,ORDER). An error is signalled if the argument is not a - Taylor kernel. -

    -

    -

    -examples:

    -

    
-    hugo := taylor(exp(x),x,0,2); 
-
-                  1  2      3
-  HUGO := 1 + X + -*X  + O(X )
-                  2
-
-
-    taylortemplate hugo; 
-
-  {{X,0,2}}  
-
-

    - - -taylortostandard -INDEX

    - - - -TAYLORTOSTANDARD _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - - This operator converts all Taylor kernels in its argument into - standard form and resimplifies the result. -

    -

    -

    -syntax:

    -taylortostandard(<expression>) or - taylortostandard <simple_expression> -

    -

    -

    -

    -examples:

    -

    
-    hugo := taylor(exp(x),x,0,2); 
-
-                  1  2      3
-  HUGO := 1 + X + -*X  + O(X )
-                  2
-
-
-    taylortostandard hugo; 
-
-   2
-  X  + 2*X + 2
-  ------------  
-       2
-
-

    - - - -Taylor series -INDEX

    -Taylor series

    -
  • TAYLOR introduction

    -

  • taylor operator

    -

  • taylorautocombine switch

    -

  • taylorautoexpand switch

    -

  • taylorcombine operator

    -

  • taylorkeeporiginal switch

    -

  • taylororiginal operator

    -

  • taylorprintorder switch

    -

  • taylorprintterms variable

    -

  • taylorrevert operator

    -

  • taylorseriesp operator

    -

  • taylortemplate operator

    -

  • taylortostandard operator

    -

    •  - - -GNUPLOT_and_REDUCE -INDEX

    - - - -GNUPLOT AND REDUCE _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -

    -

    -The GNUPLOT system provides easy to use graphics output -for curves or surfaces which are defined by -formulas and/or data sets. GNUPLOT supports -a great variety of output devices -such as X-windows, VGA screen, postscript, picTeX. -The REDUCE GNUPLOT package lets one use the GNUPLOT -graphical output directly from inside REDUCE, either for -the interactive display of curves/surfaces or for the production -of pictures on paper. -

    -

    -Note that this package may not be supported on all system -platforms. -

    -

    -For a detailed description you should read the GNUPLOT -system documentation, available together with the GNUPLOT -installation material from several servers by anonymous FTP. -

    -

    -The REDUCE developers thank the GNUPLOT people for their permission -to distribute GNUPLOT together with REDUCE. -

    -

    - - - -Axes_names -INDEX

    - - - -AXES NAMES

    -

    - -Inside REDUCE the choice of variable names for a graph is completely -free. For referring to the GNUPLOT axes the names -X and Y for 2 dimensions, X,Y and Z for 3 dimensions are used -in the usual schoolbook sense independent from the variables of -the REDUCE expression. -

    -

    -

    -examples:

    -

    - - -Pointset -INDEX

    - - - -POINTSET _ _ _ _ _ _ _ _ _ _ _ _ type

    -

    - -

    -

    -A curve can be give as a set of precomputed points (a polygon) -in 2 or 3 dimensions. Such a point set is a -list -of points, where each point is a -list 2 (or 3) -numbers. These numbers are interpreted as (x,y) -(or x,y,z) coordinates. All points of one set must have -the same dimension. -

    -

    -

    -examples:

    -

    Also a surface in 3d can be given by precomputed point -s, -but only on a logically orthogonal mesh: the surface is defined -by a list of curves (in 3d) which must have a uniform length. -GNUPLOT then will draw an orthogonal mesh by first drawing the -given lines, and second connecting the 1st point of the 1st curve -with the 1st point of the 2nd curve, that one with the 1st point -of the 3rd curve and so on for all curves and for all indexes. -

    -

    - - - -PLOT -INDEX

    - - - -PLOT _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The command plot is the main entry for drawing a -picture from inside REDUCE. -

    -

    -

    -syntax:

    -plot(<spec>,<spec>,...) -

    -

    -

    -where <spec> is a <function>, a <range> or an <option>. - -

    -

    -<function>: -

    -

    -- an expression depending -on one unknown (e.g. sin(x) or two unknowns (e.g. -sin(x+y), -

    -

    -- an equation with a function on its right-hand -side and a single name on its left-hand side (e.g. -z=sin(x+y) where the name on the left-hand side specifies -the dependent variable. -

    -

    -- a list of functions: -if in 2 dimensions the picture should have more than one -curve the expressions can be given as list (e.g. {sin(x),cos(x)}). -

    -

    -- an equation with zero left or right hand side describing - an implicit curve in two dimensions (e.g. x**3+x*y**3-9x=0). -

    -

    -- a point set: the graph can be given -as point set in 2 dimensions or a -pointset or pointset list -in 3 dimensions. -

    -

    -<range>: -

    -

    -Each dependent and independent variable can be limited -to an interval by an equation where the left-hand side specifies -the variable and the right-hand side defines the -interval, -e.g. x=( -3 .. 5). -

    -

    -If omitted the independent variables -range from -10 to 10 and the dependent variable is limited only -by the precision of the IEEE floating point arithmetic. -

    -

    -<option>: -

    -

    -An option can be an equation equating a variable -and a value (in general a string), or a keyword(GNUPLOT switch). -These have to be included in the gnuplot command arguments directly. -Strings have to be enclosed in -string quotes (see -string). Available options are: -

    -

    - -title: assign a heading (default: empty) -

    -

    - -xlabel: set label for the x axis -

    -

    - -ylabel: set label for the y axis -

    -

    - -zlabel: set label for the z axis -

    -

    - -terminal: select an output device -

    -

    - -size: rescale the picture -

    -

    - -view: set a viewpoint -

    -

    -(no) -contour: 3d: add contour lines -

    -

    -(no) -surface: 3d: draw surface (default: yes) -

    -

    -(no) -hidden3d: 3d: remove hidden lines (default: no) -

    -

    -

    -examples:

    -

    
-plot(cos x);
-
-plot(s=sin phi,phi=(-3 .. 3));
-
-plot(sin phi,cos phi,phi=(-3 .. 3));
-
-plot (cos sqrt(x**2 + y**2),x=(-3 .. 3),y=(-3 .. 3),hidden3d);
-
-plot {{0,0},{0,1},{1,1},{0,0},{1,0},{0,1},{0.5,1.5},{1,1},{1,0}};
-
-
-
-on rounded;
-
-w:=for j:=1:200 collect {1/j*sin j,1/j*cos j,j/200}$
-
-plot w; 
-

    Additional control of the plot operation: - -plotrefine, - -plot_xmesh, -plot_ymesh, -trplot, - -plotkeep, -show_grid. -

    -

    - - - -PLOTRESET -INDEX

    - - - -PLOTRESET _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The command plotreset closes the current GNUPLOT windows. -The next call to -plot will create a new one. plotreset -can also be used to reset the system status after technical problems. -

    -

    -

    -syntax:

    -plotreset; -

    -

    -

    - - - -title -INDEX

    - - - -TITLE _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    - -plotoption: -Assign a title to the GNUPLOT graph. -

    -

    -

    -syntax:

    -title= <string> -

    -

    -

    -

    -examples:

    -

    
-title="annual revenue in 1993"
-

    - - -xlabel -INDEX

    - - - -XLABEL _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    - -plotoption: -Assign a name to to the x axis (see -axes names). -

    -

    -

    -syntax:

    -xlabel= <string> -

    -

    -

    -

    -examples:

    -

    
-xlabel="month"
-

    - - -ylabel -INDEX

    - - - -YLABEL _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    - -plotoption: -Assign a name to to the x axis (see -axes names). -

    -

    -

    -syntax:

    -ylabel= <string> -

    -

    -

    -

    -examples:

    -

    
-ylabel="million forint"
-

    - - -zlabel -INDEX

    - - - -ZLABEL _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    - -plotoption: -Assign a name to to the z axis (see -axes names). -

    -

    -

    -syntax:

    -zlabel= <string> -

    -

    -

    -

    -examples:

    -

    
-zlabel="local weight"
-

    - - -terminal -INDEX

    - - - -TERMINAL _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    - -plotoption: -Select a different output device. The possible values here -depend highly on the facilities installed for your GNUPLOT -software. -

    -

    -

    -syntax:

    -terminal= <string> -

    -

    -

    -

    -examples:

    -

    
-terminal="x11"
-

    - - -size -INDEX

    - - - -SIZE _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    - -plotoption: -Rescale the graph (not the window!) in x and y direction. -Default is 1.0 (no rescaling). -

    -

    -

    -syntax:

    -size= "<sx>,<sy>" -

    -

    -

    -where <sx>,<sy> are floating point number not too -far from 1.0. -

    -examples:

    -

    
-size="0.7,1"
-

    -

    - - - -view -INDEX

    - - - -VIEW _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    - -plotoption: -Set a new viewpoint by turning the object around the x and then -around the z axis (see -axes names). -

    -

    -

    -syntax:

    -view= "<sx>,<sz>" -

    -

    -

    -where <sx>,<sz> are floating point number representing -angles in degrees. -

    -examples:

    -

    
-view="30,130"
-

    -

    - - - -contour -INDEX

    - - - -CONTOUR _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    - -plotoption: -If contour is member of the options for a 3d -plot -contour lines are projected to the z=0 plane -(see -axes names). The absence of contour lines -can be selected explicitly by including nocontour. Default -is nocontour. -

    -

    - - - -surface -INDEX

    - - - -SURFACE _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    - -plotoption: -If surface is member of the options for a 3d -plot -the surface is drawn. The absence of the surface plotting -can be selected by including nosurface, e.g. if -only the -contour should be visualized. Default is surface -. -

    -

    - - - -hidden3d -INDEX

    - - - -HIDDEN3D _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    - -plotoption: -If hidden3d is member of the options for a 3d -plot -hidden lines are removed from the picture. Otherwise a -surface is drawn as transparent object. Default is -nohidden3d. Selecting hidden3d increases the -computing time substantially. -

    -

    - - - -PLOTKEEP -INDEX

    - - - -PLOTKEEP _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -Normally all intermediate data sets are deleted after terminating -a plot session. If the switch plotkeep is set -on, -the data sets are kept for eventual post processing independent -of REDUCE. -

    -

    - - - -PLOTREFINE -INDEX

    - - - -PLOTREFINE _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -In general -plot tries to generate smooth pictures by evaluating -the functions at interior points until the distances are fine -enough. This can require a lot of computing time if the -single function evaluation is expensive. The refinement is -controlled by the switch plotrefine which is -on -by default. When you turn it -off the functions will -be evaluated only at the basic points (see -plot_xmesh, - -plot_ymesh). -

    -

    - - - -plot_xmesh -INDEX

    - - - -PLOT_XMESH _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -The integer value of the global variable plot_xmesh -defines the number of initial function evaluations in x -direction (see -axes names) for -plot. For 2d graphs additional -points will be used as long as -plotrefine is on. -For 3d graphs this number defines also the number of mesh lines -orthogonal to the x axis. -

    -

    - - - -plot_ymesh -INDEX

    - - - -PLOT_YMESH _ _ _ _ _ _ _ _ _ _ _ _ variable

    -

    - -

    -

    -The integer value of the global variable plot_ymesh -defines for 3d -plot calls the number of function evaluations in y -direction (see -axes names) and the number of mesh lines -orthogonal to the y axis. -

    -

    - - - -SHOW_GRID -INDEX

    - - - -SHOW_GRID _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -The grid for localizing an implicitly defined curve in -plot -consists of triangles. These are computed initially equally distributed -over the x-y plane controlled by -plot_xmesh. The grid is -refined adaptively in several levels. The final grid can be visualized -by setting on the switch show_grid. -

    -

    - - - -TRPLOT -INDEX

    - - - -TRPLOT _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -In general the interaction between REDUCE and GNUPLOT is performed -as silently as possible. However, sometimes it might be useful -to see the GNUPLOT commands generated by REDUCE, e.g. for a -postprocessing of generated data sets independent of REDUCE. -When the switch trplot is set on all GNUPLOT commands will -be printed to the standard output additionally. -

    -

    - - - -Gnuplot package -INDEX

    -Gnuplot package

    -
  • GNUPLOT and REDUCE introduction

    -

  • Axes names concept

    -

  • Pointset type

    -

  • PLOT command

    -

  • PLOTRESET command

    -

  • title variable

    -

  • xlabel variable

    -

  • ylabel variable

    -

  • zlabel variable

    -

  • terminal variable

    -

  • size variable

    -

  • view variable

    -

  • contour switch

    -

  • surface switch

    -

  • hidden3d switch

    -

  • PLOTKEEP switch

    -

  • PLOTREFINE switch

    -

  • plot_xmesh variable

    -

  • plot_ymesh variable

    -

  • SHOW_GRID switch

    -

  • TRPLOT switch

    -

    •  - - -Linear_Algebra_package -INDEX

    - - - -LINEAR ALGEBRA PACKAGE _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -

    -

    -This section briefly describes what's available in the Linear Algebra -package. -

    -

    -Note on examples: In the examples throughout this -document, the matrix A will be -

    
-     [1  2  3]
-     [4  5  6]
-     [7  8  9].
-

    -

    -The functions can be divided into four categories: -

    -

    -Basic matrix handling -

    -

    - -add_columns, -

    -

    - -add_rows, -

    -

    - -add_to_columns, -

    -

    - -add_to_rows, -

    -

    - -augment_columns, -

    -

    - -char_poly, -

    -

    - -column_dim, -

    -

    - -copy_into, -

    -

    - -diagonal, -

    -

    - -extend, -

    -

    - -find_companion, -

    -

    - -get_columns, -

    -

    - -get_rows, -

    -

    - -hermitian_tp, -

    -

    - -matrix_augment, -

    -

    - -matrix_stack, -

    -

    - -minor, -

    -

    - -mult_columns, -

    -

    - -mult_rows, -

    -

    - -pivot, -

    -

    - -remove_columns, -

    -

    - -remove_rows, -

    -

    - -row_dim, -

    -

    - -rows_pivot, -

    -

    - -stack_rows, -

    -

    - -sub_matrix, -

    -

    - -swap_columns, -

    -

    - -swap_entries, -

    -

    - -swap_rows. -

    -

    -Constructors -- functions that create matrices -

    -

    - -band_matrix, -

    -

    - -block_matrix, -

    -

    - -char_matrix, -

    -

    - -coeff_matrix, -

    -

    - -companion, -

    -

    - -hessian, -

    -

    - -hilbert, -

    -

    - -jacobian, -

    -

    - -jordan_block, -

    -

    - -make_identity, -

    -

    - -random_matrix, -

    -

    - -toeplitz, -

    -

    - -vandermonde. -

    -

    -High level algorithms -

    -

    - -char_poly, -

    -

    - -cholesky, -

    -

    - -gram_schmidt, -

    -

    - -lu_decom, -

    -

    - -pseudo_inverse, -

    -

    - -simplex, -

    -

    - -svd. -

    -

    -Normal Forms -

    -

    -There is a separate package, NORMFORM, for computing -the following matrix normal forms in REDUCE: -

    -

    - -smithex, -

    -

    - -smithex_int, -

    -

    - -frobenius, -

    -

    - -ratjordan, -

    -

    - -jordansymbolic, -

    -

    - -jordan. -

    -

    -Predicates -

    -

    - -matrixp, -

    -

    - -squarep, -

    -

    - -symmetricp. -

    -

    - - - -fast_la -INDEX

    - - - -FAST_LA _ _ _ _ _ _ _ _ _ _ _ _ switch

    -

    - -

    -

    -By turning the fast_la switch on, the speed of the following -functions will be increased: -

    -

    - -add_columns, -

    -

    - -add_rows, -

    -

    - -augment_columns, -

    -

    - -column_dim, -

    -

    - -copy_into, -

    -

    - -make_identity, -

    -

    - -matrix_augment, -

    -

    - -matrix_stack, -

    -

    - -minor, -

    -

    - -mult_columns, -

    -

    - -mult_rows, -

    -

    - -pivot, -

    -

    - -remove_columns, -

    -

    - -remove_rows, -

    -

    - -rows_pivot, -

    -

    - -squarep, -

    -

    - -stack_rows, -

    -

    - -sub_matrix, -

    -

    - -swap_columns, -

    -

    - -swap_entries, -

    -

    - -swap_rows, -

    -

    - -symmetricp. -

    -

    -The increase in speed will be negligible unless you are making a -significant number (i.e. thousands) of calls. When using this switch, -error checking is minimized. This means that illegal input may give -strange error messages. Beware. -

    -

    - - - -add_columns -INDEX

    - - - -ADD_COLUMNS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -Add columns, add rows: -

    -syntax:

    -

    -

    -add_columns(<matrix>,<c1>,<c2>,<expr>) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -<c1>,<c2> :- positive integers. -

    -

    -<expr> :- a scalar expression. -

    -

    -The Operator add_columns replaces column <\meta{c2>} of -<matrix> by <expr> * column(<matrix>,<c1>) + -column(<matrix>,<c2>). -

    -

    -add_rowsperforms the equivalent task on the rows of -<matrix>. -

    -

    -

    -examples:

    -

    
-
-add_columns(A,1,2,x); 
-
-  [1   x + 2   3]
-  [             ]
-  [4  4*x + 5  6]
-  [             ]
-  [7  7*x + 8  9]
-
-
-
-add_rows(A,2,3,5); 
-
-  [1   2   3 ]
-  [          ]
-  [4   5   6 ]
-  [          ]
-  [27  33  39]
-
-

    Related functions: -add_to_columns, - -add_to_rows, -mult_columns, - -mult_rows. -

    -

    - - - -add_rows -INDEX

    - - - -ADD_ROWS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - - see: -add_columns. -

    -

    - - - -add_to_columns -INDEX

    - - - -ADD_TO_COLUMNS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -Add to columns, add to rows: -

    -

    -

    -syntax:

    -add_to_columns(<matrix>,<column\_list>,<expr>) -

    -

    -

    -<matrix> :- a matrix. -

    -

    -<column\_list> :- a positive integer or a list of positive - integers. -

    -

    -<expr> :- a scalar expression. -

    -

    -add_to_columnsadds <expr> to each column specified in -<column\_list> of <matrix>. -

    -

    -add_to_rowsperforms the equivalent task on the rows of -<matrix>. -

    -

    -

    -examples:

    -

    
-
-add_to_columns(A,{1,2},10); 
-
-  [11  12  3]
-  [         ]
-  [14  15  6]
-  [         ]
-  [17  18  9]
-
-
-
-add_to_rows(A,2,-x) 
-
-   
-  [   1         2         3    ]
-  [                            ]
-  [ - x + 4   - x + 5   - x + 6]
-  [                            ]
-  [   7         8         9    ]
-
-

    Related functions: - -add_columns, -add_rows, -mult_rows, - -mult_columns. -

    -

    - - - -add_to_rows -INDEX

    - - - -ADD_TO_ROWS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - - see: -add_to_columns. -

    -

    - - - -augment_columns -INDEX

    - - - -AUGMENT_COLUMNS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -Augment columns, stack rows: -

    -

    -

    -syntax:

    -augment_columns(<matrix>,<column\_list>) -

    -

    -

    -<matrix> :- a matrix. -

    -

    -<column\_list> :- either a positive integer or a list of positive - integers. -

    -

    -augment_columnsgets hold of the columns of <matrix> -specified in column_list and sticks them together. -

    -

    -stack_rowsperforms the same task on rows of <matrix>. -

    -

    -

    -examples:

    -

    
-
-augment_columns(A,{1,2}) 
-
-   
-  [1  2]
-  [    ]
-  [4  5]
-  [    ]
-  [7  8]
-
-
-
-stack_rows(A,{1,3}) 
-
-  [1  2  3]
-  [       ]
-  [7  8  9]
-
-

    Related functions: - -get_columns, -get_rows, -sub_matrix. -

    -

    - - - -band_matrix -INDEX

    - - - -BAND_MATRIX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -band_matrix(<expr\_list>,<square\_size>) -

    -

    -

    -<expr\_list> :- either a single scalar expression or a list of - an odd number of scalar expressions. -

    -

    -<square\_size> :- a positive integer. -

    -

    -band_matrixcreates a square matrix of dimension -<square\_size>. The diagonal consists of the middle expression -of the <expr\_list>. The expressions to the left of this fill -the required number of sub_diagonals and the expressions to the right -the super_diagonals. -

    -

    -

    -examples:

    -

    
-
-band_matrix({x,y,z},6) 
-
-  [y  z  0  0  0  0]
-  [                ]
-  [x  y  z  0  0  0]
-  [                ]
-  [0  x  y  z  0  0]
-  [                ]
-  [0  0  x  y  z  0]
-  [                ]
-  [0  0  0  x  y  z]
-  [                ]
-  [0  0  0  0  x  y]
-
-

    Related functions: -diagonal. -

    -

    - - - -block_matrix -INDEX

    - - - -BLOCK_MATRIX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -block_matrix(<r>,<c>,<matrix\_list>) -

    -

    -

    -<r>,<c> :- positive integers. -

    -

    -<matrix\_list> :- a list of matrices. -

    -

    -block_matrixcreates a matrix that consists of <r> by -<c> matrices filled from the <matrix\_list> row wise. -

    -

    -

    -examples:

    -

    
-B := make_identity(2); 
-
-       [1  0]
-  b := [    ]
-       [0  1]
-
-
-
-C := mat((5),(5)); 
-
-       [5]
-  c := [ ]
-       [5]
-
-
-
-D := mat((22,33),(44,55)); 
-
-       [22  33]
-  d := [      ]
-       [44  55]
-
-
-
-block_matrix(2,3,{B,C,D,D,C,B}); 
-
-
-  [1   0   5  22  33]
-  [                 ]
-  [0   1   5  44  55]
-  [                 ]
-  [22  33  5  1   0 ]
-  [                 ]
-  [44  55  5  0   1 ]
-
-

    - - -char_matrix -INDEX

    - - - -CHAR_MATRIX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -char_matrix(<matrix>,<lambda>) -

    -

    -

    -<matrix> :- a square matrix. -<lambda> :- a symbol or algebraic expression. -

    -

    -<char\_matrix> creates the characteristic matrix C of -<matrix>. -

    -

    -This is C = <lambda> * Id - A. -Id is the identity matrix. -

    -

    -

    -examples:

    -

    
-
-char_matrix(A,x); 
-
-  [x - 1   -2     -3  ]
-  [                   ]
-  [ -4    x - 5   -6  ]
-  [                   ]
-  [ -7     -8    x - 9]
-
-

    Related functions: -char_poly. -

    -

    - - - -char_poly -INDEX

    - - - -CHAR_POLY _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -char_poly(<matrix>,<lambda>) -

    -

    -

    -<matrix> :- a square matrix. -

    -

    -<lambda> :- a symbol or algebraic expression. -

    -

    -char_polyfinds the characteristic polynomial of <matrix>. -This is the determinant of <lambda> * Id - A. -Id is the identity matrix. -

    -

    -

    -examples:

    -

    
-char_poly(A,x); 
-
-   3     2
-  x -15*x -18*x
-
-

    Related functions: -char_matrix. -

    -

    - - - -cholesky -INDEX

    - - - -CHOLESKY _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -cholesky(<matrix>) -

    -

    -

    -<matrix> :- a positive definite matrix containing numeric entries. -

    -

    -choleskycomputes the cholesky decomposition of <matrix>. -

    -

    -It returns {L,U} where L is a lower matrix, U is an upper matrix, -A = LU, and U = L^T. -

    -

    -

    -examples:

    -

    
-F := mat((1,1,0),(1,3,1),(0,1,1)); 
-
-
-       [1  1  0]
-       [       ]
-  f := [1  3  1]
-       [       ]
-       [0  1  1]
-
-
-
-on rounded; 
-
-cholesky(F); 
-
-  {
-   [1        0               0       ]
-   [                                 ]
-   [1  1.41421356237         0       ]
-   [                                 ]
-   [0  0.707106781187  0.707106781187]
-   ,
-   [1        1              0       ]
-   [                                ]
-   [0  1.41421356237  0.707106781187]
-   [                                ]
-   [0        0        0.707106781187]
-  }
-
-

    Related functions: -lu_decom. -

    -

    - - - -coeff_matrix -INDEX

    - - - -COEFF_MATRIX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -coeff_matrix({<lineq\_list>}) -

    -

    -

    -(If you are feeling lazy then the braces can be omitted.) -

    -

    -<lineq\_list> :- linear equations. Can be of the form equation = number -or just equation. -

    -

    -coeff_matrixcreates the coefficient matrix C of the linear -equations. -

    -

    -It returns {C,X,B} such that CX = B. -

    -

    -

    -examples:

    -

    
-
-coeff_matrix({x+y+4*z=10,y+x-z=20,x+y+4}); 
-
-
-  {
-   [4   1  1]
-   [        ]
-   [-1  1  1]
-   [        ]
-   [0   1  1]
-   ,
-   [z]
-   [ ]
-   [y]
-   [ ]
-   [x]
-   ,
-   [10]
-   [  ]
-   [20]
-   [  ]
-   [-4]
-  }
-
-

    - - -column_dim -INDEX

    - - - -COLUMN_DIM _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -Column dimension, row dimension: -

    -

    -

    -syntax:

    -column_dim(<matrix>) -

    -

    -

    -<matrix> :- a matrix. -

    -

    -column_dimfinds the column dimension of <matrix>. -

    -

    -row_dimfinds the row dimension of <matrix>. -

    -

    -

    -examples:

    -

    
-
-column_dim(A); 
-
-  3 
-
-
-row_dim(A); 
-
-  3 
-
-

    - - -companion -INDEX

    - - - -COMPANION _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -companion(<poly>,<x>) -

    -

    -

    -<poly> :- a monic univariate polynomial in <x>. -

    -

    -<x> :- the variable. -

    -

    -companioncreates the companion matrix C of <poly>. -

    -

    -This is the square matrix of dimension n, where n is the degree of -<poly> w.r.t. <x>. -

    -

    -The entries of C are: -

    -

    -C(i,n) = -coeffn(<poly>,<x>,i-1) for i = 1 - ... n, C(i,i-1) = 1 for i = 2 ... n and - the rest are 0. -

    -

    -

    -examples:

    -

    
-
-companion(x^4+17*x^3-9*x^2+11,x); 
-
-
-  [0  0  0  -11]
-  [            ]
-  [1  0  0   0 ]
-  [            ]
-  [0  1  0   9 ]
-  [            ]
-  [0  0  1  -17]
-
-

    Related functions: - -find_companion. -

    -

    - - - -copy_into -INDEX

    - - - -COPY_INTO _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -copy_into(<A>,<B>,<r>,<c>) -

    -

    -

    -<A>,<B> :- matrices. -

    -

    -<r>,<c> :- positive integers. -

    -

    -copy_intocopies matrix <matrix> into <B> with -<matrix>(1,1) at <B>(<r>,<c>). -

    -

    -

    -examples:

    -

     
-
-G := mat((0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0)); 
-
-
-       [0  0  0  0  0]
-       [             ]
-       [0  0  0  0  0]
-       [             ]
-  g := [0  0  0  0  0]
-       [             ]
-       [0  0  0  0  0]
-       [             ]
-       [0  0  0  0  0]
-
-
-
-copy_into(A,G,1,2); 
-
-  [0  1  2  3  0]
-  [             ]
-  [0  4  5  6  0]
-  [             ]
-  [0  7  8  9  0]
-  [             ]
-  [0  0  0  0  0]
-  [             ]
-  [0  0  0  0  0]
-
-

    Related functions: - -augment_columns, -extend, -matrix_augment, - -matrix_stack, -stack_rows, -sub_matrix. -

    -

    - - - -diagonal -INDEX

    - - - -DIAGONAL _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -diagonal({<mat\_list>}) -

    -

    -

    -(If you are feeling lazy then the braces can be omitted.) -

    -

    -<mat\_list> :- each can be either a scalar expression or a -square -matrix. -

    -

    -diagonalcreates a matrix that contains the input on the -diagonal. -

    -

    -

    -examples:

    -

    
-
-H := mat((66,77),(88,99)); 
-
-       [66  77]
-  h := [      ]
-       [88  99]
-
-
-
-diagonal({A,x,H}); 
-
-  [1  2  3  0  0   0 ]
-  [                  ]
-  [4  5  6  0  0   0 ]
-  [                  ]
-  [7  8  9  0  0   0 ]
-  [                  ]
-  [0  0  0  x  0   0 ]
-  [                  ]
-  [0  0  0  0  66  77]
-  [                  ]
-  [0  0  0  0  88  99]
-
-

    Related functions: - -jordan_block. -

    -

    - - - -extend -INDEX

    - - - -EXTEND _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -extend(<matrix>,<r>,<c>,<expr>) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -<r>,<c> :- positive integers. -

    -

    -<expr> :- algebraic expression or symbol. -

    -

    -extendreturns a copy of <matrix> that has been extended by -<r> rows and <c> columns. The new entries are made equal to -<expr>. -

    -

    -

    -examples:

    -

    
-
-extend(A,1,2,x); 
-
-  [1  2  3  x  x]
-  [             ]
-  [4  5  6  x  x]
-  [             ]
-  [7  8  9  x  x]
-  [             ]
-  [x  x  x  x  x]
-
-

    Related functions: - -copy_into, -matrix_augment, -matrix_stack, - -remove_columns, -remove_rows. -

    -

    - - - -find_companion -INDEX

    - - - -FIND_COMPANION _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -find_companion(<matrix>,<x>) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -<x> :- the variable. -

    -

    -Given a companion matrix, find_companion finds the polynomial -from which it was made. -

    -

    -

    -examples:

    -

    
-
-C := companion(x^4+17*x^3-9*x^2+11,x); 
-
-
-       [0  0  0  -11]
-       [            ]
-       [1  0  0   0 ]
-  c := [            ]
-       [0  1  0   9 ]
-       [            ]
-       [0  0  1  -17]
-
-
-
-find_companion(C,x); 
-
-   4     3    2
-  x +17*x -9*x +11
-
-

    Related functions: - -companion. -

    -

    - - - -get_columns -INDEX

    - - - -GET_COLUMNS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -Get columns, get rows: -

    -

    -

    -syntax:

    -get_columns(<matrix>,<column\_list>) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -<c> :- either a positive integer or a list of positive - integers. -

    -

    -get_columnsremoves the columns of <matrix> specified in -<column\_list> and returns them as a list of column matrices. -

    -

    -get_rowsperforms the same task on the rows of <matrix>. -

    -

    -

    -examples:

    -

    
-
-get_columns(A,{1,3}); 
-
-  {
-   [1]
-   [ ]
-   [4]
-   [ ]
-   [7]
-   ,
-   [3]
-   [ ]
-   [6]
-   [ ]
-   [9]
-  }
-
-
-
-get_rows(A,2); 
-
-  {
-   [4  5  6]
-  }
-
-

    Related functions: - -augment_columns, -stack_rows, -sub_matrix. -

    -

    - - - -get_rows -INDEX

    - - - -GET_ROWS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -see: -get_columns. -

    -

    - - - -gram_schmidt -INDEX

    - - - -GRAM_SCHMIDT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -gram_schmidt({<vec\_list>}) -

    -

    -

    -(If you are feeling lazy then the braces can be omitted.) -

    -

    -<vec\_list> :- linearly independent vectors. Each vector must be -written as a list, eg:{1,0,0}. -

    -

    -gram_schmidtperforms the gram_schmidt orthonormalization on -the input vectors. -

    -

    -It returns a list of orthogonal normalized vectors. -

    -

    -

    -examples:

    -

    
-
-gram_schmidt({{1,0,0},{1,1,0},{1,1,1}}); 
-
-
-  {{1,0,0},{0,1,0},{0,0,1}} 
-
-
-
-gram_schmidt({{1,2},{3,4}}); 
-
-
-        1         2        2*sqrt(5)   -sqrt(5)
-  {{ ------- , ------- },{ --------- , -------- }}
-     sqrt(5)   sqrt(5)         5          5
-
-

    - - -hermitian_tp -INDEX

    - - - -HERMITIAN_TP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    - hermitian_tp(<matrix>) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -hermitian_tpcomputes the hermitian transpose of <matrix>. -

    -

    -This is a -matrix in which the (i,j)'th entry is the conjugate -of the (j,i)'th entry of <matrix>. -

    -

    -

    -examples:

    -

    
-
-J := mat((i+1,i+2,i+3),(4,5,2),(1,i,0)); 
-
-
-       [i + 1  i + 2  i + 3]
-       [                   ]
-  j := [  4      5      2  ]
-       [                   ]
-       [  1      i      0  ]
-
-
-
-hermitian_tp(j); 
-
-  [ - i + 1  4   1  ]
-  [                 ]
-  [ - i + 2  5   - i]
-  [                 ]
-  [ - i + 3  2   0  ]
-
-

    Related functions: - -tp. -

    -

    - - - -hessian -INDEX

    - - - -HESSIAN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -hessian(<expr>,<variable\_list>) -

    -

    -

    -<expr> :- a scalar expression. -

    -

    -<variable\_list> :- either a single variable or a list of - variables. -

    -

    -hessiancomputes the hessian matrix of <expr> w.r.t. the -variables in <variable\_list>. -

    -

    -This is an n by n matrix where n is the number of variables and the -(i,j)'th entry is -df(<expr>,<variable\_list>(i), -<variable\_list>(j)). -

    -

    -

    -examples:

    -

    
-
-hessian(x*y*z+x^2,{w,x,y,z}); 
-
-  [0  0  0  0]
-  [          ]
-  [0  2  z  y]
-  [          ]
-  [0  z  0  x]
-  [          ]
-  [0  y  x  0]
-
-

    Related functions: -df. -

    -

    - - - -hilbert -INDEX

    - - - -HILBERT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -hilbert(<square\_size>,<expr>) -

    -

    -

    -<square\_size> :- a positive integer. -

    -

    -<expr> :- an algebraic expression. -

    -

    -hilbertcomputes the square hilbert matrix of dimension -<square\_size>. -

    -

    -This is the symmetric matrix in which the (i,j)'th entry is -1/(i+j-<expr>). -

    -

    -

    -examples:

    -

    
-
-hilbert(3,y+x); 
-
-  [    - 1          - 1          - 1    ]
-  [-----------  -----------  -----------]
-  [ x + y - 2    x + y - 3    x + y - 4 ]
-  [                                     ]
-  [    - 1          - 1          - 1    ]
-  [-----------  -----------  -----------]
-  [ x + y - 3    x + y - 4    x + y - 5 ]
-  [                                     ]
-  [    - 1          - 1          - 1    ]
-  [-----------  -----------  -----------]
-  [ x + y - 4    x + y - 5    x + y - 6 ]
-
-

    - - -jacobian -INDEX

    - - - -JACOBIAN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -jacobian(<expr\_list>,<variable\_list>) -

    -

    -

    -<expr\_list> :- either a single algebraic expression or a list - of algebraic expressions. -

    -

    -<variable\_list> :- either a single variable or a list of - variables. -

    -

    -jacobiancomputes the jacobian matrix of <expr\_list> -w.r.t. <variable\_list>. -

    -

    -This is a matrix whose (i,j)'th entry is -df(<expr\_list> -(i),<variable\_list>(j)). -

    -

    -The matrix is n by m where n is the number of variables and m the number -of expressions. -

    -

    -

    -examples:

    -

    
- 
-jacobian({x^4,x*y^2,x*y*z^3},{w,x,y,z}); 
-
-
-  [      3                 ]
-  [0  4*x     0       0    ]
-  [                        ]
-  [     2                  ]
-  [0   y    2*x*y     0    ]
-  [                        ]
-  [      3     3          2]
-  [0  y*z   x*z    3*x*y*z ]
-
-

    Related functions: - -hessian, -df. -

    -

    - - - -jordan_block -INDEX

    - - - -JORDAN_BLOCK _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -jordan_block(<expr>,<square\_size>) -

    -

    -

    -<expr> :- an algebraic expression or symbol. -

    -

    -<square\_size> :- a positive integer. -

    -

    -jordan_blockcomputes the square jordan block matrix J of -dimension <square\_size>. -

    -

    -The entries of J are: -

    -

    -J(i,i) = <expr> for i=1 - ... n, J(i,i+1) = 1 for i=1 - ... n-1, and all other entries are 0. -

    -

    -

    -examples:

    -

    
-
-jordan_block(x,5); 
-
-  [x  1  0  0  0]
-  [             ]
-  [0  x  1  0  0]
-  [             ]
-  [0  0  x  1  0]
-  [             ]
-  [0  0  0  x  1]
-  [             ]
-  [0  0  0  0  x]
-
-

    Related functions: -diagonal, -companion. -

    -

    - - - -lu_decom -INDEX

    - - - -LU_DECOM _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -lu_decom(<matrix>) -

    -

    -

    -<matrix> :- a -matrix containing either numeric entries - or imaginary entries with numeric coefficients. -

    -

    -lu_decomperforms LU decomposition on <matrix>, ie: it -returns {L,U} where L is a lower diagonal -matrix, U an -upper diagonal -matrix and A = LU. -

    -

    -Caution: -

    -

    -The algorithm used can swap the rows of <matrix> during the -calculation. This means that LU does not equal <matrix> but a row -equivalent of it. Due to this, lu_decom returns {L,U,vec}. -The call convert(meta{matrix,vec)} will return the matrix that has -been decomposed, i.e: LU = convert(<matrix>,vec). -

    -

    -

    -examples:

    -

    
-
-K := mat((1,3,5),(-4,3,7),(8,6,4)); 
-
-
-       [1   3  5]
-       [        ]
-  k := [-4  3  7]
-       [        ]
-       [8   6  4]
-
-
-
-on rounded;
-
-lu :=  lu_decom(K); 
-
-  lu := {
-         [8    0      0  ]
-         [               ]
-         [-4  6.0     0  ]
-         [               ]
-         [1   2.25  1.125]
-         ,
-         [1  0.75  0.5]
-         [            ]
-         [0   1    1.5]
-         [            ]
-         [0   0     1 ]
-         ,
-         [3 2 3]}
-
-
-
-first lu * second lu; 
-
-  [8   6.0  4.0]
-  [            ]
-  [-4  3.0  7.0]
-  [            ]
-  [1   3.0  5.0]
-
-
-
-convert(K,third lu); 
-
-  P := mat((i+1,i+2,i+3),(4,5,2),(1,i,0));  _ _ _ 
-       [i + 1  i + 2  i + 3]
-       [                   ]
-  p := [  4      5      2  ]
-       [                   ]
-       [  1      i      0  ]
-
-
-lu :=  lu_decom(P); 
-
-  lu := {
-         [  1        0                      0                ]
-         [                                                   ]
-         [  4     - 4*i + 5                 0                ]
-         [                                                   ]
-         [i + 1      3       0.414634146341*i + 2.26829268293]
-         ,
-         [1  i                 0                ]
-         [                                      ]
-         [0  1  0.19512195122*i + 0.243902439024]
-         [                                      ]
-         [0  0                 1                ]
-         ,
-         [3 2 3]}
-
-
-
-first lu * second lu; 
-
-  [  1      i       0   ]
-  [                     ]
-  [  4      5      2.0  ]
-  [                     ]
-  [i + 1  i + 2  i + 3.0]
-
-
-
-convert(P,third lu); 
-
-  [  1      i      0  ]
-  [                   ]
-  [  4      5      2  ]
-  [                   ]
-  [i + 1  i + 2  i + 3]
-
-

    -

    -

    -Related functions: -cholesky. -

    -

    - - - -make_identity -INDEX

    - - - -MAKE_IDENTITY _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -make_identity(<square\_size>) -

    -

    -

    -<square\_size> :- a positive integer. -

    -

    -make_identitycreates the identity matrix of dimension -<square\_size>. -

    -

    -

    -examples:

    -

    
-
-make_identity(4); 
-
-  [1  0  0  0]
-  [          ]
-  [0  1  0  0]
-  [          ]
-  [0  0  1  0]
-  [          ]
-  [0  0  0  1]
-
-

    Related functions: -diagonal. -

    -

    - - - -matrix_augment -INDEX

    - - - -MATRIX_AUGMENT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -Matrix augment, matrix stack: -

    -

    -

    -syntax:

    -matrix_augment{<matrix\_list>} -

    -

    -

    -(If you are feeling lazy then the braces can be omitted.) -

    -

    -<matrix\_list> :- matrices. -

    -

    -matrix_augmentsticks the matrices in <matrix\_list> -together horizontally. -

    -

    -matrix_stacksticks the matrices in <matrix\_list> -together vertically. -

    -

    -

    -examples:

    -

    
-
-matrix_augment({A,A}); 
-
-  [1  2  3  1  2  3]
-  [                ]
-  [4  5  6  4  5  6]
-  [                ]
-  [7  8  9  7  8  9]
-
-
-
-matrix_stack(A,A); 
-
-  [1  2  3]
-  [       ]
-  [4  5  6]
-  [       ]
-  [7  8  9]
-  [       ]
-  [1  2  3]
-  [       ]
-  [4  5  6]
-  [       ]
-  [7  8  9]
-
-

    Related functions: - -augment_columns, -stack_rows, -sub_matrix. -

    -

    - - - -matrixp -INDEX

    - - - -MATRIXP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -matrixp(<test\_input>) -

    -

    -

    -<test\_input> :- anything you like. -

    -

    -matrixpis a boolean function that returns t if the input is a -matrix and nil otherwise. -

    -

    -

    -examples:

    -

    
-
-matrixp A; 
-
-  t 
-
-
-matrixp(doodlesackbanana);
-
-  nil 
-
-

    Related functions: -squarep, -symmetricp. -

    -

    - - - -matrix_stack -INDEX

    - - - -MATRIX_STACK _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -see: -matrix_augment. -

    -

    - - - -minor -INDEX

    - - - -MINOR _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -minor(<matrix>,<r>,<c>) -

    -

    -

    -<matrix> :- a -matrix. -<r>,<c> :- positive integers. -

    -

    -minorcomputes the (<r>,<c>)'th minor of <matrix>. -This is created by removing the <r>'th row and the <c>'th -column from <matrix>. -

    -

    -

    -examples:

    -

    
-
-minor(A,1,3); 
-
-  [4  5]
-  [    ]
-  [7  8]
-
-

    Related functions: - -remove_columns, -remove_rows. -

    -

    - - - -mult_columns -INDEX

    - - - -MULT_COLUMNS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -Mult columns, mult rows: -

    -

    -

    -syntax:

    -mult_columns(<matrix>,<column\_list>,<expr>) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -<column\_list> :- a positive integer or a list of positive - integers. -

    -

    -<expr> :- an algebraic expression. -

    -

    -mult_columnsreturns a copy of <matrix> in which the -columns specified in <column\_list> have been multiplied by -<expr>. -

    -

    -mult_rowsperforms the same task on the rows of <matrix>. -

    -

    -

    -examples:

    -

    
-
-mult_columns(A,{1,3},x); 
-
-  [ x   2  3*x]
-  [           ]
-  [4*x  5  6*x]
-  [           ]
-  [7*x  8  9*x]
-
-
-
-mult_rows(A,2,10); 
-
-  [1   2   3 ]
-  [          ]
-  [40  50  60]
-  [          ]
-  [7   8   9 ]
-
-

    Related functions: -add_to_columns, -add_to_rows. -

    -

    - - - -mult_rows -INDEX

    - - - -MULT_ROWS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -see: -mult_columns. -

    -

    - - - -pivot -INDEX

    - - - -PIVOT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -pivot(<matrix>,<r>,<c>) -

    -

    -

    -<matrix> :- a matrix. -

    -

    -<r>,<c> :- positive integers such that <matrix>(<r>, - <c>) neq 0. -

    -

    -pivotpivots <matrix> about it's (<r>,<c>)'th -entry. -

    -

    -To do this, multiples of the <r>'th row are added to every other -row in the matrix. -

    -

    -This means that the <c>'th column will be 0 except for the -(<r>,<c>)'th entry. -

    -

    -

    -examples:

    -

    
-
-pivot(A,2,3); 
-
-  [      - 1    ]
-  [-1  ------  0]
-  [      2      ]
-  [             ]
-  [4     5     6]
-  [             ]
-  [      1      ]
-  [1    ---    0]
-  [      2      ]
-
-

    Related functions: - -rows_pivot. -

    -

    - - - -pseudo_inverse -INDEX

    - - - -PSEUDO_INVERSE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -pseudo_inverse(<matrix>) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -pseudo_inverse, also known as the Moore-Penrose inverse, -computes the pseudo inverse of <matrix>. -

    -

    -Given the singular value decomposition of <matrix>, i.e: -A = U*P*V^T, then the pseudo inverse A^-1 is defined by -A^-1 = V^T*P^-1*U. -

    -

    -Thus <matrix> * pseudo_inverse(A) = Id. -(Id is the identity matrix). -

    -

    -

    -examples:

    -

    
-
-R := mat((1,2,3,4),(9,8,7,6)); 
-
-       [1  2  3  4]
-  r := [          ]
-       [9  8  7  6]
-
-
-
-on rounded; 
-
-pseudo_inverse(R); 
-
-  [ - 0.199999999996      0.100000000013   ]
-  [                                        ]
-  [ - 0.0499999999988    0.0500000000037   ]
-  [                                        ]
-  [ 0.0999999999982     - 5.57825497203e-12]
-  [                                        ]
-  [  0.249999999995      - 0.0500000000148 ]
-
-

    Related functions: -svd. -

    -

    - - - -random_matrix -INDEX

    - - - -RANDOM_MATRIX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -random_matrix(<r>,<c>,<limit>) -

    -

    -

    -<r>,<c>,<limit> :- positive integers. -

    -

    -random_matrixcreates an <r> by <c> matrix with random -entries in the range -limit <entry <limit. -

    -

    -Switches: -

    -

    -imaginary:- if on then matrix entries are x+i*y where -limit <x,y - <<limit>. -

    -

    -not_negative:- if on then 0 <entry <<limit>. In the imagina -ry - case we have 0 <x,y <<limit>. -

    -

    -only_integer:- if on then each entry is an integer. In the imaginary - case x and y are integers. -

    -

    -symmetric:- if on then the matrix is symmetric. -

    -

    -upper_matrix:- if on then the matrix is upper triangular. -

    -

    -lower_matrix:- if on then the matrix is lower triangular. -

    -

    -

    -examples:

    -

    
-
-on rounded; 
-
-random_matrix(3,3,10); 
-
-  [ - 8.11911717343    - 5.71677292768   0.620580830035 ]
-  [                                                     ]
-  [ - 0.032596262422    7.1655452861     5.86742633837  ]
-  [                                                     ]
-  [ - 9.37155438255    - 7.55636708637   - 8.88618627557]
-
-
-
-on only_integer, not_negative, upper_matrix, imaginary; 
-
-random_matrix(4,4,10); 
-
-  [70*i + 15  28*i + 8   2*i + 79   27*i + 44]
-  [                                          ]
-  [    0      46*i + 95  9*i + 63   95*i + 50]
-  [                                          ]
-  [    0          0      31*i + 75  14*i + 65]
-  [                                          ]
-  [    0          0          0      5*i + 52 ]
-
-

    - - -remove_columns -INDEX

    - - - -REMOVE_COLUMNS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -Remove columns, remove rows: -

    -

    -

    -syntax:

    -remove_columns(<matrix>,<column\_list>) -

    -

    -

    -<matrix> :- a -matrix. -<column\_list> :- either a positive integer or a list of positive - integers. -

    -

    -remove_columnsremoves the columns specified in -<column\_list> from <matrix>. -

    -

    -remove_rowsperforms the same task on the rows of <matrix>. -

    -

    -

    -examples:

    -

     
-
-remove_columns(A,2); 
-
-  [1  3]
-  [    ]
-  [4  6]
-  [    ]
-  [7  9]
-
-
-
-remove_rows(A,{1,3}); 
-
-  [4  5  6]
-
-

    Related functions: -minor. -

    -

    - - - -remove_rows -INDEX

    - - - -REMOVE_ROWS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -see: -remove_columns. -

    -

    - - - -row_dim -INDEX

    - - - -ROW_DIM _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -see: -column_dim. -

    -

    - - - -rows_pivot -INDEX

    - - - -ROWS_PIVOT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -rows_pivot(<matrix>,<r>,<c>,{<row\_list>}) -

    -

    -

    -<matrix> :- a namerefmatrix. -

    -

    -<r>,<c> :- positive integers such that <matrix>(<r>, - <c>) neq 0. -

    -

    -<row\_list> :- positive integer or a list of positive integers. -

    -

    -rows_pivotperforms the same task as pivot but applies -the pivot only to the rows specified in <row\_list>. -

    -

    -

    -examples:

    -

    
-
-N := mat((1,2,3),(4,5,6),(7,8,9),(1,2,3),(4,5,6)); 
-
-
-       [1  2  3]
-       [       ]
-       [4  5  6]
-       [       ]
-  n := [7  8  9]
-       [       ]
-       [1  2  3]
-       [       ]
-       [4  5  6]
-
-
-
-rows_pivot(N,2,3,{4,5}); 
-
-  [1     2     3]
-  [             ]
-  [4     5     6]
-  [             ]
-  [7     8     9]
-  [             ]
-  [      - 1    ]
-  [-1  ------  0]
-  [      2      ]
-  [             ]
-  [0     0     0]
-
-

    Related functions: -pivot. -

    -

    - - - -simplex -INDEX

    - - - -SIMPLEX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -simplex(<max/min>,<objective function>, -{<linear inequalities>}) -

    -

    -

    -<max/min> :- either max or min (signifying maximize and - minimize). -

    -

    -<objective function> :- the function you are maximizing or - minimizing. -

    -

    -<linear inequalities> :- the constraint inequalities. Each one must - be of the form sum of variables ( - <=,=,>=) number. -

    -

    -simplexapplies the revised simplex algorithm to find the -optimal(either maximum or minimum) value of the -<objective function> under the linear inequality constraints. -

    -

    -It returns {optimal value,{ values of variables at this optimal}}. -

    -

    -The algorithm implies that all the variables are non-negative. -

    -

    -

    -examples:

    -

    
-
- simplex(max,x+y,{x>=10,y>=20,x+y<=25}); 
-
-
-   ***** Error in simplex: Problem has no feasible solution
-
-
-
-simplex(max,10x+5y+5.5z,{5x+3z<=200,x+0.1y+0.5z<=12,
-0.1x+0.2y+0.3z<=9, 30x+10y+50z<=1500}); 
-
-
-  {525.0,{x=40.0,y=25.0,z=0}}
-
-

    - - -squarep -INDEX

    - - - -SQUAREP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -squarep(<matrix>) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -squarepis a predicate that returns t if the <matrix> is -square and nil otherwise. -

    -

    -

    -examples:

    -

    
-
-squarep(mat((1,3,5))); 
-
-  nil 
-
-
-squarep(A);
-t
-

    Related functions: -matrixp, -symmetricp. -

    -

    - - - -stack_rows -INDEX

    - - - -STACK_ROWS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -see: -augment_columns. -

    -

    - - - -sub_matrix -INDEX

    - - - -SUB_MATRIX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -sub_matrix(<matrix>,<row\_list>,<column\_list>) -

    -

    -

    -<matrix> :- a matrix. -<row\_list>, <column\_list> :- either a positive integer or a - list of positive integers. -

    -

    -namesub_matrix produces the matrix consisting of the intersection of -the rows specified in <row\_list> and the columns specified in -<column\_list>. -

    -

    -

    -examples:

    -

    
-
-sub_matrix(A,{1,3},{2,3}); 
-
-  [2  3]
-  [    ]
-  [8  9]
-
-

    Related functions: - -augment_columns, -stack_rows. -

    -

    - - - -svd -INDEX

    - - - -SVD _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -Singular value decomposition: -

    -

    -

    -syntax:

    -svd(<matrix>) -

    -

    -

    -<matrix> :- a -matrix containing only numeric entries. -

    -

    -svdcomputes the singular value decomposition of <matrix>. -

    -

    -It returns -

    -

    -{U,P,V} -

    -

    -where A = U*P*V^T -

    -

    -and P = diag(sigma(1) ... sigma(n)). -

    -

    -sigma(i) for i= 1 ... n are the singular values of -<matrix>. -

    -

    -n is the column dimension of <matrix>. -

    -

    -The singular values of <matrix> are the non-negative square roots -of the eigenvalues of A^T*A. -

    -

    -U and V are such that U*U^T = V*V^T = V^T*V = Id. -Id is the identity matrix. -

    -

    -

    -examples:

    -

    
-
-Q := mat((1,3),(-4,3)); 
-
-       [1   3]
-  q := [     ]
-       [-4  3]
-
-
-
-on rounded; 
-
-svd(Q); 
-
-  {
-   [ 0.289784137735    0.957092029805]
-   [                                 ]
-   [ - 0.957092029805  0.289784137735]
-   ,
-   [5.1491628629       0      ]
-   [                          ]
-   [     0        2.9130948854]
-   ,
-   [ - 0.687215403194   0.726453707825  ]
-   [                                    ]
-   [ - 0.726453707825   - 0.687215403194]
-  }
-
-

    - - -swap_columns -INDEX

    - - - -SWAP_COLUMNS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -Swap columns, swap rows: -

    -

    -

    -syntax:

    -swap_columns(<matrix>,<c1>,<c2>) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -<c1>,<c1> :- positive integers. -

    -

    -swap_columnsswaps column <c1> of <matrix> with -column <c2>. -

    -

    -swap_rowsperforms the same task on two rows of <matrix>. -

    -

    -

    -examples:

    -

    
-
-swap_columns(A,2,3); 
-
-  [1  3  2]
-  [       ]
-  [4  6  5]
-  [       ]
-  [7  9  8]
-
-
-
-swap_rows(A,1,3); 
-
-  [7  8  9]
-  [       ]
-  [4  5  6]
-  [       ]
-  [1  2  3]
-
-

    Related functions: -swap_entries. -

    -

    - - - -swap_entries -INDEX

    - - - -SWAP_ENTRIES _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -swap_entries(<matrix>,{<r1>,<c1>},{<r2>, -<c2>}) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -<r1>,<c1>,<r2>,<c2> :- positive integers. -

    -

    -swap_entriesswaps <matrix>(<r1>,<c1>) with -<matrix>(<r2>,<c2>). -

    -

    -

    -examples:

    -

    
-
-swap_entries(A,{1,1},{3,3}); 
-
-  [9  2  3]
-  [       ]
-  [4  5  6]
-  [       ]
-  [7  8  1]
-
-

    Related functions: -swap_columns, -swap_rows. -

    -

    - - - -swap_rows -INDEX

    - - - -SWAP_ROWS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -see: -swap_columns. -

    -

    - - - -symmetricp -INDEX

    - - - -SYMMETRICP _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -symmetricp(<matrix>) -

    -

    -

    -<matrix> :- a -matrix. -

    -

    -symmetricpis a predicate that returns t if the matrix is symmetric -and nil otherwise. -

    -

    -

    -examples:

    -

    
-
-symmetricp(make_identity(11)); 
-
-  t 
-
-
-symmetricp(A); 
-
-  nil
-
-

    Related functions: -matrixp, -squarep. -

    -

    - - - -toeplitz -INDEX

    - - - -TOEPLITZ _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -toeplitz(<expr\_list>) -

    -

    -

    -(If you are feeling lazy then the braces can be omitted.) -

    -

    -<expr\_list> :- list of algebraic expressions. -

    -

    -toeplitzcreates the toeplitz matrix from the <expr\_list>. -

    -

    -This is a square symmetric matrix in which the first expression is -placed on the diagonal and the i'th expression is placed on the (i-1)'th -sub and super diagonals. -

    -

    -It has dimension n where n is the number of expressions. -

    -

    -

    -examples:

    -

    
-
-toeplitz({w,x,y,z}); 
-
-  [w  x  y  z]
-  [          ]
-  [x  w  x  y]
-  [          ]
-  [y  x  w  x]
-  [          ]
-  [z  y  x  w]
-
-

    - - -vandermonde -INDEX

    - - - -VANDERMONDE _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -

    -

    -

    -syntax:

    -vandermonde({<expr\_list>}) -

    -

    -

    -(If you are feeling lazy then the braces can be omitted.) -

    -

    -<expr\_list> :- list of algebraic expressions. -

    -

    -vandermondecreates the vandermonde matrix from the -<expr\_list>. -

    -

    -This is the square matrix in which the (i,j)'th entry is -<expr\_list>(i)^(j-1). -

    -

    -It has dimension n where n is the number of expressions. -

    -

    -

    -examples:

    -

                 
-vandermonde({x,2*y,3*z}); 
-
-
-  [          2 ]
-  [1   x    x  ]
-  [            ]
-  [           2]
-  [1  2*y  4*y ]
-  [            ]
-  [           2]
-  [1  3*z  9*z ]
-
-

    - - -Linear Algebra package -INDEX

    -Linear Algebra package

    -
  • Linear Algebra package introduction

    -

  • fast_la switch

    -

  • add_columns operator

    -

  • add_rows operator

    -

  • add_to_columns operator

    -

  • add_to_rows operator

    -

  • augment_columns operator

    -

  • band_matrix operator

    -

  • block_matrix operator

    -

  • char_matrix operator

    -

  • char_poly operator

    -

  • cholesky operator

    -

  • coeff_matrix operator

    -

  • column_dim operator

    -

  • companion operator

    -

  • copy_into operator

    -

  • diagonal operator

    -

  • extend operator

    -

  • find_companion operator

    -

  • get_columns operator

    -

  • get_rows operator

    -

  • gram_schmidt operator

    -

  • hermitian_tp operator

    -

  • hessian operator

    -

  • hilbert operator

    -

  • jacobian operator

    -

  • jordan_block operator

    -

  • lu_decom operator

    -

  • make_identity operator

    -

  • matrix_augment operator

    -

  • matrixp operator

    -

  • matrix_stack operator

    -

  • minor operator

    -

  • mult_columns operator

    -

  • mult_rows operator

    -

  • pivot operator

    -

  • pseudo_inverse operator

    -

  • random_matrix operator

    -

  • remove_columns operator

    -

  • remove_rows operator

    -

  • row_dim operator

    -

  • rows_pivot operator

    -

  • simplex operator

    -

  • squarep operator

    -

  • stack_rows operator

    -

  • sub_matrix operator

    -

  • svd operator

    -

  • swap_columns operator

    -

  • swap_entries operator

    -

  • swap_rows operator

    -

  • symmetricp operator

    -

  • toeplitz operator

    -

  • vandermonde operator

    -

    •  - - -Smithex -INDEX

    - - - -SMITHEX _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The operator smithex computes the Smith normal form S of a - -matrix A (say). It returns {S,P,P^-1} where P*S*P^-1 = - A. -

    -

    -

    -syntax:

    -smithex(<matrix>,<variable>) -

    -

    -<matrix> :- a rectangular -matrix of univariate polynomials in - <variable>. -<variable> :- the variable. -

    -

    -

    -

    -examples:

    -

    
- a := mat((x,x+1),(0,3*x^2)); 
-
-        [x  x + 1]
-        [        ]
-   a := [      2 ]
-        [0  3*x  ]
-
-
-
- smithex(a,x); 
-
-     [1  0 ]    [1    0]    [x   x + 1]
-  {  [     ],   [      ],   [         ]  }
-     [    3]    [   2  ]    [         ]
-     [0  x ]    [3*x  1]    [-3    -3 ]
-
-

    - - -Smithex_int -INDEX

    - - - -SMITHEX\_INT _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The operator smithex_int performs the same task as smithex -but on matrices containing only integer entries. Namely, -smithex_int returns {S,P,P^-1} where S is the smith normal -form of the input -matrix (A say), and P*S*P^-1 = A. -

    -

    -

    -syntax:

    -smithex_int(<matrix>) -

    -

    -<matrix> :- a rectangular -matrix of integer entries. -

    -

    -

    -

    -examples:

    -

    
- a := mat((9,-36,30),(-36,192,-180),(30,-180,180)); 
-
-
-       [ 9   -36    30 ]
-       [               ]
-  a := [-36  192   -180]
-       [               ]
-       [30   -180  180 ]
-
-
-
- smithex_int(a); 
-
-    [3  0   0 ]    [-17  -5   -4 ]    [1   -24  30 ]
-    [         ]    [             ]    [            ]
-  { [0  12  0 ],   [64   19   15 ],   [-1  25   -30] }
-    [         ]    [             ]    [            ] 
-    [0  0   60]    [-50  -15  -12]    [0   -1    1 ] 
-
-

    - - -Frobenius -INDEX

    - - - -FROBENIUS _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The operator frobenius computes the frobenius normal form F of - a - -matrix (A say). It returns {F,P,P^-1} where P*F*P^-1 = - A. -

    -

    -

    -syntax:

    -frobenius(<matrix>) -

    -

    -<matrix> :- a square -matrix. -

    -

    -

    -Field Extensions: -

    -

    -By default, calculations are performed in the rational numbers. To -extend this field the -arnum package can be used. The package must -first be loaded by load_package arnum;. The field can now be extended -by using the defpoly command. For example, defpoly sqrt2**2-2; will -extend the field to include the square root of 2 (now defined by sqrt2). -

    -

    -Modular Arithmetic: -

    -

    -Frobeniuscan also be calculated in a modular base. To do this -first type on modular;. Then setmod p; (where p is a prime) will set -the modular base of calculation to p. By further typing on balanced_mod -the answer will appear using a symmetric modular representation. See - -ratjordan for an example. -

    -

    -

    -examples:

    -

    
- a := mat((x,x^2),(3,5*x)); 
-
-       [    2 ]
-       [x  x  ]
-  a := [      ]
-       [3  5*x]
-
-
- frobenius(a);
-
-     [         2]    [1  x]    [       - x ]
-  {  [0   - 2*x ],   [    ],   [1     -----]  }
-     [          ]    [0  3]    [        3  ]
-     [1    6*x  ]              [           ]
-                               [        1  ]
-                               [0      --- ]
-                               [        3  ]
-
-
- load_package arnum;
-
- defpoly sqrt2**2-2;
-
- a := mat((sqrt2,5),(7*sqrt2,sqrt2));
-
-
-       [ sqrt2     5  ]
-  a := [              ]
-       [7*sqrt2  sqrt2]
-
-
-
- frobenius(a); 
-
-    [0  35*sqrt2 - 2]    [1   sqrt2 ]    [           1  ]
-  { [               ],   [          ],   [1       - --- ]  }
-    [1    2*sqrt2   ]    [1  7*sqrt2]    [           7  ]
-                                         [              ]
-                                         [     1        ]
-                                         [0   ----*sqrt2]
-                                         [     14       ]
-
-

    - - -Ratjordan -INDEX

    - - - -RATJORDAN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The operator ratjordan computes the rational Jordan normal form R -of a -matrix (A say). It returns {R,P,P^-1} where P*R*P^-1 = - A. -

    -

    -

    -syntax:

    -ratjordan(<matrix>) -

    -

    -<matrix> :- a square -matrix. -

    -

    -

    -Field Extensions: -

    -

    -By default, calculations are performed in the rational numbers. To -extend this field the arnum package can be used. The package must -first be loaded by load_package arnum;. The field can now be extended -by using the defpoly command. For example, defpoly sqrt2**2-2; will -extend the field to include the square root of 2 (now defined by sqrt2). -See -frobenius for an example. -

    -

    -Modular Arithmetic: -

    -

    -ratjordancan also be calculated in a modular base. To do this -first type on modular;. Then setmod p; (where p is a prime) will set -the modular base of calculation to p. By further typing on balanced_mod -the answer will appear using a symmetric modular representation. -

    -

    -

    -examples:

    -

    
- a := mat((5,4*x),(2,x^2));
-
-       [5  4*x]
-       [      ]
-  a := [    2 ]
-       [2  x  ]
-
-
-
- ratjordan(a); 
-
-    [0  x*( - 5*x + 8)]   [1  5]    [        -5 ]  
-  { [                 ],  [    ],   [1     -----] }
-    [        2        ]   [0  2]    [        2  ]
-    [1      x  + 5    ]             [           ]
-                                    [        1  ]
-                                    [0     -----]
-                                    [        2  ]
-
-
- on modular; 
-
- setmod 23; 
-
- a := mat((12,34),(56,78)); 
-
-       [12  11]
-  a := [      ]
-       [10  9 ]
-
-
-
- ratjordan(a); 
-
-    [15  0]   [16  8]   [1  21]
-  { [     ],  [     ],  [     ]  }
-    [0   6]   [19  4]   [1  4 ]
-
-
-
- on balanced_mod;
-
- ratjordan(a);
-
-    [- 8  0]   [ - 7  8]   [1  - 2]
-  { [      ],  [       ],  [      ]  }
-    [ 0   6]   [ - 4  4]   [1   4 ]
-
-

    - - -Jordansymbolic -INDEX

    - - - -JORDANSYMBOLIC _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The operator jordansymbolic computes the Jordan normal form J -of a -matrix (A say). It returns {J,L,P,P^-1} where -P*J*P^-1 = A. L = {ll,mm} where mm is a name and ll is a list of -irreducible factors of p(mm). -

    -

    -

    -syntax:

    -jordansymbolic(<matrix>) -

    -

    -<matrix> :- a square -matrix. -

    -

    -

    -Field Extensions: -

    -

    -By default, calculations are performed in the rational numbers. To -extend this field the -arnum package can be used. The package must -first be loaded by load_package arnum;. The field can now be extended -by using the defpoly command. For example, defpoly sqrt2**2-2; will -extend the field to include the square root of 2 (now defined by sqrt2). -See -frobenius for an example. -

    -

    -Modular Arithmetic: -

    -

    -jordansymboliccan also be calculated in a modular base. To do this -first type on modular;. Then setmod p; (where p is a prime) will set -the modular base of calculation to p. By further typing on balanced_mod -the answer will appear using a symmetric modular representation. See - -ratjordan for an example. -

    -

    -

    -examples:

    -

    
-
- a := mat((1,y),(2,5*y)); 
-
-       [1   y ]
-  a := [      ]
-       [2  5*y]
-
-
-
- jordansymbolic(a); 
-
-  {
-   [lambda11     0    ]
-   [                  ]
-   [   0      lambda12]
-   ,
-           2
-   lambda  - 5*lambda*y - lambda + 3*y,lambda,
-   [lambda11 - 5*y  lambda12 - 5*y]
-   [                              ]
-   [      2               2       ]
-   ,
-   [ 2*lambda11 - 5*y - 1    5*lambda11*y - lambda11 - y + 1 ]
-   [----------------------  ---------------------------------]
-   [       2                              2                  ]
-   [   25*y  - 2*y + 1             2*(25*y  - 2*y + 1)       ]
-   [                                                         ]
-   [ 2*lambda12 - 5*y - 1    5*lambda12*y - lambda12 - y + 1 ]
-   [----------------------  ---------------------------------]
-   [       2                              2                  ]
-   [   25*y  - 2*y + 1             2*(25*y  - 2*y + 1)       ]
-   }
-
-

    - - -Jordan -INDEX

    - - - -JORDAN _ _ _ _ _ _ _ _ _ _ _ _ operator

    -

    - -The operator jordan computes the Jordan normal form J -of a -matrix (A say). It returns {J,P,P^-1} where P*J*P^-1 = - A. -

    -

    -

    -syntax:

    -jordan(<matrix>) -

    -

    -<matrix> :- a square -matrix. -

    -

    -

    -Field Extensions: -By default, calculations are performed in the rational numbers. To -extend this field the arnum package can be used. The package must -first be loaded by load_package arnum;. The field can now be extended -by using the defpoly command. For example, defpoly sqrt2**2-2; will -extend the field to include the square root of 2 (now defined by sqrt2). -See -frobenius for an example. -

    -

    -Modular Arithmetic: -Jordan can also be calculated in a modular base. To do this -first type on modular;. Then setmod p; (where p is a prime) will set -the modular base of calculation to p. By further typing on balanced_mod -the answer will appear using a symmetric modular representation. See - -ratjordan for an example. -

    -

    -

    -examples:

    -

    
-
- a := mat((1,x),(0,x)); 
-
-       [1  x]
-  a := [    ]
-       [0  x]
-
-
-
- jordan(a);
-
-  {
-   [1  0]
-   [    ]
-   [0  x]
-   ,
-   [   1           x       ]
-   [-------  --------------]
-   [ x - 1     2           ]
-   [          x  - 2*x + 1 ]
-   [                       ]
-   [               1       ]
-   [   0        -------    ]
-   [             x - 1     ]
-   ,
-   [x - 1   - x ]
-   [            ]
-   [  0    x - 1]
-   }
-
-

    - - -Matrix Normal Forms -INDEX

    -Matrix Normal Forms

    -
  • Smithex operator

    -

  • Smithex\_int operator

    -

  • Frobenius operator

    -

  • Ratjordan operator

    -

  • Jordansymbolic operator

    -

  • Jordan operator

    -

    •  - - -Miscellaneous_Packages -INDEX

    - - - -MISCELLANEOUS PACKAGES _ _ _ _ _ _ _ _ _ _ _ _ introduction

    -

    - -REDUCE includes a large number of packages that have been contributed by -users from various fields. Some of these, together with their relevant -commands, switches and so on (e.g., the NUMERIC package), have -been described elsewhere. This section describes those packages for which -no separate help material exists. Each has its own switches, commands, -and operators, and some redefine special characters to aid in their -notation. However, the brief descriptions given here do not include all -such information. Readers are referred to the general package -documentation in this case, which can be found, along with the source -code, under the subdirectories doc and src in the -reduce directory. The -load_package command is used to -load the files you wish into your system. There will be a short delay -while the package is loaded. A package cannot be unloaded. Once it -is in your system, it stays there until you end the session. Each package -also has a test file, which you will find under its name in the -$reduce/xmpl directory. -

    -

    -Finally, it should be mentioned that such user-contributed packages are -unsupported; any questions or problems should be directed to their -authors. -

    -

    - - - -ALGINT_package -INDEX

    - - - -ALGINT _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: James H. Davenport -

    -

    -The algint package provides indefinite integration of square roots. -This package, which is an extension of the basic integration package -distributed with REDUCE, will analytically integrate a wide range of -expressions involving square roots. The -algint switch provides for -the use of the facilities given by the package, and is automatically turned -on when the package is loaded. If you want to return to the standard -integration algorithms, turn -algint off. An error message is given -if you try to turn the -algint switch on when its package is not -loaded. -

    -

    - - - -APPLYSYM -INDEX

    - - - -APPLYSYM _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Thomas Wolf -

    -

    -This package provides programs APPLYSYM, QUASILINPDE and DETRAFO for -computing with infinitesimal symmetries of differential equations. -

    -

    - - - -ARNUM -INDEX

    - - - -ARNUM _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Eberhard Schruefer -

    -

    -This package provides facilities for handling algebraic numbers as polynomial -coefficients in REDUCE calculations. It includes facilities for introducing -indeterminates to represent algebraic numbers, for calculating splitting -fields, and for factoring and finding greatest common divisors in such -domains. -

    -

    - - - -ASSIST -INDEX

    - - - -ASSIST _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Hubert Caprasse -

    -

    -ASSIST contains a large number of additional general purpose functions -that allow a user to better adapt REDUCE to various calculational -strategies and to make the programming task more straightforward and more -efficient. -

    -

    - - - -AVECTOR -INDEX

    - - - -AVECTOR _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: David Harper -

    -

    -This package provides REDUCE with the ability to perform vector algebra -using the same notation as scalar algebra. The basic algebraic operations -are supported, as are differentiation and integration of vectors with -respect to scalar variables, cross product and dot product, component -manipulation and application of scalar functions (e.g. cosine) to a vector -to yield a vector result. -

    -

    - - - -BOOLEAN -INDEX

    - - - -BOOLEAN _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Herbert Melenk -

    -

    -This package supports the computation with boolean expressions in the -propositional calculus. The data objects are composed from algebraic -expressions connected by the infix boolean operators and, or, - implies, equiv, and the unary prefix operator not. - Boolean allows you to simplify expressions built from these -operators, and to test properties like equivalence, subset property etc. -

    -

    - - - -CALI -INDEX

    - - - -CALI _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Hans-Gert Gr"abe -

    -

    -This package contains algorithms for computations in commutative algebra -closely related to the Groebner algorithm for ideals and modules. Its -heart is a new implementation of the Groebner algorithm that also allows -for the computation of syzygies. This implementation is also applicable to -submodules of free modules with generators represented as rows of a matrix. -

    -

    - - - -CAMAL -INDEX

    - - - -CAMAL _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: John P. Fitch -

    -

    -This package implements in REDUCE the Fourier transform procedures of the -CAMAL package for celestial mechanics. -

    -

    - - - -CHANGEVR -INDEX

    - - - -CHANGEVR _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -Author: G. Ucoluk -

    -

    -This package provides facilities for changing the independent variables in -a differential equation. It is basically the application of the chain rule. -

    -

    - - - -COMPACT -INDEX

    - - - -COMPACT _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Anthony C. Hearn -

    -

    -COMPACT is a package of functions for the reduction of a polynomial in the -presence of side relations. COMPACT applies the side relations to the -polynomial so that an equivalent expression results with as few terms as -possible. For example, the evaluation of -

    -

    -

    
-     compact(s*(1-sin x^2)+c*(1-cos x^2)+sin x^2+cos x^2,
-             {cos x^2+sin x^2=1});
-
-

    yields the result -

    
-
-              2           2
-        SIN(X) *C + COS(X) *S + 1
-

    -

    -The first argument to the operator compact is the expression -and the second is a list of side relations that can be -equations or simple expressions (implicitly equated to zero). The -kernels in the side relations may also be free variables with the -same meaning as in rules, e.g. -

    
-     sin_cos_identity :=  {cos ~w^2+sin ~w^2=1}$
-     compact(u,in_cos_identity);
-

    -

    -Also the full rule syntax with the replacement operator is allowed here. -

    -

    - - - -CRACK -INDEX

    - - - -CRACK _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Authors: Andreas Brand, Thomas Wolf -

    -

    -CRACK is a package for solving overdetermined systems of partial or -ordinary differential equations (PDEs, ODEs). Examples of programs which -make use of CRACK for investigating ODEs (finding symmetries, first -integrals, an equivalent Lagrangian or a ``differential factorization'') are -included. -

    -

    - - - -CVIT -INDEX

    - - - -CVIT _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Authors: V.Ilyin, A.Kryukov, A.Rodionov, A.Taranov -

    -

    -This package provides an alternative method for computing traces of Dirac -gamma matrices, based on an algorithm by Cvitanovich that treats gamma -matrices as 3-j symbols. -

    -

    - - - -DEFINT -INDEX

    - - - -DEFINT _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Authors: Kerry Gaskell, Stanley M. Kameny, Winfried Neun -

    -

    -This package finds the definite integral of an expression in a stated -interval. It uses several techniques, including an innovative approach -based on the Meijer G-function, and contour integration. -

    -

    - - - -DESIR -INDEX

    - - - -DESIR _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Authors: C. Dicrescenzo, F. Richard-Jung, E. Tournier -

    -

    -This package enables the basis of formal solutions to be computed for an -ordinary homogeneous differential equation with polynomial coefficients -over Q of any order, in the neighborhood of zero (regular or irregular -singular point, or ordinary point). -

    -

    - - - -DFPART -INDEX

    - - - -DFPART _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Herbert Melenk -

    -

    -This package supports computations with total and partial derivatives of -formal function objects. Such computations can be useful in the context -of differential equations or power series expansions. -

    -

    - - - -DUMMY -INDEX

    - - - -DUMMY _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Alain Dresse -

    -

    -This package allows a user to find the canonical form of expressions -involving dummy variables. In that way, the simplification of -polynomial expressions can be fully done. The indeterminates are general -operator objects endowed with as few properties as possible. In that way -the package may be used in a large spectrum of applications. -

    -

    - - - -EXCALC -INDEX

    - - - -EXCALC _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Eberhard Schruefer -

    -

    -The excalc package is designed for easy use by all who are familiar -with the calculus of Modern Differential Geometry. The program is currently -able to handle scalar-valued exterior forms, vectors and operations between -them, as well as non-scalar valued forms (indexed forms). It is thus an ideal -tool for studying differential equations, doing calculations in general -relativity and field theories, or doing simple things such as calculating the -Laplacian of a tensor field for an arbitrary given frame. -

    -

    - - - -FPS -INDEX

    - - - -FPS _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Authors: Wolfram Koepf, Winfried Neun -

    -

    -This package can expand a specific class of functions into their -corresponding Laurent-Puiseux series. -

    -

    - - - -FIDE -INDEX

    - - - -FIDE _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -Author: Richard Liska -

    -

    -This package performs automation of the process of numerically -solving partial differential equations systems (PDES) by means of -computer algebra. For PDES solving, the finite difference method is applied. -The computer algebra system REDUCE and the numerical programming -language FORTRAN are used in the presented methodology. The main aim of -this methodology is to speed up the process of preparing numerical -programs for solving PDES. This process is quite often, especially for -complicated systems, a tedious and time consuming task. -

    -

    - - - -GENTRAN -INDEX

    - - - -GENTRAN _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Barbara L. Gates -

    -

    -This package is an automatic code GENerator and TRANslator. It constructs -complete numerical programs based on sets of algorithmic specifications and -symbolic expressions. Formatted FORTRAN, RATFOR or C code can be generated -through a series of interactive commands or under the control of a template -processing routine. Large expressions can be automatically segmented into -subexpressions of manageable size, and a special file-handling mechanism -maintains stacks of open I/O channels to allow output to be sent to any -number of files simultaneously and to facilitate recursive invocation of the -whole code generation process. -

    -

    - - - -IDEALS -INDEX

    - - - -IDEALS _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Herbert Melenk -

    -

    -This package implements the basic arithmetic for polynomial ideals by -exploiting the Groebner bases package of REDUCE. In order to save -computing time all intermediate Groebner bases are stored internally such -that time consuming repetitions are inhibited. -

    -

    - - - -INEQ -INDEX

    - - - -INEQ _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Herbert Melenk -

    -

    -This package supports the operator ineq_solve that -attempts to solve single inequalities and sets of coupled inequalities. -

    -

    - - - -INVBASE -INDEX

    - - - -INVBASE _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -Authors: A.Yu. Zharkov and Yu.A. Blinkov -

    -

    -Involutive bases are a new tool for solving problems in connection with -multivariate polynomials, such as solving systems of polynomial equations -and analyzing polynomial ideals. An involutive basis of a polynomial ideal -is nothing more than a special form of a redundant Groebner basis. The -construction of involutive bases reduces the problem of solving polynomial -systems to simple linear algebra. -

    -

    - - - -LAPLACE -INDEX

    - - - -LAPLACE _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Authors: C. Kazasov, M. Spiridonova, V. Tomov -

    -

    -This package can calculate ordinary and inverse Laplace transforms of -expressions. -

    -

    - - - -LIE -INDEX

    - - - -LIE _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -Authors: Carsten and Franziska Sch"obel -

    -

    -Lieis a package of functions for the classification of real -n-dimensional Lie algebras. It consists of two modules: liendmc1 -and lie1234. With the help of the functions in the liendmcl -module, real n-dimensional Lie algebras L with a derived algebra -L^(1) of dimension 1 can be classified. -

    -

    - - - -MODSR -INDEX

    - - - -MODSR _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Herbert Melenk -

    -

    -This package supports solve (M_SOLVE) and roots (M_ROOTS) operators for -modular polynomials and modular polynomial systems. The moduli need not -be primes. M_SOLVE requires a modulus to be set. M_ROOTS takes the -modulus as a second argument. For example: -

    -

    -

    
-on modular; setmod 8;
-m_solve(2x=4);            ->  {{X=2},{X=6}}
-m_solve({x^2-y^3=3});
-    ->  {{X=0,Y=5}, {X=2,Y=1}, {X=4,Y=5}, {X=6,Y=1}}
-m_solve({x=2,x^2-y^3=3}); ->  {{X=2,Y=1}}
-off modular;
-m_roots(x^2-1,8);         ->  {1,3,5,7}
-m_roots(x^3-x,7);         ->  {0,1,6}
-

    - - -NCPOLY -INDEX

    - - - -NCPOLY _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Authors: Herbert Melenk, Joachim Apel -

    -

    -This package allows the user to set up automatically a consistent -environment for computing in an algebra where the non--commutativity is -defined by Lie-bracket commutators. The package uses the REDUCE -noncom mechanism for elementary polynomial arithmetic; the commutator -rules are automatically computed from the Lie brackets. -

    -

    - - - -ORTHOVEC -INDEX

    - - - -ORTHOVEC _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: James W. Eastwood -

    -

    -orthovecis a collection of REDUCE procedures and operations which -provide a simple-to-use environment for the manipulation of scalars and -vectors. Operations include addition, subtraction, dot and cross -products, division, modulus, div, grad, curl, laplacian, differentiation, -integration, and Taylor expansion. -

    -

    - - - -PHYSOP -INDEX

    - - - -PHYSOP _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -Author: Mathias Warns -

    -

    -This package has been designed to meet the requirements of theoretical -physicists looking for a computer algebra tool to perform complicated -calculations in quantum theory with expressions containing operators. -These operations consist mainly of the calculation of commutators between -operator expressions and in the evaluations of operator matrix elements in -some abstract space. -

    -

    - - - -PM -INDEX

    - - - -PM _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Kevin McIsaac -

    -

    -PM is a general pattern matcher similar in style to those found in systems -such as SMP and Mathematica, and is based on the pattern matcher described -in Kevin McIsaac, ``Pattern Matching Algebraic Identities'', SIGSAM Bulletin, -19 (1985), 4-13. -

    -

    - - - -RANDPOLY -INDEX

    - - - -RANDPOLY _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Francis J. Wright -

    -

    -This package is based on a port of the Maple random polynomial -generator together with some support facilities for the generation -of random numbers and anonymous procedures. -

    -

    - - - -REACTEQN -INDEX

    - - - -REACTEQN _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Herbert Melenk -

    -

    -This package allows a user to transform chemical reaction systems into -ordinary differential equation systems (ODE) corresponding to the laws of -pure mass action. -

    -

    - - - -RESET -INDEX

    - - - -RESET _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -Author: John Fitch -

    -

    -This package defines a command command RESETREDUCE that works through the -history of previous commands, and clears any values which have been -assigned, plus any rules, arrays and the like. It also sets the various -switches to their initial values. It is not complete, but does work for -most things that cause a gradual loss of space. It would be relatively -easy to make it interactive, so allowing for selective resetting. -

    -

    - - - -RESIDUE -INDEX

    - - - -RESIDUE _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -Author: Wolfram Koepf -

    -

    -This package supports the calculation of residues of arbitrary -expressions. -

    -

    - - - -RLFI -INDEX

    - - - -RLFI _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Richard Liska -

    -

    -This package -adds LaTeX syntax -to REDUCE. Text generated by REDUCE in this mode can be directly -used in LaTeX source -documents. Various -mathematical constructions are supported by the interface including -subscripts, superscripts, font changing, Greek letters, divide-bars, -integral and sum signs, derivatives, and so on. -

    -

    - - - -SCOPE -INDEX

    - - - -SCOPE _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: J.A. van Hulzen -

    -

    -SCOPE is a package for the production of an optimized form of a set of -expressions. It applies an heuristic search for common (sub)expressions -to almost any set of proper REDUCE assignment statements. The output is -obtained as a sequence of assignment statements. gentran is used to -facilitate expression output. -

    -

    - - - -SETS -INDEX

    - - - -SETS _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -Author: Francis J. Wright -

    -

    -The SETS package provides algebraic-mode support for set operations on -lists regarded as sets (or representing explicit sets) and on implicit -sets represented by identifiers. -

    -

    - - - -SPDE -INDEX

    - - - -SPDE _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Fritz Schwartz -

    -

    -The package spde provides a set of functions which may be used to -determine the symmetry group of Lie- or point-symmetries of a given system of -partial differential equations. In many cases the determining system is -solved completely automatically. In other cases the user has to provide -additional input information for the solution algorithm to terminate. -

    -

    - - - -SYMMETRY -INDEX

    - - - -SYMMETRY _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -Author: Karin Gatermann -

    -

    -This package computes symmetry-adapted bases and block diagonal forms of -matrices which have the symmetry of a group. The package is the -implementation of the theory of linear representations for small finite -groups such as the dihedral groups. -

    -

    - - - -TPS -INDEX

    - - - -TPS _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Authors: Alan Barnes, Julian Padget -

    -

    -This package implements formal Laurent series expansions in one variable -using the domain mechanism of REDUCE. This means that power series -objects can be added, multiplied, differentiated etc., like other first -class objects in the system. A lazy evaluation scheme is used and thus -terms of the series are not evaluated until they are required for printing -or for use in calculating terms in other power series. The series are -extendible giving the user the impression that the full infinite series is -being manipulated. The errors that can sometimes occur using series that -are truncated at some fixed depth (for example when a term in the required -series depends on terms of an intermediate series beyond the truncation -depth) are thus avoided. -

    -

    - - - -TRI -INDEX

    - - - -TRI _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Werner Antweiler -

    -

    -This package provides facilities written in REDUCE-Lisp for typesetting -REDUCE formulas -using TeX. The -TeX-REDUCE-Interface incorporates three levels -of TeX output: -without line breaking, with line breaking, and -with line breaking plus indentation. -

    -

    - - - -TRIGSIMP -INDEX

    - - - -TRIGSIMP _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Wolfram Koepf -

    -

    -TRIGSIMP is a useful tool for all kinds of trigonometric and hyperbolic -simplification and factorization. There are three procedures included in -TRIGSIMP: trigsimp, trigfactorize and triggcd. The -first is for finding simplifications of trigonometric or hyperbolic -expressions with many options, the second for factorizing them and the -third for finding the greatest common divisor of two trigonometric or -hyperbolic polynomials. -

    -

    - - - -XCOLOR -INDEX

    - - - -XCOLOR _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: A. Kryukov -

    -

    -This package calculates the color factor in non-abelian gauge field -theories using an algorithm due to Cvitanovich. -

    -

    - - - -XIDEAL -INDEX

    - - - -XIDEAL _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: David Hartley -

    -

    -xidealconstructs Groebner bases for solving the left ideal -membership problem: Groebner left ideal bases or GLIBs. For graded -ideals, where each form is homogeneous in degree, the distinction between -left and right ideals vanishes. Furthermore, if the generating forms are -all homogeneous, then the Groebner bases for the non-graded and graded -ideals are identical. In this case, xideal is able to save time by -truncating the Groebner basis at some maximum degree if desired. -

    -

    - - - -WU -INDEX

    - - - -WU _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Author: Russell Bradford -

    -

    -This is a simple implementation of the Wu algorithm implemented in REDUCE -working directly from ``A Zero Structure Theorem for -Polynomial-Equations-Solving,'' Wu Wen-tsun, Institute of Systems Science, -Academia Sinica, Beijing. -

    -

    - - - -ZEILBERG -INDEX

    - - - -ZEILBERG _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -

    -

    -Authors: Gregor St"olting and Wolfram Koepf -

    -

    -This package is a careful implementation of the Gosper and Zeilberger -algorithms for indefinite and definite summation of hypergeometric terms, -respectively. Extensions of these algorithms are also included that are -valid for ratios of products of powers, -factorials, gamma function -terms, binomial coefficients, and shifted factorials that are -rational-linear in their arguments. -

    -

    - - - -ZTRANS -INDEX

    - - - -ZTRANS _ _ _ _ _ _ _ _ _ _ _ _ package

    -

    - -Authors: Wolfram Koepf, Lisa Temme -

    -

    -This package is an implementation of the Z-transform of a sequence. -This is the discrete analogue of the Laplace Transform. -

    -

    - - - -Miscellaneous Packages -INDEX

    -Miscellaneous Packages

    -
  • Miscellaneous Packages introduction

    -

  • ALGINT package

    -

  • APPLYSYM package

    -

  • ARNUM package

    -

  • ASSIST package

    -

  • AVECTOR package

    -

  • BOOLEAN package

    -

  • CALI package

    -

  • CAMAL package

    -

  • CHANGEVR package

    -

  • COMPACT package

    -

  • CRACK package

    -

  • CVIT package

    -

  • DEFINT package

    -

  • DESIR package

    -

  • DFPART package

    -

  • DUMMY package

    -

  • EXCALC package

    -

  • FPS package

    -

  • FIDE package

    -

  • GENTRAN package

    -

  • IDEALS package

    -

  • INEQ package

    -

  • INVBASE package

    -

  • LAPLACE package

    -

  • LIE package

    -

  • MODSR package

    -

  • NCPOLY package

    -

  • ORTHOVEC package

    -

  • PHYSOP package

    -

  • PM package

    -

  • RANDPOLY package

    -

  • REACTEQN package

    -

  • RESET package

    -

  • RESIDUE package

    -

  • RLFI package

    -

  • SCOPE package

    -

  • SETS package

    -

  • SPDE package

    -

  • SYMMETRY package

    -

  • TPS package

    -

  • TRI package

    -

  • TRIGSIMP package

    -

  • XCOLOR package

    -

  • XIDEAL package

    -

  • WU package

    -

  • ZEILBERG package

    -

  • ZTRANS package

    -

    •  - - -ED -INDEX

    - - - -ED _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -The ed command invokes a simple line editor for REDUCE input -statements. -

    -

    -

    -syntax:

    -ed<integer> or ed -

    -

    -

    -edcalled with no argument edits the last input statement. If -<integer> is greater than or equal to the current line number, an error -message is printed. Reenter a proper ed command or return to the -top level with a semicolon. -

    -

    -The editor formats REDUCE's version of the desired input statement, -dividing it into lines at semicolons and dollar signs. The statement is -printed at the beginning of the edit session. The editor works on one -line at a time, and has a pointer (shown by ^) to the current -character of that line. When the session begins, the pointer is at the -left hand side of the first line. The editing prompt is >. -

    -

    -The following commands are available. They may be entered in either upper -or lower case. All commands are activated by the carriage return, which -also prints out the current line after changes. Several commands can be -placed on a single line, except that commands terminated by an ESC -must be the last command before the carriage return. -

    -

    - _ _ _ b -Move pointer to beginning of current line. -

    -

    - _ _ _ d<digit> -Delete current character and next (digit-1) characters. An error message -is printed if anything other than a single digit follows d. If there are -fewer than <digit> characters left on the line, all but the final -dollar sign or semicolon is removed. To delete a line completely, use the -k command. -

    -

    - _ _ _ e -End the current session, causing the edited expression to be reparsed by -REDUCE. -

    -

    - _ _ _ f<char> -Find the next occurrence of the character <char> to the right of the -pointer on the current line and move the pointer to it. If the character is -not found, an error message is printed and the pointer remains in its -original position. Other lines are not searched. The f command is not -case-sensitive. -

    -

    - _ _ _ i<string>ESC -Insert <string> in front of pointer. The ESC key is your -delimiter for the input string. No other command may follow this one on -the same line. -

    -

    - _ _ _ k -Kill rest of the current line, including the semicolon or dollar sign -terminator. If there are characters remaining on the current line, and it -is the last line of the input statement, a semicolon is added to the line -as a terminator for REDUCE. If the current line is now empty, one of the -following actions is performed: If there is a following line, it becomes -the current line and the pointer is placed at its first character. If the -current line was the final line of the statement, and there is a previous -line, the previous line becomes the current line. If the current line was -the only line of the statement, and it is empty, a single semicolon is -inserted for REDUCE to parse. -

    -

    - _ _ _ l -Finish editing this line and move to the last previous line. An error message -is printed if there is no previous line. -

    -

    - _ _ _ n -Finish editing this line and move to the next line. An error message is -printed if there is no next line. -

    -

    - _ _ _ p -Print out all the lines of the statement. Then a dotted line is printed, and -the current line is reprinted, with the pointer under it. -

    -

    - _ _ _ q -Quit the editing session without saving the changes. If a semicolon is -entered after q, a new line prompt is given, otherwise REDUCE prompts you -for another command. Whatever you type in to the prompt appearing after -the q is entered is stored as the input for the line number in which you -called the edit. Thus if you enter a semicolon, neither -input -ed will find anything under the current number. -

    -

    - _ _ _ r<char> -Replace the character at the pointer by <char>. -

    -

    - _ _ _ s<string>ESC -Search for the first occurrence of <string> to the right of the -pointer on the current line and move the pointer to its first character. -The ESC key is your delimiter for the input string. The s function -does not search other lines of the statement. If the string is not found, -an error message is printed and the pointer remains in its original -position. The s command is not case-sensitive. No other command may -follow this one on the same line. -

    -

    - _ _ _ x <or space> -Move the pointer one character to the right. If the pointer is already at -the end of the line, an error message is printed. -

    -

    - _ _ _ - <(minus)> -Move the pointer one character to the left. If the pointer is already at the -beginning of the line, an error message is printed. -

    -

    - _ _ _ ? -Display the Help menu, showing the commands and their actions. -

    -

    -

    -examples:

    -

    (Line numbers are shown in the following examples)

    -

     
-
-2: >>x**2 + y; 
-
-X^{2} + Y 
-
-3: >>ed 2; 
-
-  X**2 + Y; 
-
-  ^ 
-
-For help, type '?' 
-
-?-                  (Enter three spaces and key{Return}) 
-
-  X**2 + Y; 
-
-     ^ 
-
-?- r5 
-
-  X**5 + Y; 
-
-     ^ 
-
-?- fY 
-
-  X**5 + Y; 
-
-	 ^ 
-
-?- iabc (Terminate with key{ESC} and key{Return}) 
-
-  X**5 + abcY; 
-
-	    ^ 
-
-?- ---- 
-
-  X**5 + abcY; 
-
-	^ 
-
-?- fbd2 
-
-  X**5 + aY; 
-
-	  ^ 
-
-?- b 
-
-  X**5 + aY; 
-
-  ^ 
-
-?- e 
-
-AY + X^{5} 
-
-4: >>procedure dumb(a); 
-
->>write a; 
-
-DUMB 
-
-5: >>dumb(17); 
-
-17 
-
-6: >>ed 4; 
-
-  PROCEDURE DUMB (A); 
-
-  ^ 
-
-WRITE A; 
-
-?- fArBn 
-
-  WRITE A; 
-
-  ^ 
-
-?- ibegin scalar a; a := b + 10; (Type a space, key{ESC}, and key{Return}) 
-
-  begin scalar a; a := b + 10; WRITE A; 
-
-?- f;i end key{ESC}, key{Return} 
-
-  begin scalar b; b := a + 10; WRITE A end; 
-
-					  ^ 
-
-?- p 
-
- PROCEDURE DUMB (B); 
-
- begin scalar b; b := a + 10; WRITE A end; 
-
- -  -  -  -  -  -  -  -  -  - 
-
-  begin scalar b; b := a + 10; WRITE A end; 
-
-					  ^ 
-
-?- e 
-
-DUMB 
-
-7: >>dumb(17); 
-
-27 
-
-8: >> 
-
-

    -

    -

    -Note that REDUCE reparsed the procedure dumb and updated the -definition. -

    -

    -Since REDUCE divides the expression to be edited into lines at semicolons or -dollar sign terminators, some lines may occupy more than one line of screen -space. If the pointer is directly beneath the last line of text, it -refers to the top line of text. If there is a blank line between the -last line of text and the pointer, it refers to the second line -of text, and likewise for cases of greater than two lines of text. In other -words, the entire REDUCE statement up to the next terminator is printed, even -if it runs to several lines, then the pointer line is printed. -

    -

    -You can insert new statements which contain semicolons of their own into the -current line. They are run into the current line where you placed them -until you edit the statement again. REDUCE will understand the set of -statements if the syntax is correct. -

    -

    -If you leave out needed closing brackets when you exit the editor, a message -is printed allowing you to redo the edit (you can edit the previous line -number and return to where you were). If you leave out a closing -double-quotation mark, an error message is printed, and the editing must be -redone from the original version; the edited version has been destroyed. -Most syntax errors which you inadvertently leave in an edited statement are -caught as usual by the REDUCE parser, and you will be able to re-edit the -statement. -

    -

    -When the editor processes a previous statement for your editing, escape -characters are removed. Most special characters that you may use in -identifiers are printed in legal fashion, prefixed by the exclamation -point. Be sure to treat the special character and its escape as a pair in -your editing. The characters ( ) # ; ' ` are different. Since -they have special meaning in Lisp, they are double-escaped in the editor. -It is unwise to use these characters inside identifiers anyway, due to the -probability of confusion. -

    -

    -If you see a Lisp error message during editing, the edit has been aborted. -Enter a semicolon and you will see a new line prompt. -

    -

    -Since the editor has no dependence on any window system, it can be used if you -are running REDUCE without windows. -

    -

    -

    - - - -EDITDEF -INDEX

    - - - -EDITDEF _ _ _ _ _ _ _ _ _ _ _ _ command

    -

    - -

    -

    -The interactive editor -ed may be used to edit a user-defined -procedure that has not been compiled. -

    -syntax:

    -

    -

    -editdef(identifier) -

    -

    -

    -where identifier is the name of the procedure. When editdef -is invoked, the procedure definition will be displayed in editing mode, -and may then be edited and redefined on exiting from the editor using -standard -ed commands. -

    -

    - - - -Outmoded Operations -INDEX

    -Outmoded Operations

    -
  • ED command

    -

  • EDITDEF command

    -

    • r38 search index -
    - -
    absolute value: ABS -
    ABS operator: ABS -
    accuracy: ROOTACC -
    ACOSH operator: ACOSH -
    ACOS operator: ACOS -
    ACOTH operator: ACOTH -
    ACOT operator: ACOT -
    ACSCH operator: ACSCH -
    ACSC operator: ACSC -
    add columns operator: add columns -
    add rows operator: add rows -
    add to columns operator: add to columns -
    add to rows operator: add to rows -
    ADJPREC switch: ADJPREC -
    AGM FUNCTION operator: AGM FUNCTION -
    Airy Ai operator: Airy Ai -
    Airy Aiprime operator: Airy Aiprime -
    Airy Bi operator: Airy Bi -
    Airy Biprime operator: Airy Biprime -
    algebraic: EVAL MODE -
    ALGEBRAIC command: ALGEBRAIC -
    algebraic numbers: ARNUM -
    ALGINT package: ALGINT package -
    ALGINT switch: ALGINT -
    ALLBRANCH switch: ALLBRANCH -
    ALLFAC switch: ALLFAC -
    AND operator: AND -
    ANTISYMMETRIC declaration: ANTISYMMETRIC -
    APPEND operator: APPEND -
    APPLYSYM package: APPLYSYM -
    approximation: CONTINUED FRACTION -
    approximation: INTERPOL -
    approximation: Chebyshev fit -
    approximation: num fit -
    ARBCOMPLEX operator: ARBCOMPLEX -
    ARBINT operator: ARBINT -
    arbitrary value: ARBINT -
    arbitrary value: ARBCOMPLEX -
    ARBVARS switch: ARBVARS -
    arccosecant: ACSC -
    arccosecant: ACSCH -
    arccosecant: ASEC -
    arccosine: ACOS -
    arccotangent: ACOT -
    arcsine: ASIN -
    arctangent: ATAN -
    ARGLENGTH operator: ARGLENGTH -
    ARG operator: ARG -
    argument: ARGLENGTH -
    argument: LISTARGP -
    argument: LISTARGS -
    arithmetic: EQUATION -
    ARITHMETIC OPERATIONS introduction: ARITHMETIC OPERATI -ONS -
    ARNUM package: ARNUM -
    ARRAY declaration: ARRAY -
    ASECH operator: ASECH -
    ASEC operator: ASEC -
    ASINH operator: ASINH -
    ASIN operator: ASIN -
    assign: SET -
    assign: SETQ -
    assign operator: assign -
    ASSIST package: ASSIST -
    assumptions variable: assumptions -
    asterisk operator: asterisk -
    ATAN2 operator: ATAN2 -
    ATANH operator: ATANH -
    ATAN operator: ATAN -
    augment columns operator: augment columns -
    AVECTOR package: AVECTOR -
    Axes names concept: Axes names -
    BALANCED MOD switch: BALANCED MOD -
    band matrix operator: band matrix -
    BEGIN command: BEGIN -
    BERNOULLI operator: BERNOULLI -
    BERNOULLIP operator: BERNOULLIP -
    BESSELI operator: BESSELI -
    BESSELJ operator: BESSELJ -
    BESSELK operator: BESSELK -
    BESSELY operator: BESSELY -
    BETA operator: BETA -
    BFSPACE switch: BFSPACE -
    BINOMIAL operator: BINOMIAL -
    block command: block -
    block matrix operator: block matrix -
    boolean expressions: BOOLEAN -
    BOOLEAN package: BOOLEAN -
    boolean value concept: boolean value -
    bounds operator: bounds -
    Buchberger algorithm: Groebner bases -
    Buchberger algorithm: groebner -
    BYE command: BYE -
    CALI package: CALI -
    CAMAL package: CAMAL -
    CARD NO variable: CARD NO -
    caret operator: caret -
    Catalan's constant: Constants -
    CEILING operator: CEILING -
    celestial mechanics: CAMAL -
    CHANGEVR package: CHANGEVR -
    character: RAISE -
    char matrix operator: char matrix -
    char poly operator: char poly -
    Chebyshev fit concept: Chebyshev fit -
    ChebyshevT operator: ChebyshevT -
    ChebyshevU operator: ChebyshevU -
    chemical reaction: REACTEQN -
    Chi operator: Chi -
    cholesky operator: cholesky -
    CHOOSE operator: CHOOSE -
    Ci operator: Ci -
    CLEAR command: CLEAR -
    CLEARRULES command: CLEARRULES -
    Clebsch Gordan operator: Clebsch Gordan -
    close: SHUT -
    code generation: GENTRAN -
    code generation: SCOPE -
    coefficient: COEFF -
    coefficient: COEFFN -
    coefficient: LCOF -
    coeff matrix operator: coeff matrix -
    COEFFN operator: COEFFN -
    COEFF operator: COEFF -
    COFACTOR operator: COFACTOR -
    column dim operator: column dim -
    COMBINEEXPT switch: COMBINEEXPT -
    COMBINELOGS switch: COMBINELOGS -
    command: semicolon -
    command: dollar -
    command: percent -
    command: group -
    command: BEGIN -
    command: block -
    command: COMMENT -
    command: END -
    command: FOR -
    command: FOREACH -
    command: GOTO -
    command: IF -
    command: PROCEDURE -
    command: REPEAT -
    command: RETURN -
    command: SETMOD -
    command: BYE -
    command: CONT -
    command: DISPLAY -
    command: LOAD PACKAGE -
    command: PAUSE -
    command: QUIT -
    command: REDERR -
    command: RETRY -
    command: SAVEAS -
    command: SHOWTIME -
    command: WRITE -
    command: MKID -
    command: ALGEBRAIC -
    command: CLEAR -
    command: CLEARRULES -
    command: DEFINE -
    command: FORALL -
    command: LET -
    command: LISP -
    command: MATCH -
    command: OFF -
    command: ON -
    command: SYMBOLIC -
    command: WEIGHT -
    command: WHILE -
    command: WTLEVEL -
    command: IN -
    command: INPUT -
    command: OUT -
    command: SHUT -
    command: MASS -
    command: MSHELL -
    command: VECDIM -
    command: PLOT -
    command: PLOTRESET -
    command: ED -
    command: EDITDEF -
    COMMENT command: COMMENT -
    commutative: NONCOM -
    commutative algebra: CALI -
    commutative algebra: IDEALS -
    COMPACT package: COMPACT -
    companion operator: companion -
    compiler: COMP -
    complementary error function: erfc -
    complex: I -
    complex: ARG -
    complex: NORM -
    complex: CONJ -
    complex: IMPART -
    complex: REPART -
    complex: RATIONALIZE -
    complex: ROOTSCOMPLEX -
    complex: ROOTSREAL -
    COMPLEX switch: COMPLEX -
    composite structure: MAP -
    COMP switch: COMP -
    concept: boolean value -
    concept: false -
    concept: TRUE -
    concept: Ideal Parameters -
    concept: lex term order -
    concept: gradlex term order -
    concept: revgradlex term order -
    concept: gradlexgradlex term order -
    concept: gradlexrevgradlex term order -
    concept: lexgradlex term order -
    concept: lexrevgradlex term order -
    concept: weighted term order -
    concept: graded term order -
    concept: matrix term order -
    concept: Module -
    concept: numeric accuracy -
    concept: Chebyshev fit -
    concept: Constants -
    concept: Axes names -
    Confluent Hypergeometric function: KummerM -
    Confluent Hypergeometric function: KummerU -
    Confluent Hypergeometric function: WhittakerW -
    CONJ operator: CONJ -
    conjugate: CONJ -
    CONS operator: CONS -
    constant: E -
    constant: I -
    constant: INFINITY -
    constant: NIL -
    constant: PI -
    constant: T -
    Constants concept: Constants -
    CONT command: CONT -
    CONTINUED FRACTION operator: CONTINUED FRACTION -
    contour switch: contour -
    copy into operator: copy into -
    cosecant: CSC -
    COSH operator: COSH -
    cosine integral function: Ci -
    COS operator: COS -
    COTH operator: COTH -
    COT operator: COT -
    CRACK package: CRACK -
    CRAMER switch: CRAMER -
    CREF switch: CREF -
    cross product: AVECTOR -
    cross product: ORTHOVEC -
    cross reference: CREF -
    CSCH operator: CSCH -
    CSC operator: CSC -
    curl: ORTHOVEC -
    CVIT package: CVIT -
    dd groebner operator: dd groebner -
    declaration: ANTISYMMETRIC -
    declaration: ARRAY -
    declaration: DEPEND -
    declaration: EVEN -
    declaration: FACTOR declaration -
    declaration: INFIX -
    declaration: INTEGER -
    declaration: KORDER -
    declaration: LINEAR -
    declaration: LINELENGTH -
    declaration: LISTARGP -
    declaration: NODEPEND -
    declaration: NONCOM -
    declaration: NONZERO -
    declaration: ODD -
    declaration: OPERATOR -
    declaration: ORDER -
    declaration: PRECEDENCE -
    declaration: PRECISION -
    declaration: PRINT PRECISION -
    declaration: REAL -
    declaration: REMFAC -
    declaration: SCALAR -
    declaration: SCIENTIFIC NOTATION -
    declaration: SHARE -
    declaration: SYMMETRIC -
    declaration: TR -
    declaration: UNTR -
    declaration: VARNAME -
    declaration: MATRIX -
    declaration: INDEX -
    declaration: NOSPUR -
    declaration: REMIND -
    declaration: SPUR -
    declaration: VECTOR -
    DECOMPOSE operator: DECOMPOSE -
    decomposition: FIRST -
    decomposition: REST -
    decomposition: SECOND -
    decomposition: THIRD -
    decomposition: DECOMPOSE -
    decomposition: PART -
    decomposition: STRUCTR -
    DEFINE command: DEFINE -
    definite integration: DEFINT -
    DEFINT package: DEFINT -
    DEFN switch: DEFN -
    DEG2DMS operator: DEG2DMS -
    DEG2RAD operator: DEG2RAD -
    DEG operator: DEG -
    degree: HIGH POW -
    degree: LOW POW -
    degree: DEG -
    degrees: DEG2DMS -
    degrees: DEG2RAD -
    degrees: DMS2DEG -
    degrees: DMS2RAD -
    degrees: RAD2DEG -
    degrees: RAD2DMS -
    DEMO switch: DEMO -
    denominator: DEN -
    DEN operator: DEN -
    depend: NODEPEND -
    DEPEND declaration: DEPEND -
    dependency: DEPEND -
    derivative: DF -
    derivative: DFPRINT -
    derivative: NOARG -
    DESIR package: DESIR -
    determinant: DET -
    DET operator: DET -
    DF operator: DF -
    DFPART package: DFPART -
    DFPRINT switch: DFPRINT -
    diagonal operator: diagonal -
    DIFFERENCE operator: DIFFERENCE -
    differential calculus: EXCALC -
    differential equation: ODESOLVE -
    differential equation: CRACK -
    differential equation: DESIR -
    differential equation: SPDE -
    differential equations: APPLYSYM -
    differential form: EXCALC -
    dilogarithm function: DILOG -
    dilogarithm function: DILOG extended -
    DILOG extended operator: DILOG extended -
    DILOG operator: DILOG -
    Dirac algebra: CVIT -
    DISPLAY command: DISPLAY -
    distributive polynomials: Term order -
    distributive polynomials: gsort -
    distributive polynomials: gsplit -
    distributive polynomials: gspoly -
    div: ORTHOVEC -
    DIV switch: DIV -
    DMS2DEG operator: DMS2DEG -
    DMS2RAD operator: DMS2RAD -
    dollar command: dollar -
    dot operator: dot -
    dot product: AVECTOR -
    dot product: ORTHOVEC -
    DUMMY package: DUMMY -
    dummy variable: DUMMY -
    ECHO switch: ECHO -
    E constant: E -
    ED command: ED -
    EDITDEF command: EDITDEF -
    eigenvalue: MATEIGEN -
    Ei operator: Ei -
    EllipticE operator: EllipticE -
    EllipticF operator: EllipticF -
    EllipticK operator: EllipticK -
    EllipticKprime operator: EllipticKprime -
    EllipticTHETA operator: EllipticTHETA -
    else: IF -
    END command: END -
    EPS operator: EPS -
    equal: EQUATION -
    EQUAL operator: EQUAL -
    equalsign operator: equalsign -
    equation: EQUAL -
    equation: LHS -
    equation: RHS -
    equation: EVALLHSEQP -
    equation solving: SOLVE -
    equation solving: num solve -
    equation system: SOLVE -
    equation system: num solve -
    EQUATION type: EQUATION -
    erfc operator: erfc -
    ERF extended operator: ERF extended -
    erfi operator: erfi -
    ERF operator: ERF -
    ERRCONT switch: ERRCONT -
    error function: ERF -
    error function: ERF extended -
    error function: erfc -
    error handling: REDERR -
    error handling: ERRCONT -
    EULER operator: EULER -
    EULERP operator: EULERP -
    Euler's constant: Constants -
    Euler's constant: PSI -
    EVALLHSEQP switch: EVALLHSEQP -
    EVAL MODE variable: EVAL MODE -
    evaluation: ALGEBRAIC -
    EVEN declaration: EVEN -
    EVENP operator: EVENP -
    EXCALC package: EXCALC -
    EXPAND CASES operator: EXPAND CASES -
    EXPANDLOGS switch: EXPANDLOGS -
    exponential function: EXP -
    exponential integral function: Ei -
    exponent simplification: COMBINEEXPT -
    EXP operator: EXP -
    EXPREAD operator: EXPREAD -
    EXP switch: EXP switch -
    EXPT operator: EXPT -
    extend operator: extend -
    exterior calculus: EXCALC -
    EZGCD switch: EZGCD -
    factor: REMFAC -
    FACTOR declaration: FACTOR declaration -
    FACTORIAL operator: FACTORIAL -
    factorize: IFACTOR -
    factorize: LIMITEDFACTORS -
    factorize: OVERVIEW -
    factorize: TRALLFAC -
    factorize: TRFAC -
    FACTORIZE operator: FACTORIZE -
    FACTOR switch: FACTOR -
    FAILHARD switch: FAILHARD -
    false: NIL -
    false: TRUE -
    false concept: false -
    fast la switch: fast la -
    Faugere-Gianni-Lazard-Mora algorithm: Groebner bases - -
    FIDE package: FIDE -
    find companion operator: find companion -
    FIRST operator: FIRST -
    firstroot: Roots Package -
    FIX operator: FIX -
    FIXP operator: FIXP -
    Fletcher Reeves: num min -
    floating point: PRECISION -
    floating point: PRINT PRECISION -
    floating point: SCIENTIFIC NOTATION -
    floating point: BFSPACE -
    floating point: ROUNDALL -
    floating point: ROUNDED -
    FLOOR operator: FLOOR -
    FORALL command: FORALL -
    FOR command: FOR -
    FOREACH command: FOREACH -
    FORTRAN: CARD NO -
    FORTRAN: FORT WIDTH -
    FORTRAN: FORT -
    FORTRAN: FORTUPPER -
    FORTRAN: GENTRAN -
    FORT switch: FORT -
    FORTUPPER switch: FORTUPPER -
    FORT WIDTH variable: FORT WIDTH -
    Fourier series: CAMAL -
    FPS package: FPS -
    FREEOF operator: FREEOF -
    Free Variable type: Free Variable -
    Fresnel C operator: Fresnel C -
    Fresnel S operator: Fresnel S -
    Frobenius operator: Frobenius -
    FULLPREC switch: FULLPREC -
    FULLROOTS switch: FULLROOTS -
    gamma: FACTORIAL -
    GAMMA operator: GAMMA -
    GCD operator: GCD -
    GCD switch: GCD switch -
    GC switch: GC -
    gdimension operator: gdimension -
    GegenbauerP operator: GegenbauerP -
    generalized hypergeometric function: HYPERGEOMETRIC - -
    GENTRAN package: GENTRAN -
    GEQ operator: GEQ -
    geqsign operator: geqsign -
    get columns operator: get columns -
    getroot: Roots Package -
    get rows operator: get rows -
    gindependent sets operator: gindependent sets -
    glexconvert operator: glexconvert -
    gltbasis switch: gltbasis -
    gltb variable: gltb -
    glterms variable: glterms -
    gmodule variable: gmodule -
    GNUPLOT and REDUCE introduction: GNUPLOT and REDUCE - -
    Golden Ratio: Constants -
    G operator: G -
    Gosper algorithm: PROD -
    Gosper algorithm: SUM -
    GOTO command: GOTO -
    grad: ORTHOVEC -
    graded term order concept: graded term order -
    gradlexgradlex term order concept: gradlexgradlex term - order -
    gradlexrevgradlex term order concept: gradlexrevgradle -x term order -
    gradlex term order concept: gradlex term order -
    gram schmidt operator: gram schmidt -
    graphics: PLOT -
    greater operator: greater -
    GREATERP operator: GREATERP -
    greatest common divisor: GCD -
    greatest common divisor: EZGCD -
    greatest common divisor: GCD switch -
    greduce operator: greduce -
    groebfullreduction switch: groebfullreduction -
    groebmonfac variable: groebmonfac -
    groebner: gdimension -
    groebner: gindependent sets -
    Groebner: CALI -
    Groebner: IDEALS -
    Groebner bases introduction: Groebner bases -
    Groebner basis: XIDEAL -
    groebnerf operator: groebnerf -
    groebner operator: groebner -
    groebnert operator: groebnert -
    groebner walk operator: groebner walk -
    groebopt switch: groebopt -
    groebprereduce switch: groebprereduce -
    groebprotfile variable: groebprotfile -
    groebprot switch: groebprot -
    groebresmax variable: groebresmax -
    groebrestriction variable: groebrestriction -
    groebstat switch: groebstat -
    group command: group -
    gsort operator: gsort -
    gsplit operator: gsplit -
    gspoly operator: gspoly -
    gvarslast variable: gvarslast -
    gvars operator: gvars -
    gzerodim? operator: gzerodim -
    HANKEL1 operator: HANKEL1 -
    HANKEL2 operator: HANKEL2 -
    Heaviside operator: Heaviside -
    HE-dot operator: HE dot -
    HEPHYS introduction: HEPHYS -
    HermiteP operator: HermiteP -
    hermitian tp operator: hermitian tp -
    hessian operator: hessian -
    hidden3d switch: hidden3d -
    high energy physics: XCOLOR -
    HIGH POW variable: HIGH POW -
    hilbert operator: hilbert -
    hilbertpolynomial operator: hilbertpolynomial -
    history: DISPLAY -
    Hollmann algorithm: Groebner bases -
    Hollmann algorithm: hilbertpolynomial -
    HORNER switch: HORNER -
    hyperbolic arccosecant: ASECH -
    hyperbolic arccosine: ACOSH -
    hyperbolic arcsine: ASINH -
    hyperbolic arctangent: ATANH -
    hyperbolic cosecan: CSCH -
    hyperbolic cosine: COSH -
    hyperbolic cosine integral function: Chi -
    hyperbolic cotangent: ACOTH -
    hyperbolic cotangent: COTH -
    hyperbolic secant: SECH -
    hyperbolic sine: SINH -
    hyperbolic sine integral function: Shi -
    hyperbolic tangent: TANH -
    hypergeometric function: HYPERGEOMETRIC -
    HYPERGEOMETRIC operator: HYPERGEOMETRIC -
    HYPOT operator: HYPOT -
    I constant: I -
    ideal dimension: gdimension -
    ideal dimension: gindependent sets -
    Ideal Parameters concept: Ideal Parameters -
    idealquotient operator: idealquotient -
    IDEALS package: IDEALS -
    ideal variables: gindependent sets -
    ideal variables: glexconvert -
    identifier: MKID -
    IDENTIFIER type: IDENTIFIER -
    IFACTOR switch: IFACTOR -
    IF command: IF -
    imaginary part: IMPART -
    IMPART operator: IMPART -
    IN command: IN -
    INDEX declaration: INDEX -
    INEQ package: INEQ -
    inequality: INEQ -
    INFINITY constant: INFINITY -
    INFIX declaration: INFIX -
    initial value problem: num odesolve -
    input: ADJPREC -
    input: EXPREAD -
    input: IN -
    input: RAISE -
    INPUT command: INPUT -
    integer: CEILING -
    integer: FIX -
    integer: FIXP -
    integer: FLOOR -
    integer: ROUND -
    integer: IFACTOR -
    integer: PERIOD -
    INTEGER declaration: INTEGER -
    integral function: Si -
    integral function: Shi -
    integral function: s i -
    integral function: Chi -
    integration: INT -
    integration: ALGINT -
    integration: FAILHARD -
    integration: NOLNR -
    integration: TRINT -
    integration: num int -
    integration of square roots: ALGINT package -
    interactive: DISPLAY -
    interactive: PAUSE -
    interactive: RETRY -
    interactive: WS -
    interactive: INPUT -
    interactive: DEMO -
    interactive: INT switch -
    interpolation: INTERPOL -
    interpolation: MKPOLY -
    INTERPOL operator: INTERPOL -
    Interval type: Interval -
    INT operator: INT -
    introduction: ARITHMETIC OPERATIONS -
    introduction: SWITCHES -
    introduction: Groebner bases -
    introduction: Term order -
    introduction: HEPHYS -
    introduction: Numeric Package -
    introduction: Roots Package -
    introduction: Special Function Package -
    introduction: TAYLOR introduction -
    introduction: GNUPLOT and REDUCE -
    introduction: Linear Algebra package -
    introduction: Miscellaneous Packages -
    INTSTR switch: INTSTR -
    INT switch: INT switch -
    INVBASE package: INVBASE -
    isolater: Roots Package -
    JacobiAMPLITUDE operator: JacobiAMPLITUDE -
    Jacobian matrix: num solve -
    jacobian operator: jacobian -
    JacobiCD operator: JacobiCD -
    JacobiCN operator: JacobiCN -
    JacobiCS operator: JacobiCS -
    JacobiDC operator: JacobiDC -
    JacobiDN operator: JacobiDN -
    JacobiDS operator: JacobiDS -
    JacobiNC operator: JacobiNC -
    JacobiND operator: JacobiND -
    JacobiNS operator: JacobiNS -
    JacobiP operator: JacobiP -
    JacobiSC operator: JacobiSC -
    JacobiSD operator: JacobiSD -
    JacobiSN operator: JacobiSN -
    JacobiZETA operator: JacobiZETA -
    jordan block operator: jordan block -
    Jordan operator: Jordan -
    Jordansymbolic operator: Jordansymbolic -
    kernel order: KORDER -
    KERNEL type: KERNEL -
    Khinchin's constant: Constants -
    KORDER declaration: KORDER -
    Kredel-Weispfenning algorithm: Groebner bases -
    Kredel-Weispfenning algorithm: gindependent sets -
    KummerM operator: KummerM -
    KummerU operator: KummerU -
    LaguerreP operator: LaguerreP -
    Lambert W function operator: Lambert W function -
    LANDENTRANS operator: LANDENTRANS -
    LAPLACE package: LAPLACE -
    Laplacian: ORTHOVEC -
    Laurent-Puiseux series: FPS -
    LCM switch: LCM -
    LCOF operator: LCOF -
    leading power: LPOWER -
    leading term: LTERM -
    least squares: num fit -
    left-hand side: LHS -
    LegendreP operator: LegendreP -
    LENGTH operator: LENGTH -
    LEQ operator: LEQ -
    leqsign operator: leqsign -
    less operator: less -
    LESSP operator: LESSP -
    LESSSPACE switch: LESSSPACE -
    LET command: LET -
    lexgradlex term order concept: lexgradlex term order - -
    lexrevgradlex term order concept: lexrevgradlex term o -rder -
    lex term order concept: lex term order -
    l'Hopital's rule: LIMIT -
    LHS operator: LHS -
    LIE package: LIE -
    Lie symmetry: SPDE -
    LIMITEDFACTORS switch: LIMITEDFACTORS -
    LIMIT operator: LIMIT -
    Linear Algebra package introduction: Linear Algebra pa -ckage -
    LINEAR declaration: LINEAR -
    linear system: CRAMER -
    LINELENGTH declaration: LINELENGTH -
    lisp: DEFN -
    lisp: RLISP88 -
    LISP command: LISP -
    list: dot -
    list: FIRST -
    list: REST -
    list: REVERSE -
    list: SECOND -
    list: THIRD -
    list: MEMBER -
    list: APPEND -
    list: LENGTH -
    list: SELECT -
    LISTARGP declaration: LISTARGP -
    LISTARGS switch: LISTARGS -
    LIST operator: LIST -
    LIST switch: LIST switch -
    LN operator: LN -
    LOAD PACKAGE command: LOAD PACKAGE -
    logarithm: LN -
    logarithm: LOG -
    logarithm: LOGB -
    logarithm: COMBINELOGS -
    logarithm: EXPANDLOGS -
    LOGB operator: LOGB -
    LOG operator: LOG -
    loop: FOR -
    loop: FOREACH -
    loop: REPEAT -
    loop: WHILE -
    LOW POW variable: LOW POW -
    LPOWER operator: LPOWER -
    LTERM operator: LTERM -
    lu decom operator: lu decom -
    main variable: MAINVAR -
    MAINVAR operator: MAINVAR -
    make identity operator: make identity -
    map: SELECT -
    MAP operator: MAP -
    MASS command: MASS -
    MATCH command: MATCH -
    MATEIGEN operator: MATEIGEN -
    MAT operator: MAT -
    matrix: CRAMER -
    matrix: COFACTOR -
    matrix: DET -
    matrix: MAT -
    matrix: MATEIGEN -
    matrix: NULLSPACE -
    matrix: RANK -
    matrix: TP -
    matrix: TRACE -
    matrix augment operator: matrix augment -
    MATRIX declaration: MATRIX -
    matrixp operator: matrixp -
    matrix stack operator: matrix stack -
    matrix term order concept: matrix term order -
    maximum: MAX -
    MAX operator: MAX -
    MCD switch: MCD -
    MeijerG operator: MeijerG -
    MEMBER operator: MEMBER -
    memory: RECLAIM -
    memory: GC -
    minimum: MIN -
    minimum: num min -
    MIN operator: MIN -
    minor operator: minor -
    MINUS operator: MINUS -
    minussign operator: minussign -
    Miscellaneous Packages introduction: Miscellaneous Pac -kages -
    MKID command: MKID -
    MKPOLY operator: MKPOLY -
    MODSR package: MODSR -
    modular: SETMOD -
    modular: BALANCED MOD -
    modular polynomial: MODSR -
    MODULAR switch: MODULAR -
    Module concept: Module -
    MSG switch: MSG -
    MSHELL command: MSHELL -
    mult columns operator: mult columns -
    MULTIPLICITIES switch: MULTIPLICITIES -
    mult rows operator: mult rows -
    NAT switch: NAT -
    NCPOLY package: NCPOLY -
    NEARESTROOT operator: NEARESTROOT -
    NEQ operator: NEQ -
    NERO switch: NERO -
    Newton iteration: num solve -
    NEXTPRIME operator: NEXTPRIME -
    NIL constant: NIL -
    NOARG switch: NOARG -
    NOCONVERT switch: NOCONVERT -
    NODEPEND declaration: NODEPEND -
    NOLNR switch: NOLNR -
    NONCOM declaration: NONCOM -
    non commutative: NONCOM -
    non-commutativity: NCPOLY -
    NONZERO declaration: NONZERO -
    NORM operator: NORM -
    NOSPLIT switch: NOSPLIT -
    NOSPUR declaration: NOSPUR -
    NOT operator: NOT -
    NPRIMITIVE operator: NPRIMITIVE -
    NULLSPACE operator: NULLSPACE -
    NUMBERP operator: NUMBERP -
    numerator: NUM -
    numeric accuracy concept: numeric accuracy -
    Numeric Package introduction: Numeric Package -
    num fit operator: num fit -
    num int operator: num int -
    num min operator: num min -
    num odesolve operator: num odesolve -
    NUM operator: NUM -
    num solve operator: num solve -
    NUMVAL switch: NUMVAL -
    ODD declaration: ODD -
    ODE: num odesolve -
    ODESOLVE operator: ODESOLVE -
    OFF command: OFF -
    ON command: ON -
    ONE OF type: ONE OF -
    open: OUT -
    operator: dot -
    operator: assign -
    operator: equalsign -
    operator: replace -
    operator: plussign -
    operator: minussign -
    operator: asterisk -
    operator: slash -
    operator: power -
    operator: caret -
    operator: geqsign -
    operator: greater -
    operator: leqsign -
    operator: less -
    operator: tilde -
    operator: AND -
    operator: CONS -
    operator: FIRST -
    operator: GEQ -
    operator: GREATERP -
    operator: LIST -
    operator: OR -
    operator: REST -
    operator: REVERSE -
    operator: SECOND -
    operator: SET -
    operator: SETQ -
    operator: THIRD -
    operator: WHEN -
    operator: ABS -
    operator: ARG -
    operator: CEILING -
    operator: CHOOSE -
    operator: DEG2DMS -
    operator: DEG2RAD -
    operator: DIFFERENCE -
    operator: DILOG -
    operator: DMS2DEG -
    operator: DMS2RAD -
    operator: FACTORIAL -
    operator: FIX -
    operator: FIXP -
    operator: FLOOR -
    operator: EXPT -
    operator: GCD -
    operator: LN -
    operator: LOG -
    operator: LOGB -
    operator: MAX -
    operator: MIN -
    operator: MINUS -
    operator: NEXTPRIME -
    operator: NORM -
    operator: PERM -
    operator: PLUS -
    operator: QUOTIENT -
    operator: RAD2DEG -
    operator: RAD2DMS -
    operator: RECIP -
    operator: REMAINDER -
    operator: ROUND -
    operator: SIGN -
    operator: SQRT -
    operator: TIMES -
    operator: EQUAL -
    operator: EVENP -
    operator: FREEOF -
    operator: LEQ -
    operator: LESSP -
    operator: MEMBER -
    operator: NEQ -
    operator: NOT -
    operator: NUMBERP -
    operator: ORDP -
    operator: PRIMEP -
    operator: RECLAIM -
    operator: APPEND -
    operator: ARBINT -
    operator: ARBCOMPLEX -
    operator: ARGLENGTH -
    operator: COEFF -
    operator: COEFFN -
    operator: CONJ -
    operator: CONTINUED FRACTION -
    operator: DECOMPOSE -
    operator: DEG -
    operator: DEN -
    operator: DF -
    operator: EXPAND CASES -
    operator: EXPREAD -
    operator: FACTORIZE -
    operator: HYPOT -
    operator: IMPART -
    operator: INT -
    operator: INTERPOL -
    operator: LCOF -
    operator: LENGTH -
    operator: LHS -
    operator: LIMIT -
    operator: LPOWER -
    operator: LTERM -
    operator: MAINVAR -
    operator: MAP -
    operator: NPRIMITIVE -
    operator: NUM -
    operator: ODESOLVE -
    operator: PART -
    operator: PF -
    operator: PROD -
    operator: REDUCT -
    operator: REPART -
    operator: RESULTANT -
    operator: RHS -
    operator: ROOT OF -
    operator: SELECT -
    operator: SHOWRULES -
    operator: SOLVE -
    operator: SORT -
    operator: STRUCTR -
    operator: SUB -
    operator: SUM -
    operator: WS -
    operator: INFIX -
    operator: LINEAR -
    operator: NONCOM -
    operator: NONZERO -
    operator: ODD -
    operator: PRECEDENCE -
    operator: SYMMETRIC -
    operator: WHERE -
    operator: ACOS -
    operator: ACOSH -
    operator: ACOT -
    operator: ACOTH -
    operator: ACSC -
    operator: ACSCH -
    operator: ASEC -
    operator: ASECH -
    operator: ASIN -
    operator: ASINH -
    operator: ATAN -
    operator: ATANH -
    operator: ATAN2 -
    operator: COS -
    operator: COSH -
    operator: COT -
    operator: COTH -
    operator: CSC -
    operator: CSCH -
    operator: ERF -
    operator: EXP -
    operator: SEC -
    operator: SECH -
    operator: SIN -
    operator: SINH -
    operator: TAN -
    operator: TANH -
    operator: LISTARGS -
    operator: COFACTOR -
    operator: DET -
    operator: MAT -
    operator: MATEIGEN -
    operator: NULLSPACE -
    operator: RANK -
    operator: TP -
    operator: TRACE -
    operator: torder -
    operator: torder compile -
    operator: gvars -
    operator: groebner -
    operator: groebner walk -
    operator: gzerodim -
    operator: gdimension -
    operator: gindependent sets -
    operator: dd groebner -
    operator: glexconvert -
    operator: greduce -
    operator: preduce -
    operator: idealquotient -
    operator: hilbertpolynomial -
    operator: saturation -
    operator: groebnerf -
    operator: groebnert -
    operator: preducet -
    operator: gsort -
    operator: gsplit -
    operator: gspoly -
    operator: HE dot -
    operator: EPS -
    operator: G -
    operator: num min -
    operator: num solve -
    operator: num int -
    operator: num odesolve -
    operator: bounds -
    operator: num fit -
    operator: MKPOLY -
    operator: NEARESTROOT -
    operator: REALROOTS -
    operator: ROOTACC -
    operator: ROOTS -
    operator: ROOT VAL -
    operator: BERNOULLI -
    operator: BERNOULLIP -
    operator: EULER -
    operator: EULERP -
    operator: ZETA -
    operator: BESSELJ -
    operator: BESSELY -
    operator: HANKEL1 -
    operator: HANKEL2 -
    operator: BESSELI -
    operator: BESSELK -
    operator: StruveH -
    operator: StruveL -
    operator: KummerM -
    operator: KummerU -
    operator: WhittakerW -
    operator: Airy Ai -
    operator: Airy Bi -
    operator: Airy Aiprime -
    operator: Airy Biprime -
    operator: JacobiSN -
    operator: JacobiCN -
    operator: JacobiDN -
    operator: JacobiCD -
    operator: JacobiSD -
    operator: JacobiND -
    operator: JacobiDC -
    operator: JacobiNC -
    operator: JacobiSC -
    operator: JacobiNS -
    operator: JacobiDS -
    operator: JacobiCS -
    operator: JacobiAMPLITUDE -
    operator: AGM FUNCTION -
    operator: LANDENTRANS -
    operator: EllipticF -
    operator: EllipticK -
    operator: EllipticKprime -
    operator: EllipticE -
    operator: EllipticTHETA -
    operator: JacobiZETA -
    operator: POCHHAMMER -
    operator: GAMMA -
    operator: BETA -
    operator: PSI -
    operator: POLYGAMMA -
    operator: DILOG extended -
    operator: Lambert W function -
    operator: ChebyshevT -
    operator: ChebyshevU -
    operator: HermiteP -
    operator: LaguerreP -
    operator: LegendreP -
    operator: JacobiP -
    operator: GegenbauerP -
    operator: SolidHarmonicY -
    operator: SphericalHarmonicY -
    operator: Si -
    operator: Shi -
    operator: s i -
    operator: Ci -
    operator: Chi -
    operator: ERF extended -
    operator: erfc -
    operator: Ei -
    operator: Fresnel C -
    operator: Fresnel S -
    operator: BINOMIAL -
    operator: STIRLING1 -
    operator: STIRLING2 -
    operator: ThreejSymbol -
    operator: Clebsch Gordan -
    operator: SixjSymbol -
    operator: HYPERGEOMETRIC -
    operator: MeijerG -
    operator: Heaviside -
    operator: erfi -
    operator: taylor -
    operator: taylorcombine -
    operator: taylororiginal -
    operator: taylorrevert -
    operator: taylorseriesp -
    operator: taylortemplate -
    operator: taylortostandard -
    operator: add columns -
    operator: add rows -
    operator: add to columns -
    operator: add to rows -
    operator: augment columns -
    operator: band matrix -
    operator: block matrix -
    operator: char matrix -
    operator: char poly -
    operator: cholesky -
    operator: coeff matrix -
    operator: column dim -
    operator: companion -
    operator: copy into -
    operator: diagonal -
    operator: extend -
    operator: find companion -
    operator: get columns -
    operator: get rows -
    operator: gram schmidt -
    operator: hermitian tp -
    operator: hessian -
    operator: hilbert -
    operator: jacobian -
    operator: jordan block -
    operator: lu decom -
    operator: make identity -
    operator: matrix augment -
    operator: matrixp -
    operator: matrix stack -
    operator: minor -
    operator: mult columns -
    operator: mult rows -
    operator: pivot -
    operator: pseudo inverse -
    operator: random matrix -
    operator: remove columns -
    operator: remove rows -
    operator: row dim -
    operator: rows pivot -
    operator: simplex -
    operator: squarep -
    operator: stack rows -
    operator: sub matrix -
    operator: svd -
    operator: swap columns -
    operator: swap entries -
    operator: swap rows -
    operator: symmetricp -
    operator: toeplitz -
    operator: vandermonde -
    operator: Smithex -
    operator: Smithex int -
    operator: Frobenius -
    operator: Ratjordan -
    operator: Jordansymbolic -
    operator: Jordan -
    OPERATOR declaration: OPERATOR -
    optimization: SCOPE -
    Optional Free Variable type: Optional Free Variable - -
    order: ORDP -
    order: KORDER -
    ORDER declaration: ORDER -
    ORDP operator: ORDP -
    OR operator: OR -
    ORTHOVEC package: ORTHOVEC -
    OUT command: OUT -
    Outmoded Operations: EDITDEF -
    output: CARD NO -
    output: FORT WIDTH -
    output: WRITE -
    output: SHOWRULES -
    output: FACTOR declaration -
    output: LINELENGTH -
    output: ORDER -
    output: PRINT PRECISION -
    output: REMFAC -
    output: SCIENTIFIC NOTATION -
    output: OUT -
    output: SHUT -
    output: ALLFAC -
    output: BFSPACE -
    output: DEMO -
    output: DFPRINT -
    output: DIV -
    output: ECHO -
    output: FACTOR -
    output: HORNER -
    output: INTSTR -
    output: LESSSPACE -
    output: MSG -
    output: NAT -
    output: NERO -
    output: NOARG -
    output: NOSPLIT -
    output: PERIOD -
    output: PRET -
    output: PRI -
    output: RAT -
    output: RATPRI -
    output: REVPRI -
    output: RLFI -
    output: TRI -
    OUTPUT switch: OUTPUT -
    OVERVIEW switch: OVERVIEW -
    package: LOAD PACKAGE -
    package: ALGINT package -
    package: APPLYSYM -
    package: ARNUM -
    package: ASSIST -
    package: AVECTOR -
    package: BOOLEAN -
    package: CALI -
    package: CAMAL -
    package: CHANGEVR -
    package: COMPACT -
    package: CRACK -
    package: CVIT -
    package: DEFINT -
    package: DESIR -
    package: DFPART -
    package: DUMMY -
    package: EXCALC -
    package: FPS -
    package: FIDE -
    package: GENTRAN -
    package: IDEALS -
    package: INEQ -
    package: INVBASE -
    package: LAPLACE -
    package: LIE -
    package: MODSR -
    package: NCPOLY -
    package: ORTHOVEC -
    package: PHYSOP -
    package: PM -
    package: RANDPOLY -
    package: REACTEQN -
    package: RESET -
    package: RESIDUE -
    package: RLFI -
    package: SCOPE -
    package: SETS -
    package: SPDE -
    package: SYMMETRY -
    package: TPS -
    package: TRI -
    package: TRIGSIMP -
    package: XCOLOR -
    package: XIDEAL -
    package: WU -
    package: ZEILBERG -
    package: ZTRANS -
    partial derivative: DF -
    partial derivative: DFPART -
    partial fraction: PF -
    PART operator: PART -
    pattern matching: PM -
    PAUSE command: PAUSE -
    percent command: percent -
    PERIOD switch: PERIOD -
    PERM operator: PERM -
    permutation: PERM -
    PF operator: PF -
    PHYSOP package: PHYSOP -
    PI constant: PI -
    pivot operator: pivot -
    plot: Pointset -
    plot: title -
    plot: xlabel -
    plot: ylabel -
    plot: zlabel -
    plot: terminal -
    plot: size -
    plot: view -
    plot: contour -
    plot: surface -
    plot: hidden3d -
    plot: SHOW GRID -
    plot: TRPLOT -
    PLOT command: PLOT -
    PLOTKEEP switch: PLOTKEEP -
    PLOTREFINE switch: PLOTREFINE -
    PLOTRESET command: PLOTRESET -
    plot xmesh variable: plot xmesh -
    plot ymesh variable: plot ymesh -
    PLUS operator: PLUS -
    plussign operator: plussign -
    PM package: PM -
    POCHHAMMER operator: POCHHAMMER -
    Pointset type: Pointset -
    polar angle: ARG -
    POLYGAMMA operator: POLYGAMMA -
    polynomial: HIGH POW -
    polynomial: LOW POW -
    polynomial: ROOT MULTIPLICITIES -
    polynomial: GCD -
    polynomial: REMAINDER -
    polynomial: DECOMPOSE -
    polynomial: DEG -
    polynomial: FACTORIZE -
    polynomial: INTERPOL -
    polynomial: LCOF -
    polynomial: LPOWER -
    polynomial: LTERM -
    polynomial: MAINVAR -
    polynomial: NPRIMITIVE -
    polynomial: REDUCT -
    polynomial: RESULTANT -
    polynomial: EZGCD -
    polynomial: FULLROOTS -
    polynomial: HORNER -
    polynomial: LIMITEDFACTORS -
    polynomial: RATARG -
    polynomial: RATIONAL -
    polynomial: TRIGFORM -
    polynomial: Ideal Parameters -
    polynomial: Roots Package -
    polynomial: MKPOLY -
    polynomial: ROOTS -
    polynomial: ROOT VAL -
    polynomial: CALI -
    polynomial: IDEALS -
    polynomial: WU -
    power operator: power -
    power series: FPS -
    power series: TPS -
    PRECEDENCE declaration: PRECEDENCE -
    PRECISE switch: PRECISE -
    precision: ADJPREC -
    precision: FULLPREC -
    PRECISION declaration: PRECISION -
    preduce operator: preduce -
    preducet operator: preducet -
    PRET switch: PRET -
    prime number: NEXTPRIME -
    prime number: PRIMEP -
    PRIMEP operator: PRIMEP -
    primitive part: NPRIMITIVE -
    PRINT PRECISION declaration: PRINT PRECISION -
    PRI switch: PRI -
    PROCEDURE command: PROCEDURE -
    PROD operator: PROD -
    product: PROD -
    pseudo inverse operator: pseudo inverse -
    PSI operator: PSI -
    QUIT command: QUIT -
    QUOTIENT operator: QUOTIENT -
    RAD2DEG operator: RAD2DEG -
    RAD2DMS operator: RAD2DMS -
    radians: DEG2DMS -
    radians: DEG2RAD -
    radians: DMS2DEG -
    radians: DMS2RAD -
    radians: RAD2DEG -
    radians: RAD2DMS -
    RAISE switch: RAISE -
    random matrix operator: random matrix -
    random polynomial: RANDPOLY -
    RANDPOLY package: RANDPOLY -
    RANK operator: RANK -
    RATARG switch: RATARG -
    rational expression: DEN -
    rational expression: NUM -
    rational expression: PF -
    rational expression: GCD switch -
    rational expression: LCM -
    rational expression: MCD -
    rational expression: RATARG -
    rational expression: RATIONAL -
    rational expression: RATIONALIZE -
    rational expression: RATPRI -
    rational expression: ROUNDALL -
    RATIONALIZE switch: RATIONALIZE -
    rational numbers: CONTINUED FRACTION -
    RATIONAL switch: RATIONAL -
    Ratjordan operator: Ratjordan -
    RATPRI switch: RATPRI -
    RAT switch: RAT -
    REACTEQN package: REACTEQN -
    REAL declaration: REAL -
    real part: REPART -
    REALROOTS operator: REALROOTS -
    RECIP operator: RECIP -
    RECLAIM operator: RECLAIM -
    REDERR command: REDERR -
    REDUCT operator: REDUCT -
    reductum: REDUCT -
    REMAINDER operator: REMAINDER -
    REMFAC declaration: REMFAC -
    REMIND declaration: REMIND -
    remove columns operator: remove columns -
    remove rows operator: remove rows -
    REPART operator: REPART -
    REPEAT command: REPEAT -
    replace operator: replace -
    requirements variable: requirements -
    RESET package: RESET -
    RESIDUE package: RESIDUE -
    REST operator: REST -
    RESULTANT operator: RESULTANT -
    RETRY command: RETRY -
    RETURN command: RETURN -
    REVERSE operator: REVERSE -
    revgradlex term order concept: revgradlex term order - -
    REVPRI switch: REVPRI -
    RHS operator: RHS -
    right-hand side: RHS -
    RLFI package: RLFI -
    RLISP88 switch: RLISP88 -
    rlrootno: Roots Package -
    root: SOLVE -
    root: num solve -
    ROOTACC operator: ROOTACC -
    ROOT MULTIPLICITIES variable: ROOT MULTIPLICITIES -
    ROOT OF operator: ROOT OF -
    roots: ROOT OF -
    roots: MKPOLY -
    roots: NEARESTROOT -
    roots: REALROOTS -
    roots: ROOTACC -
    roots: ROOT VAL -
    rootsat-prec: Roots Package -
    ROOTSCOMPLEX variable: ROOTSCOMPLEX -
    ROOTS operator: ROOTS -
    Roots Package introduction: Roots Package -
    ROOTSREAL variable: ROOTSREAL -
    rootval: Roots Package -
    ROOT VAL operator: ROOT VAL -
    ROUNDALL switch: ROUNDALL -
    ROUNDBF switch: ROUNDBF -
    rounded: PRECISION -
    rounded: PRINT PRECISION -
    rounded: SCIENTIFIC NOTATION -
    rounded: FULLPREC -
    rounded: NUMVAL -
    rounded: ROUNDALL -
    ROUNDED switch: ROUNDED -
    ROUND operator: ROUND -
    row dim operator: row dim -
    rows pivot operator: rows pivot -
    rule: WHEN -
    rule: SHOWRULES -
    rule: CLEARRULES -
    rule: LET -
    rule list: RULE -
    RULE type: RULE -
    Runge-Kutta: num odesolve -
    saturation operator: saturation -
    SAVEAS command: SAVEAS -
    SAVESTRUCTR switch: SAVESTRUCTR -
    SCALAR declaration: SCALAR -
    SCIENTIFIC NOTATION declaration: SCIENTIFIC NOTATION - -
    SCOPE package: SCOPE -
    SECH operator: SECH -
    SECOND operator: SECOND -
    SEC operator: SEC -
    SELECT operator: SELECT -
    semicolon command: semicolon -
    SETMOD command: SETMOD -
    SET operator: SET -
    SETQ operator: SETQ -
    SETS package: SETS -
    SHARE declaration: SHARE -
    Shi operator: Shi -
    SHOW GRID switch: SHOW GRID -
    SHOWRULES operator: SHOWRULES -
    SHOWTIME command: SHOWTIME -
    SHUT command: SHUT -
    SIGN operator: SIGN -
    simplex operator: simplex -
    simplification: EXP switch -
    simplification: PRECISE -
    simplification: RATIONALIZE -
    simplification: COMPACT -
    simplification: TRIGSIMP -
    sine: SIN -
    Sine integral function: Si -
    sine integral function: s i -
    singular value decomposition: svd -
    SINH operator: SINH -
    SIN operator: SIN -
    Si operator: Si -
    s i operator: s i -
    SixjSymbol operator: SixjSymbol -
    size variable: size -
    slash operator: slash -
    Smithex int operator: Smithex int -
    Smithex operator: Smithex -
    Solid harmonic polynomials: SolidHarmonicY -
    SolidHarmonicY operator: SolidHarmonicY -
    solve: assumptions -
    solve: requirements -
    solve: ROOT MULTIPLICITIES -
    solve: EXPAND CASES -
    solve: ODESOLVE -
    solve: ROOT OF -
    solve: ARBVARS -
    solve: CRAMER -
    solve: FULLROOTS -
    solve: MULTIPLICITIES -
    solve: TRIGFORM -
    solve: TRNONLNR -
    solve: VAROPT -
    solve: NEARESTROOT -
    solve: REALROOTS -
    solve: ROOTS -
    solve: ROOT VAL -
    SOLVE operator: SOLVE -
    SOLVESINGULAR switch: SOLVESINGULAR -
    sorting: SORT -
    SORT operator: SORT -
    SPDE package: SPDE -
    Special Function Package introduction: Special Functio -n Package -
    Spence's Integral: DILOG extended -
    Spherical harmonic polynomials: SphericalHarmonicY -
    SphericalHarmonicY operator: SphericalHarmonicY -
    SPUR declaration: SPUR -
    SQRT operator: SQRT -
    squarep operator: squarep -
    square root: SQRT -
    square root: PRECISE -
    stack rows operator: stack rows -
    steepest descent: num min -
    STIRLING1 operator: STIRLING1 -
    STIRLING2 operator: STIRLING2 -
    STRING type: STRING -
    STRUCTR operator: STRUCTR -
    STRUCTR OPERATOR: SAVESTRUCTR -
    StruveH operator: StruveH -
    StruveL operator: StruveL -
    sub matrix operator: sub matrix -
    SUB operator: SUB -
    substitution: SUB -
    substitution: FORALL -
    substitution: LET -
    substitution: MATCH -
    substitution: WHERE -
    summation: SUM -
    summation: ZEILBERG -
    SUM operator: SUM -
    surface switch: surface -
    svd operator: svd -
    swap columns operator: swap columns -
    swap entries operator: swap entries -
    swap rows operator: swap rows -
    switch: ADJPREC -
    switch: NOCONVERT -
    switch: OFF -
    switch: ON -
    switch: ALGINT -
    switch: ALLBRANCH -
    switch: ALLFAC -
    switch: ARBVARS -
    switch: BALANCED MOD -
    switch: BFSPACE -
    switch: COMBINEEXPT -
    switch: COMBINELOGS -
    switch: COMP -
    switch: COMPLEX -
    switch: CREF -
    switch: CRAMER -
    switch: DEFN -
    switch: DEMO -
    switch: DFPRINT -
    switch: DIV -
    switch: ECHO -
    switch: ERRCONT -
    switch: EVALLHSEQP -
    switch: EXP switch -
    switch: EXPANDLOGS -
    switch: EZGCD -
    switch: FACTOR -
    switch: FAILHARD -
    switch: FORT -
    switch: FORTUPPER -
    switch: FULLPREC -
    switch: FULLROOTS -
    switch: GC -
    switch: GCD switch -
    switch: HORNER -
    switch: IFACTOR -
    switch: INT switch -
    switch: INTSTR -
    switch: LCM -
    switch: LESSSPACE -
    switch: LIMITEDFACTORS -
    switch: LIST switch -
    switch: LISTARGS -
    switch: MCD -
    switch: MODULAR -
    switch: MSG -
    switch: MULTIPLICITIES -
    switch: NAT -
    switch: NERO -
    switch: NOARG -
    switch: NOLNR -
    switch: NOSPLIT -
    switch: NUMVAL -
    switch: OUTPUT -
    switch: OVERVIEW -
    switch: PERIOD -
    switch: PRECISE -
    switch: PRET -
    switch: PRI -
    switch: RAISE -
    switch: RAT -
    switch: RATARG -
    switch: RATIONAL -
    switch: RATIONALIZE -
    switch: RATPRI -
    switch: REVPRI -
    switch: RLISP88 -
    switch: ROUNDALL -
    switch: ROUNDBF -
    switch: ROUNDED -
    switch: SAVESTRUCTR -
    switch: SOLVESINGULAR -
    switch: TIME -
    switch: TRALLFAC -
    switch: TRFAC -
    switch: TRIGFORM -
    switch: TRINT -
    switch: TRNONLNR -
    switch: VAROPT -
    switch: groebopt -
    switch: groebprereduce -
    switch: groebfullreduction -
    switch: gltbasis -
    switch: groebstat -
    switch: trgroeb -
    switch: trgroebs -
    switch: groebprot -
    switch: TRNUMERIC -
    switch: taylorautocombine -
    switch: taylorautoexpand -
    switch: taylorkeeporiginal -
    switch: taylorprintorder -
    switch: contour -
    switch: surface -
    switch: hidden3d -
    switch: PLOTKEEP -
    switch: PLOTREFINE -
    switch: SHOW GRID -
    switch: TRPLOT -
    switch: fast la -
    SWITCHES introduction: SWITCHES -
    symbolic: EVAL MODE -
    SYMBOLIC command: SYMBOLIC -
    SYMMETRIC declaration: SYMMETRIC -
    symmetricp operator: symmetricp -
    symmetries: APPLYSYM -
    SYMMETRY package: SYMMETRY -
    TANH operator: TANH -
    TAN operator: TAN -
    Taylor: ORTHOVEC -
    taylorautocombine switch: taylorautocombine -
    taylorautoexpand switch: taylorautoexpand -
    taylorcombine operator: taylorcombine -
    TAYLOR introduction: TAYLOR introduction -
    taylorkeeporiginal switch: taylorkeeporiginal -
    taylor operator: taylor -
    taylororiginal operator: taylororiginal -
    taylorprintorder switch: taylorprintorder -
    taylorprintterms variable: taylorprintterms -
    taylorrevert operator: taylorrevert -
    Taylor series: TPS -
    taylorseriesp operator: taylorseriesp -
    taylortemplate operator: taylortemplate -
    taylortostandard operator: taylortostandard -
    T constant: T -
    terminal variable: terminal -
    term order: torder compile -
    term order: lex term order -
    term order: gradlex term order -
    term order: revgradlex term order -
    term order: gradlexgradlex term order -
    term order: gradlexrevgradlex term order -
    term order: lexgradlex term order -
    term order: lexrevgradlex term order -
    term order: weighted term order -
    term order: graded term order -
    term order: matrix term order -
    term order: glexconvert -
    Term order introduction: Term order -
    TEX: RLFI -
    TEX: TRI -
    then: IF -
    THIRD operator: THIRD -
    ThreejSymbol operator: ThreejSymbol -
    tilde operator: tilde -
    time: SHOWTIME -
    TIMES operator: TIMES -
    TIME switch: TIME -
    title variable: title -
    toeplitz operator: toeplitz -
    Top: EDITDEF -
    torder compile operator: torder compile -
    torder operator: torder -
    TP operator: TP -
    TPS package: TPS -
    trace: TR -
    trace: UNTR -
    TRACE operator: TRACE -
    TRALLFAC switch: TRALLFAC -
    transform: LAPLACE -
    transpose: TP -
    TR declaration: TR -
    TRFAC switch: TRFAC -
    trgroebs switch: trgroebs -
    trgroeb switch: trgroeb -
    TRIGFORM switch: TRIGFORM -
    TRIGSIMP package: TRIGSIMP -
    TRINT switch: TRINT -
    TRI package: TRI -
    TRNONLNR switch: TRNONLNR -
    TRNUMERIC switch: TRNUMERIC -
    TRPLOT switch: TRPLOT -
    TRUE concept: TRUE -
    type: IDENTIFIER -
    type: KERNEL -
    type: STRING -
    type: EQUATION -
    type: RULE -
    type: Free Variable -
    type: Optional Free Variable -
    type: ONE OF -
    type: Interval -
    type: Pointset -
    ultraspherical polynomials: GegenbauerP -
    univariate polynomial: glexconvert -
    until: REPEAT -
    UNTR declaration: UNTR -
    utilities: ASSIST -
    vandermonde operator: vandermonde -
    variable: assumptions -
    variable: CARD NO -
    variable: EVAL MODE -
    variable: FORT WIDTH -
    variable: HIGH POW -
    variable: LOW POW -
    variable: requirements -
    variable: ROOT MULTIPLICITIES -
    variable: Free Variable -
    variable: Optional Free Variable -
    variable: gvarslast -
    variable: gltb -
    variable: glterms -
    variable: groebmonfac -
    variable: groebresmax -
    variable: groebrestriction -
    variable: groebprotfile -
    variable: gmodule -
    variable: ROOTSCOMPLEX -
    variable: ROOTSREAL -
    variable: taylorprintterms -
    variable: title -
    variable: xlabel -
    variable: ylabel -
    variable: zlabel -
    variable: terminal -
    variable: size -
    variable: view -
    variable: plot xmesh -
    variable: plot ymesh -
    variable elimination: lex term order -
    variable order: KORDER -
    variable order: ORDER -
    VARNAME declaration: VARNAME -
    VAROPT switch: VAROPT -
    VECDIM command: VECDIM -
    vector algebra: AVECTOR -
    vector algebra: ORTHOVEC -
    vector calculus: ORTHOVEC -
    VECTOR declaration: VECTOR -
    view variable: view -
    Weber's function: BESSELY -
    WEIGHT command: WEIGHT -
    weighted term order concept: weighted term order -
    WHEN operator: WHEN -
    WHERE operator: WHERE -
    WHILE command: WHILE -
    WhittakerW operator: WhittakerW -
    work space: WS -
    WRITE command: WRITE -
    WS operator: WS -
    WTLEVEL command: WTLEVEL -
    WU package: WU -
    Wu-Wen-Tsun algorithm: WU -
    XCOLOR package: XCOLOR -
    XIDEAL package: XIDEAL -
    xlabel variable: xlabel -
    ylabel variable: ylabel -
    ZEILBERG package: ZEILBERG -
    ZETA operator: ZETA -
    zlabel variable: zlabel -
    ZTRANS package: ZTRANS -
     DELETED r38/help/hcrtf.exe Index: r38/help/hcrtf.exe ================================================================== --- r38/help/hcrtf.exe +++ r38/help/hcrtf.exe cannot compute difference between binary files DELETED r38/help/htmlhelp.exe Index: r38/help/htmlhelp.exe ================================================================== --- r38/help/htmlhelp.exe +++ r38/help/htmlhelp.exe cannot compute difference between binary files DELETED r38/help/mkhelpw.log Index: r38/help/mkhelpw.log ================================================================== --- r38/help/mkhelpw.log +++ r38/help/mkhelpw.log @@ -1,2563 +0,0 @@ -+++ Transcript to mkhelpw.log started at Fri Mar 26 11:27:59 2004 +++ -Codemist Standard Lisp 5.00 for Windows: Mar 25 2004 -Created: Tue Mar 23 16:08:03 2004 - -REDUCE 3.8, 15-Oct-03 ... -Memory allocation: 54522624 bytes - -+++ About to read file "mkhelpw.red" - -% This file runs the "*.tex" to "redhelp.rtf" conversion code in CSL. -% As well as providing the top level direction to the process it patches -% up for at least some of the places where the conversion code had been -% written in a manner not strongly related to the portability objectives -% of Standard Lisp... - -symbolic; -nil - - -off echo; -nil - - -on backtrace; -nil - - -on comp; -nil - - -!*windows := t; -t - - - -fluid '(package); -nil - - - -package := 'redhelp; -redhelp - - - -symbolic procedure inf x; - char!-code x;+++ inf compiled as link to char-code - -inf - - - -symbolic procedure channellinelength(f, l); - begin - f := wrs f; - l := linelength l; - wrs f; - return l - end;+++ channellinelength compiled, 12 + 4 bytes - -channellinelength - - - -symbolic procedure channelprin2(f, x); - begin - f := wrs f; - prin2 x; - wrs f; - return x - end;+++ channelprin2 compiled, 10 + 8 bytes - -channelprin2 - - - -symbolic macro procedure channelprintf u; - begin - scalar g; - g := gensym(); - return list('prog, list g, - list('setq, g, list('wrs, cadr u)), - 'printf . cddr u, - list('wrs, g)) - end;+++ channelprintf_76z7r3p3pl4b compiled, 33 + 24 bytes - -channelprintf - - - -symbolic procedure channelterpri f; - begin - f := wrs f; - terpri(); - wrs f; - end;+++ channelterpri compiled, 10 + 4 bytes - -channelterpri - - - -symbolic procedure channelreadch f; - begin - scalar c; - f := rds f; - c := readch(); - rds f; - return c - end;+++ channelreadch compiled, 12 + 4 bytes - -channelreadch - - - -in "comphelp.red"$ -nil - -nil - -nil - -nil - -nil - -nil - -nil - -nil - -nil - -nil - -0 - -nil -+++++ global !*raise converted to fluid -+++ job compiled, 113 + 148 bytes - -job - -nil - -nil - -nil -+++ rdch compiled, 3 + 8 bytes - -rdch -+++ rdch!* compiled, 3 + 12 bytes - -rdch!* -+++ rdchr0 compiled, 54 + 64 bytes - -rdchr0 -+++ unrdch compiled, 3 + 12 bytes - -unrdch -+++ myskip compiled, 5 + 8 bytes - -myskip -+++ myskipl compiled, 9 + 8 bytes - -myskipl -+++ myskipstring compiled, 41 + 24 bytes - -myskipstring - -nil - -lower -+++ mytoken compiled, 83 + 44 bytes - -mytoken -+++ mystring compiled, 28 + 16 bytes - -mystring -+++ mystring2 compiled, 14 + 12 bytes - -mystring2 -+++ mystring2!] compiled, 14 + 12 bytes - -mystring2!] -+++ mystring_nodename compiled, 41 + 40 bytes - -mystring_nodename -+++ mystring3 compiled, 23 + 16 bytes - -mystring3 -+++ raisestring compiled, 67 + 20 bytes - -raisestring -+++ lowerstring compiled, 75 + 20 bytes - -lowerstring -+++ mycompress compiled, 12 + 8 bytes - -mycompress -+++ mainloop compiled, 200 + 156 bytes - -mainloop - -*** local variable u in procedure include not used -+++ include compiled, 129 + 100 bytes - -include - -include - -include - -null - -null -+++ print_indent compiled, 21 + 20 bytes - -print_indent - -nil - -0 - -type - -seq - -lab - -count - -name -+++ reset compiled, 51 + 40 bytes - -reset -+++ sectappend compiled, 11 + 8 bytes - -sectappend -+++ section compiled, 10 + 20 bytes - -section -+++ close_section compiled, 39 + 24 bytes - -close_section -+++ close_section1 compiled, 40 + 40 bytes - -close_section1 -+++ write_sections compiled, 15 + 12 bytes - -write_sections -+++ write_section compiled, 67 + 52 bytes - -write_section -+++ make_dir_entry compiled, 8 + 8 bytes - -make_dir_entry -+++ help_gensym compiled, 10 + 12 bytes - -help_gensym -+++ open_section compiled, 172 + 112 bytes - -open_section - -section - -section - -section - -*** local variable u in procedure beg not used -+++ beg compiled, 87 + 68 bytes - -beg - -beg - -*** local variable u in procedure mmain not used -+++ mmain compiled as link to mainloop - -mmain - -mmain - -nil -+++ clean_name compiled, 27 + 16 bytes - -clean_name - -(((!,) . comma_sign) ((!.) . dot_sign) ((!;) . semicolon_sign) ((!%) . -percent_sign) ((!$) . dollar_sign) ((!: !=) . assign_sign) ((!=) . equal_sign) ( -(!+) . plus_sign) ((!-) . minus_sign) ((!*) . times_sign) ((!/) . slash_sign) (( -!* !*) . power_sign) ((!$ !> != !$) . geq_sign) ((!> !=) . geq_sign) ((!>) . -greater_sign) ((!$ !< != !$) . leq_sign) ((!< !=) . leq_sign) ((!<) . less_sign) -((!< !<) . block)) -+++ make_label compiled, 64 + 40 bytes - -make_label -+++ get_label compiled, 8 + 8 bytes - -get_label -+++ patch_ compiled, 29 + 16 bytes - -patch_ -+++ get_label1 compiled, 32 + 40 bytes - -get_label1 -+++ get_label2 compiled, 103 + 56 bytes - -get_label2 -+++ update_labels compiled, 47 + 36 bytes - -update_labels -+++ node compiled, 197 + 136 bytes - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node -+++ part compiled, 63 + 68 bytes - -part -+++ par_heading compiled, 26 + 16 bytes - -par_heading - -*** local variable type in procedure vpart not used -+++ vpart compiled, 13 + 32 bytes - -vpart -+++ vpart0 compiled, 81 + 72 bytes - -vpart0 -+++ compareahead compiled, 6 + 8 bytes - -compareahead -+++ compareahead1 compiled, 28 + 12 bytes - -compareahead1 -+++ look_ahead_suctx941k0ih compiled, 11 + 16 bytes - -look_ahead -REHASHING: chunks = 1, grow=1 - -*** local variable type in procedure examples_part not used -+++ examples_part compiled, 381 + 204 bytes - -examples_part -+++ non_verb_block compiled, 15 + 36 bytes - -non_verb_block -+++ make_multi_out compiled, 121 + 60 bytes - -make_multi_out -+++ cut_lines compiled, 37 + 8 bytes - -cut_lines -+++ matchleft compiled, 22 + 16 bytes - -matchleft -+++ matcharb compiled, 12 + 8 bytes - -matcharb -+++ read_one_line compiled, 15 + 12 bytes - -read_one_line - -part - -part - -part - -part - -part - -part - -part - -part - -*** local variable type in procedure do-itemize not used -+++ do!-itemize compiled, 6 + 20 bytes - -do!-itemize - -do!-itemize -+++ context_error compiled, 11 + 20 bytes - -context_error - -*** local variable u in procedure verb not used -+++ verb compiled, 12 + 24 bytes - -verb - -verb - -*** local variable u in procedure ldots not used -+++ ldots compiled, 3 + 12 bytes - -ldots - -ldots - -nil - -*** local variable u in procedure cdots not used -+++ cdots compiled, 3 + 12 bytes - -cdots - -cdots - -nil - -*** local variable u in procedure cdot not used -+++ cdot compiled, 3 + 12 bytes - -cdot - -cdot - -nil - -*** local variable u in procedure write_pi not used -+++ write_pi compiled, 3 + 12 bytes - -write_pi - -write_pi - -nil - -*** local variable u in procedure emphase not used -+++ emphase compiled, 3 + 12 bytes - -emphase - -emphase - -*** local variable u in procedure meta not used -+++ meta compiled, 18 + 20 bytes - -meta - -meta - -*** local variable u in procedure italic not used -+++ italic compiled, 8 + 24 bytes - -italic - -*** local variable u in procedure switchitalic not used -+++ switchitalic compiled, 1 + 4 bytes - -switchitalic - -italic - -italic - -italic - -*** local variable u in procedure nameref not used -+++ nameref compiled, 3 + 12 bytes - -nameref - -nameref - -*** local variable u in procedure ref not used -+++ ref compiled, 3 + 12 bytes - -ref - -ref -+++ see compiled, 4 + 12 bytes - -see - -see - -*** local variable u in procedure myname not used -+++ myname compiled, 3 + 12 bytes - -myname - -myname - -*** local variable u in procedure myindex not used -+++ myindex compiled, 5 + 20 bytes - -myindex - -myindex - -*** local variable u in procedure nameindex not used -+++ nameindex compiled, 10 + 24 bytes - -nameindex - -nameindex - -*** local variable u in procedure reduce not used -+++ reduce compiled, 3 + 12 bytes - -reduce - -reduce - -nil - -*** local variable u in procedure rept not used -+++ rept compiled, 3 + 12 bytes - -rept - -rept - -nil - -*** local variable u in procedure optional not used -+++ optional compiled, 3 + 12 bytes - -optional - -optional - -nil - -*** local variable u in procedure myexp not used -+++ myexp compiled, 5 + 16 bytes - -myexp - -myexp - -*** local variable u in procedure formula not used -+++ formula compiled, 3 + 12 bytes - -formula - -formula - -formula - -*** local variable u in procedure rfrac not used -+++ rfrac compiled, 8 + 24 bytes - -rfrac - -rfrac - -*** local variable u in procedure item not used -+++ item compiled, 14 + 32 bytes - -item - -item - -*** local variable u in procedure texonly1 not used -+++ texonly1 compiled, 48 + 44 bytes - -texonly1 - -texonly1 - -*** local variable u in procedure texonly2 not used -+++ texonly2 compiled, 67 + 68 bytes - -texonly2 - -texonly2 - -*** local variable u in procedure infoonly not used -+++ infoonly compiled, 6 + 12 bytes - -infoonly - -infoonly -+++ reporttopic compiled, 12 + 20 bytes - -reporttopic -+++ substipq compiled, 32 + 12 bytes - -substipq - -nil - - -in "helpwin.red"$ -nil - -t - -"f2" - -"f4" -+++ initoutput compiled, 404 + 520 bytes - -initoutput -+++ endoutput compiled, 11 + 16 bytes - -endoutput -+++ verbatim compiled, 3 + 8 bytes - -verbatim -+++ newfont compiled, 10 + 16 bytes - -newfont -+++ fontoff compiled, 11 + 24 bytes - -fontoff -+++ fonton compiled, 22 + 32 bytes - -fonton -+++ myprin2 compiled, 6 + 16 bytes - -myprin2 -+++ myprin2_protected compiled, 10 + 16 bytes - -myprin2_protected - -nil -+++ emit_start_verbatim compiled, 1 + 4 bytes - -emit_start_verbatim -+++ emit_end_verbatim compiled, 1 + 4 bytes - -emit_end_verbatim -+++ verbprin2 compiled, 56 + 60 bytes - -verbprin2 -+++ myterpri compiled, 3 + 12 bytes - -myterpri -+++ number4out compiled, 24 + 32 bytes - -number4out -+++ textout compiled, 132 + 84 bytes - -textout - -+++ par_heading redefined -+++ par_heading compiled, 24 + 16 bytes - -par_heading - -*** local variable name in procedure base_new_dir not used -+++ base_new_dir compiled, 1 + 4 bytes - -base_new_dir -+++ emit_dir_new compiled, 1 + 4 bytes - -emit_dir_new -+++ emit_dir_key compiled as link to emit_node_key - -emit_dir_key -+++ emit_dir_separator compiled as link to emit_node_separator - -emit_dir_separator -+++ emit_dir_label compiled as link to emit_node_label - -emit_dir_label -+++ emit_dir_title compiled, 6 + 12 bytes - -emit_dir_title -+++ emit_dir_browse compiled as link to emit_node_browse - -emit_dir_browse -+++ emit_node_separator compiled, 16 + 40 bytes - -emit_node_separator -+++ set_tab compiled, 3 + 12 bytes - -set_tab -+++ release_tab compiled, 3 + 12 bytes - -release_tab -+++ textoutl compiled, 26 + 16 bytes - -textoutl -+++ textout2 compiled, 26 + 16 bytes - -textout2 -+++ printem compiled, 18 + 16 bytes - -printem - -+++ printem redefined -+++ printem compiled, 19 + 24 bytes - -printem -+++ printref compiled, 49 + 40 bytes - -printref -+++ printnameref compiled as link to printref - -printnameref - -nil -+++ emit_node_keys compiled, 33 + 36 bytes - -emit_node_keys -+++ emit_node_key compiled as link to emit_hidden_node_key - -emit_node_key -+++ emit_hidden_node_key compiled, 33 + 16 bytes - -emit_hidden_node_key -+++ emit_node_label compiled, 10 + 28 bytes - -emit_node_label - -*** local variable dummy in procedure emit_node_title not used - -*** local variable type in procedure emit_node_title not used -+++ emit_node_title compiled, 10 + 28 bytes - -emit_node_title -+++ emit_node_browse compiled, 16 + 36 bytes - -emit_node_browse -+++ print_bold compiled, 19 + 20 bytes - -print_bold -+++ emit_dir_header compiled, 5 + 20 bytes - -emit_dir_header -+++ emit_dir_entry compiled, 45 + 48 bytes - -emit_dir_entry -+++ print_newline compiled, 13 + 32 bytes - -print_newline -+++ second_newline compiled, 4 + 12 bytes - -second_newline -+++ print_tab compiled, 4 + 16 bytes - -print_tab -+++ printstruct compiled, 7 + 16 bytes - -printstruct -+++ printstruct1 compiled, 58 + 16 bytes - -printstruct1 - -nil - - -in "minitex.red"$ -nil - -nil - -nil - -!\ - -!^ - -!_ - -2 - -3 - -4 -+++ mintex_convert0 compiled, 38 + 12 bytes - -mintex_convert0 -+++ mintex_convert compiled as link to mintex_convert0 - -mintex_convert -+++ minitex compiled, 102 + 40 bytes - -minitex -+++ minitex_pop_char compiled, 11 + 8 bytes - -minitex_pop_char -+++ minitex_skip compiled, 11 + 8 bytes - -minitex_skip -+++ minitex_next_char compiled, 6 + 8 bytes - -minitex_next_char -+++ struct compiled, 9 + 8 bytes - -struct -+++ make_chain compiled, 317 + 120 bytes - -make_chain - -*** local variable font in procedure make_char not used - -*** local variable cs in procedure make_char not used -+++ make_char compiled, 23 + 48 bytes - -make_char -+++ make_frac compiled, 211 + 76 bytes - -make_frac - -*** local variable y1 in procedure make_line not used -+++ make_line compiled, 22 + 32 bytes - -make_line -+++ make_multi compiled, 63 + 60 bytes - -make_multi - -*** local variable font in procedure make_end not used -+++ make_end compiled, 4 + 16 bytes - -make_end - -nil - -(chain 0 0 0) - -nil - -*** local variable term in procedure make_escape not used -+++ make_escape compiled, 77 + 72 bytes - -make_escape -+++ my_compare compiled, 23 + 16 bytes - -my_compare -+++ minitex_collect compiled, 8 + 12 bytes - -minitex_collect -+++ minitex_do compiled, 25 + 20 bytes - -minitex_do - -minitex_chain -+++ minitex_chain compiled, 37 + 24 bytes - -minitex_chain - -minitex_char - -*** local variable font in procedure minitex_char not used -+++ minitex_char compiled, 23 + 24 bytes - -minitex_char - -minitex_line - -*** local variable font in procedure minitex_line not used -+++ minitex_line compiled, 38 + 28 bytes - -minitex_line -+++ minitex_putchar compiled, 31 + 24 bytes - -minitex_putchar - -nil - - - -dir_src := "$reduce/doc/help/"; -"$reduce/doc/help/" - - - -job(bldmsg("%w.tex",package),"null.fil"); -**** unknown token: documentclass -**** unknown token: usepackage -**** unknown token: usepackage - ---- input file $reduce/doc/help/concept.tex -section 2 g2 Concepts - type IDENTIFIER 1 - type KERNEL 2 - type STRING 3 - ---- return from file $reduce/doc/help/concept.tex - ---- input file $reduce/doc/help/variable.tex - section end: 4 -section 3 g3 Variables - variable assumptions 5 - variable CARD\_NO 6 - constant E 7 - variable EVAL\_MODE 8 - variable FORT\_WIDTH 9 - variable HIGH\_POW 10 - constant I 11 - constant INFINITY 12 - variable LOW\_POW 13 - constant NIL 14 - constant PI 15 - variable requirements 16 - variable ROOT\_MULTIPLICITIES 17 - constant T 18 - ---- return from file $reduce/doc/help/variable.tex - ---- input file $reduce/doc/help/syntax.tex - section end: 19 -section 4 g4 Syntax - command semicolon 20 - command dollar 21 - command percent 22 - operator dot 23 - operator assign 24 - operator equalsign 25 - operator replace 26 - operator plussign 27 - operator minussign 28 - operator asterisk 29 - operator slash 30 - operator power 31 - operator caret 32 - operator geqsign 33 - operator greater 34 - operator leqsign 35 - operator less 36 - operator tilde 37 - command group 38 - operator AND 39 - command BEGIN 40 - command block 41 - command COMMENT 42 - operator CONS 43 - command END 44 - type EQUATION 45 - operator FIRST 46 - command FOR 47 - command FOREACH 48 - operator GEQ 49 - command GOTO 50 - operator GREATERP 51 - command IF 52 - operator LIST 53 - operator OR 54 - command PROCEDURE 55 - command REPEAT 56 - operator REST 57 - command RETURN 58 - operator REVERSE 59 - type RULE 60 - type Free Variable 61 - type Optional Free Variable 62 - operator SECOND 63 - operator SET 64 - operator SETQ 65 - operator THIRD 66 - operator WHEN 67 - ---- return from file $reduce/doc/help/syntax.tex - ---- input file $reduce/doc/help/arith.tex - section end: 68 -section 5 g5 Arithmetic Operations - introduction ARITHMETIC\_OPERATIONS 69 - operator ABS 70 - switch ADJPREC 71 - operator ARG 72 - operator CEILING 73 - operator CHOOSE 74 - operator DEG2DMS 75 - operator DEG2RAD 76 - operator DIFFERENCE 77 - operator DILOG 78 - operator DMS2DEG 79 - operator DMS2RAD 80 - operator FACTORIAL 81 - operator FIX 82 - operator FIXP 83 - operator FLOOR 84 - operator EXPT 85 - operator GCD 86 - operator LN 87 - operator LOG 88 - operator LOGB 89 - operator MAX 90 - operator MIN 91 - operator MINUS 92 - operator NEXTPRIME 93 - switch NOCONVERT 94 - operator NORM 95 - operator PERM 96 - operator PLUS 97 - operator QUOTIENT 98 - operator RAD2DEG 99 - operator RAD2DMS 100 - operator RECIP 101 - operator REMAINDER 102 - operator ROUND 103 - command SETMOD 104 - operator SIGN 105 - operator SQRT 106 - operator TIMES 107 - ---- return from file $reduce/doc/help/arith.tex - ---- input file $reduce/doc/help/boolean.tex - section end: 108 -section 6 g6 Boolean Operators - concept boolean value 109 - operator EQUAL 110 - operator EVENP 111 - concept false 112 - operator FREEOF 113 - operator LEQ 114 - operator LESSP 115 - operator MEMBER 116 - operator NEQ 117 - operator NOT 118 - operator NUMBERP 119 - operator ORDP 120 - operator PRIMEP 121 - concept TRUE 122 - ---- return from file $reduce/doc/help/boolean.tex - ---- input file $reduce/doc/help/command.tex - section end: 123 -section 7 g7 General Commands - command BYE 124 - command CONT 125 - command DISPLAY 126 - command LOAD\_PACKAGE 127 - command PAUSE 128 - command QUIT 129 - operator RECLAIM 130 - command REDERR 131 - command RETRY 132 - command SAVEAS 133 - command SHOWTIME 134 - command WRITE 135 - ---- return from file $reduce/doc/help/command.tex - ---- input file $reduce/doc/help/algebra.tex - section end: 136 -section 8 g8 Algebraic Operators - operator APPEND 137 - operator ARBINT 138 - operator ARBCOMPLEX 139 - operator ARGLENGTH 140 - operator COEFF 141 - operator COEFFN 142 - operator CONJ 143 - operator CONTINUED_FRACTION 144 - operator DECOMPOSE 145 - operator DEG 146 - operator DEN 147 - operator DF 148 - operator EXPAND\_CASES 149 - operator EXPREAD 150 - operator FACTORIZE 151 - operator HYPOT 152 - operator IMPART 153 - operator INT 154 - operator INTERPOL 155 - operator LCOF 156 - operator LENGTH 157 - operator LHS 158 - operator LIMIT 159 - operator LPOWER 160 - operator LTERM 161 - operator MAINVAR 162 - operator MAP 163 - command MKID 164 - operator NPRIMITIVE 165 - operator NUM 166 - operator ODESOLVE 167 - type ONE\_OF 168 - operator PART 169 - operator PF 170 - operator PROD 171 - operator REDUCT 172 - operator REPART 173 - operator RESULTANT 174 - operator RHS 175 - operator ROOT\_OF 176 - operator SELECT 177 - operator SHOWRULES 178 - operator SOLVE 179 - operator SORT 180 - operator STRUCTR 181 - operator SUB 182 - operator SUM 183 - operator WS 184 - ---- return from file $reduce/doc/help/algebra.tex - ---- input file $reduce/doc/help/declare.tex - section end: 185 -section 9 g9 Declarations - command ALGEBRAIC 186 - declaration ANTISYMMETRIC 187 - declaration ARRAY 188 - command CLEAR 189 - command CLEARRULES 190 - command DEFINE 191 - declaration DEPEND 192 - declaration EVEN 193 - declaration FACTOR 194 - command FORALL 195 - declaration INFIX 196 - declaration INTEGER 197 - declaration KORDER 198 - command LET 199 - declaration LINEAR 200 - declaration LINELENGTH 201 - command LISP 202 - declaration LISTARGP 203 - declaration NODEPEND 204 - command MATCH 205 - declaration NONCOM 206 - declaration NONZERO 207 - declaration ODD 208 - command OFF 209 - command ON 210 - declaration OPERATOR 211 - declaration ORDER 212 - declaration PRECEDENCE 213 - declaration PRECISION 214 - declaration PRINT\_PRECISION 215 - declaration REAL 216 - declaration REMFAC 217 - declaration SCALAR 218 - declaration SCIENTIFIC\_NOTATION 219 - declaration SHARE 220 - command SYMBOLIC 221 - declaration SYMMETRIC 222 - declaration TR 223 - declaration UNTR 224 - declaration VARNAME 225 - command WEIGHT 226 - operator WHERE 227 - command WHILE 228 - command WTLEVEL 229 - ---- return from file $reduce/doc/help/declare.tex - ---- input file $reduce/doc/help/io.tex - section end: 230 -section 10 g10 Input and Output - command IN 231 - command INPUT 232 - command OUT 233 - command SHUT 234 - ---- return from file $reduce/doc/help/io.tex - ---- input file $reduce/doc/help/elemfn.tex - section end: 235 -section 11 g11 Elementary Functions - operator ACOS 236 - operator ACOSH 237 - operator ACOT 238 - operator ACOTH 239 - operator ACSC 240 - operator ACSCH 241 - operator ASEC 242 - operator ASECH 243 - operator ASIN 244 - operator ASINH 245 - operator ATAN 246 - operator ATANH 247 - operator ATAN2 248 - operator COS 249 - operator COSH 250 - operator COT 251 - operator COTH 252 - operator CSC 253 - operator CSCH 254 - operator ERF 255 - operator EXP 256 - operator SEC 257 - operator SECH 258 - operator SIN 259 - operator SINH 260 - operator TAN 261 - operator TANH 262 - ---- return from file $reduce/doc/help/elemfn.tex - ---- input file $reduce/doc/help/switch.tex - section end: 263 -section 12 g12 General Switches - introduction SWITCHES 264 - switch ALGINT 265 - switch ALLBRANCH 266 - switch ALLFAC 267 - switch ARBVARS 268 - switch BALANCED\_MOD 269 - switch BFSPACE 270 - switch COMBINEEXPT 271 - switch COMBINELOGS 272 - switch COMP 273 - switch COMPLEX 274 - switch CREF 275 - switch CRAMER 276 - switch DEFN 277 - switch DEMO 278 - switch DFPRINT 279 - switch DIV 280 - switch ECHO 281 - switch ERRCONT 282 - switch EVALLHSEQP 283 - switch EXP 284 - switch EXPANDLOGS 285 - switch EZGCD 286 - switch FACTOR 287 - switch FAILHARD 288 - switch FORT 289 - switch FORTUPPER 290 - switch FULLPREC 291 - switch FULLROOTS 292 - switch GC 293 - switch GCD 294 - switch HORNER 295 - switch IFACTOR 296 - switch INT 297 - switch INTSTR 298 - switch LCM 299 - switch LESSSPACE 300 - switch LIMITEDFACTORS 301 - switch LIST 302 - switch LISTARGS 303 - switch MCD 304 - switch MODULAR 305 - switch MSG 306 - switch MULTIPLICITIES 307 - switch NAT 308 - switch NERO 309 - switch NOARG 310 - switch NOLNR 311 - switch NOSPLIT 312 - switch NUMVAL 313 - switch OUTPUT 314 - switch OVERVIEW 315 - switch PERIOD 316 - switch PRECISE 317 - switch PRET 318 - switch PRI 319 - switch RAISE 320 - switch RAT 321 - switch RATARG 322 - switch RATIONAL 323 - switch RATIONALIZE 324 - switch RATPRI 325 - switch REVPRI 326 - switch RLISP88 327 - switch ROUNDALL 328 - switch ROUNDBF 329 - switch ROUNDED 330 - switch SAVESTRUCTR 331 - switch SOLVESINGULAR 332 - switch TIME 333 - switch TRALLFAC 334 - switch TRFAC 335 - switch TRIGFORM 336 - switch TRINT 337 - switch TRNONLNR 338 - switch VAROPT 339 - ---- return from file $reduce/doc/help/switch.tex - ---- input file $reduce/doc/help/matrix.tex - section end: 340 -section 13 g13 Matrix Operations - operator COFACTOR 341 - operator DET 342 - operator MAT 343 - operator MATEIGEN 344 - declaration MATRIX 345 - operator NULLSPACE 346 - operator RANK 347 - operator TP 348 - operator TRACE 349 - ---- return from file $reduce/doc/help/matrix.tex - ---- input file $reduce/doc/help/pk-groeb.tex - section end: 350 -section 14 g14 Groebner package - introduction Groebner bases 351 - concept Ideal Parameters 352 - subsection 15 g15 Term order - introduction Term order 353 - operator torder 354 - operator torder_compile 355 - concept lex term order 356 - concept gradlex term order 357 - concept revgradlex term order 358 - concept gradlexgradlex term order 359 - concept gradlexrevgradlex term order 360 - concept lexgradlex term order 361 - concept lexrevgradlex term order 362 - concept weighted term order 363 - concept graded term order 364 - concept matrix term order 365 - section end: 366 - subsection 16 g16 Basic Groebner operators - operator gvars 367 - operator groebner 368 - operator groebner\_walk 369 - switch groebopt 370 - variable gvarslast 371 - switch groebprereduce 372 - switch groebfullreduction 373 - switch gltbasis 374 - variable gltb 375 - variable glterms 376 - switch groebstat 377 - switch trgroeb 378 - switch trgroebs 379 - operator gzerodim? 380 - operator gdimension 381 - operator gindependent\_sets 382 - operator dd_groebner 383 - operator glexconvert 384 - operator greduce 385 - operator preduce 386 - operator idealquotient 387 - operator hilbertpolynomial 388 - operator saturation 389 - section end: 390 - subsection 17 g17 Factorizing Groebner bases - operator groebnerf 391 - variable groebmonfac 392 - variable groebresmax 393 - variable groebrestriction 394 - section end: 395 - subsection 18 g18 Tracing Groebner bases - switch groebprot 396 - variable groebprotfile 397 - operator groebnert 398 - operator preducet 399 - section end: 400 - subsection 19 g19 Groebner Bases for Modules - concept Module 401 - variable gmodule 402 - section end: 403 - subsection 20 g20 Computing with distributive polynomials - operator gsort 404 - operator gsplit 405 - operator gspoly 406 - ---- return from file $reduce/doc/help/pk-groeb.tex - ---- input file $reduce/doc/help/hephys.tex - section end: 407 - section end: 408 -section 21 g21 High Energy Physics - introduction HEPHYS 409 - operator HE-dot 410 - operator EPS 411 - operator G 412 - declaration INDEX 413 - command MASS 414 - command MSHELL 415 - declaration NOSPUR 416 - declaration REMIND 417 - declaration SPUR 418 - command VECDIM 419 - declaration VECTOR 420 - ---- return from file $reduce/doc/help/hephys.tex - ---- input file $reduce/doc/help/pk-numer.tex - section end: 421 -section 22 g22 Numeric Package - introduction Numeric Package 422 - type Interval 423 - concept numeric accuracy 424 - switch TRNUMERIC 425 - operator num_min 426 - operator num_solve 427 - operator num_int 428 - operator num_odesolve 429 - operator bounds 430 - concept Chebyshev fit 431 - operator num_fit 432 - ---- return from file $reduce/doc/help/pk-numer.tex - ---- input file $reduce/doc/help/pk-roots.tex - section end: 433 -section 23 g23 Roots Package - introduction Roots Package 434 - operator MKPOLY 435 - operator NEARESTROOT 436 - operator REALROOTS 437 - operator ROOTACC 438 - operator ROOTS 439 - operator ROOT\_VAL 440 - variable ROOTSCOMPLEX 441 - variable ROOTSREAL 442 - ---- return from file $reduce/doc/help/pk-roots.tex - ---- input file $reduce/doc/help/pk-specf.tex - section end: 443 -section 24 g24 Special Functions - introduction Special Function Package 444 - concept Constants 445 - subsection 25 g25 Bernoulli Euler Zeta - operator BERNOULLI 446 - operator BERNOULLIP 447 - operator EULER 448 - operator EULERP 449 - operator ZETA 450 - section end: 451 - subsection 26 g26 Bessel Functions - operator BESSELJ 452 - operator BESSELY 453 - operator HANKEL1 454 - operator HANKEL2 455 - operator BESSELI 456 - operator BESSELK 457 - operator StruveH 458 - operator StruveL 459 - operator KummerM 460 - operator KummerU 461 - operator WhittakerW 462 - section end: 463 - subsection 27 g27 Airy Functions - operator Airy_Ai 464 - operator Airy_Bi 465 - operator Airy_Aiprime 466 - operator Airy_Biprime 467 - section end: 468 - subsection 28 g28 Jacobi's Elliptic Functions and Elliptic Integrals - operator JacobiSN 469 - operator JacobiCN 470 - operator JacobiDN 471 - operator JacobiCD 472 - operator JacobiSD 473 - operator JacobiND 474 - operator JacobiDC 475 - operator JacobiNC 476 - operator JacobiSC 477 - operator JacobiNS 478 - operator JacobiDS 479 - operator JacobiCS 480 - operator JacobiAMPLITUDE 481 - operator AGM_FUNCTION 482 - operator LANDENTRANS 483 - operator EllipticF 484 - operator EllipticK 485 - operator EllipticKprime 486 - operator EllipticE 487 - operator EllipticTHETA 488 - operator JacobiZETA 489 - section end: 490 - subsection 29 g29 Gamma and Related Functions - operator POCHHAMMER 491 - operator GAMMA 492 - operator BETA 493 - operator PSI 494 - operator POLYGAMMA 495 - section end: 496 - subsection 30 g30 Miscellaneous Functions - operator DILOG extended 497 - operator Lambert\_W function 498 - section end: 499 - subsection 31 g31 Orthogonal Polynomials - operator ChebyshevT 500 - operator ChebyshevU 501 - operator HermiteP 502 - operator LaguerreP 503 - operator LegendreP 504 - operator JacobiP 505 - operator GegenbauerP 506 - operator SolidHarmonicY 507 - operator SphericalHarmonicY 508 - section end: 509 - subsection 32 g32 Integral Functions - operator Si 510 - operator Shi 511 - operator s_i 512 - operator Ci 513 - operator Chi 514 - operator ERF extended 515 - operator erfc 516 - operator Ei 517 - operator Fresnel_C 518 - operator Fresnel_S 519 - section end: 520 - subsection 33 g33 Combinatorial Operators - operator BINOMIAL 521 - operator STIRLING1 522 - operator STIRLING2 523 - section end: 524 - subsection 34 g34 3j and 6j symbols - operator ThreejSymbol 525 - operator Clebsch_Gordan 526 - operator SixjSymbol 527 - section end: 528 - subsection 35 g35 Miscellaneous - operator HYPERGEOMETRIC 529 - operator MeijerG 530 - operator Heaviside 531 - operator erfi 532 - ---- return from file $reduce/doc/help/pk-specf.tex - ---- input file $reduce/doc/help/taylor.tex - section end: 533 - section end: 534 -section 36 g36 Taylor series - introduction TAYLOR 535 - operator taylor 536 - switch taylorautocombine 537 - switch taylorautoexpand 538 - operator taylorcombine 539 - switch taylorkeeporiginal 540 - operator taylororiginal 541 - switch taylorprintorder 542 - variable taylorprintterms 543 - operator taylorrevert 544 - operator taylorseriesp 545 - operator taylortemplate 546 - operator taylortostandard 547 - ---- return from file $reduce/doc/help/taylor.tex - ---- input file $reduce/doc/help/pk-gplot.tex - section end: 548 -section 37 g37 Gnuplot package - introduction GNUPLOT and REDUCE 549 - concept Axes names 550 - type Pointset 551 - command PLOT 552 - command PLOTRESET 553 - variable title 554 - variable xlabel 555 - variable ylabel 556 - variable zlabel 557 - variable terminal 558 - variable size 559 - variable view 560 - switch contour 561 - switch surface 562 - switch hidden3d 563 - switch PLOTKEEP 564 - switch PLOTREFINE 565 - variable plot_xmesh 566 - variable plot_ymesh 567 - switch SHOW_GRID 568 - switch TRPLOT 569 - ---- return from file $reduce/doc/help/pk-gplot.tex - ---- input file $reduce/doc/help/linalg.tex - section end: 570 -section 38 g38 Linear Algebra package - introduction Linear Algebra package 571 - switch fast_la 572 - operator add_columns 573 - operator add_rows 574 - operator add_to_columns 575 - operator add_to_rows 576 - operator augment_columns 577 - operator band_matrix 578 - operator block_matrix 579 - operator char_matrix 580 - operator char_poly 581 - operator cholesky 582 - operator coeff_matrix 583 - operator column_dim 584 - operator companion 585 - operator copy_into 586 - operator diagonal 587 - operator extend 588 - operator find_companion 589 - operator get_columns 590 - operator get_rows 591 - operator gram_schmidt 592 - operator hermitian_tp 593 - operator hessian 594 - operator hilbert 595 - operator jacobian 596 - operator jordan_block 597 - operator lu_decom 598 - operator make_identity 599 - operator matrix_augment 600 - operator matrixp 601 - operator matrix_stack 602 - operator minor 603 - operator mult_columns 604 - operator mult_rows 605 - operator pivot 606 - operator pseudo_inverse 607 - operator random_matrix 608 - operator remove_columns 609 - operator remove_rows 610 - operator row_dim 611 - operator rows_pivot 612 - operator simplex 613 - operator squarep 614 - operator stack_rows 615 - operator sub_matrix 616 - operator svd 617 - operator swap_columns 618 - operator swap_entries 619 - operator swap_rows 620 - operator symmetricp 621 - operator toeplitz 622 - operator vandermonde 623 - ---- return from file $reduce/doc/help/linalg.tex - ---- input file $reduce/doc/help/normform.tex - section end: 624 -section 39 g39 Matrix Normal Forms - operator Smithex 625 - operator Smithex\_int 626 - operator Frobenius 627 - operator Ratjordan 628 - operator Jordansymbolic 629 - operator Jordan 630 - ---- return from file $reduce/doc/help/normform.tex - ---- input file $reduce/doc/help/pk-misc.tex - section end: 631 -section 40 g40 Miscellaneous Packages - introduction Miscellaneous Packages 632 - package ALGINT 633 - package APPLYSYM 634 - package ARNUM 635 - package ASSIST 636 - package AVECTOR 637 - package BOOLEAN 638 - package CALI 639 - package CAMAL 640 - package CHANGEVR 641 - package COMPACT 642 - package CRACK 643 - package CVIT 644 - package DEFINT 645 - package DESIR 646 - package DFPART 647 - package DUMMY 648 - package EXCALC 649 - package FPS 650 - package FIDE 651 - package GENTRAN 652 - package IDEALS 653 - package INEQ 654 - package INVBASE 655 - package LAPLACE 656 - package LIE 657 - package MODSR 658 - package NCPOLY 659 - package ORTHOVEC 660 - package PHYSOP 661 - package PM 662 - package RANDPOLY 663 - package REACTEQN 664 - package RESET 665 - package RESIDUE 666 - package RLFI 667 - package SCOPE 668 - package SETS 669 - package SPDE 670 - package SYMMETRY 671 - package TPS 672 - package TRI 673 - package TRIGSIMP 674 - package XCOLOR 675 - package XIDEAL 676 - package WU 677 - package ZEILBERG 678 - package ZTRANS 679 - ---- return from file $reduce/doc/help/pk-misc.tex - ---- input file $reduce/doc/help/outmode.tex - section end: 680 -section 41 g41 Outmoded Operations - command ED 681 - command EDITDEF 682 - ---- return from file $reduce/doc/help/outmode.tex - section end: 683 - section end: 684 - -nil - - - -delete!-file "null.fil"; -t - - - -job(bldmsg("%w.tex",package),bldmsg("%w.rtf",package)); - ------- updating node labels ----- ------- updating done ------------ -**** unknown token: documentclass -**** unknown token: usepackage -**** unknown token: usepackage - ---- input file $reduce/doc/help/concept.tex -section 2 g2 Concepts - type IDENTIFIER 1 - type KERNEL 2 - type STRING 3 - ---- return from file $reduce/doc/help/concept.tex - ---- input file $reduce/doc/help/variable.tex - section end: 4 -section 3 g3 Variables - variable assumptions 5 - variable CARD\_NO 6 - constant E 7 - variable EVAL\_MODE 8 - variable FORT\_WIDTH 9 - variable HIGH\_POW 10 - constant I 11 - constant INFINITY 12 - variable LOW\_POW 13 - constant NIL 14 - constant PI 15 - variable requirements 16 - variable ROOT\_MULTIPLICITIES 17 - constant T 18 - ---- return from file $reduce/doc/help/variable.tex - ---- input file $reduce/doc/help/syntax.tex - section end: 19 -section 4 g4 Syntax - command semicolon 20 - command dollar 21 - command percent 22 - operator dot 23 - operator assign 24 - operator equalsign 25 - operator replace 26 - operator plussign 27 - operator minussign 28 - operator asterisk 29 - operator slash 30 - operator power 31 - operator caret 32 - operator geqsign 33 - operator greater 34 - operator leqsign 35 - operator less 36 - operator tilde 37 - command group 38 - operator AND 39 - command BEGIN 40 - command block 41 - command COMMENT 42 - operator CONS 43 - command END 44 - type EQUATION 45 - operator FIRST 46 - command FOR 47 - command FOREACH 48 - operator GEQ 49 - command GOTO 50 - operator GREATERP 51 - command IF 52 - operator LIST 53 - operator OR 54 - command PROCEDURE 55 - command REPEAT 56 - operator REST 57 - command RETURN 58 - operator REVERSE 59 - type RULE 60 - type Free Variable 61 - type Optional Free Variable 62 - operator SECOND 63 - operator SET 64 - operator SETQ 65 - operator THIRD 66 - operator WHEN 67 - ---- return from file $reduce/doc/help/syntax.tex - ---- input file $reduce/doc/help/arith.tex - section end: 68 -section 5 g5 Arithmetic Operations - introduction ARITHMETIC\_OPERATIONS 69 - operator ABS 70 - switch ADJPREC 71 - operator ARG 72 - operator CEILING 73 - operator CHOOSE 74 - operator DEG2DMS 75 - operator DEG2RAD 76 - operator DIFFERENCE 77 - operator DILOG 78 - operator DMS2DEG 79 - operator DMS2RAD 80 - operator FACTORIAL 81 - operator FIX 82 - operator FIXP 83 - operator FLOOR 84 - operator EXPT 85 - operator GCD 86 - operator LN 87 - operator LOG 88 - operator LOGB 89 - operator MAX 90 - operator MIN 91 - operator MINUS 92 - operator NEXTPRIME 93 - switch NOCONVERT 94 - operator NORM 95 - operator PERM 96 - operator PLUS 97 - operator QUOTIENT 98 - operator RAD2DEG 99 - operator RAD2DMS 100 - operator RECIP 101 - operator REMAINDER 102 - operator ROUND 103 - command SETMOD 104 - operator SIGN 105 - operator SQRT 106 - operator TIMES 107 - ---- return from file $reduce/doc/help/arith.tex - ---- input file $reduce/doc/help/boolean.tex - section end: 108 -section 6 g6 Boolean Operators - concept boolean value 109 - operator EQUAL 110 - operator EVENP 111 - concept false 112 - operator FREEOF 113 - operator LEQ 114 - operator LESSP 115 - operator MEMBER 116 - operator NEQ 117 - operator NOT 118 - operator NUMBERP 119 - operator ORDP 120 - operator PRIMEP 121 - concept TRUE 122 - ---- return from file $reduce/doc/help/boolean.tex - ---- input file $reduce/doc/help/command.tex - section end: 123 -section 7 g7 General Commands - command BYE 124 - command CONT 125 - command DISPLAY 126 - command LOAD\_PACKAGE 127 - command PAUSE 128 - command QUIT 129 - operator RECLAIM 130 - command REDERR 131 - command RETRY 132 - command SAVEAS 133 - command SHOWTIME 134 - command WRITE 135 - ---- return from file $reduce/doc/help/command.tex - ---- input file $reduce/doc/help/algebra.tex - section end: 136 -section 8 g8 Algebraic Operators - operator APPEND 137 - operator ARBINT 138 - operator ARBCOMPLEX 139 - operator ARGLENGTH 140 - operator COEFF 141 - operator COEFFN 142 - operator CONJ 143 - operator CONTINUED_FRACTION 144 - operator DECOMPOSE 145 - operator DEG 146 - operator DEN 147 - operator DF 148 - operator EXPAND\_CASES 149 - operator EXPREAD 150 - operator FACTORIZE 151 - operator HYPOT 152 - operator IMPART 153 - operator INT 154 - operator INTERPOL 155 - operator LCOF 156 - operator LENGTH 157 - operator LHS 158 - operator LIMIT 159 - operator LPOWER 160 - operator LTERM 161 - operator MAINVAR 162 - operator MAP 163 - command MKID 164 - operator NPRIMITIVE 165 - operator NUM 166 - operator ODESOLVE 167 - type ONE\_OF 168 - operator PART 169 - operator PF 170 - operator PROD 171 - operator REDUCT 172 - operator REPART 173 - operator RESULTANT 174 - operator RHS 175 - operator ROOT\_OF 176 - operator SELECT 177 - operator SHOWRULES 178 - operator SOLVE 179 - operator SORT 180 - operator STRUCTR 181 - operator SUB 182 - operator SUM 183 - operator WS 184 - ---- return from file $reduce/doc/help/algebra.tex - ---- input file $reduce/doc/help/declare.tex - section end: 185 -section 9 g9 Declarations - command ALGEBRAIC 186 - declaration ANTISYMMETRIC 187 - declaration ARRAY 188 - command CLEAR 189 - command CLEARRULES 190 - command DEFINE 191 - declaration DEPEND 192 - declaration EVEN 193 - declaration FACTOR 194 - command FORALL 195 - declaration INFIX 196 - declaration INTEGER 197 - declaration KORDER 198 - command LET 199 - declaration LINEAR 200 - declaration LINELENGTH 201 - command LISP 202 - declaration LISTARGP 203 - declaration NODEPEND 204 - command MATCH 205 - declaration NONCOM 206 - declaration NONZERO 207 - declaration ODD 208 - command OFF 209 - command ON 210 - declaration OPERATOR 211 - declaration ORDER 212 - declaration PRECEDENCE 213 - declaration PRECISION 214 - declaration PRINT\_PRECISION 215 - declaration REAL 216 - declaration REMFAC 217 - declaration SCALAR 218 - declaration SCIENTIFIC\_NOTATION 219 - declaration SHARE 220 - command SYMBOLIC 221 - declaration SYMMETRIC 222 - declaration TR 223 - declaration UNTR 224 - declaration VARNAME 225 - command WEIGHT 226 - operator WHERE 227 - command WHILE 228 - command WTLEVEL 229 - ---- return from file $reduce/doc/help/declare.tex - ---- input file $reduce/doc/help/io.tex - section end: 230 -section 10 g10 Input and Output - command IN 231 - command INPUT 232 - command OUT 233 - command SHUT 234 - ---- return from file $reduce/doc/help/io.tex - ---- input file $reduce/doc/help/elemfn.tex - section end: 235 -section 11 g11 Elementary Functions - operator ACOS 236 - operator ACOSH 237 - operator ACOT 238 - operator ACOTH 239 - operator ACSC 240 - operator ACSCH 241 - operator ASEC 242 - operator ASECH 243 - operator ASIN 244 - operator ASINH 245 - operator ATAN 246 - operator ATANH 247 - operator ATAN2 248 - operator COS 249 - operator COSH 250 - operator COT 251 - operator COTH 252 - operator CSC 253 - operator CSCH 254 - operator ERF 255 - operator EXP 256 - operator SEC 257 - operator SECH 258 - operator SIN 259 - operator SINH 260 - operator TAN 261 - operator TANH 262 - ---- return from file $reduce/doc/help/elemfn.tex - ---- input file $reduce/doc/help/switch.tex - section end: 263 -section 12 g12 General Switches - introduction SWITCHES 264 - switch ALGINT 265 - switch ALLBRANCH 266 - switch ALLFAC 267 - switch ARBVARS 268 - switch BALANCED\_MOD 269 - switch BFSPACE 270 - switch COMBINEEXPT 271 - switch COMBINELOGS 272 - switch COMP 273 - switch COMPLEX 274 - switch CREF 275 - switch CRAMER 276 - switch DEFN 277 - switch DEMO 278 - switch DFPRINT 279 - switch DIV 280 - switch ECHO 281 - switch ERRCONT 282 - switch EVALLHSEQP 283 - switch EXP 284 - switch EXPANDLOGS 285 - switch EZGCD 286 - switch FACTOR 287 - switch FAILHARD 288 - switch FORT 289 - switch FORTUPPER 290 - switch FULLPREC 291 - switch FULLROOTS 292 - switch GC 293 - switch GCD 294 - switch HORNER 295 - switch IFACTOR 296 - switch INT 297 - switch INTSTR 298 - switch LCM 299 - switch LESSSPACE 300 - switch LIMITEDFACTORS 301 - switch LIST 302 - switch LISTARGS 303 - switch MCD 304 - switch MODULAR 305 - switch MSG 306 - switch MULTIPLICITIES 307 - switch NAT 308 - switch NERO 309 - switch NOARG 310 - switch NOLNR 311 - switch NOSPLIT 312 - switch NUMVAL 313 - switch OUTPUT 314 - switch OVERVIEW 315 - switch PERIOD 316 - switch PRECISE 317 - switch PRET 318 - switch PRI 319 - switch RAISE 320 - switch RAT 321 - switch RATARG 322 - switch RATIONAL 323 - switch RATIONALIZE 324 - switch RATPRI 325 - switch REVPRI 326 - switch RLISP88 327 - switch ROUNDALL 328 - switch ROUNDBF 329 - switch ROUNDED 330 - switch SAVESTRUCTR 331 - switch SOLVESINGULAR 332 - switch TIME 333 - switch TRALLFAC 334 - switch TRFAC 335 - switch TRIGFORM 336 - switch TRINT 337 - switch TRNONLNR 338 - switch VAROPT 339 - ---- return from file $reduce/doc/help/switch.tex - ---- input file $reduce/doc/help/matrix.tex - section end: 340 -section 13 g13 Matrix Operations - operator COFACTOR 341 - operator DET 342 - operator MAT 343 - operator MATEIGEN 344 - declaration MATRIX 345 - operator NULLSPACE 346 - operator RANK 347 - operator TP 348 - operator TRACE 349 - ---- return from file $reduce/doc/help/matrix.tex - ---- input file $reduce/doc/help/pk-groeb.tex - section end: 350 -section 14 g14 Groebner package - introduction Groebner bases 351 - concept Ideal Parameters 352 - subsection 15 g15 Term order - introduction Term order 353 - operator torder 354 - operator torder_compile 355 - concept lex term order 356 - concept gradlex term order 357 - concept revgradlex term order 358 - concept gradlexgradlex term order 359 - concept gradlexrevgradlex term order 360 - concept lexgradlex term order 361 - concept lexrevgradlex term order 362 - concept weighted term order 363 - concept graded term order 364 - concept matrix term order 365 - section end: 366 - subsection 16 g16 Basic Groebner operators - operator gvars 367 - operator groebner 368 - operator groebner\_walk 369 - ######## reference to (l e x) not found, - ######## reference to (g r a d e d) not found, - ######## reference to (w e i g h t e d) not found, - ######## reference to (g r a d e d) not found, - ######## reference to (w e i g h t e d) not found, - ######## reference to (k e x) not found, - ######## reference to (l e x) not found, - switch groebopt 370 - variable gvarslast 371 - switch groebprereduce 372 - switch groebfullreduction 373 - switch gltbasis 374 - variable gltb 375 - variable glterms 376 - switch groebstat 377 - switch trgroeb 378 - switch trgroebs 379 - operator gzerodim? 380 - operator gdimension 381 - operator gindependent\_sets 382 - operator dd_groebner 383 - operator glexconvert 384 - operator greduce 385 - operator preduce 386 - operator idealquotient 387 - operator hilbertpolynomial 388 - operator saturation 389 - section end: 390 - subsection 17 g17 Factorizing Groebner bases - operator groebnerf 391 - variable groebmonfac 392 - variable groebresmax 393 - variable groebrestriction 394 - section end: 395 - subsection 18 g18 Tracing Groebner bases - switch groebprot 396 - variable groebprotfile 397 - operator groebnert 398 - operator preducet 399 - section end: 400 - subsection 19 g19 Groebner Bases for Modules - concept Module 401 - variable gmodule 402 - section end: 403 - subsection 20 g20 Computing with distributive polynomials - operator gsort 404 - operator gsplit 405 - operator gspoly 406 - ---- return from file $reduce/doc/help/pk-groeb.tex - ---- input file $reduce/doc/help/hephys.tex - section end: 407 - section end: 408 -section 21 g21 High Energy Physics - introduction HEPHYS 409 - operator HE-dot 410 - operator EPS 411 - operator G 412 - declaration INDEX 413 - command MASS 414 - command MSHELL 415 - declaration NOSPUR 416 - declaration REMIND 417 - declaration SPUR 418 - command VECDIM 419 - declaration VECTOR 420 - ---- return from file $reduce/doc/help/hephys.tex - ---- input file $reduce/doc/help/pk-numer.tex - section end: 421 -section 22 g22 Numeric Package - introduction Numeric Package 422 - type Interval 423 - concept numeric accuracy 424 - switch TRNUMERIC 425 - operator num_min 426 - operator num_solve 427 - operator num_int 428 - operator num_odesolve 429 - operator bounds 430 - concept Chebyshev fit 431 - operator num_fit 432 - ---- return from file $reduce/doc/help/pk-numer.tex - ---- input file $reduce/doc/help/pk-roots.tex - section end: 433 -section 23 g23 Roots Package - introduction Roots Package 434 - operator MKPOLY 435 - operator NEARESTROOT 436 - operator REALROOTS 437 - operator ROOTACC 438 - operator ROOTS 439 - operator ROOT\_VAL 440 - variable ROOTSCOMPLEX 441 - variable ROOTSREAL 442 - ---- return from file $reduce/doc/help/pk-roots.tex - ---- input file $reduce/doc/help/pk-specf.tex - section end: 443 -section 24 g24 Special Functions - introduction Special Function Package 444 - concept Constants 445 - subsection 25 g25 Bernoulli Euler Zeta - operator BERNOULLI 446 - operator BERNOULLIP 447 - operator EULER 448 - operator EULERP 449 - operator ZETA 450 - section end: 451 - subsection 26 g26 Bessel Functions - operator BESSELJ 452 - operator BESSELY 453 - operator HANKEL1 454 - operator HANKEL2 455 - operator BESSELI 456 - operator BESSELK 457 - operator StruveH 458 - operator StruveL 459 - operator KummerM 460 - operator KummerU 461 - operator WhittakerW 462 - section end: 463 - subsection 27 g27 Airy Functions - operator Airy_Ai 464 - operator Airy_Bi 465 - operator Airy_Aiprime 466 - operator Airy_Biprime 467 - section end: 468 - subsection 28 g28 Jacobi's Elliptic Functions and Elliptic Integrals - operator JacobiSN 469 - operator JacobiCN 470 - operator JacobiDN 471 - operator JacobiCD 472 - operator JacobiSD 473 - operator JacobiND 474 - operator JacobiDC 475 - operator JacobiNC 476 - operator JacobiSC 477 - operator JacobiNS 478 - operator JacobiDS 479 - operator JacobiCS 480 - operator JacobiAMPLITUDE 481 - operator AGM_FUNCTION 482 - operator LANDENTRANS 483 - operator EllipticF 484 - operator EllipticK 485 - operator EllipticKprime 486 - operator EllipticE 487 - operator EllipticTHETA 488 - operator JacobiZETA 489 - section end: 490 - subsection 29 g29 Gamma and Related Functions - operator POCHHAMMER 491 - operator GAMMA 492 - operator BETA 493 - operator PSI 494 - operator POLYGAMMA 495 - section end: 496 - subsection 30 g30 Miscellaneous Functions - operator DILOG extended 497 - operator Lambert\_W function 498 - section end: 499 - subsection 31 g31 Orthogonal Polynomials - operator ChebyshevT 500 - operator ChebyshevU 501 - operator HermiteP 502 - operator LaguerreP 503 - operator LegendreP 504 - operator JacobiP 505 - operator GegenbauerP 506 - operator SolidHarmonicY 507 - operator SphericalHarmonicY 508 - section end: 509 - subsection 32 g32 Integral Functions - operator Si 510 - operator Shi 511 - operator s_i 512 - operator Ci 513 - operator Chi 514 - operator ERF extended 515 - operator erfc 516 - operator Ei 517 - operator Fresnel_C 518 - operator Fresnel_S 519 - section end: 520 - subsection 33 g33 Combinatorial Operators - operator BINOMIAL 521 - operator STIRLING1 522 - operator STIRLING2 523 - section end: 524 - subsection 34 g34 3j and 6j symbols - operator ThreejSymbol 525 - operator Clebsch_Gordan 526 - operator SixjSymbol 527 - section end: 528 - subsection 35 g35 Miscellaneous - operator HYPERGEOMETRIC 529 - operator MeijerG 530 - operator Heaviside 531 - operator erfi 532 - ---- return from file $reduce/doc/help/pk-specf.tex - ---- input file $reduce/doc/help/taylor.tex - section end: 533 - section end: 534 -section 36 g36 Taylor series - introduction TAYLOR 535 - operator taylor 536 - switch taylorautocombine 537 - switch taylorautoexpand 538 - operator taylorcombine 539 - switch taylorkeeporiginal 540 - operator taylororiginal 541 - switch taylorprintorder 542 - variable taylorprintterms 543 - operator taylorrevert 544 - operator taylorseriesp 545 - operator taylortemplate 546 - operator taylortostandard 547 - ---- return from file $reduce/doc/help/taylor.tex - ---- input file $reduce/doc/help/pk-gplot.tex - section end: 548 -section 37 g37 Gnuplot package - introduction GNUPLOT and REDUCE 549 - concept Axes names 550 - type Pointset 551 - command PLOT 552 - command PLOTRESET 553 - variable title 554 - variable xlabel 555 - variable ylabel 556 - variable zlabel 557 - variable terminal 558 - variable size 559 - variable view 560 - switch contour 561 - switch surface 562 - switch hidden3d 563 - switch PLOTKEEP 564 - switch PLOTREFINE 565 - variable plot_xmesh 566 - variable plot_ymesh 567 - switch SHOW_GRID 568 - switch TRPLOT 569 - ---- return from file $reduce/doc/help/pk-gplot.tex - ---- input file $reduce/doc/help/linalg.tex - section end: 570 -section 38 g38 Linear Algebra package - introduction Linear Algebra package 571 - switch fast_la 572 - operator add_columns 573 - operator add_rows 574 - operator add_to_columns 575 - operator add_to_rows 576 - operator augment_columns 577 - operator band_matrix 578 - operator block_matrix 579 - operator char_matrix 580 - operator char_poly 581 - operator cholesky 582 - operator coeff_matrix 583 - operator column_dim 584 - operator companion 585 - operator copy_into 586 - operator diagonal 587 - operator extend 588 - operator find_companion 589 - operator get_columns 590 - operator get_rows 591 - operator gram_schmidt 592 - operator hermitian_tp 593 - operator hessian 594 - operator hilbert 595 - operator jacobian 596 - operator jordan_block 597 - operator lu_decom 598 - operator make_identity 599 - operator matrix_augment 600 - operator matrixp 601 - operator matrix_stack 602 - operator minor 603 - operator mult_columns 604 - operator mult_rows 605 - operator pivot 606 - operator pseudo_inverse 607 - operator random_matrix 608 - operator remove_columns 609 - operator remove_rows 610 - operator row_dim 611 - operator rows_pivot 612 - operator simplex 613 - operator squarep 614 - operator stack_rows 615 - operator sub_matrix 616 - operator svd 617 - operator swap_columns 618 - operator swap_entries 619 - operator swap_rows 620 - operator symmetricp 621 - operator toeplitz 622 - operator vandermonde 623 - ---- return from file $reduce/doc/help/linalg.tex - ---- input file $reduce/doc/help/normform.tex - section end: 624 -section 39 g39 Matrix Normal Forms - operator Smithex 625 - operator Smithex\_int 626 - operator Frobenius 627 - operator Ratjordan 628 - operator Jordansymbolic 629 - operator Jordan 630 - ---- return from file $reduce/doc/help/normform.tex - ---- input file $reduce/doc/help/pk-misc.tex - section end: 631 -section 40 g40 Miscellaneous Packages - introduction Miscellaneous Packages 632 - package ALGINT 633 - package APPLYSYM 634 - package ARNUM 635 - package ASSIST 636 - package AVECTOR 637 - package BOOLEAN 638 - package CALI 639 - package CAMAL 640 - package CHANGEVR 641 - package COMPACT 642 - package CRACK 643 - package CVIT 644 - package DEFINT 645 - package DESIR 646 - package DFPART 647 - package DUMMY 648 - package EXCALC 649 - package FPS 650 - package FIDE 651 - package GENTRAN 652 - package IDEALS 653 - package INEQ 654 - package INVBASE 655 - package LAPLACE 656 - package LIE 657 - package MODSR 658 - package NCPOLY 659 - package ORTHOVEC 660 - package PHYSOP 661 - package PM 662 - package RANDPOLY 663 - package REACTEQN 664 - package RESET 665 - package RESIDUE 666 - package RLFI 667 - package SCOPE 668 - package SETS 669 - package SPDE 670 - package SYMMETRY 671 - package TPS 672 - package TRI 673 - package TRIGSIMP 674 - package XCOLOR 675 - package XIDEAL 676 - package WU 677 - package ZEILBERG 678 - package ZTRANS 679 - ---- return from file $reduce/doc/help/pk-misc.tex - ---- input file $reduce/doc/help/outmode.tex - section end: 680 -section 41 g41 Outmoded Operations - command ED 681 - command EDITDEF 682 - ---- return from file $reduce/doc/help/outmode.tex - section end: 683 - section end: 684 - -nil - - - -bye; - -End of Lisp run after 6.61+0.35 seconds - -+++ Transcript closed at end of run +++ DELETED r38/help/mkhtml.log Index: r38/help/mkhtml.log ================================================================== --- r38/help/mkhtml.log +++ r38/help/mkhtml.log @@ -1,2671 +0,0 @@ -+++ Transcript to mkhtml.log started at Fri Apr 16 10:24:02 2004 +++ -Codemist Standard Lisp 5.00 for Windows: Apr 13 2004 -Created: Tue Apr 13 21:59:08 2004 - -REDUCE 3.8, 15-Apr-04 ... -Memory allocation: 54522624 bytes - -+++ About to read file "mkhtml.red" - -% -% This file runs the "*.tex" to "redhelp.html" conversion code. As -% well as providing the top level direction to the process it patches up -% for at least some of the places where the conversion code had been -% written in a manner not strongly related to the portability objectives of -% Standard Lisp... -% - -symbolic; -nil - - -off echo; -nil - - -on backtrace; -nil - - -on comp; -nil - - -!*windows := t; -t - - - -fluid '(package); -nil - - - -package := 'redhelp; -redhelp - - - -symbolic procedure deletip(a, b); - delete(a, b);+++ deletip compiled as link to delete - -deletip - - - -symbolic procedure inf x; - char!-code x;+++ inf compiled as link to char-code - -inf - - - -symbolic procedure channellinelength(f, l); - begin - f := wrs f; - l := linelength l; - wrs f; - return l - end;+++ channellinelength compiled, 12 + 4 bytes - -channellinelength - - - -symbolic procedure channelprin2(f, x); - begin - f := wrs f; - prin2 x; - wrs f; - return x - end;+++ channelprin2 compiled, 10 + 8 bytes - -channelprin2 - - - -symbolic procedure channelprin2t(f, x); - begin - f := wrs f; - prin2t x; - wrs f; - return x - end;+++ channelprin2t compiled, 10 + 8 bytes - -channelprin2t - - - -symbolic macro procedure channelprintf u; - begin - scalar g; - g := gensym(); - return list('prog, list g, - list('setq, g, list('wrs, cadr u)), - 'printf . cddr u, - list('wrs, g)) - end;+++ channelprintf_76z7r3p3pl4b compiled, 33 + 24 bytes - -channelprintf - - - -symbolic procedure channelterpri f; - begin - f := wrs f; - terpri(); - wrs f; - end;+++ channelterpri compiled, 10 + 4 bytes - -channelterpri - - - -symbolic procedure channelreadch f; - begin - scalar c; - f := rds f; - c := readch(); - rds f; - return c - end;+++ channelreadch compiled, 12 + 4 bytes - -channelreadch - - - -in "comphelp.red"$ -nil - -nil - -nil - -nil - -nil - -nil - -nil - -nil - -nil - -nil - -0 - -nil -+++++ global !*raise converted to fluid -+++ job compiled, 113 + 148 bytes - -job - -nil - -nil - -nil -+++ rdch compiled, 3 + 8 bytes - -rdch -+++ rdch!* compiled, 3 + 12 bytes - -rdch!* -+++ rdchr0 compiled, 54 + 64 bytes - -rdchr0 -+++ unrdch compiled, 3 + 12 bytes - -unrdch -+++ myskip compiled, 5 + 8 bytes - -myskip -+++ myskipl compiled, 9 + 8 bytes - -myskipl -+++ myskipstring compiled, 41 + 24 bytes - -myskipstring - -nil - -lower -+++ mytoken compiled, 83 + 44 bytes - -mytoken -+++ mystring compiled, 28 + 16 bytes - -mystring -+++ mystring2 compiled, 14 + 12 bytes - -mystring2 -+++ mystring2!] compiled, 14 + 12 bytes - -mystring2!] -+++ mystring_nodename compiled, 41 + 40 bytes - -mystring_nodename -+++ mystring3 compiled, 23 + 16 bytes - -mystring3 -+++ raisestring compiled, 67 + 20 bytes - -raisestring -+++ lowerstring compiled, 75 + 20 bytes - -lowerstring -+++ mycompress compiled, 12 + 8 bytes - -mycompress -+++ mainloop compiled, 200 + 156 bytes - -mainloop - -*** local variable u in procedure include not used -+++ include compiled, 129 + 100 bytes - -include - -include - -include - -null - -null -+++ print_indent compiled, 21 + 20 bytes - -print_indent - -nil - -0 - -type - -seq - -lab - -count - -name -+++ reset compiled, 51 + 40 bytes - -reset -+++ sectappend compiled, 11 + 8 bytes - -sectappend -+++ section compiled, 10 + 20 bytes - -section -+++ close_section compiled, 39 + 24 bytes - -close_section -+++ close_section1 compiled, 40 + 40 bytes - -close_section1 -+++ write_sections compiled, 15 + 12 bytes - -write_sections -+++ write_section compiled, 67 + 52 bytes - -write_section -+++ make_dir_entry compiled, 8 + 8 bytes - -make_dir_entry -+++ help_gensym compiled, 10 + 12 bytes - -help_gensym -+++ open_section compiled, 172 + 112 bytes - -open_section - -section - -section - -section - -*** local variable u in procedure beg not used -+++ beg compiled, 87 + 68 bytes - -beg - -beg - -*** local variable u in procedure mmain not used -+++ mmain compiled as link to mainloop - -mmain - -mmain - -nil -+++ clean_name compiled, 27 + 16 bytes - -clean_name - -(((!,) . comma_sign) ((!.) . dot_sign) ((!;) . semicolon_sign) ((!%) . -percent_sign) ((!$) . dollar_sign) ((!: !=) . assign_sign) ((!=) . equal_sign) ( -(!+) . plus_sign) ((!-) . minus_sign) ((!*) . times_sign) ((!/) . slash_sign) (( -!* !*) . power_sign) ((!$ !> != !$) . geq_sign) ((!> !=) . geq_sign) ((!>) . -greater_sign) ((!$ !< != !$) . leq_sign) ((!< !=) . leq_sign) ((!<) . less_sign) -((!< !<) . block)) -+++ make_label compiled, 64 + 40 bytes - -make_label -+++ get_label compiled, 8 + 8 bytes - -get_label -+++ patch_ compiled, 29 + 16 bytes - -patch_ -+++ get_label1 compiled, 32 + 40 bytes - -get_label1 -+++ get_label2 compiled, 103 + 56 bytes - -get_label2 -+++ update_labels compiled, 47 + 36 bytes - -update_labels -+++ node compiled, 197 + 136 bytes - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node -+++ part compiled, 63 + 68 bytes - -part -+++ par_heading compiled, 26 + 16 bytes - -par_heading - -*** local variable type in procedure vpart not used -+++ vpart compiled, 13 + 32 bytes - -vpart -+++ vpart0 compiled, 81 + 72 bytes - -vpart0 -+++ compareahead compiled, 6 + 8 bytes - -compareahead -+++ compareahead1 compiled, 28 + 12 bytes - -compareahead1 -+++ look_ahead_suctx941k0ih compiled, 11 + 16 bytes - -look_ahead - -*** local variable type in procedure examples_part not used -+++ examples_part compiled, 381 + 204 bytes - -examples_part -+++ non_verb_block compiled, 15 + 36 bytes - -non_verb_block -+++ make_multi_out compiled, 121 + 60 bytes - -make_multi_out -+++ cut_lines compiled, 37 + 8 bytes - -cut_lines -+++ matchleft compiled, 22 + 16 bytes - -matchleft -+++ matcharb compiled, 12 + 8 bytes - -matcharb -+++ read_one_line compiled, 15 + 12 bytes - -read_one_line - -part - -part - -part - -part - -part - -part - -part - -part - -*** local variable type in procedure do-itemize not used -+++ do!-itemize compiled, 6 + 20 bytes - -do!-itemize - -do!-itemize -+++ context_error compiled, 11 + 20 bytes - -context_error - -*** local variable u in procedure verb not used -+++ verb compiled, 12 + 24 bytes - -verb - -verb - -*** local variable u in procedure ldots not used -+++ ldots compiled, 3 + 12 bytes - -ldots - -ldots - -nil - -*** local variable u in procedure cdots not used -+++ cdots compiled, 3 + 12 bytes - -cdots - -cdots - -nil - -*** local variable u in procedure cdot not used -+++ cdot compiled, 3 + 12 bytes - -cdot - -cdot - -nil - -*** local variable u in procedure write_pi not used -+++ write_pi compiled, 3 + 12 bytes - -write_pi - -write_pi - -nil - -*** local variable u in procedure emphase not used -+++ emphase compiled, 3 + 12 bytes - -emphase - -emphase - -*** local variable u in procedure meta not used -+++ meta compiled, 18 + 20 bytes - -meta - -meta - -*** local variable u in procedure italic not used -+++ italic compiled, 8 + 24 bytes - -italic - -*** local variable u in procedure switchitalic not used -+++ switchitalic compiled, 1 + 4 bytes - -switchitalic - -italic - -italic - -italic - -*** local variable u in procedure nameref not used -+++ nameref compiled, 3 + 12 bytes - -nameref - -nameref - -*** local variable u in procedure ref not used -+++ ref compiled, 3 + 12 bytes - -ref - -ref -+++ see compiled, 4 + 12 bytes - -see - -see - -*** local variable u in procedure myname not used -+++ myname compiled, 3 + 12 bytes - -myname - -myname - -*** local variable u in procedure myindex not used -+++ myindex compiled, 5 + 20 bytes - -myindex - -myindex - -*** local variable u in procedure nameindex not used -+++ nameindex compiled, 10 + 24 bytes - -nameindex - -nameindex - -*** local variable u in procedure reduce not used -+++ reduce compiled, 3 + 12 bytes - -reduce - -reduce - -nil - -*** local variable u in procedure rept not used -+++ rept compiled, 3 + 12 bytes - -rept - -rept - -nil - -*** local variable u in procedure optional not used -+++ optional compiled, 3 + 12 bytes - -optional - -optional - -nil - -*** local variable u in procedure myexp not used -+++ myexp compiled, 5 + 16 bytes - -myexp - -myexp - -*** local variable u in procedure formula not used -+++ formula compiled, 3 + 12 bytes - -formula - -formula - -formula - -*** local variable u in procedure rfrac not used -+++ rfrac compiled, 8 + 24 bytes - -rfrac - -rfrac - -*** local variable u in procedure item not used -+++ item compiled, 14 + 32 bytes - -item - -item - -*** local variable u in procedure texonly1 not used -+++ texonly1 compiled, 48 + 44 bytes - -texonly1 - -texonly1 - -*** local variable u in procedure texonly2 not used -+++ texonly2 compiled, 67 + 68 bytes - -texonly2 - -texonly2 - -*** local variable u in procedure infoonly not used -+++ infoonly compiled, 6 + 12 bytes - -infoonly - -infoonly -+++ reporttopic compiled, 12 + 20 bytes - -reporttopic -+++ substipq compiled, 32 + 12 bytes - -substipq - -nil - - -in "helphtml.red"$ -nil - -nil - -t - -t - -"R" - -"TT" - -nil -+++ rootname compiled, 13 + 16 bytes - -rootname -+++ dest_directory compiled, 13 + 16 bytes - -dest_directory - -nil - -0 -+++ reset_html compiled, 8 + 24 bytes - -reset_html -+++ html_open compiled, 15 + 20 bytes - -html_open -+++ html_close compiled, 17 + 24 bytes - -html_close - -*** local variable u in procedure open_current_base_dir not used -+++ open_current_base_dir compiled, 1 + 4 bytes - -open_current_base_dir -+++ close_current_base_dir compiled, 1 + 4 bytes - -close_current_base_dir -+++ make_html_file_name compiled, 74 + 52 bytes - -make_html_file_name -+++ open_node_file compiled, 73 + 64 bytes - -open_node_file -+++ close_node_file compiled, 11 + 12 bytes - -close_node_file -+++ node_file_name compiled, 2 + 8 bytes - -node_file_name -+++ initoutput compiled, 1 + 4 bytes - -initoutput -+++ endoutput compiled, 1 + 4 bytes - -endoutput -+++ verbatim compiled, 3 + 8 bytes - -verbatim -+++ newfont compiled, 10 + 16 bytes - -newfont -+++ fontoff compiled, 5 + 12 bytes - -fontoff -+++ fonton compiled, 5 + 12 bytes - -fonton -+++ myprin2 compiled, 6 + 16 bytes - -myprin2 - -(!< !> !" !&) - -(!< !> !" !&) -+++ myprin2_protected compiled, 14 + 16 bytes - -myprin2_protected - -nil -+++ emit_start_verbatim compiled, 7 + 20 bytes - -emit_start_verbatim -+++ emit_end_verbatim compiled, 7 + 24 bytes - -emit_end_verbatim -+++ verbprin2 compiled, 55 + 56 bytes - -verbprin2 -+++ myterpri compiled, 3 + 12 bytes - -myterpri -+++ number4out compiled, 24 + 32 bytes - -number4out -+++ textout compiled, 149 + 84 bytes - -textout - -+++ par_heading redefined -+++ par_heading compiled, 26 + 24 bytes - -par_heading -+++ base_new_dir compiled, 6 + 16 bytes - -base_new_dir -+++ emit_dir_new compiled, 1 + 4 bytes - -emit_dir_new -+++ emit_dir_key compiled as link to emit_node_key - -emit_dir_key -+++ emit_dir_separator compiled as link to emit_node_separator - -emit_dir_separator -+++ emit_dir_label compiled as link to emit_node_label - -emit_dir_label -+++ emit_dir_title compiled, 6 + 12 bytes - -emit_dir_title -+++ emit_dir_browse compiled as link to emit_node_browse - -emit_dir_browse -+++ emit_node_separator compiled, 15 + 40 bytes - -emit_node_separator -+++ set_tab compiled, 1 + 4 bytes - -set_tab -+++ release_tab compiled, 1 + 4 bytes - -release_tab -+++ textout_name compiled, 26 + 16 bytes - -textout_name -+++ textout2 compiled, 29 + 16 bytes - -textout2 -+++ printem compiled, 18 + 24 bytes - -printem - -+++ printem redefined -+++ printem compiled, 19 + 28 bytes - -printem -+++ printref compiled, 234 + 124 bytes - -printref -+++ printnameref compiled as link to printref - -printnameref - -nil -+++ emit_node_keys compiled, 28 + 24 bytes - -emit_node_keys -+++ emit_node_key compiled as link to emit_hidden_node_key - -emit_node_key -+++ emit_hidden_node_key compiled, 33 + 16 bytes - -emit_hidden_node_key -+++ emit_node_label compiled, 24 + 40 bytes - -emit_node_label - -*** local variable dummy in procedure emit_node_title not used - -*** local variable type in procedure emit_node_title not used -+++ emit_node_title compiled, 33 + 52 bytes - -emit_node_title -+++ emit_node_browse compiled, 11 + 20 bytes - -emit_node_browse -+++ print_bold compiled, 19 + 28 bytes - -print_bold -+++ emit_dir_header compiled, 8 + 28 bytes - -emit_dir_header -+++ emit_dir_entry compiled, 66 + 68 bytes - -emit_dir_entry -+++ print_newline compiled, 13 + 32 bytes - -print_newline -+++ second_newline compiled, 4 + 12 bytes - -second_newline -+++ print_tab compiled, 4 + 16 bytes - -print_tab -+++ html_indexfile compiled, 327 + 124 bytes - -html_indexfile -+++ sort_term compiled, 60 + 12 bytes - -sort_term -+++ html_indexfile_sort compiled, 4 + 8 bytes - -html_indexfile_sort -+++ html_indexfile_sort1 compiled, 27 + 12 bytes - -html_indexfile_sort1 -+++ html_indexfile_subsetp compiled, 18 + 4 bytes - -html_indexfile_subsetp -+++ lisp_indexfile compiled, 329 + 136 bytes - -lisp_indexfile -+++ printstruct compiled, 7 + 16 bytes - -printstruct -+++ printstruct1 compiled, 58 + 16 bytes - -printstruct1 - -nil - - -in "minitex.red"$ -nil - -nil - -nil - -!\ - -!^ - -!_ - -2 - -3 - -4 -+++ mintex_convert0 compiled, 38 + 12 bytes - -mintex_convert0 -+++ mintex_convert compiled as link to mintex_convert0 - -mintex_convert -+++ minitex compiled, 102 + 40 bytes - -minitex -+++ minitex_pop_char compiled, 11 + 8 bytes - -minitex_pop_char -+++ minitex_skip compiled, 11 + 8 bytes - -minitex_skip -+++ minitex_next_char compiled, 6 + 8 bytes - -minitex_next_char -+++ struct compiled, 9 + 8 bytes - -struct -+++ make_chain compiled, 317 + 120 bytes - -make_chain - -*** local variable font in procedure make_char not used - -*** local variable cs in procedure make_char not used -+++ make_char compiled, 23 + 48 bytes - -make_char -+++ make_frac compiled, 211 + 76 bytes - -make_frac - -*** local variable y1 in procedure make_line not used -+++ make_line compiled, 22 + 32 bytes - -make_line -+++ make_multi compiled, 63 + 60 bytes - -make_multi - -*** local variable font in procedure make_end not used -+++ make_end compiled, 4 + 16 bytes - -make_end - -nil - -(chain 0 0 0) - -nil - -*** local variable term in procedure make_escape not used -+++ make_escape compiled, 77 + 72 bytes - -make_escape -+++ my_compare compiled, 23 + 16 bytes - -my_compare -+++ minitex_collect compiled, 8 + 12 bytes - -minitex_collect -+++ minitex_do compiled, 25 + 20 bytes - -minitex_do - -minitex_chain -+++ minitex_chain compiled, 37 + 24 bytes - -minitex_chain - -minitex_char - -*** local variable font in procedure minitex_char not used -+++ minitex_char compiled, 23 + 24 bytes - -minitex_char - -minitex_line - -*** local variable font in procedure minitex_line not used -+++ minitex_line compiled, 38 + 28 bytes - -minitex_line -+++ minitex_putchar compiled, 31 + 24 bytes - -minitex_putchar - -nil - - - -dir_src := "../doc/help/"; -"../doc/help/" - - -% dir_src := "~/reduce/doc/help/"; - -on backtrace; -nil - - - -reset_html(); -nil - - - -job(bldmsg("%w.tex",package), "null.fil"); -**** unknown token: documentclass -**** unknown token: usepackage -**** unknown token: usepackage - ---- input file ../doc/help/concept.tex -section 2 g2 Concepts - type IDENTIFIER 1 - type KERNEL 2 - type STRING 3 - ---- return from file ../doc/help/concept.tex - ---- input file ../doc/help/variable.tex - section end: 4 -section 3 g3 Variables - variable assumptions 5 - variable CARD\_NO 6 - constant E 7 - variable EVAL\_MODE 8 - variable FORT\_WIDTH 9 - variable HIGH\_POW 10 - constant I 11 - constant INFINITY 12 - variable LOW\_POW 13 - constant NIL 14 - constant PI 15 - variable requirements 16 - variable ROOT\_MULTIPLICITIES 17 - constant T 18 - ---- return from file ../doc/help/variable.tex - ---- input file ../doc/help/syntax.tex - section end: 19 -section 4 g4 Syntax - command semicolon 20 - command dollar 21 - command percent 22 - operator dot 23 - operator assign 24 - operator equalsign 25 - operator replace 26 - operator plussign 27 - operator minussign 28 - operator asterisk 29 - operator slash 30 - operator power 31 - operator caret 32 - operator geqsign 33 - operator greater 34 - operator leqsign 35 - operator less 36 - operator tilde 37 - command group 38 - operator AND 39 - command BEGIN 40 - command block 41 - command COMMENT 42 - operator CONS 43 - command END 44 - type EQUATION 45 - operator FIRST 46 - command FOR 47 - command FOREACH 48 - operator GEQ 49 - command GOTO 50 - operator GREATERP 51 - command IF 52 - operator LIST 53 - operator OR 54 - command PROCEDURE 55 - command REPEAT 56 - operator REST 57 - command RETURN 58 - operator REVERSE 59 - type RULE 60 - type Free Variable 61 - type Optional Free Variable 62 - operator SECOND 63 - operator SET 64 - operator SETQ 65 - operator THIRD 66 - operator WHEN 67 - ---- return from file ../doc/help/syntax.tex - ---- input file ../doc/help/arith.tex - section end: 68 -section 5 g5 Arithmetic Operations - introduction ARITHMETIC\_OPERATIONS 69 - operator ABS 70 - switch ADJPREC 71 - operator ARG 72 - operator CEILING 73 - operator CHOOSE 74 - operator DEG2DMS 75 - operator DEG2RAD 76 - operator DIFFERENCE 77 - operator DILOG 78 - operator DMS2DEG 79 - operator DMS2RAD 80 - operator FACTORIAL 81 - operator FIX 82 - operator FIXP 83 - operator FLOOR 84 - operator EXPT 85 - operator GCD 86 - operator LN 87 - operator LOG 88 - operator LOGB 89 - operator MAX 90 - operator MIN 91 - operator MINUS 92 - operator NEXTPRIME 93 - switch NOCONVERT 94 - operator NORM 95 - operator PERM 96 - operator PLUS 97 - operator QUOTIENT 98 - operator RAD2DEG 99 - operator RAD2DMS 100 - operator RECIP 101 - operator REMAINDER 102 - operator ROUND 103 - command SETMOD 104 - operator SIGN 105 - operator SQRT 106 - operator TIMES 107 - ---- return from file ../doc/help/arith.tex - ---- input file ../doc/help/boolean.tex - section end: 108 -section 6 g6 Boolean Operators - concept boolean value 109 - operator EQUAL 110 - operator EVENP 111 - concept false 112 - operator FREEOF 113 - operator LEQ 114 - operator LESSP 115 - operator MEMBER 116 - operator NEQ 117 - operator NOT 118 - operator NUMBERP 119 - operator ORDP 120 - operator PRIMEP 121 - concept TRUE 122 - ---- return from file ../doc/help/boolean.tex - ---- input file ../doc/help/command.tex - section end: 123 -section 7 g7 General Commands - command BYE 124 - command CONT 125 - command DISPLAY 126 - command LOAD\_PACKAGE 127 - command PAUSE 128 - command QUIT 129 - operator RECLAIM 130 - command REDERR 131 - command RETRY 132 - command SAVEAS 133 - command SHOWTIME 134 - command WRITE 135 - ---- return from file ../doc/help/command.tex - ---- input file ../doc/help/algebra.tex - section end: 136 -section 8 g8 Algebraic Operators - operator APPEND 137 - operator ARBINT 138 - operator ARBCOMPLEX 139 - operator ARGLENGTH 140 - operator COEFF 141 - operator COEFFN 142 - operator CONJ 143 - operator CONTINUED_FRACTION 144 - operator DECOMPOSE 145 - operator DEG 146 - operator DEN 147 - operator DF 148 - operator EXPAND\_CASES 149 - operator EXPREAD 150 - operator FACTORIZE 151 - operator HYPOT 152 - operator IMPART 153 - operator INT 154 - operator INTERPOL 155 - operator LCOF 156 - operator LENGTH 157 - operator LHS 158 - operator LIMIT 159 - operator LPOWER 160 - operator LTERM 161 - operator MAINVAR 162 - operator MAP 163 - command MKID 164 - operator NPRIMITIVE 165 - operator NUM 166 - operator ODESOLVE 167 - type ONE\_OF 168 - operator PART 169 - operator PF 170 - operator PROD 171 - operator REDUCT 172 - operator REPART 173 - operator RESULTANT 174 - operator RHS 175 - operator ROOT\_OF 176 - operator SELECT 177 - operator SHOWRULES 178 - operator SOLVE 179 - operator SORT 180 - operator STRUCTR 181 - operator SUB 182 - operator SUM 183 - operator WS 184 - ---- return from file ../doc/help/algebra.tex - ---- input file ../doc/help/declare.tex - section end: 185 -section 9 g9 Declarations - command ALGEBRAIC 186 - declaration ANTISYMMETRIC 187 - declaration ARRAY 188 - command CLEAR 189 - command CLEARRULES 190 - command DEFINE 191 - declaration DEPEND 192 - declaration EVEN 193 - declaration FACTOR 194 - command FORALL 195 - declaration INFIX 196 - declaration INTEGER 197 - declaration KORDER 198 - command LET 199 - declaration LINEAR 200 - declaration LINELENGTH 201 - command LISP 202 - declaration LISTARGP 203 - declaration NODEPEND 204 - command MATCH 205 - declaration NONCOM 206 - declaration NONZERO 207 - declaration ODD 208 - command OFF 209 - command ON 210 - declaration OPERATOR 211 - declaration ORDER 212 - declaration PRECEDENCE 213 - declaration PRECISION 214 - declaration PRINT\_PRECISION 215 - declaration REAL 216 - declaration REMFAC 217 - declaration SCALAR 218 - declaration SCIENTIFIC\_NOTATION 219 - declaration SHARE 220 - command SYMBOLIC 221 - declaration SYMMETRIC 222 - declaration TR 223 - declaration UNTR 224 - declaration VARNAME 225 - command WEIGHT 226 - operator WHERE 227 - command WHILE 228 - command WTLEVEL 229 - ---- return from file ../doc/help/declare.tex - ---- input file ../doc/help/io.tex - section end: 230 -section 10 g10 Input and Output - command IN 231 - command INPUT 232 - command OUT 233 - command SHUT 234 - ---- return from file ../doc/help/io.tex - ---- input file ../doc/help/elemfn.tex - section end: 235 -section 11 g11 Elementary Functions - operator ACOS 236 - operator ACOSH 237 - operator ACOT 238 - operator ACOTH 239 - operator ACSC 240 - operator ACSCH 241 - operator ASEC 242 - operator ASECH 243 - operator ASIN 244 - operator ASINH 245 - operator ATAN 246 - operator ATANH 247 - operator ATAN2 248 - operator COS 249 - operator COSH 250 - operator COT 251 - operator COTH 252 - operator CSC 253 - operator CSCH 254 - operator ERF 255 - operator EXP 256 - operator SEC 257 - operator SECH 258 - operator SIN 259 - operator SINH 260 - operator TAN 261 - operator TANH 262 - ---- return from file ../doc/help/elemfn.tex - ---- input file ../doc/help/switch.tex - section end: 263 -section 12 g12 General Switches - introduction SWITCHES 264 - switch ALGINT 265 - switch ALLBRANCH 266 - switch ALLFAC 267 - switch ARBVARS 268 - switch BALANCED\_MOD 269 - switch BFSPACE 270 - switch COMBINEEXPT 271 - switch COMBINELOGS 272 - switch COMP 273 - switch COMPLEX 274 - switch CREF 275 - switch CRAMER 276 - switch DEFN 277 - switch DEMO 278 - switch DFPRINT 279 - switch DIV 280 - switch ECHO 281 - switch ERRCONT 282 - switch EVALLHSEQP 283 - switch EXP 284 - switch EXPANDLOGS 285 - switch EZGCD 286 - switch FACTOR 287 - switch FAILHARD 288 - switch FORT 289 - switch FORTUPPER 290 - switch FULLPREC 291 - switch FULLROOTS 292 - switch GC 293 - switch GCD 294 - switch HORNER 295 - switch IFACTOR 296 - switch INT 297 - switch INTSTR 298 - switch LCM 299 - switch LESSSPACE 300 - switch LIMITEDFACTORS 301 - switch LIST 302 - switch LISTARGS 303 - switch MCD 304 - switch MODULAR 305 - switch MSG 306 - switch MULTIPLICITIES 307 - switch NAT 308 - switch NERO 309 - switch NOARG 310 - switch NOLNR 311 - switch NOSPLIT 312 - switch NUMVAL 313 - switch OUTPUT 314 - switch OVERVIEW 315 - switch PERIOD 316 - switch PRECISE 317 - switch PRET 318 - switch PRI 319 - switch RAISE 320 - switch RAT 321 - switch RATARG 322 - switch RATIONAL 323 - switch RATIONALIZE 324 - switch RATPRI 325 - switch REVPRI 326 - switch RLISP88 327 - switch ROUNDALL 328 - switch ROUNDBF 329 - switch ROUNDED 330 - switch SAVESTRUCTR 331 - switch SOLVESINGULAR 332 - switch TIME 333 - switch TRALLFAC 334 - switch TRFAC 335 - switch TRIGFORM 336 - switch TRINT 337 - switch TRNONLNR 338 - switch VAROPT 339 - ---- return from file ../doc/help/switch.tex - ---- input file ../doc/help/matrix.tex - section end: 340 -section 13 g13 Matrix Operations - operator COFACTOR 341 - operator DET 342 - operator MAT 343 - operator MATEIGEN 344 - declaration MATRIX 345 - operator NULLSPACE 346 - operator RANK 347 - operator TP 348 - operator TRACE 349 - ---- return from file ../doc/help/matrix.tex - ---- input file ../doc/help/pk-groeb.tex - section end: 350 -section 14 g14 Groebner package - introduction Groebner bases 351 - concept Ideal Parameters 352 - subsection 15 g15 Term order - introduction Term order 353 - operator torder 354 - operator torder_compile 355 - concept lex term order 356 - concept gradlex term order 357 - concept revgradlex term order 358 - concept gradlexgradlex term order 359 - concept gradlexrevgradlex term order 360 - concept lexgradlex term order 361 - concept lexrevgradlex term order 362 - concept weighted term order 363 - concept graded term order 364 - concept matrix term order 365 - section end: 366 - subsection 16 g16 Basic Groebner operators - operator gvars 367 - operator groebner 368 - operator groebner\_walk 369 - switch groebopt 370 - variable gvarslast 371 - switch groebprereduce 372 - switch groebfullreduction 373 - switch gltbasis 374 - variable gltb 375 - variable glterms 376 - switch groebstat 377 - switch trgroeb 378 - switch trgroebs 379 - operator gzerodim? 380 - operator gdimension 381 - operator gindependent\_sets 382 - operator dd_groebner 383 - operator glexconvert 384 - operator greduce 385 - operator preduce 386 - operator idealquotient 387 - operator hilbertpolynomial 388 - operator saturation 389 - section end: 390 - subsection 17 g17 Factorizing Groebner bases - operator groebnerf 391 - variable groebmonfac 392 - variable groebresmax 393 - variable groebrestriction 394 - section end: 395 - subsection 18 g18 Tracing Groebner bases - switch groebprot 396 - variable groebprotfile 397 - operator groebnert 398 - operator preducet 399 - section end: 400 - subsection 19 g19 Groebner Bases for Modules - concept Module 401 - variable gmodule 402 - section end: 403 - subsection 20 g20 Computing with distributive polynomials - operator gsort 404 - operator gsplit 405 - operator gspoly 406 - ---- return from file ../doc/help/pk-groeb.tex - ---- input file ../doc/help/hephys.tex - section end: 407 - section end: 408 -section 21 g21 High Energy Physics - introduction HEPHYS 409 - operator HE-dot 410 - operator EPS 411 - operator G 412 - declaration INDEX 413 - command MASS 414 - command MSHELL 415 - declaration NOSPUR 416 - declaration REMIND 417 - declaration SPUR 418 - command VECDIM 419 - declaration VECTOR 420 - ---- return from file ../doc/help/hephys.tex - ---- input file ../doc/help/pk-numer.tex - section end: 421 -section 22 g22 Numeric Package - introduction Numeric Package 422 - type Interval 423 - concept numeric accuracy 424 - switch TRNUMERIC 425 - operator num_min 426 - operator num_solve 427 - operator num_int 428 - operator num_odesolve 429 - operator bounds 430 - concept Chebyshev fit 431 - operator num_fit 432 - ---- return from file ../doc/help/pk-numer.tex - ---- input file ../doc/help/pk-roots.tex - section end: 433 -section 23 g23 Roots Package - introduction Roots Package 434 - operator MKPOLY 435 - operator NEARESTROOT 436 - operator REALROOTS 437 - operator ROOTACC 438 - operator ROOTS 439 - operator ROOT\_VAL 440 - variable ROOTSCOMPLEX 441 - variable ROOTSREAL 442 - ---- return from file ../doc/help/pk-roots.tex - ---- input file ../doc/help/pk-specf.tex - section end: 443 -section 24 g24 Special Functions - introduction Special Function Package 444 - concept Constants 445 - subsection 25 g25 Bernoulli Euler Zeta - operator BERNOULLI 446 - operator BERNOULLIP 447 - operator EULER 448 - operator EULERP 449 - operator ZETA 450 - section end: 451 - subsection 26 g26 Bessel Functions - operator BESSELJ 452 - operator BESSELY 453 - operator HANKEL1 454 - operator HANKEL2 455 - operator BESSELI 456 - operator BESSELK 457 - operator StruveH 458 - operator StruveL 459 - operator KummerM 460 - operator KummerU 461 - operator WhittakerW 462 - section end: 463 - subsection 27 g27 Airy Functions - operator Airy_Ai 464 - operator Airy_Bi 465 - operator Airy_Aiprime 466 - operator Airy_Biprime 467 - section end: 468 - subsection 28 g28 Jacobi's Elliptic Functions and Elliptic Integrals - operator JacobiSN 469 - operator JacobiCN 470 - operator JacobiDN 471 - operator JacobiCD 472 - operator JacobiSD 473 - operator JacobiND 474 - operator JacobiDC 475 - operator JacobiNC 476 - operator JacobiSC 477 - operator JacobiNS 478 - operator JacobiDS 479 - operator JacobiCS 480 - operator JacobiAMPLITUDE 481 - operator AGM_FUNCTION 482 - operator LANDENTRANS 483 - operator EllipticF 484 - operator EllipticK 485 - operator EllipticKprime 486 - operator EllipticE 487 - operator EllipticTHETA 488 - operator JacobiZETA 489 - section end: 490 - subsection 29 g29 Gamma and Related Functions - operator POCHHAMMER 491 - operator GAMMA 492 - operator BETA 493 - operator PSI 494 - operator POLYGAMMA 495 - section end: 496 - subsection 30 g30 Miscellaneous Functions - operator DILOG extended 497 - operator Lambert\_W function 498 - section end: 499 - subsection 31 g31 Orthogonal Polynomials - operator ChebyshevT 500 - operator ChebyshevU 501 - operator HermiteP 502 - operator LaguerreP 503 - operator LegendreP 504 - operator JacobiP 505 - operator GegenbauerP 506 - operator SolidHarmonicY 507 - operator SphericalHarmonicY 508 - section end: 509 - subsection 32 g32 Integral Functions - operator Si 510 - operator Shi 511 - operator s_i 512 - operator Ci 513 - operator Chi 514 - operator ERF extended 515 - operator erfc 516 - operator Ei 517 - operator Fresnel_C 518 - operator Fresnel_S 519 - section end: 520 - subsection 33 g33 Combinatorial Operators - operator BINOMIAL 521 - operator STIRLING1 522 - operator STIRLING2 523 - section end: 524 - subsection 34 g34 3j and 6j symbols - operator ThreejSymbol 525 - operator Clebsch_Gordan 526 - operator SixjSymbol 527 - section end: 528 - subsection 35 g35 Miscellaneous - operator HYPERGEOMETRIC 529 - operator MeijerG 530 - operator Heaviside 531 - operator erfi 532 - ---- return from file ../doc/help/pk-specf.tex - ---- input file ../doc/help/taylor.tex - section end: 533 - section end: 534 -section 36 g36 Taylor series - introduction TAYLOR 535 - operator taylor 536 - switch taylorautocombine 537 - switch taylorautoexpand 538 - operator taylorcombine 539 - switch taylorkeeporiginal 540 - operator taylororiginal 541 - switch taylorprintorder 542 - variable taylorprintterms 543 - operator taylorrevert 544 - operator taylorseriesp 545 - operator taylortemplate 546 - operator taylortostandard 547 - ---- return from file ../doc/help/taylor.tex - ---- input file ../doc/help/pk-gplot.tex - section end: 548 -section 37 g37 Gnuplot package - introduction GNUPLOT and REDUCE 549 - concept Axes names 550 - type Pointset 551 - command PLOT 552 - command PLOTRESET 553 - variable title 554 - variable xlabel 555 - variable ylabel 556 - variable zlabel 557 - variable terminal 558 - variable size 559 - variable view 560 - switch contour 561 - switch surface 562 - switch hidden3d 563 - switch PLOTKEEP 564 - switch PLOTREFINE 565 - variable plot_xmesh 566 - variable plot_ymesh 567 - switch SHOW_GRID 568 - switch TRPLOT 569 - ---- return from file ../doc/help/pk-gplot.tex - ---- input file ../doc/help/linalg.tex - section end: 570 -section 38 g38 Linear Algebra package - introduction Linear Algebra package 571 - switch fast_la 572 - operator add_columns 573 - operator add_rows 574 - operator add_to_columns 575 - operator add_to_rows 576 - operator augment_columns 577 - operator band_matrix 578 - operator block_matrix 579 - operator char_matrix 580 - operator char_poly 581 - operator cholesky 582 - operator coeff_matrix 583 - operator column_dim 584 - operator companion 585 - operator copy_into 586 - operator diagonal 587 - operator extend 588 - operator find_companion 589 - operator get_columns 590 - operator get_rows 591 - operator gram_schmidt 592 - operator hermitian_tp 593 - operator hessian 594 - operator hilbert 595 - operator jacobian 596 - operator jordan_block 597 - operator lu_decom 598 - operator make_identity 599 - operator matrix_augment 600 - operator matrixp 601 - operator matrix_stack 602 - operator minor 603 - operator mult_columns 604 - operator mult_rows 605 - operator pivot 606 - operator pseudo_inverse 607 - operator random_matrix 608 - operator remove_columns 609 - operator remove_rows 610 - operator row_dim 611 - operator rows_pivot 612 - operator simplex 613 - operator squarep 614 - operator stack_rows 615 - operator sub_matrix 616 - operator svd 617 - operator swap_columns 618 - operator swap_entries 619 - operator swap_rows 620 - operator symmetricp 621 - operator toeplitz 622 - operator vandermonde 623 - ---- return from file ../doc/help/linalg.tex - ---- input file ../doc/help/normform.tex - section end: 624 -section 39 g39 Matrix Normal Forms - operator Smithex 625 - operator Smithex\_int 626 - operator Frobenius 627 - operator Ratjordan 628 - operator Jordansymbolic 629 - operator Jordan 630 - ---- return from file ../doc/help/normform.tex - ---- input file ../doc/help/pk-misc.tex - section end: 631 -section 40 g40 Miscellaneous Packages - introduction Miscellaneous Packages 632 - package ALGINT 633 - package APPLYSYM 634 - package ARNUM 635 - package ASSIST 636 - package AVECTOR 637 - package BOOLEAN 638 - package CALI 639 - package CAMAL 640 - package CHANGEVR 641 - package COMPACT 642 - package CRACK 643 - package CVIT 644 - package DEFINT 645 - package DESIR 646 - package DFPART 647 - package DUMMY 648 - package EXCALC 649 - package FPS 650 - package FIDE 651 - package GENTRAN 652 - package IDEALS 653 - package INEQ 654 - package INVBASE 655 - package LAPLACE 656 - package LIE 657 - package MODSR 658 - package NCPOLY 659 - package ORTHOVEC 660 - package PHYSOP 661 - package PM 662 - package RANDPOLY 663 - package REACTEQN 664 - package RESET 665 - package RESIDUE 666 - package RLFI 667 - package SCOPE 668 - package SETS 669 - package SPDE 670 - package SYMMETRY 671 - package TPS 672 - package TRI 673 - package TRIGSIMP 674 - package XCOLOR 675 - package XIDEAL 676 - package WU 677 - package ZEILBERG 678 - package ZTRANS 679 - ---- return from file ../doc/help/pk-misc.tex - ---- input file ../doc/help/outmode.tex - section end: 680 -section 41 g41 Outmoded Operations - command ED 681 - command EDITDEF 682 - ---- return from file ../doc/help/outmode.tex - section end: 683 - section end: 684 - -nil - - - -reset_html(); -nil - - - -job(bldmsg("%w.tex",package), "null.fil"); - ------- updating node labels ----- ------- updating done ------------ -**** unknown token: documentclass -**** unknown token: usepackage -**** unknown token: usepackage - ---- input file ../doc/help/concept.tex -section 2 g2 Concepts - type IDENTIFIER 1 - type KERNEL 2 - type STRING 3 - ---- return from file ../doc/help/concept.tex - ---- input file ../doc/help/variable.tex - section end: 4 -section 3 g3 Variables - variable assumptions 5 - variable CARD\_NO 6 - constant E 7 - variable EVAL\_MODE 8 - variable FORT\_WIDTH 9 - variable HIGH\_POW 10 - constant I 11 - constant INFINITY 12 - variable LOW\_POW 13 - constant NIL 14 - constant PI 15 - variable requirements 16 - variable ROOT\_MULTIPLICITIES 17 - constant T 18 - ---- return from file ../doc/help/variable.tex - ---- input file ../doc/help/syntax.tex - section end: 19 -section 4 g4 Syntax - command semicolon 20 - command dollar 21 - command percent 22 - operator dot 23 - operator assign 24 - operator equalsign 25 - operator replace 26 - operator plussign 27 - operator minussign 28 - operator asterisk 29 - operator slash 30 - operator power 31 - operator caret 32 - operator geqsign 33 - operator greater 34 - operator leqsign 35 - operator less 36 - operator tilde 37 - command group 38 - operator AND 39 - command BEGIN 40 - command block 41 - command COMMENT 42 - operator CONS 43 - command END 44 - type EQUATION 45 - operator FIRST 46 - command FOR 47 - command FOREACH 48 - operator GEQ 49 - command GOTO 50 - operator GREATERP 51 - command IF 52 - operator LIST 53 - operator OR 54 - command PROCEDURE 55 - command REPEAT 56 - operator REST 57 - command RETURN 58 - operator REVERSE 59 - type RULE 60 - type Free Variable 61 - type Optional Free Variable 62 - operator SECOND 63 - operator SET 64 - operator SETQ 65 - operator THIRD 66 - operator WHEN 67 - ---- return from file ../doc/help/syntax.tex - ---- input file ../doc/help/arith.tex - section end: 68 -section 5 g5 Arithmetic Operations - introduction ARITHMETIC\_OPERATIONS 69 - operator ABS 70 - switch ADJPREC 71 - operator ARG 72 - operator CEILING 73 - operator CHOOSE 74 - operator DEG2DMS 75 - operator DEG2RAD 76 - operator DIFFERENCE 77 - operator DILOG 78 - operator DMS2DEG 79 - operator DMS2RAD 80 - operator FACTORIAL 81 - operator FIX 82 - operator FIXP 83 - operator FLOOR 84 - operator EXPT 85 - operator GCD 86 - operator LN 87 - operator LOG 88 - operator LOGB 89 - operator MAX 90 - operator MIN 91 - operator MINUS 92 - operator NEXTPRIME 93 - switch NOCONVERT 94 - operator NORM 95 - operator PERM 96 - operator PLUS 97 - operator QUOTIENT 98 - operator RAD2DEG 99 - operator RAD2DMS 100 - operator RECIP 101 - operator REMAINDER 102 - operator ROUND 103 - command SETMOD 104 - operator SIGN 105 - operator SQRT 106 - operator TIMES 107 - ---- return from file ../doc/help/arith.tex - ---- input file ../doc/help/boolean.tex - section end: 108 -section 6 g6 Boolean Operators - concept boolean value 109 - operator EQUAL 110 - operator EVENP 111 - concept false 112 - operator FREEOF 113 - operator LEQ 114 - operator LESSP 115 - operator MEMBER 116 - operator NEQ 117 - operator NOT 118 - operator NUMBERP 119 - operator ORDP 120 - operator PRIMEP 121 - concept TRUE 122 - ---- return from file ../doc/help/boolean.tex - ---- input file ../doc/help/command.tex - section end: 123 -section 7 g7 General Commands - command BYE 124 - command CONT 125 - command DISPLAY 126 - command LOAD\_PACKAGE 127 - command PAUSE 128 - command QUIT 129 - operator RECLAIM 130 - command REDERR 131 - command RETRY 132 - command SAVEAS 133 - command SHOWTIME 134 - command WRITE 135 - ---- return from file ../doc/help/command.tex - ---- input file ../doc/help/algebra.tex - section end: 136 -section 8 g8 Algebraic Operators - operator APPEND 137 - operator ARBINT 138 - operator ARBCOMPLEX 139 - operator ARGLENGTH 140 - operator COEFF 141 - operator COEFFN 142 - operator CONJ 143 - operator CONTINUED_FRACTION 144 - operator DECOMPOSE 145 - operator DEG 146 - operator DEN 147 - operator DF 148 - operator EXPAND\_CASES 149 - operator EXPREAD 150 - operator FACTORIZE 151 - operator HYPOT 152 - operator IMPART 153 - operator INT 154 - operator INTERPOL 155 - operator LCOF 156 - operator LENGTH 157 - operator LHS 158 - operator LIMIT 159 - operator LPOWER 160 - operator LTERM 161 - operator MAINVAR 162 - operator MAP 163 - command MKID 164 - operator NPRIMITIVE 165 - operator NUM 166 - operator ODESOLVE 167 - type ONE\_OF 168 - operator PART 169 - operator PF 170 - operator PROD 171 - operator REDUCT 172 - operator REPART 173 - operator RESULTANT 174 - operator RHS 175 - operator ROOT\_OF 176 - operator SELECT 177 - operator SHOWRULES 178 - operator SOLVE 179 - operator SORT 180 - operator STRUCTR 181 - operator SUB 182 - operator SUM 183 - operator WS 184 - ---- return from file ../doc/help/algebra.tex - ---- input file ../doc/help/declare.tex - section end: 185 -section 9 g9 Declarations - command ALGEBRAIC 186 - declaration ANTISYMMETRIC 187 - declaration ARRAY 188 - command CLEAR 189 - command CLEARRULES 190 - command DEFINE 191 - declaration DEPEND 192 - declaration EVEN 193 - declaration FACTOR 194 - command FORALL 195 - declaration INFIX 196 - declaration INTEGER 197 - declaration KORDER 198 - command LET 199 - declaration LINEAR 200 - declaration LINELENGTH 201 - command LISP 202 - declaration LISTARGP 203 - declaration NODEPEND 204 - command MATCH 205 - declaration NONCOM 206 - declaration NONZERO 207 - declaration ODD 208 - command OFF 209 - command ON 210 - declaration OPERATOR 211 - declaration ORDER 212 - declaration PRECEDENCE 213 - declaration PRECISION 214 - declaration PRINT\_PRECISION 215 - declaration REAL 216 - declaration REMFAC 217 - declaration SCALAR 218 - declaration SCIENTIFIC\_NOTATION 219 - declaration SHARE 220 - command SYMBOLIC 221 - declaration SYMMETRIC 222 - declaration TR 223 - declaration UNTR 224 - declaration VARNAME 225 - command WEIGHT 226 - operator WHERE 227 - command WHILE 228 - command WTLEVEL 229 - ---- return from file ../doc/help/declare.tex - ---- input file ../doc/help/io.tex - section end: 230 -section 10 g10 Input and Output - command IN 231 - command INPUT 232 - command OUT 233 - command SHUT 234 - ---- return from file ../doc/help/io.tex - ---- input file ../doc/help/elemfn.tex - section end: 235 -section 11 g11 Elementary Functions - operator ACOS 236 - operator ACOSH 237 - operator ACOT 238 - operator ACOTH 239 - operator ACSC 240 - operator ACSCH 241 - operator ASEC 242 - operator ASECH 243 - operator ASIN 244 - operator ASINH 245 - operator ATAN 246 - operator ATANH 247 - operator ATAN2 248 - operator COS 249 - operator COSH 250 - operator COT 251 - operator COTH 252 - operator CSC 253 - operator CSCH 254 - operator ERF 255 - operator EXP 256 - operator SEC 257 - operator SECH 258 - operator SIN 259 - operator SINH 260 - operator TAN 261 - operator TANH 262 - ---- return from file ../doc/help/elemfn.tex - ---- input file ../doc/help/switch.tex - section end: 263 -section 12 g12 General Switches - introduction SWITCHES 264 - switch ALGINT 265 - switch ALLBRANCH 266 - switch ALLFAC 267 - switch ARBVARS 268 - switch BALANCED\_MOD 269 - switch BFSPACE 270 - switch COMBINEEXPT 271 - switch COMBINELOGS 272 - switch COMP 273 - switch COMPLEX 274 - switch CREF 275 - switch CRAMER 276 - switch DEFN 277 - switch DEMO 278 - switch DFPRINT 279 - switch DIV 280 - switch ECHO 281 - switch ERRCONT 282 - switch EVALLHSEQP 283 - switch EXP 284 - switch EXPANDLOGS 285 - switch EZGCD 286 - switch FACTOR 287 - switch FAILHARD 288 - switch FORT 289 - switch FORTUPPER 290 - switch FULLPREC 291 - switch FULLROOTS 292 - switch GC 293 - switch GCD 294 - switch HORNER 295 - switch IFACTOR 296 - switch INT 297 - switch INTSTR 298 - switch LCM 299 - switch LESSSPACE 300 - switch LIMITEDFACTORS 301 - switch LIST 302 - switch LISTARGS 303 - switch MCD 304 - switch MODULAR 305 - switch MSG 306 - switch MULTIPLICITIES 307 - switch NAT 308 - switch NERO 309 - switch NOARG 310 - switch NOLNR 311 - switch NOSPLIT 312 - switch NUMVAL 313 - switch OUTPUT 314 - switch OVERVIEW 315 - switch PERIOD 316 - switch PRECISE 317 - switch PRET 318 - switch PRI 319 - switch RAISE 320 - switch RAT 321 - switch RATARG 322 - switch RATIONAL 323 - switch RATIONALIZE 324 - switch RATPRI 325 - switch REVPRI 326 - switch RLISP88 327 - switch ROUNDALL 328 - switch ROUNDBF 329 - switch ROUNDED 330 - switch SAVESTRUCTR 331 - switch SOLVESINGULAR 332 - switch TIME 333 - switch TRALLFAC 334 - switch TRFAC 335 - switch TRIGFORM 336 - switch TRINT 337 - switch TRNONLNR 338 - switch VAROPT 339 - ---- return from file ../doc/help/switch.tex - ---- input file ../doc/help/matrix.tex - section end: 340 -section 13 g13 Matrix Operations - operator COFACTOR 341 - operator DET 342 - operator MAT 343 - operator MATEIGEN 344 - declaration MATRIX 345 - operator NULLSPACE 346 - operator RANK 347 - operator TP 348 - operator TRACE 349 - ---- return from file ../doc/help/matrix.tex - ---- input file ../doc/help/pk-groeb.tex - section end: 350 -section 14 g14 Groebner package - introduction Groebner bases 351 - concept Ideal Parameters 352 - subsection 15 g15 Term order - introduction Term order 353 - operator torder 354 - operator torder_compile 355 - concept lex term order 356 - concept gradlex term order 357 - concept revgradlex term order 358 - concept gradlexgradlex term order 359 - concept gradlexrevgradlex term order 360 - concept lexgradlex term order 361 - concept lexrevgradlex term order 362 - concept weighted term order 363 - concept graded term order 364 - concept matrix term order 365 - section end: 366 - subsection 16 g16 Basic Groebner operators - operator gvars 367 - operator groebner 368 - operator groebner\_walk 369 - ######## reference to (l e x) not found, - ######## reference to (g r a d e d) not found, - ######## reference to (w e i g h t e d) not found, - ######## reference to (g r a d e d) not found, - ######## reference to (w e i g h t e d) not found, - ######## reference to (k e x) not found, - ######## reference to (l e x) not found, - switch groebopt 370 - variable gvarslast 371 - switch groebprereduce 372 - switch groebfullreduction 373 - switch gltbasis 374 - variable gltb 375 - variable glterms 376 - switch groebstat 377 - switch trgroeb 378 - switch trgroebs 379 - operator gzerodim? 380 - operator gdimension 381 - operator gindependent\_sets 382 - operator dd_groebner 383 - operator glexconvert 384 - operator greduce 385 - operator preduce 386 - operator idealquotient 387 - operator hilbertpolynomial 388 - operator saturation 389 - section end: 390 - subsection 17 g17 Factorizing Groebner bases - operator groebnerf 391 - variable groebmonfac 392 - variable groebresmax 393 - variable groebrestriction 394 - section end: 395 - subsection 18 g18 Tracing Groebner bases - switch groebprot 396 - variable groebprotfile 397 - operator groebnert 398 - operator preducet 399 - section end: 400 - subsection 19 g19 Groebner Bases for Modules - concept Module 401 - variable gmodule 402 - section end: 403 - subsection 20 g20 Computing with distributive polynomials - operator gsort 404 - operator gsplit 405 - operator gspoly 406 - ---- return from file ../doc/help/pk-groeb.tex - ---- input file ../doc/help/hephys.tex - section end: 407 - section end: 408 -section 21 g21 High Energy Physics - introduction HEPHYS 409 - operator HE-dot 410 - operator EPS 411 - operator G 412 - declaration INDEX 413 - command MASS 414 - command MSHELL 415 - declaration NOSPUR 416 - declaration REMIND 417 - declaration SPUR 418 - command VECDIM 419 - declaration VECTOR 420 - ---- return from file ../doc/help/hephys.tex - ---- input file ../doc/help/pk-numer.tex - section end: 421 -section 22 g22 Numeric Package - introduction Numeric Package 422 - type Interval 423 - concept numeric accuracy 424 - switch TRNUMERIC 425 - operator num_min 426 - operator num_solve 427 - operator num_int 428 - operator num_odesolve 429 - operator bounds 430 - concept Chebyshev fit 431 - operator num_fit 432 - ---- return from file ../doc/help/pk-numer.tex - ---- input file ../doc/help/pk-roots.tex - section end: 433 -section 23 g23 Roots Package - introduction Roots Package 434 - operator MKPOLY 435 - operator NEARESTROOT 436 - operator REALROOTS 437 - operator ROOTACC 438 - operator ROOTS 439 - operator ROOT\_VAL 440 - variable ROOTSCOMPLEX 441 - variable ROOTSREAL 442 - ---- return from file ../doc/help/pk-roots.tex - ---- input file ../doc/help/pk-specf.tex - section end: 443 -section 24 g24 Special Functions - introduction Special Function Package 444 - concept Constants 445 - subsection 25 g25 Bernoulli Euler Zeta - operator BERNOULLI 446 - operator BERNOULLIP 447 - operator EULER 448 - operator EULERP 449 - operator ZETA 450 - section end: 451 - subsection 26 g26 Bessel Functions - operator BESSELJ 452 - operator BESSELY 453 - operator HANKEL1 454 - operator HANKEL2 455 - operator BESSELI 456 - operator BESSELK 457 - operator StruveH 458 - operator StruveL 459 - operator KummerM 460 - operator KummerU 461 - operator WhittakerW 462 - section end: 463 - subsection 27 g27 Airy Functions - operator Airy_Ai 464 - operator Airy_Bi 465 - operator Airy_Aiprime 466 - operator Airy_Biprime 467 - section end: 468 - subsection 28 g28 Jacobi's Elliptic Functions and Elliptic Integrals - operator JacobiSN 469 - operator JacobiCN 470 - operator JacobiDN 471 - operator JacobiCD 472 - operator JacobiSD 473 - operator JacobiND 474 - operator JacobiDC 475 - operator JacobiNC 476 - operator JacobiSC 477 - operator JacobiNS 478 - operator JacobiDS 479 - operator JacobiCS 480 - operator JacobiAMPLITUDE 481 - operator AGM_FUNCTION 482 - operator LANDENTRANS 483 - operator EllipticF 484 - operator EllipticK 485 - operator EllipticKprime 486 - operator EllipticE 487 - operator EllipticTHETA 488 - operator JacobiZETA 489 - section end: 490 - subsection 29 g29 Gamma and Related Functions - operator POCHHAMMER 491 - operator GAMMA 492 - operator BETA 493 - operator PSI 494 - operator POLYGAMMA 495 - section end: 496 - subsection 30 g30 Miscellaneous Functions - operator DILOG extended 497 - operator Lambert\_W function 498 - section end: 499 - subsection 31 g31 Orthogonal Polynomials - operator ChebyshevT 500 - operator ChebyshevU 501 - operator HermiteP 502 - operator LaguerreP 503 - operator LegendreP 504 - operator JacobiP 505 - operator GegenbauerP 506 - operator SolidHarmonicY 507 - operator SphericalHarmonicY 508 - section end: 509 - subsection 32 g32 Integral Functions - operator Si 510 - operator Shi 511 - operator s_i 512 - operator Ci 513 - operator Chi 514 - operator ERF extended 515 - operator erfc 516 - operator Ei 517 - operator Fresnel_C 518 - operator Fresnel_S 519 - section end: 520 - subsection 33 g33 Combinatorial Operators - operator BINOMIAL 521 - operator STIRLING1 522 - operator STIRLING2 523 - section end: 524 - subsection 34 g34 3j and 6j symbols - operator ThreejSymbol 525 - operator Clebsch_Gordan 526 - operator SixjSymbol 527 - section end: 528 - subsection 35 g35 Miscellaneous - operator HYPERGEOMETRIC 529 - operator MeijerG 530 - operator Heaviside 531 - operator erfi 532 - ---- return from file ../doc/help/pk-specf.tex - ---- input file ../doc/help/taylor.tex - section end: 533 - section end: 534 -section 36 g36 Taylor series - introduction TAYLOR 535 - operator taylor 536 - switch taylorautocombine 537 - switch taylorautoexpand 538 - operator taylorcombine 539 - switch taylorkeeporiginal 540 - operator taylororiginal 541 - switch taylorprintorder 542 - variable taylorprintterms 543 - operator taylorrevert 544 - operator taylorseriesp 545 - operator taylortemplate 546 - operator taylortostandard 547 - ---- return from file ../doc/help/taylor.tex - ---- input file ../doc/help/pk-gplot.tex - section end: 548 -section 37 g37 Gnuplot package - introduction GNUPLOT and REDUCE 549 - concept Axes names 550 - type Pointset 551 - command PLOT 552 - command PLOTRESET 553 - variable title 554 - variable xlabel 555 - variable ylabel 556 - variable zlabel 557 - variable terminal 558 - variable size 559 - variable view 560 - switch contour 561 - switch surface 562 - switch hidden3d 563 - switch PLOTKEEP 564 - switch PLOTREFINE 565 - variable plot_xmesh 566 - variable plot_ymesh 567 - switch SHOW_GRID 568 - switch TRPLOT 569 - ---- return from file ../doc/help/pk-gplot.tex - ---- input file ../doc/help/linalg.tex - section end: 570 -section 38 g38 Linear Algebra package - introduction Linear Algebra package 571 - switch fast_la 572 - operator add_columns 573 - operator add_rows 574 - operator add_to_columns 575 - operator add_to_rows 576 - operator augment_columns 577 - operator band_matrix 578 - operator block_matrix 579 - operator char_matrix 580 - operator char_poly 581 - operator cholesky 582 - operator coeff_matrix 583 - operator column_dim 584 - operator companion 585 - operator copy_into 586 - operator diagonal 587 - operator extend 588 - operator find_companion 589 - operator get_columns 590 - operator get_rows 591 - operator gram_schmidt 592 - operator hermitian_tp 593 - operator hessian 594 - operator hilbert 595 - operator jacobian 596 - operator jordan_block 597 - operator lu_decom 598 - operator make_identity 599 - operator matrix_augment 600 - operator matrixp 601 - operator matrix_stack 602 - operator minor 603 - operator mult_columns 604 - operator mult_rows 605 - operator pivot 606 - operator pseudo_inverse 607 - operator random_matrix 608 - operator remove_columns 609 - operator remove_rows 610 - operator row_dim 611 - operator rows_pivot 612 - operator simplex 613 - operator squarep 614 - operator stack_rows 615 - operator sub_matrix 616 - operator svd 617 - operator swap_columns 618 - operator swap_entries 619 - operator swap_rows 620 - operator symmetricp 621 - operator toeplitz 622 - operator vandermonde 623 - ---- return from file ../doc/help/linalg.tex - ---- input file ../doc/help/normform.tex - section end: 624 -section 39 g39 Matrix Normal Forms - operator Smithex 625 - operator Smithex\_int 626 - operator Frobenius 627 - operator Ratjordan 628 - operator Jordansymbolic 629 - operator Jordan 630 - ---- return from file ../doc/help/normform.tex - ---- input file ../doc/help/pk-misc.tex - section end: 631 -section 40 g40 Miscellaneous Packages - introduction Miscellaneous Packages 632 - package ALGINT 633 - package APPLYSYM 634 - package ARNUM 635 - package ASSIST 636 - package AVECTOR 637 - package BOOLEAN 638 - package CALI 639 - package CAMAL 640 - package CHANGEVR 641 - package COMPACT 642 - package CRACK 643 - package CVIT 644 - package DEFINT 645 - package DESIR 646 - package DFPART 647 - package DUMMY 648 - package EXCALC 649 - package FPS 650 - package FIDE 651 - package GENTRAN 652 - package IDEALS 653 - package INEQ 654 - package INVBASE 655 - package LAPLACE 656 - package LIE 657 - package MODSR 658 - package NCPOLY 659 - package ORTHOVEC 660 - package PHYSOP 661 - package PM 662 - package RANDPOLY 663 - package REACTEQN 664 - package RESET 665 - package RESIDUE 666 - package RLFI 667 - package SCOPE 668 - package SETS 669 - package SPDE 670 - package SYMMETRY 671 - package TPS 672 - package TRI 673 - package TRIGSIMP 674 - package XCOLOR 675 - package XIDEAL 676 - package WU 677 - package ZEILBERG 678 - package ZTRANS 679 - ---- return from file ../doc/help/pk-misc.tex - ---- input file ../doc/help/outmode.tex - section end: 680 -section 41 g41 Outmoded Operations - command ED 681 - command EDITDEF 682 - ---- return from file ../doc/help/outmode.tex - section end: 683 - section end: 684 - -nil - - - -html_indexfile();..... compiling html index file - -nil - - - -LISP_indexfile();..... compiling independent index file - -nil - - - -bye; - -End of Lisp run after 24.73+0.68 seconds - -+++ Transcript closed at end of run +++ DELETED r38/help/mkhtml1.log Index: r38/help/mkhtml1.log ================================================================== --- r38/help/mkhtml1.log +++ r38/help/mkhtml1.log @@ -1,2680 +0,0 @@ -+++ Transcript to mkhtml1.log started at Sat Aug 07 16:05:54 2004 +++ -Codemist Standard Lisp 6.00 for Windows: Aug 5 2004 -Created: Thu Aug 05 11:50:03 2004 - -REDUCE 3.8, 15-Apr-04 ... -Memory allocation: 54522624 bytes - -+++ About to read file "mkhtml1.red" - -% -% This file runs the "*.tex" to "redhelp.html" conversion code. As -% well as providing the top level direction to the process it patches up -% for at least some of the places where the conversion code had been -% written in a manner not strongly related to the portability objectives of -% Standard Lisp... -% - -symbolic; -nil - - -off echo; -nil - - -on backtrace; -nil - - -on comp; -nil - - -!*windows := t; -t - - - -fluid '(package); -nil - - - -package := 'redhelp; -redhelp - - - -symbolic procedure deletip(a, b); - delete(a, b);+++ deletip compiled as link to delete - -deletip - - - -symbolic procedure inf x; - char!-code x;+++ inf compiled as link to char-code - -inf - - - -symbolic procedure channellinelength(f, l); - begin - f := wrs f; - l := linelength l; - wrs f; - return l - end;+++ channellinelength compiled, 12 + 4 bytes - -channellinelength - - - -symbolic procedure channelprin2(f, x); - begin - f := wrs f; - prin2 x; - wrs f; - return x - end;+++ channelprin2 compiled, 10 + 8 bytes - -channelprin2 - - - -symbolic procedure channelprin2t(f, x); - begin - f := wrs f; - prin2t x; - wrs f; - return x - end;+++ channelprin2t compiled, 10 + 8 bytes - -channelprin2t - - - -symbolic macro procedure channelprintf u; - begin - scalar g; - g := gensym(); - return list('prog, list g, - list('setq, g, list('wrs, cadr u)), - 'printf . cddr u, - list('wrs, g)) - end;+++ channelprintf_76z7r3p3pl4b compiled, 33 + 24 bytes - -channelprintf - - - -symbolic procedure channelterpri f; - begin - f := wrs f; - terpri(); - wrs f; - end;+++ channelterpri compiled, 10 + 4 bytes - -channelterpri - - - -symbolic procedure channelreadch f; - begin - scalar c; - f := rds f; - c := readch(); - rds f; - return c - end;+++ channelreadch compiled, 12 + 4 bytes - -channelreadch - - - -in "comphelp.red"$ -nil - -nil - -nil - -nil - -nil - -nil - -nil - -nil - -nil - -nil - -0 - -nil -+++++ global !*raise converted to fluid -+++ job compiled, 113 + 148 bytes - -job - -nil - -nil - -nil -+++ rdch compiled, 3 + 8 bytes - -rdch -+++ rdch!* compiled, 3 + 12 bytes - -rdch!* -+++ rdchr0 compiled, 54 + 64 bytes - -rdchr0 -+++ unrdch compiled, 3 + 12 bytes - -unrdch -+++ myskip compiled, 5 + 8 bytes - -myskip -+++ myskipl compiled, 9 + 8 bytes - -myskipl -+++ myskipstring compiled, 41 + 24 bytes - -myskipstring - -nil - -lower -+++ mytoken compiled, 83 + 44 bytes - -mytoken -+++ mystring compiled, 28 + 16 bytes - -mystring -+++ mystring2 compiled, 14 + 12 bytes - -mystring2 -+++ mystring2!] compiled, 14 + 12 bytes - -mystring2!] -+++ mystring_nodename compiled, 41 + 40 bytes - -mystring_nodename -+++ mystring3 compiled, 23 + 16 bytes - -mystring3 -+++ raisestring compiled, 67 + 20 bytes - -raisestring -+++ lowerstring compiled, 75 + 20 bytes - -lowerstring -+++ mycompress compiled, 12 + 8 bytes - -mycompress -+++ mainloop compiled, 200 + 156 bytes - -mainloop - -*** local variable u in procedure include not used -+++ include compiled, 129 + 100 bytes - -include - -include - -include - -null - -null -+++ print_indent compiled, 21 + 20 bytes - -print_indent - -nil - -0 - -type - -seq - -lab - -count - -name -+++ reset compiled, 51 + 40 bytes - -reset -+++ sectappend compiled, 11 + 8 bytes - -sectappend -+++ section compiled, 10 + 20 bytes - -section -+++ close_section compiled, 39 + 24 bytes - -close_section -+++ close_section1 compiled, 40 + 40 bytes - -close_section1 -+++ write_sections compiled, 15 + 12 bytes - -write_sections -+++ write_section compiled, 67 + 52 bytes - -write_section -+++ make_dir_entry compiled, 8 + 8 bytes - -make_dir_entry -+++ help_gensym compiled, 10 + 12 bytes - -help_gensym -+++ open_section compiled, 172 + 112 bytes - -open_section - -section - -section - -section - -*** local variable u in procedure beg not used -+++ beg compiled, 87 + 68 bytes - -beg - -beg - -*** local variable u in procedure mmain not used -+++ mmain compiled as link to mainloop - -mmain - -mmain - -nil -+++ clean_name compiled, 27 + 16 bytes - -clean_name - -(((!,) . comma_sign) ((!.) . dot_sign) ((!;) . semicolon_sign) ((!%) . -percent_sign) ((!$) . dollar_sign) ((!: !=) . assign_sign) ((!=) . equal_sign) ( -(!+) . plus_sign) ((!-) . minus_sign) ((!*) . times_sign) ((!/) . slash_sign) (( -!* !*) . power_sign) ((!$ !> != !$) . geq_sign) ((!> !=) . geq_sign) ((!>) . -greater_sign) ((!$ !< != !$) . leq_sign) ((!< !=) . leq_sign) ((!<) . less_sign) -((!< !<) . block)) -+++ make_label compiled, 64 + 40 bytes - -make_label -+++ get_label compiled, 8 + 8 bytes - -get_label -+++ patch_ compiled, 29 + 16 bytes - -patch_ -+++ get_label1 compiled, 32 + 40 bytes - -get_label1 -+++ get_label2 compiled, 103 + 56 bytes - -get_label2 -+++ update_labels compiled, 47 + 36 bytes - -update_labels -+++ node compiled, 197 + 136 bytes - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node - -node -+++ part compiled, 63 + 68 bytes - -part -+++ par_heading compiled, 26 + 16 bytes - -par_heading - -*** local variable type in procedure vpart not used -+++ vpart compiled, 13 + 32 bytes - -vpart -+++ vpart0 compiled, 81 + 72 bytes - -vpart0 -+++ compareahead compiled, 6 + 8 bytes - -compareahead -+++ compareahead1 compiled, 28 + 12 bytes - -compareahead1 -+++ look_ahead_suctx941k0ih compiled, 11 + 16 bytes - -look_ahead - -*** local variable type in procedure examples_part not used -+++ examples_part compiled, 381 + 204 bytes - -examples_part -+++ non_verb_block compiled, 15 + 36 bytes - -non_verb_block -+++ make_multi_out compiled, 121 + 60 bytes - -make_multi_out -+++ cut_lines compiled, 37 + 8 bytes - -cut_lines -+++ matchleft compiled, 22 + 16 bytes - -matchleft -+++ matcharb compiled, 12 + 8 bytes - -matcharb -+++ read_one_line compiled, 15 + 12 bytes - -read_one_line - -part - -part - -part - -part - -part - -part - -part - -part - -*** local variable type in procedure do-itemize not used -+++ do!-itemize compiled, 6 + 20 bytes - -do!-itemize - -do!-itemize -+++ context_error compiled, 11 + 20 bytes - -context_error - -*** local variable u in procedure verb not used -+++ verb compiled, 12 + 24 bytes - -verb - -verb - -*** local variable u in procedure ldots not used -+++ ldots compiled, 3 + 12 bytes - -ldots - -ldots - -nil - -*** local variable u in procedure cdots not used -+++ cdots compiled, 3 + 12 bytes - -cdots - -cdots - -nil - -*** local variable u in procedure cdot not used -+++ cdot compiled, 3 + 12 bytes - -cdot - -cdot - -nil - -*** local variable u in procedure write_pi not used -+++ write_pi compiled, 3 + 12 bytes - -write_pi - -write_pi - -nil - -*** local variable u in procedure emphase not used -+++ emphase compiled, 3 + 12 bytes - -emphase - -emphase - -*** local variable u in procedure meta not used -+++ meta compiled, 18 + 20 bytes - -meta - -meta - -*** local variable u in procedure italic not used -+++ italic compiled, 8 + 24 bytes - -italic - -*** local variable u in procedure switchitalic not used -+++ switchitalic compiled, 1 + 4 bytes - -switchitalic - -italic - -italic - -italic - -*** local variable u in procedure nameref not used -+++ nameref compiled, 3 + 12 bytes - -nameref - -nameref - -*** local variable u in procedure ref not used -+++ ref compiled, 3 + 12 bytes - -ref - -ref -+++ see compiled, 4 + 12 bytes - -see - -see - -*** local variable u in procedure myname not used -+++ myname compiled, 3 + 12 bytes - -myname - -myname - -*** local variable u in procedure myindex not used -+++ myindex compiled, 5 + 20 bytes - -myindex - -myindex - -*** local variable u in procedure nameindex not used -+++ nameindex compiled, 10 + 24 bytes - -nameindex - -nameindex - -*** local variable u in procedure reduce not used -+++ reduce compiled, 3 + 12 bytes - -reduce - -reduce - -nil - -*** local variable u in procedure rept not used -+++ rept compiled, 3 + 12 bytes - -rept - -rept - -nil - -*** local variable u in procedure optional not used -+++ optional compiled, 3 + 12 bytes - -optional - -optional - -nil - -*** local variable u in procedure myexp not used -+++ myexp compiled, 5 + 16 bytes - -myexp - -myexp - -*** local variable u in procedure formula not used -+++ formula compiled, 3 + 12 bytes - -formula - -formula - -formula - -*** local variable u in procedure rfrac not used -+++ rfrac compiled, 8 + 24 bytes - -rfrac - -rfrac - -*** local variable u in procedure item not used -+++ item compiled, 14 + 32 bytes - -item - -item - -*** local variable u in procedure texonly1 not used -+++ texonly1 compiled, 48 + 44 bytes - -texonly1 - -texonly1 - -*** local variable u in procedure texonly2 not used -+++ texonly2 compiled, 67 + 68 bytes - -texonly2 - -texonly2 - -*** local variable u in procedure infoonly not used -+++ infoonly compiled, 6 + 12 bytes - -infoonly - -infoonly -+++ reporttopic compiled, 12 + 20 bytes - -reporttopic -+++ substipq compiled, 32 + 12 bytes - -substipq - -nil - - -in "helphtml1.red"$ -nil - -nil - -t - -t - -"R" - -"TT" - -nil -+++ rootname compiled, 13 + 16 bytes - -rootname -+++ dest_directory compiled, 13 + 16 bytes - -dest_directory - -nil - -0 -+++ reset_html compiled, 8 + 24 bytes - -reset_html -+++ html_open compiled, 15 + 20 bytes - -html_open -+++ html_close compiled, 17 + 24 bytes - -html_close - -*** local variable u in procedure open_current_base_dir not used -+++ open_current_base_dir compiled, 1 + 4 bytes - -open_current_base_dir -+++ close_current_base_dir compiled, 1 + 4 bytes - -close_current_base_dir -+++ make_html_file_name compiled, 72 + 56 bytes - -make_html_file_name -+++ open_node_file compiled, 92 + 80 bytes - -open_node_file -+++ close_node_file compiled, 1 + 4 bytes - -close_node_file -+++ node_file_name compiled, 2 + 8 bytes - -node_file_name -+++ initoutput compiled, 1 + 4 bytes - -initoutput -+++ endoutput compiled, 1 + 4 bytes - -endoutput -+++ verbatim compiled, 3 + 8 bytes - -verbatim -+++ newfont compiled, 10 + 16 bytes - -newfont -+++ fontoff compiled, 5 + 12 bytes - -fontoff -+++ fonton compiled, 5 + 12 bytes - -fonton -+++ myprin2 compiled, 6 + 16 bytes - -myprin2 - -(!< !> !" !&) - -(!< !> !" !&) -+++ myprin2_protected compiled, 14 + 16 bytes - -myprin2_protected - -nil -+++ emit_start_verbatim compiled, 7 + 20 bytes - -emit_start_verbatim -+++ emit_end_verbatim compiled, 7 + 24 bytes - -emit_end_verbatim -+++ verbprin2 compiled, 55 + 56 bytes - -verbprin2 -+++ myterpri compiled, 3 + 12 bytes - -myterpri -+++ number4out compiled, 24 + 32 bytes - -number4out -+++ textout compiled, 149 + 84 bytes - -textout - -+++ par_heading redefined -+++ par_heading compiled, 26 + 24 bytes - -par_heading -+++ base_new_dir compiled, 6 + 16 bytes - -base_new_dir -+++ emit_dir_new compiled, 1 + 4 bytes - -emit_dir_new -+++ emit_dir_key compiled as link to emit_node_key - -emit_dir_key -+++ emit_dir_separator compiled as link to emit_node_separator - -emit_dir_separator -+++ emit_dir_label compiled as link to emit_node_label - -emit_dir_label -+++ emit_dir_title compiled, 6 + 12 bytes - -emit_dir_title -+++ emit_dir_browse compiled as link to emit_node_browse - -emit_dir_browse -+++ emit_node_separator compiled, 15 + 40 bytes - -emit_node_separator -+++ set_tab compiled, 1 + 4 bytes - -set_tab -+++ release_tab compiled, 1 + 4 bytes - -release_tab -+++ textout_name compiled, 26 + 16 bytes - -textout_name -+++ textout2 compiled, 29 + 16 bytes - -textout2 -+++ printem compiled, 18 + 24 bytes - -printem - -+++ printem redefined -+++ printem compiled, 19 + 28 bytes - -printem -+++ printref compiled, 235 + 128 bytes - -printref -+++ printnameref compiled as link to printref - -printnameref - -nil -+++ emit_node_keys compiled, 28 + 24 bytes - -emit_node_keys -+++ emit_node_key compiled as link to emit_hidden_node_key - -emit_node_key -+++ emit_hidden_node_key compiled, 33 + 16 bytes - -emit_hidden_node_key -+++ remove_html compiled, 13 + 4 bytes - -remove_html -+++ emit_node_label compiled, 25 + 40 bytes - -emit_node_label - -*** local variable dummy in procedure emit_node_title not used - -*** local variable type in procedure emit_node_title not used -+++ emit_node_title compiled, 49 + 56 bytes - -emit_node_title -+++ emit_node_browse compiled, 11 + 20 bytes - -emit_node_browse -+++ print_bold compiled, 19 + 28 bytes - -print_bold -+++ emit_dir_header compiled, 8 + 28 bytes - -emit_dir_header -+++ dig4 compiled, 14 + 16 bytes - -dig4 -+++ refout compiled, 39 + 32 bytes - -refout -+++ emit_dir_entry compiled, 66 + 68 bytes - -emit_dir_entry -+++ print_newline compiled, 13 + 32 bytes - -print_newline -+++ second_newline compiled, 4 + 12 bytes - -second_newline -+++ print_tab compiled, 4 + 16 bytes - -print_tab -+++ html_indexfile compiled, 353 + 144 bytes - -html_indexfile -+++ sort_term compiled, 60 + 12 bytes - -sort_term -+++ html_indexfile_sort compiled, 4 + 8 bytes - -html_indexfile_sort -+++ html_indexfile_sort1 compiled, 27 + 12 bytes - -html_indexfile_sort1 -+++ html_indexfile_subsetp compiled, 18 + 4 bytes - -html_indexfile_subsetp -+++ lisp_indexfile compiled, 329 + 136 bytes - -lisp_indexfile -+++ printstruct compiled, 7 + 16 bytes - -printstruct -+++ printstruct1 compiled, 58 + 16 bytes - -printstruct1 - -nil - - -in "minitex.red"$ -nil - -nil - -nil - -!\ - -!^ - -!_ - -2 - -3 - -4 -+++ mintex_convert0 compiled, 38 + 12 bytes - -mintex_convert0 -+++ mintex_convert compiled as link to mintex_convert0 - -mintex_convert -+++ minitex compiled, 102 + 40 bytes - -minitex -+++ minitex_pop_char compiled, 11 + 8 bytes - -minitex_pop_char -+++ minitex_skip compiled, 11 + 8 bytes - -minitex_skip -+++ minitex_next_char compiled, 6 + 8 bytes - -minitex_next_char -+++ struct compiled, 9 + 8 bytes - -struct -+++ make_chain compiled, 317 + 120 bytes - -make_chain - -*** local variable font in procedure make_char not used - -*** local variable cs in procedure make_char not used -+++ make_char compiled, 23 + 48 bytes - -make_char -+++ make_frac compiled, 211 + 76 bytes - -make_frac - -*** local variable y1 in procedure make_line not used -+++ make_line compiled, 22 + 32 bytes - -make_line -+++ make_multi compiled, 63 + 60 bytes - -make_multi - -*** local variable font in procedure make_end not used -+++ make_end compiled, 4 + 16 bytes - -make_end - -nil - -(chain 0 0 0) - -nil - -*** local variable term in procedure make_escape not used -+++ make_escape compiled, 77 + 72 bytes - -make_escape -+++ my_compare compiled, 23 + 16 bytes - -my_compare -+++ minitex_collect compiled, 8 + 12 bytes - -minitex_collect -+++ minitex_do compiled, 25 + 20 bytes - -minitex_do - -minitex_chain -+++ minitex_chain compiled, 37 + 24 bytes - -minitex_chain - -minitex_char - -*** local variable font in procedure minitex_char not used -+++ minitex_char compiled, 23 + 24 bytes - -minitex_char - -minitex_line - -*** local variable font in procedure minitex_line not used -+++ minitex_line compiled, 38 + 28 bytes - -minitex_line -+++ minitex_putchar compiled, 31 + 24 bytes - -minitex_putchar - -nil - - - -dir_src := "../doc/help/"; -"../doc/help/" - - -% dir_src := "~/reduce/doc/help/"; - -on backtrace; -nil - - - -reset_html(); -nil - - - -job(bldmsg("%w.tex",package), "null.fil"); -**** unknown token: documentclass -**** unknown token: usepackage -**** unknown token: usepackage - ---- input file ../doc/help/concept.tex -section 2 g2 Concepts - type IDENTIFIER 1 - type KERNEL 2 - type STRING 3 - ---- return from file ../doc/help/concept.tex - ---- input file ../doc/help/variable.tex - section end: 4 -section 3 g3 Variables - variable assumptions 5 - variable CARD\_NO 6 - constant E 7 - variable EVAL\_MODE 8 - variable FORT\_WIDTH 9 - variable HIGH\_POW 10 - constant I 11 - constant INFINITY 12 - variable LOW\_POW 13 - constant NIL 14 - constant PI 15 - variable requirements 16 - variable ROOT\_MULTIPLICITIES 17 - constant T 18 - ---- return from file ../doc/help/variable.tex - ---- input file ../doc/help/syntax.tex - section end: 19 -section 4 g4 Syntax - command semicolon 20 - command dollar 21 - command percent 22 - operator dot 23 - operator assign 24 - operator equalsign 25 - operator replace 26 - operator plussign 27 - operator minussign 28 - operator asterisk 29 - operator slash 30 - operator power 31 - operator caret 32 - operator geqsign 33 - operator greater 34 - operator leqsign 35 - operator less 36 - operator tilde 37 - command group 38 - operator AND 39 - command BEGIN 40 - command block 41 - command COMMENT 42 - operator CONS 43 - command END 44 - type EQUATION 45 - operator FIRST 46 - command FOR 47 - command FOREACH 48 - operator GEQ 49 - command GOTO 50 - operator GREATERP 51 - command IF 52 - operator LIST 53 - operator OR 54 - command PROCEDURE 55 - command REPEAT 56 - operator REST 57 - command RETURN 58 - operator REVERSE 59 - type RULE 60 - type Free Variable 61 - type Optional Free Variable 62 - operator SECOND 63 - operator SET 64 - operator SETQ 65 - operator THIRD 66 - operator WHEN 67 - ---- return from file ../doc/help/syntax.tex - ---- input file ../doc/help/arith.tex - section end: 68 -section 5 g5 Arithmetic Operations - introduction ARITHMETIC\_OPERATIONS 69 - operator ABS 70 - switch ADJPREC 71 - operator ARG 72 - operator CEILING 73 - operator CHOOSE 74 - operator DEG2DMS 75 - operator DEG2RAD 76 - operator DIFFERENCE 77 - operator DILOG 78 - operator DMS2DEG 79 - operator DMS2RAD 80 - operator FACTORIAL 81 - operator FIX 82 - operator FIXP 83 - operator FLOOR 84 - operator EXPT 85 - operator GCD 86 - operator LN 87 - operator LOG 88 - operator LOGB 89 - operator MAX 90 - operator MIN 91 - operator MINUS 92 - operator NEXTPRIME 93 - switch NOCONVERT 94 - operator NORM 95 - operator PERM 96 - operator PLUS 97 - operator QUOTIENT 98 - operator RAD2DEG 99 - operator RAD2DMS 100 - operator RECIP 101 - operator REMAINDER 102 - operator ROUND 103 - command SETMOD 104 - operator SIGN 105 - operator SQRT 106 - operator TIMES 107 - ---- return from file ../doc/help/arith.tex - ---- input file ../doc/help/boolean.tex - section end: 108 -section 6 g6 Boolean Operators - concept boolean value 109 - operator EQUAL 110 - operator EVENP 111 - concept false 112 - operator FREEOF 113 - operator LEQ 114 - operator LESSP 115 - operator MEMBER 116 - operator NEQ 117 - operator NOT 118 - operator NUMBERP 119 - operator ORDP 120 - operator PRIMEP 121 - concept TRUE 122 - ---- return from file ../doc/help/boolean.tex - ---- input file ../doc/help/command.tex - section end: 123 -section 7 g7 General Commands - command BYE 124 - command CONT 125 - command DISPLAY 126 - command LOAD\_PACKAGE 127 - command PAUSE 128 - command QUIT 129 - operator RECLAIM 130 - command REDERR 131 - command RETRY 132 - command SAVEAS 133 - command SHOWTIME 134 - command WRITE 135 - ---- return from file ../doc/help/command.tex - ---- input file ../doc/help/algebra.tex - section end: 136 -section 8 g8 Algebraic Operators - operator APPEND 137 - operator ARBINT 138 - operator ARBCOMPLEX 139 - operator ARGLENGTH 140 - operator COEFF 141 - operator COEFFN 142 - operator CONJ 143 - operator CONTINUED_FRACTION 144 - operator DECOMPOSE 145 - operator DEG 146 - operator DEN 147 - operator DF 148 - operator EXPAND\_CASES 149 - operator EXPREAD 150 - operator FACTORIZE 151 - operator HYPOT 152 - operator IMPART 153 - operator INT 154 - operator INTERPOL 155 - operator LCOF 156 - operator LENGTH 157 - operator LHS 158 - operator LIMIT 159 - operator LPOWER 160 - operator LTERM 161 - operator MAINVAR 162 - operator MAP 163 - command MKID 164 - operator NPRIMITIVE 165 - operator NUM 166 - operator ODESOLVE 167 - type ONE\_OF 168 - operator PART 169 - operator PF 170 - operator PROD 171 - operator REDUCT 172 - operator REPART 173 - operator RESULTANT 174 - operator RHS 175 - operator ROOT\_OF 176 - operator SELECT 177 - operator SHOWRULES 178 - operator SOLVE 179 - operator SORT 180 - operator STRUCTR 181 - operator SUB 182 - operator SUM 183 - operator WS 184 - ---- return from file ../doc/help/algebra.tex - ---- input file ../doc/help/declare.tex - section end: 185 -section 9 g9 Declarations - command ALGEBRAIC 186 - declaration ANTISYMMETRIC 187 - declaration ARRAY 188 - command CLEAR 189 - command CLEARRULES 190 - command DEFINE 191 - declaration DEPEND 192 - declaration EVEN 193 - declaration FACTOR 194 - command FORALL 195 - declaration INFIX 196 - declaration INTEGER 197 - declaration KORDER 198 - command LET 199 - declaration LINEAR 200 - declaration LINELENGTH 201 - command LISP 202 - declaration LISTARGP 203 - declaration NODEPEND 204 - command MATCH 205 - declaration NONCOM 206 - declaration NONZERO 207 - declaration ODD 208 - command OFF 209 - command ON 210 - declaration OPERATOR 211 - declaration ORDER 212 - declaration PRECEDENCE 213 - declaration PRECISION 214 - declaration PRINT\_PRECISION 215 - declaration REAL 216 - declaration REMFAC 217 - declaration SCALAR 218 - declaration SCIENTIFIC\_NOTATION 219 - declaration SHARE 220 - command SYMBOLIC 221 - declaration SYMMETRIC 222 - declaration TR 223 - declaration UNTR 224 - declaration VARNAME 225 - command WEIGHT 226 - operator WHERE 227 - command WHILE 228 - command WTLEVEL 229 - ---- return from file ../doc/help/declare.tex - ---- input file ../doc/help/io.tex - section end: 230 -section 10 g10 Input and Output - command IN 231 - command INPUT 232 - command OUT 233 - command SHUT 234 - ---- return from file ../doc/help/io.tex - ---- input file ../doc/help/elemfn.tex - section end: 235 -section 11 g11 Elementary Functions - operator ACOS 236 - operator ACOSH 237 - operator ACOT 238 - operator ACOTH 239 - operator ACSC 240 - operator ACSCH 241 - operator ASEC 242 - operator ASECH 243 - operator ASIN 244 - operator ASINH 245 - operator ATAN 246 - operator ATANH 247 - operator ATAN2 248 - operator COS 249 - operator COSH 250 - operator COT 251 - operator COTH 252 - operator CSC 253 - operator CSCH 254 - operator ERF 255 - operator EXP 256 - operator SEC 257 - operator SECH 258 - operator SIN 259 - operator SINH 260 - operator TAN 261 - operator TANH 262 - ---- return from file ../doc/help/elemfn.tex - ---- input file ../doc/help/switch.tex - section end: 263 -section 12 g12 General Switches - introduction SWITCHES 264 - switch ALGINT 265 - switch ALLBRANCH 266 - switch ALLFAC 267 - switch ARBVARS 268 - switch BALANCED\_MOD 269 - switch BFSPACE 270 - switch COMBINEEXPT 271 - switch COMBINELOGS 272 - switch COMP 273 - switch COMPLEX 274 - switch CREF 275 - switch CRAMER 276 - switch DEFN 277 - switch DEMO 278 - switch DFPRINT 279 - switch DIV 280 - switch ECHO 281 - switch ERRCONT 282 - switch EVALLHSEQP 283 - switch EXP 284 - switch EXPANDLOGS 285 - switch EZGCD 286 - switch FACTOR 287 - switch FAILHARD 288 - switch FORT 289 - switch FORTUPPER 290 - switch FULLPREC 291 - switch FULLROOTS 292 - switch GC 293 - switch GCD 294 - switch HORNER 295 - switch IFACTOR 296 - switch INT 297 - switch INTSTR 298 - switch LCM 299 - switch LESSSPACE 300 - switch LIMITEDFACTORS 301 - switch LIST 302 - switch LISTARGS 303 - switch MCD 304 - switch MODULAR 305 - switch MSG 306 - switch MULTIPLICITIES 307 - switch NAT 308 - switch NERO 309 - switch NOARG 310 - switch NOLNR 311 - switch NOSPLIT 312 - switch NUMVAL 313 - switch OUTPUT 314 - switch OVERVIEW 315 - switch PERIOD 316 - switch PRECISE 317 - switch PRET 318 - switch PRI 319 - switch RAISE 320 - switch RAT 321 - switch RATARG 322 - switch RATIONAL 323 - switch RATIONALIZE 324 - switch RATPRI 325 - switch REVPRI 326 - switch RLISP88 327 - switch ROUNDALL 328 - switch ROUNDBF 329 - switch ROUNDED 330 - switch SAVESTRUCTR 331 - switch SOLVESINGULAR 332 - switch TIME 333 - switch TRALLFAC 334 - switch TRFAC 335 - switch TRIGFORM 336 - switch TRINT 337 - switch TRNONLNR 338 - switch VAROPT 339 - ---- return from file ../doc/help/switch.tex - ---- input file ../doc/help/matrix.tex - section end: 340 -section 13 g13 Matrix Operations - operator COFACTOR 341 - operator DET 342 - operator MAT 343 - operator MATEIGEN 344 - declaration MATRIX 345 - operator NULLSPACE 346 - operator RANK 347 - operator TP 348 - operator TRACE 349 - ---- return from file ../doc/help/matrix.tex - ---- input file ../doc/help/pk-groeb.tex - section end: 350 -section 14 g14 Groebner package - introduction Groebner bases 351 - concept Ideal Parameters 352 - subsection 15 g15 Term order - introduction Term order 353 - operator torder 354 - operator torder_compile 355 - concept lex term order 356 - concept gradlex term order 357 - concept revgradlex term order 358 - concept gradlexgradlex term order 359 - concept gradlexrevgradlex term order 360 - concept lexgradlex term order 361 - concept lexrevgradlex term order 362 - concept weighted term order 363 - concept graded term order 364 - concept matrix term order 365 - section end: 366 - subsection 16 g16 Basic Groebner operators - operator gvars 367 - operator groebner 368 - operator groebner\_walk 369 - switch groebopt 370 - variable gvarslast 371 - switch groebprereduce 372 - switch groebfullreduction 373 - switch gltbasis 374 - variable gltb 375 - variable glterms 376 - switch groebstat 377 - switch trgroeb 378 - switch trgroebs 379 - operator gzerodim? 380 - operator gdimension 381 - operator gindependent\_sets 382 - operator dd_groebner 383 - operator glexconvert 384 - operator greduce 385 - operator preduce 386 - operator idealquotient 387 - operator hilbertpolynomial 388 - operator saturation 389 - section end: 390 - subsection 17 g17 Factorizing Groebner bases - operator groebnerf 391 - variable groebmonfac 392 - variable groebresmax 393 - variable groebrestriction 394 - section end: 395 - subsection 18 g18 Tracing Groebner bases - switch groebprot 396 - variable groebprotfile 397 - operator groebnert 398 - operator preducet 399 - section end: 400 - subsection 19 g19 Groebner Bases for Modules - concept Module 401 - variable gmodule 402 - section end: 403 - subsection 20 g20 Computing with distributive polynomials - operator gsort 404 - operator gsplit 405 - operator gspoly 406 - ---- return from file ../doc/help/pk-groeb.tex - ---- input file ../doc/help/hephys.tex - section end: 407 - section end: 408 -section 21 g21 High Energy Physics - introduction HEPHYS 409 - operator HE-dot 410 - operator EPS 411 - operator G 412 - declaration INDEX 413 - command MASS 414 - command MSHELL 415 - declaration NOSPUR 416 - declaration REMIND 417 - declaration SPUR 418 - command VECDIM 419 - declaration VECTOR 420 - ---- return from file ../doc/help/hephys.tex - ---- input file ../doc/help/pk-numer.tex - section end: 421 -section 22 g22 Numeric Package - introduction Numeric Package 422 - type Interval 423 - concept numeric accuracy 424 - switch TRNUMERIC 425 - operator num_min 426 - operator num_solve 427 - operator num_int 428 - operator num_odesolve 429 - operator bounds 430 - concept Chebyshev fit 431 - operator num_fit 432 - ---- return from file ../doc/help/pk-numer.tex - ---- input file ../doc/help/pk-roots.tex - section end: 433 -section 23 g23 Roots Package - introduction Roots Package 434 - operator MKPOLY 435 - operator NEARESTROOT 436 - operator REALROOTS 437 - operator ROOTACC 438 - operator ROOTS 439 - operator ROOT\_VAL 440 - variable ROOTSCOMPLEX 441 - variable ROOTSREAL 442 - ---- return from file ../doc/help/pk-roots.tex - ---- input file ../doc/help/pk-specf.tex - section end: 443 -section 24 g24 Special Functions - introduction Special Function Package 444 - concept Constants 445 - subsection 25 g25 Bernoulli Euler Zeta - operator BERNOULLI 446 - operator BERNOULLIP 447 - operator EULER 448 - operator EULERP 449 - operator ZETA 450 - section end: 451 - subsection 26 g26 Bessel Functions - operator BESSELJ 452 - operator BESSELY 453 - operator HANKEL1 454 - operator HANKEL2 455 - operator BESSELI 456 - operator BESSELK 457 - operator StruveH 458 - operator StruveL 459 - operator KummerM 460 - operator KummerU 461 - operator WhittakerW 462 - section end: 463 - subsection 27 g27 Airy Functions - operator Airy_Ai 464 - operator Airy_Bi 465 - operator Airy_Aiprime 466 - operator Airy_Biprime 467 - section end: 468 - subsection 28 g28 Jacobi's Elliptic Functions and Elliptic Integrals - operator JacobiSN 469 - operator JacobiCN 470 - operator JacobiDN 471 - operator JacobiCD 472 - operator JacobiSD 473 - operator JacobiND 474 - operator JacobiDC 475 - operator JacobiNC 476 - operator JacobiSC 477 - operator JacobiNS 478 - operator JacobiDS 479 - operator JacobiCS 480 - operator JacobiAMPLITUDE 481 - operator AGM_FUNCTION 482 - operator LANDENTRANS 483 - operator EllipticF 484 - operator EllipticK 485 - operator EllipticKprime 486 - operator EllipticE 487 - operator EllipticTHETA 488 - operator JacobiZETA 489 - section end: 490 - subsection 29 g29 Gamma and Related Functions - operator POCHHAMMER 491 - operator GAMMA 492 - operator BETA 493 - operator PSI 494 - operator POLYGAMMA 495 - section end: 496 - subsection 30 g30 Miscellaneous Functions - operator DILOG extended 497 - operator Lambert\_W function 498 - section end: 499 - subsection 31 g31 Orthogonal Polynomials - operator ChebyshevT 500 - operator ChebyshevU 501 - operator HermiteP 502 - operator LaguerreP 503 - operator LegendreP 504 - operator JacobiP 505 - operator GegenbauerP 506 - operator SolidHarmonicY 507 - operator SphericalHarmonicY 508 - section end: 509 - subsection 32 g32 Integral Functions - operator Si 510 - operator Shi 511 - operator s_i 512 - operator Ci 513 - operator Chi 514 - operator ERF extended 515 - operator erfc 516 - operator Ei 517 - operator Fresnel_C 518 - operator Fresnel_S 519 - section end: 520 - subsection 33 g33 Combinatorial Operators - operator BINOMIAL 521 - operator STIRLING1 522 - operator STIRLING2 523 - section end: 524 - subsection 34 g34 3j and 6j symbols - operator ThreejSymbol 525 - operator Clebsch_Gordan 526 - operator SixjSymbol 527 - section end: 528 - subsection 35 g35 Miscellaneous - operator HYPERGEOMETRIC 529 - operator MeijerG 530 - operator Heaviside 531 - operator erfi 532 - ---- return from file ../doc/help/pk-specf.tex - ---- input file ../doc/help/taylor.tex - section end: 533 - section end: 534 -section 36 g36 Taylor series - introduction TAYLOR 535 - operator taylor 536 - switch taylorautocombine 537 - switch taylorautoexpand 538 - operator taylorcombine 539 - switch taylorkeeporiginal 540 - operator taylororiginal 541 - switch taylorprintorder 542 - variable taylorprintterms 543 - operator taylorrevert 544 - operator taylorseriesp 545 - operator taylortemplate 546 - operator taylortostandard 547 - ---- return from file ../doc/help/taylor.tex - ---- input file ../doc/help/pk-gplot.tex - section end: 548 -section 37 g37 Gnuplot package - introduction GNUPLOT and REDUCE 549 - concept Axes names 550 - type Pointset 551 - command PLOT 552 - command PLOTRESET 553 - variable title 554 - variable xlabel 555 - variable ylabel 556 - variable zlabel 557 - variable terminal 558 - variable size 559 - variable view 560 - switch contour 561 - switch surface 562 - switch hidden3d 563 - switch PLOTKEEP 564 - switch PLOTREFINE 565 - variable plot_xmesh 566 - variable plot_ymesh 567 - switch SHOW_GRID 568 - switch TRPLOT 569 - ---- return from file ../doc/help/pk-gplot.tex - ---- input file ../doc/help/linalg.tex - section end: 570 -section 38 g38 Linear Algebra package - introduction Linear Algebra package 571 - switch fast_la 572 - operator add_columns 573 - operator add_rows 574 - operator add_to_columns 575 - operator add_to_rows 576 - operator augment_columns 577 - operator band_matrix 578 - operator block_matrix 579 - operator char_matrix 580 - operator char_poly 581 - operator cholesky 582 - operator coeff_matrix 583 - operator column_dim 584 - operator companion 585 - operator copy_into 586 - operator diagonal 587 - operator extend 588 - operator find_companion 589 - operator get_columns 590 - operator get_rows 591 - operator gram_schmidt 592 - operator hermitian_tp 593 - operator hessian 594 - operator hilbert 595 - operator jacobian 596 - operator jordan_block 597 - operator lu_decom 598 - operator make_identity 599 - operator matrix_augment 600 - operator matrixp 601 - operator matrix_stack 602 - operator minor 603 - operator mult_columns 604 - operator mult_rows 605 - operator pivot 606 - operator pseudo_inverse 607 - operator random_matrix 608 - operator remove_columns 609 - operator remove_rows 610 - operator row_dim 611 - operator rows_pivot 612 - operator simplex 613 - operator squarep 614 - operator stack_rows 615 - operator sub_matrix 616 - operator svd 617 - operator swap_columns 618 - operator swap_entries 619 - operator swap_rows 620 - operator symmetricp 621 - operator toeplitz 622 - operator vandermonde 623 - ---- return from file ../doc/help/linalg.tex - ---- input file ../doc/help/normform.tex - section end: 624 -section 39 g39 Matrix Normal Forms - operator Smithex 625 - operator Smithex\_int 626 - operator Frobenius 627 - operator Ratjordan 628 - operator Jordansymbolic 629 - operator Jordan 630 - ---- return from file ../doc/help/normform.tex - ---- input file ../doc/help/pk-misc.tex - section end: 631 -section 40 g40 Miscellaneous Packages - introduction Miscellaneous Packages 632 - package ALGINT 633 - package APPLYSYM 634 - package ARNUM 635 - package ASSIST 636 - package AVECTOR 637 - package BOOLEAN 638 - package CALI 639 - package CAMAL 640 - package CHANGEVR 641 - package COMPACT 642 - package CRACK 643 - package CVIT 644 - package DEFINT 645 - package DESIR 646 - package DFPART 647 - package DUMMY 648 - package EXCALC 649 - package FPS 650 - package FIDE 651 - package GENTRAN 652 - package IDEALS 653 - package INEQ 654 - package INVBASE 655 - package LAPLACE 656 - package LIE 657 - package MODSR 658 - package NCPOLY 659 - package ORTHOVEC 660 - package PHYSOP 661 - package PM 662 - package RANDPOLY 663 - package REACTEQN 664 - package RESET 665 - package RESIDUE 666 - package RLFI 667 - package SCOPE 668 - package SETS 669 - package SPDE 670 - package SYMMETRY 671 - package TPS 672 - package TRI 673 - package TRIGSIMP 674 - package XCOLOR 675 - package XIDEAL 676 - package WU 677 - package ZEILBERG 678 - package ZTRANS 679 - ---- return from file ../doc/help/pk-misc.tex - ---- input file ../doc/help/outmode.tex - section end: 680 -section 41 g41 Outmoded Operations - command ED 681 - command EDITDEF 682 - ---- return from file ../doc/help/outmode.tex - section end: 683 - section end: 684 - -nil - - - -reset_html(); -nil - - - -job(bldmsg("%w.tex",package), "null.fil"); - ------- updating node labels ----- ------- updating done ------------ -**** unknown token: documentclass -**** unknown token: usepackage -**** unknown token: usepackage - ---- input file ../doc/help/concept.tex -section 2 g2 Concepts - type IDENTIFIER 1 - type KERNEL 2 - type STRING 3 - ---- return from file ../doc/help/concept.tex - ---- input file ../doc/help/variable.tex - section end: 4 -section 3 g3 Variables - variable assumptions 5 - variable CARD\_NO 6 - constant E 7 - variable EVAL\_MODE 8 - variable FORT\_WIDTH 9 - variable HIGH\_POW 10 - constant I 11 - constant INFINITY 12 - variable LOW\_POW 13 - constant NIL 14 - constant PI 15 - variable requirements 16 - variable ROOT\_MULTIPLICITIES 17 - constant T 18 - ---- return from file ../doc/help/variable.tex - ---- input file ../doc/help/syntax.tex - section end: 19 -section 4 g4 Syntax - command semicolon 20 - command dollar 21 - command percent 22 - operator dot 23 - operator assign 24 - operator equalsign 25 - operator replace 26 - operator plussign 27 - operator minussign 28 - operator asterisk 29 - operator slash 30 - operator power 31 - operator caret 32 - operator geqsign 33 - operator greater 34 - operator leqsign 35 - operator less 36 - operator tilde 37 - command group 38 - operator AND 39 - command BEGIN 40 - command block 41 - command COMMENT 42 - operator CONS 43 - command END 44 - type EQUATION 45 - operator FIRST 46 - command FOR 47 - command FOREACH 48 - operator GEQ 49 - command GOTO 50 - operator GREATERP 51 - command IF 52 - operator LIST 53 - operator OR 54 - command PROCEDURE 55 - command REPEAT 56 - operator REST 57 - command RETURN 58 - operator REVERSE 59 - type RULE 60 - type Free Variable 61 - type Optional Free Variable 62 - operator SECOND 63 - operator SET 64 - operator SETQ 65 - operator THIRD 66 - operator WHEN 67 - ---- return from file ../doc/help/syntax.tex - ---- input file ../doc/help/arith.tex - section end: 68 -section 5 g5 Arithmetic Operations - introduction ARITHMETIC\_OPERATIONS 69 - operator ABS 70 - switch ADJPREC 71 - operator ARG 72 - operator CEILING 73 - operator CHOOSE 74 - operator DEG2DMS 75 - operator DEG2RAD 76 - operator DIFFERENCE 77 - operator DILOG 78 - operator DMS2DEG 79 - operator DMS2RAD 80 - operator FACTORIAL 81 - operator FIX 82 - operator FIXP 83 - operator FLOOR 84 - operator EXPT 85 - operator GCD 86 - operator LN 87 - operator LOG 88 - operator LOGB 89 - operator MAX 90 - operator MIN 91 - operator MINUS 92 - operator NEXTPRIME 93 - switch NOCONVERT 94 - operator NORM 95 - operator PERM 96 - operator PLUS 97 - operator QUOTIENT 98 - operator RAD2DEG 99 - operator RAD2DMS 100 - operator RECIP 101 - operator REMAINDER 102 - operator ROUND 103 - command SETMOD 104 - operator SIGN 105 - operator SQRT 106 - operator TIMES 107 - ---- return from file ../doc/help/arith.tex - ---- input file ../doc/help/boolean.tex - section end: 108 -section 6 g6 Boolean Operators - concept boolean value 109 - operator EQUAL 110 - operator EVENP 111 - concept false 112 - operator FREEOF 113 - operator LEQ 114 - operator LESSP 115 - operator MEMBER 116 - operator NEQ 117 - operator NOT 118 - operator NUMBERP 119 - operator ORDP 120 - operator PRIMEP 121 - concept TRUE 122 - ---- return from file ../doc/help/boolean.tex - ---- input file ../doc/help/command.tex - section end: 123 -section 7 g7 General Commands - command BYE 124 - command CONT 125 - command DISPLAY 126 - command LOAD\_PACKAGE 127 - command PAUSE 128 - command QUIT 129 - operator RECLAIM 130 - command REDERR 131 - command RETRY 132 - command SAVEAS 133 - command SHOWTIME 134 - command WRITE 135 - ---- return from file ../doc/help/command.tex - ---- input file ../doc/help/algebra.tex - section end: 136 -section 8 g8 Algebraic Operators - operator APPEND 137 - operator ARBINT 138 - operator ARBCOMPLEX 139 - operator ARGLENGTH 140 - operator COEFF 141 - operator COEFFN 142 - operator CONJ 143 - operator CONTINUED_FRACTION 144 - operator DECOMPOSE 145 - operator DEG 146 - operator DEN 147 - operator DF 148 - operator EXPAND\_CASES 149 - operator EXPREAD 150 - operator FACTORIZE 151 - operator HYPOT 152 - operator IMPART 153 - operator INT 154 - operator INTERPOL 155 - operator LCOF 156 - operator LENGTH 157 - operator LHS 158 - operator LIMIT 159 - operator LPOWER 160 - operator LTERM 161 - operator MAINVAR 162 - operator MAP 163 - command MKID 164 - operator NPRIMITIVE 165 - operator NUM 166 - operator ODESOLVE 167 - type ONE\_OF 168 - operator PART 169 - operator PF 170 - operator PROD 171 - operator REDUCT 172 - operator REPART 173 - operator RESULTANT 174 - operator RHS 175 - operator ROOT\_OF 176 - operator SELECT 177 - operator SHOWRULES 178 - operator SOLVE 179 - operator SORT 180 - operator STRUCTR 181 - operator SUB 182 - operator SUM 183 - operator WS 184 - ---- return from file ../doc/help/algebra.tex - ---- input file ../doc/help/declare.tex - section end: 185 -section 9 g9 Declarations - command ALGEBRAIC 186 - declaration ANTISYMMETRIC 187 - declaration ARRAY 188 - command CLEAR 189 - command CLEARRULES 190 - command DEFINE 191 - declaration DEPEND 192 - declaration EVEN 193 - declaration FACTOR 194 - command FORALL 195 - declaration INFIX 196 - declaration INTEGER 197 - declaration KORDER 198 - command LET 199 - declaration LINEAR 200 - declaration LINELENGTH 201 - command LISP 202 - declaration LISTARGP 203 - declaration NODEPEND 204 - command MATCH 205 - declaration NONCOM 206 - declaration NONZERO 207 - declaration ODD 208 - command OFF 209 - command ON 210 - declaration OPERATOR 211 - declaration ORDER 212 - declaration PRECEDENCE 213 - declaration PRECISION 214 - declaration PRINT\_PRECISION 215 - declaration REAL 216 - declaration REMFAC 217 - declaration SCALAR 218 - declaration SCIENTIFIC\_NOTATION 219 - declaration SHARE 220 - command SYMBOLIC 221 - declaration SYMMETRIC 222 - declaration TR 223 - declaration UNTR 224 - declaration VARNAME 225 - command WEIGHT 226 - operator WHERE 227 - command WHILE 228 - command WTLEVEL 229 - ---- return from file ../doc/help/declare.tex - ---- input file ../doc/help/io.tex - section end: 230 -section 10 g10 Input and Output - command IN 231 - command INPUT 232 - command OUT 233 - command SHUT 234 - ---- return from file ../doc/help/io.tex - ---- input file ../doc/help/elemfn.tex - section end: 235 -section 11 g11 Elementary Functions - operator ACOS 236 - operator ACOSH 237 - operator ACOT 238 - operator ACOTH 239 - operator ACSC 240 - operator ACSCH 241 - operator ASEC 242 - operator ASECH 243 - operator ASIN 244 - operator ASINH 245 - operator ATAN 246 - operator ATANH 247 - operator ATAN2 248 - operator COS 249 - operator COSH 250 - operator COT 251 - operator COTH 252 - operator CSC 253 - operator CSCH 254 - operator ERF 255 - operator EXP 256 - operator SEC 257 - operator SECH 258 - operator SIN 259 - operator SINH 260 - operator TAN 261 - operator TANH 262 - ---- return from file ../doc/help/elemfn.tex - ---- input file ../doc/help/switch.tex - section end: 263 -section 12 g12 General Switches - introduction SWITCHES 264 - switch ALGINT 265 - switch ALLBRANCH 266 - switch ALLFAC 267 - switch ARBVARS 268 - switch BALANCED\_MOD 269 - switch BFSPACE 270 - switch COMBINEEXPT 271 - switch COMBINELOGS 272 - switch COMP 273 - switch COMPLEX 274 - switch CREF 275 - switch CRAMER 276 - switch DEFN 277 - switch DEMO 278 - switch DFPRINT 279 - switch DIV 280 - switch ECHO 281 - switch ERRCONT 282 - switch EVALLHSEQP 283 - switch EXP 284 - switch EXPANDLOGS 285 - switch EZGCD 286 - switch FACTOR 287 - switch FAILHARD 288 - switch FORT 289 - switch FORTUPPER 290 - switch FULLPREC 291 - switch FULLROOTS 292 - switch GC 293 - switch GCD 294 - switch HORNER 295 - switch IFACTOR 296 - switch INT 297 - switch INTSTR 298 - switch LCM 299 - switch LESSSPACE 300 - switch LIMITEDFACTORS 301 - switch LIST 302 - switch LISTARGS 303 - switch MCD 304 - switch MODULAR 305 - switch MSG 306 - switch MULTIPLICITIES 307 - switch NAT 308 - switch NERO 309 - switch NOARG 310 - switch NOLNR 311 - switch NOSPLIT 312 - switch NUMVAL 313 - switch OUTPUT 314 - switch OVERVIEW 315 - switch PERIOD 316 - switch PRECISE 317 - switch PRET 318 - switch PRI 319 - switch RAISE 320 - switch RAT 321 - switch RATARG 322 - switch RATIONAL 323 - switch RATIONALIZE 324 - switch RATPRI 325 - switch REVPRI 326 - switch RLISP88 327 - switch ROUNDALL 328 - switch ROUNDBF 329 - switch ROUNDED 330 - switch SAVESTRUCTR 331 - switch SOLVESINGULAR 332 - switch TIME 333 - switch TRALLFAC 334 - switch TRFAC 335 - switch TRIGFORM 336 - switch TRINT 337 - switch TRNONLNR 338 - switch VAROPT 339 - ---- return from file ../doc/help/switch.tex - ---- input file ../doc/help/matrix.tex - section end: 340 -section 13 g13 Matrix Operations - operator COFACTOR 341 - operator DET 342 - operator MAT 343 - operator MATEIGEN 344 - declaration MATRIX 345 - operator NULLSPACE 346 - operator RANK 347 - operator TP 348 - operator TRACE 349 - ---- return from file ../doc/help/matrix.tex - ---- input file ../doc/help/pk-groeb.tex - section end: 350 -section 14 g14 Groebner package - introduction Groebner bases 351 - concept Ideal Parameters 352 - subsection 15 g15 Term order - introduction Term order 353 - operator torder 354 - operator torder_compile 355 - concept lex term order 356 - concept gradlex term order 357 - concept revgradlex term order 358 - concept gradlexgradlex term order 359 - concept gradlexrevgradlex term order 360 - concept lexgradlex term order 361 - concept lexrevgradlex term order 362 - concept weighted term order 363 - concept graded term order 364 - concept matrix term order 365 - section end: 366 - subsection 16 g16 Basic Groebner operators - operator gvars 367 - operator groebner 368 - operator groebner\_walk 369 - ######## reference to (l e x) not found, - ######## reference to (g r a d e d) not found, - ######## reference to (w e i g h t e d) not found, - ######## reference to (g r a d e d) not found, - ######## reference to (w e i g h t e d) not found, - ######## reference to (k e x) not found, - ######## reference to (l e x) not found, - switch groebopt 370 - variable gvarslast 371 - switch groebprereduce 372 - switch groebfullreduction 373 - switch gltbasis 374 - variable gltb 375 - variable glterms 376 - switch groebstat 377 - switch trgroeb 378 - switch trgroebs 379 - operator gzerodim? 380 - operator gdimension 381 - operator gindependent\_sets 382 - operator dd_groebner 383 - operator glexconvert 384 - operator greduce 385 - operator preduce 386 - operator idealquotient 387 - operator hilbertpolynomial 388 - operator saturation 389 - section end: 390 - subsection 17 g17 Factorizing Groebner bases - operator groebnerf 391 - variable groebmonfac 392 - variable groebresmax 393 - variable groebrestriction 394 - section end: 395 - subsection 18 g18 Tracing Groebner bases - switch groebprot 396 - variable groebprotfile 397 - operator groebnert 398 - operator preducet 399 - section end: 400 - subsection 19 g19 Groebner Bases for Modules - concept Module 401 - variable gmodule 402 - section end: 403 - subsection 20 g20 Computing with distributive polynomials - operator gsort 404 - operator gsplit 405 - operator gspoly 406 - ---- return from file ../doc/help/pk-groeb.tex - ---- input file ../doc/help/hephys.tex - section end: 407 - section end: 408 -section 21 g21 High Energy Physics - introduction HEPHYS 409 - operator HE-dot 410 - operator EPS 411 - operator G 412 - declaration INDEX 413 - command MASS 414 - command MSHELL 415 - declaration NOSPUR 416 - declaration REMIND 417 - declaration SPUR 418 - command VECDIM 419 - declaration VECTOR 420 - ---- return from file ../doc/help/hephys.tex - ---- input file ../doc/help/pk-numer.tex - section end: 421 -section 22 g22 Numeric Package - introduction Numeric Package 422 - type Interval 423 - concept numeric accuracy 424 - switch TRNUMERIC 425 - operator num_min 426 - operator num_solve 427 - operator num_int 428 - operator num_odesolve 429 - operator bounds 430 - concept Chebyshev fit 431 - operator num_fit 432 - ---- return from file ../doc/help/pk-numer.tex - ---- input file ../doc/help/pk-roots.tex - section end: 433 -section 23 g23 Roots Package - introduction Roots Package 434 - operator MKPOLY 435 - operator NEARESTROOT 436 - operator REALROOTS 437 - operator ROOTACC 438 - operator ROOTS 439 - operator ROOT\_VAL 440 - variable ROOTSCOMPLEX 441 - variable ROOTSREAL 442 - ---- return from file ../doc/help/pk-roots.tex - ---- input file ../doc/help/pk-specf.tex - section end: 443 -section 24 g24 Special Functions - introduction Special Function Package 444 - concept Constants 445 - subsection 25 g25 Bernoulli Euler Zeta - operator BERNOULLI 446 - operator BERNOULLIP 447 - operator EULER 448 - operator EULERP 449 - operator ZETA 450 - section end: 451 - subsection 26 g26 Bessel Functions - operator BESSELJ 452 - operator BESSELY 453 - operator HANKEL1 454 - operator HANKEL2 455 - operator BESSELI 456 - operator BESSELK 457 - operator StruveH 458 - operator StruveL 459 - operator KummerM 460 - operator KummerU 461 - operator WhittakerW 462 - section end: 463 - subsection 27 g27 Airy Functions - operator Airy_Ai 464 - operator Airy_Bi 465 - operator Airy_Aiprime 466 - operator Airy_Biprime 467 - section end: 468 - subsection 28 g28 Jacobi's Elliptic Functions and Elliptic Integrals - operator JacobiSN 469 - operator JacobiCN 470 - operator JacobiDN 471 - operator JacobiCD 472 - operator JacobiSD 473 - operator JacobiND 474 - operator JacobiDC 475 - operator JacobiNC 476 - operator JacobiSC 477 - operator JacobiNS 478 - operator JacobiDS 479 - operator JacobiCS 480 - operator JacobiAMPLITUDE 481 - operator AGM_FUNCTION 482 - operator LANDENTRANS 483 - operator EllipticF 484 - operator EllipticK 485 - operator EllipticKprime 486 - operator EllipticE 487 - operator EllipticTHETA 488 - operator JacobiZETA 489 - section end: 490 - subsection 29 g29 Gamma and Related Functions - operator POCHHAMMER 491 - operator GAMMA 492 - operator BETA 493 - operator PSI 494 - operator POLYGAMMA 495 - section end: 496 - subsection 30 g30 Miscellaneous Functions - operator DILOG extended 497 - operator Lambert\_W function 498 - section end: 499 - subsection 31 g31 Orthogonal Polynomials - operator ChebyshevT 500 - operator ChebyshevU 501 - operator HermiteP 502 - operator LaguerreP 503 - operator LegendreP 504 - operator JacobiP 505 - operator GegenbauerP 506 - operator SolidHarmonicY 507 - operator SphericalHarmonicY 508 - section end: 509 - subsection 32 g32 Integral Functions - operator Si 510 - operator Shi 511 - operator s_i 512 - operator Ci 513 - operator Chi 514 - operator ERF extended 515 - operator erfc 516 - operator Ei 517 - operator Fresnel_C 518 - operator Fresnel_S 519 - section end: 520 - subsection 33 g33 Combinatorial Operators - operator BINOMIAL 521 - operator STIRLING1 522 - operator STIRLING2 523 - section end: 524 - subsection 34 g34 3j and 6j symbols - operator ThreejSymbol 525 - operator Clebsch_Gordan 526 - operator SixjSymbol 527 - section end: 528 - subsection 35 g35 Miscellaneous - operator HYPERGEOMETRIC 529 - operator MeijerG 530 - operator Heaviside 531 - operator erfi 532 - ---- return from file ../doc/help/pk-specf.tex - ---- input file ../doc/help/taylor.tex - section end: 533 - section end: 534 -section 36 g36 Taylor series - introduction TAYLOR 535 - operator taylor 536 - switch taylorautocombine 537 - switch taylorautoexpand 538 - operator taylorcombine 539 - switch taylorkeeporiginal 540 - operator taylororiginal 541 - switch taylorprintorder 542 - variable taylorprintterms 543 - operator taylorrevert 544 - operator taylorseriesp 545 - operator taylortemplate 546 - operator taylortostandard 547 - ---- return from file ../doc/help/taylor.tex - ---- input file ../doc/help/pk-gplot.tex - section end: 548 -section 37 g37 Gnuplot package - introduction GNUPLOT and REDUCE 549 - concept Axes names 550 - type Pointset 551 - command PLOT 552 - command PLOTRESET 553 - variable title 554 - variable xlabel 555 - variable ylabel 556 - variable zlabel 557 - variable terminal 558 - variable size 559 - variable view 560 - switch contour 561 - switch surface 562 - switch hidden3d 563 - switch PLOTKEEP 564 - switch PLOTREFINE 565 - variable plot_xmesh 566 - variable plot_ymesh 567 - switch SHOW_GRID 568 - switch TRPLOT 569 - ---- return from file ../doc/help/pk-gplot.tex - ---- input file ../doc/help/linalg.tex - section end: 570 -section 38 g38 Linear Algebra package - introduction Linear Algebra package 571 - switch fast_la 572 - operator add_columns 573 - operator add_rows 574 - operator add_to_columns 575 - operator add_to_rows 576 - operator augment_columns 577 - operator band_matrix 578 - operator block_matrix 579 - operator char_matrix 580 - operator char_poly 581 - operator cholesky 582 - operator coeff_matrix 583 - operator column_dim 584 - operator companion 585 - operator copy_into 586 - operator diagonal 587 - operator extend 588 - operator find_companion 589 - operator get_columns 590 - operator get_rows 591 - operator gram_schmidt 592 - operator hermitian_tp 593 - operator hessian 594 - operator hilbert 595 - operator jacobian 596 - operator jordan_block 597 - operator lu_decom 598 - operator make_identity 599 - operator matrix_augment 600 - operator matrixp 601 - operator matrix_stack 602 - operator minor 603 - operator mult_columns 604 - operator mult_rows 605 - operator pivot 606 - operator pseudo_inverse 607 - operator random_matrix 608 - operator remove_columns 609 - operator remove_rows 610 - operator row_dim 611 - operator rows_pivot 612 - operator simplex 613 - operator squarep 614 - operator stack_rows 615 - operator sub_matrix 616 - operator svd 617 - operator swap_columns 618 - operator swap_entries 619 - operator swap_rows 620 - operator symmetricp 621 - operator toeplitz 622 - operator vandermonde 623 - ---- return from file ../doc/help/linalg.tex - ---- input file ../doc/help/normform.tex - section end: 624 -section 39 g39 Matrix Normal Forms - operator Smithex 625 - operator Smithex\_int 626 - operator Frobenius 627 - operator Ratjordan 628 - operator Jordansymbolic 629 - operator Jordan 630 - ---- return from file ../doc/help/normform.tex - ---- input file ../doc/help/pk-misc.tex - section end: 631 -section 40 g40 Miscellaneous Packages - introduction Miscellaneous Packages 632 - package ALGINT 633 - package APPLYSYM 634 - package ARNUM 635 - package ASSIST 636 - package AVECTOR 637 - package BOOLEAN 638 - package CALI 639 - package CAMAL 640 - package CHANGEVR 641 - package COMPACT 642 - package CRACK 643 - package CVIT 644 - package DEFINT 645 - package DESIR 646 - package DFPART 647 - package DUMMY 648 - package EXCALC 649 - package FPS 650 - package FIDE 651 - package GENTRAN 652 - package IDEALS 653 - package INEQ 654 - package INVBASE 655 - package LAPLACE 656 - package LIE 657 - package MODSR 658 - package NCPOLY 659 - package ORTHOVEC 660 - package PHYSOP 661 - package PM 662 - package RANDPOLY 663 - package REACTEQN 664 - package RESET 665 - package RESIDUE 666 - package RLFI 667 - package SCOPE 668 - package SETS 669 - package SPDE 670 - package SYMMETRY 671 - package TPS 672 - package TRI 673 - package TRIGSIMP 674 - package XCOLOR 675 - package XIDEAL 676 - package WU 677 - package ZEILBERG 678 - package ZTRANS 679 - ---- return from file ../doc/help/pk-misc.tex - ---- input file ../doc/help/outmode.tex - section end: 680 -section 41 g41 Outmoded Operations - command ED 681 - command EDITDEF 682 - ---- return from file ../doc/help/outmode.tex - section end: 683 - section end: 684 - -nil - - - -html_indexfile();..... compiling html index file - -nil - - - -LISP_indexfile();..... compiling independent index file - -nil - - - -bye; - -End of Lisp run after 11.39+0.61 seconds - -+++ Transcript closed at end of run +++ DELETED r38/help/redhelp.rtf Index: r38/help/redhelp.rtf ================================================================== --- r38/help/redhelp.rtf +++ r38/help/redhelp.rtf @@ -1,37984 +0,0 @@ -{\rtf1\ansi \deff0{\fonttbl{\f0\froman Tms Rmn;} -{\f1\fdecor Symbol;} -{\f2\fswiss Helv;} -{\f3\fmodern pica;} -{\f4\fmodern Courier;} -{\f5\fmodern elite;} -{\f6\fmodern prestige;} -{\f7\fmodern lettergothic;} -{\f8\fmodern gothicPS;} -{\f9\fmodern cubicPS;} -{\f10\fmodern lineprinter;} -{\f11\fswiss Helvetica;} -{\f12\fmodern avantegarde;} -{\f13\fmodern spartan;} -{\f14\fmodern metro;} -{\f15\fmodern presentation;} -{\f16\fmodern APL;} -{\f17\fmodern OCRA;} -{\f18\fmodern OCRB;} -{\f19\froman boldPS;} -{\f20\froman emperorPS;} -{\f21\froman madaleine;} -{\f22\froman zapf humanist;} -{\f23\froman classic;} -{\f24\froman roman f;} -{\f25\froman roman g;} -{\f26\froman roman h;} -{\f27\froman timesroman;} -{\f28\froman century;} -{\f29\froman palantino;} -{\f30\froman souvenir;} -{\f31\froman garamond;} -{\f32\froman caledonia;} -{\f33\froman bodini;} -{\f34\froman university;} -{\f35\fscript Script;} -{\f36\fscript scriptPS;} -{\f37\fscript script c;} -{\f38\fscript script d;} -{\f39\fscript commercial script;} -{\f40\fscript park avenue;} -{\f41\fscript coronet;} -{\f42\fscript script h;} -{\f43\fscript greek;} -{\f44\froman kana;} -{\f45\froman hebrew;} -{\f46\froman roman s;} -{\f47\froman russian;} -{\f48\froman roman u;} -{\f49\froman roman v;} -{\f50\froman roman w;} -{\f51\fdecor narrator;} -{\f52\fdecor emphasis;} -{\f53\fdecor zapf chancery;} -{\f54\fdecor decor d;} -{\f55\fdecor old english;} -{\f56\fdecor decor f;} -{\f57\fdecor decor g;} -{\f58\fdecor cooper black;} -{\f59\fnil linedraw;} -{\f60\fnil math7;} -{\f61\fnil math8;} -{\f62\fnil bar3of9;} -{\f63\fnil EAN;} -{\f64\fnil pcline;} -{\f65\fnil tech h;} -{\f66\fswiss Helvetica-Narrow;} -{\f67\fmodern Modern;} -{\f68\froman Roman;}} - 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-{\info{\author Dan Davids}{\operator Dan Davids}{\creatim\yr2137\mo8\dy7} -{\revtim\yr1990\mo5\dy9\hr16\min54}{\version3}{\edmins3134}{\nofpages0} -{\nofwords65536}{\nofchars69885}{\vern8310}} - -\ftnbj \sectd \linex576\endnhere -\pard\plain \sl240 \fs20 - - - -{\f2 -#{\footnote \pard\plain \sl240 \fs20 # IDENTIFIER} - -${\footnote \pard\plain \sl240 \fs20 $ IDENTIFIER} - -+{\footnote \pard\plain \sl240 \fs20 + g2:0644} - - K{\footnote \pard\plain \sl240 \fs20 K IDENTIFIER type;type} - -}{\b\f2 IDENTIFIER}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par -\par - -Identifiers in REDUCE consist of one or more alphanumeric characters, of -which the first must be alphabetical. The maximum number of characters -allowed is system dependent, but is usually over 100. However, printing -is simplified if they are kept under 25 characters. -\par -\par -You can also use special characters in your identifiers, but each must be -preceded by an exclamation point }{\f3 !} {\f2 as an escape character. Useful -special characters are }{\f3 # $ % ^ & * - + = ? < > ~ | / !} {\f2 and -the space. Note that the use of the exclamation point as a special -character requires a second exclamation point as an escape character. -The underscore }{\f3 _} {\f2 is special in this regard. It must be preceded -by an escape character in the first position in an identifier, but is -treated like a normal letter within an identifier. -\par -\par -Other characters, such as }{\f3 ( ) # ; ` ' "} {\f2 can also be used if -preceded by a }{\f3 !} {\f2 , but as they have special meanings to the Lisp -reader it is best to avoid them to avoid confusion. -\par -\par -Many system identifiers have * before or after their names, or - between -words. If you accidentally pick one of these names for your own identifier, -it could have disastrous effects. For this reason it is wise not to include -* or - anywhere in your identifiers. -\par -\par -You will notice that REDUCE does not use the escape characters when it prints -identifiers containing special characters; however, you still must use them -when you refer to these identifiers. Be careful when editing statements -containing escaped special characters to treat the character and its escape -as an inseparable pair. -\par -\par -Identifiers are used for variable names, labels for }{\f3 go to} {\f2 statements, -and names of arrays, matrices, operators, and procedures. Once an identifier is -used as a matrix, array, scalar or operator identifier, it may not be used -again as a matrix, array or operator. An operator or array identifier may -later be used as a scalar without problems, but a matrix identifier cannot be -used as a scalar. All procedures are entered into the system as operators, so -the name of a procedure may not be used as a matrix, array, or operator -identifier either. - \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # KERNEL} - -${\footnote \pard\plain \sl240 \fs20 $ KERNEL} - -+{\footnote \pard\plain \sl240 \fs20 + g2:0645} - - K{\footnote \pard\plain \sl240 \fs20 K KERNEL type;type} - -}{\b\f2 KERNEL}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par -\par - -A }{\f3 kernel} {\f2 is a form that cannot be modified further by the REDUCE -canonical simplifier. Scalar variables are always kernels. The -other important class of kernels are operators with their arguments. -Some examples should help clarify this concept: -\par -\par -\pard \tx3420 }{\f4 \par - Expression Kernel? \par - \par - x Yes \par - varname Yes \par - cos(a) Yes \par - log(sin(x**2)) Yes \par - a*b No \par - (x+y)**4 No \par - matrix-identifier No \par -\pard \sl240 }{\f2 Many REDUCE operators expect kernels among their arguments. Error messages -result from attempts to use non-kernel expressions for these arguments. - \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # STRING} - -${\footnote \pard\plain \sl240 \fs20 $ STRING} - -+{\footnote \pard\plain \sl240 \fs20 + g2:0646} - - K{\footnote \pard\plain \sl240 \fs20 K STRING type;type} - -}{\b\f2 STRING}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par -\par - -A }{\f3 string} {\f2 is any collection of characters enclosed in double quotation -marks (}{\f3 "} {\f2 ). It may be used as an argument for a variety of commands -and operators, such as }{\f3 in} {\f2 , }{\f3 rederr} {\f2 and }{\f3 write} {\f2 . - \par -examples: \par -\pard \tx3420 }{\f4 \par -write "this is a string"; \par - \par - this is a string \par - \par - \par -write a, " ", b, " ",c,"!"; \par - \par - A B C! \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g2} - -${\footnote \pard\plain \sl240 \fs20 $ Concepts} - -+{\footnote \pard\plain \sl240 \fs20 + index:0002} -}{\b\f2 Concepts}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb IDENTIFIER type} -{\v\f2 IDENTIFIER}{\f2 \par -}{\f2 \tab}{\f2\uldb KERNEL type} -{\v\f2 KERNEL}{\f2 \par -}{\f2 \tab}{\f2\uldb STRING type} -{\v\f2 STRING}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # assumptions} - -${\footnote \pard\plain \sl240 \fs20 $ assumptions} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0647} - - K{\footnote \pard\plain \sl240 \fs20 K solve;assumptions variable;variable} - -}{\b\f2 ASSUMPTIONS}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -After solving a linear or polynomial equation system -with parameters, the variable }{\f3 assumptions} {\f2 contains a list -of side relations for the parameters. The solution is valid only -as long as none of these expression is zero. - \par -examples: \par -\pard \tx3420 }{\f4 \par -solve(\{a*x-b*y+x,y-c\},\{x,y\}); \par - \par - b*c \par - \{\{x=-----,y=c\}\} \par - a + 1 \par - \par - \par -assumptions; \par - \par - \{a + 1\} \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CARD\_NO} - -${\footnote \pard\plain \sl240 \fs20 $ CARD_NO} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0648} - - K{\footnote \pard\plain \sl240 \fs20 K output;FORTRAN;CARD_NO variable;variable} - -}{\b\f2 CARD\_NO}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -}{\f3 card_no} {\f2 sets the total number of cards allowed in a Fortran -output statement when }{\f3 fort} {\f2 is on. Default is 20. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on fort; \par - \par -card_no := 4; \par - \par - CARD_NO=4. \par - \par - \par -z := (x + y)**15; \par - \par - ANS1=5005.*X**6*Y**9+3003.*X**5*Y**10+1365.*X**4*Y** \par - . 11+455.*X**3*Y**12+105.*X**2*Y**13+15.*X*Y**14+Y**15 \par - Z=X**15+15.*X**14*Y+105.*X**13*Y**2+455.*X**12*Y**3+ \par - . 1365.*X**11*Y**4+3003.*X**10*Y**5+5005.*X**9*Y**6+ \par - . 6435.*X**8*Y**7+6435.*X**7*Y**8+ANS1 \par - \par -\pard \sl240 }{\f2 Twenty total cards means 19 continuation cards. You may set it for more -if your Fortran system allows more. Expressions are broken apart in a -Fortran-compatible way if they extend for more than }{\f3 card_no} {\f2 -continuation cards. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # E} - -${\footnote \pard\plain \sl240 \fs20 $ E} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0649} - - K{\footnote \pard\plain \sl240 \fs20 K E constant;constant} - -}{\b\f2 E}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - -The constant }{\f3 e} {\f2 is reserved for use as the base of the natural -logarithm. Its value is approximately 2.71828284590, which REDUCE gives -to the current decimal precision when the switch } -{\f2\uldb rounded}{\v\f2 ROUNDED} -{\f2 is on. -\par -\par -}{\f3 e} {\f2 may be used as an iterative variable in a } -{\f2\uldb for}{\v\f2 FOR} -{\f2 statement, -or as a local variable or a } -{\f2\uldb procedure}{\v\f2 PROCEDURE} -{\f2 . If }{\f3 e} {\f2 is defined -as a local -variable inside the procedure, the normal definition as the base of the -natural logarithm would be suspended inside the procedure. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EVAL\_MODE} - -${\footnote \pard\plain \sl240 \fs20 $ EVAL_MODE} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0650} - - K{\footnote \pard\plain \sl240 \fs20 K symbolic;algebraic;EVAL_MODE variable;variable} - -}{\b\f2 EVAL\_MODE}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -The system variable }{\f3 eval_mode} {\f2 contains the current mode, either -} -{\f2\uldb algebraic}{\v\f2 ALGEBRAIC} -{\f2 or } -{\f2\uldb symbolic}{\v\f2 SYMBOLIC} -{\f2 . -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -EVAL\_MODE; \par - \par - ALGEBRAIC \par - \par -\pard \sl240 }{\f2 Some commands do not behave the same way in algebraic and symbolic modes. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FORT\_WIDTH} - -${\footnote \pard\plain \sl240 \fs20 $ FORT_WIDTH} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0651} - - K{\footnote \pard\plain \sl240 \fs20 K FORTRAN;output;FORT_WIDTH variable;variable} - -}{\b\f2 FORT\_WIDTH}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -The }{\f3 fort_width} {\f2 variable sets the number of characters in a line of -Fortran-compatible output produced when the } -{\f2\uldb fort}{\v\f2 FORT} -{\f2 switch is on. -Default is 70. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -fort_width := 30; \par - \par - FORT_WIDTH := 30 \par - \par - \par -on fort; \par - \par -df(sin(x**3*y),x); \par - \par - ANS=3.*COS(X \par - . **3*Y)*X**2* \par - . Y \par - \par -\pard \sl240 }{\f2 }{\f3 fort_width} {\f2 includes the usually blank characters at the beginning -of the card. As you may notice above, it is conservative and makes the -lines even shorter than it was told. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # HIGH\_POW} - -${\footnote \pard\plain \sl240 \fs20 $ HIGH_POW} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0652} - - K{\footnote \pard\plain \sl240 \fs20 K degree;polynomial;HIGH_POW variable;variable} - -}{\b\f2 HIGH\_POW}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -The variable }{\f3 high_pow} {\f2 is set by } -{\f2\uldb coeff}{\v\f2 COEFF} -{\f2 to the highest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -coeff((x+1)^5*(x*(y+3)^2)^2,x); \par - \par - \{0, \par - 0, \par - 4 3 2 \par - Y + 12*Y + 54*Y + 108*Y + 81, \par - 4 3 2 \par - 5*(Y + 12*Y + 54*Y + 108*Y + 81), \par - 4 3 2 \par - 10*(Y + 12*Y + 54*Y + 108*Y + 81), \par - 4 3 2 \par - 10*(Y + 12*Y + 54*Y + 108*Y + 81), \par - 4 3 2 \par - 5*(Y + 12*Y + 54*Y + 108*Y + 81), \par - 4 3 2 \par - Y + 12*Y + 54*Y + 108*Y + 81\} \par - \par - \par -high_pow; \par - \par - 7 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # I} - -${\footnote \pard\plain \sl240 \fs20 $ I} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0653} - - K{\footnote \pard\plain \sl240 \fs20 K complex;I constant;constant} - -}{\b\f2 I}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - - \par -\par -REDUCE knows }{\f3 i} {\f2 is the square root of -1, - and that i^2 = -1. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -(a + b*i)*(c + d*i); \par - \par - A*C + A*D*I + B*C*I - B*D \par - \par - \par -i**2; \par - \par - -1 \par - \par -\pard \sl240 }{\f2 }{\f3 i} {\f2 cannot be used as an identifier. It is all right to use }{\f3 i} {\f2 -as an index variable in a }{\f3 for} {\f2 loop, or as a local (}{\f3 scalar} {\f2 ) -variable inside a }{\f3 begin...end} {\f2 block, but it loses its definition as -the square root of -1 inside the block in that case. -\par -\par -Only the simplest properties of i are known by REDUCE unless -the switch } -{\f2\uldb complex}{\v\f2 COMPLEX} -{\f2 is turned on, which implements full complex -arithmetic in factoring, simplification, and functional values. -}{\f3 complex} {\f2 is ordinarily off. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # INFINITY} - -${\footnote \pard\plain \sl240 \fs20 $ INFINITY} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0654} - - K{\footnote \pard\plain \sl240 \fs20 K INFINITY constant;constant} - -}{\b\f2 INFINITY}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - -The name }{\f3 infinity} {\f2 is used to represent the infinite positive number. -However, at the present time, arithmetic in terms of this operator reflects -finite arithmetic, rather than true operations on infinity. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # LOW\_POW} - -${\footnote \pard\plain \sl240 \fs20 $ LOW_POW} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0655} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;degree;LOW_POW variable;variable} - -}{\b\f2 LOW\_POW}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -The variable }{\f3 low_pow} {\f2 is set by } -{\f2\uldb coeff}{\v\f2 COEFF} -{\f2 to the lowest power -of the variable of interest in the given expression. You can access this -variable for use in further computation or display. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -coeff((x+2*y)**6,y); \par - \par - 6 \par - \{X , \par - 5 \par - 12*X , \par - 4 \par - 60*X , \par - 3 \par - 160*X , \par - 2 \par - 240*X , \par - 192*X, \par - 64\} \par - \par - \par -low_pow; \par - \par - 0 \par - \par - \par -coeff(x**2*(x*sin(y) + 1),x); \par - \par - \par - \par - \{0,0,1,SIN(Y)\} \par - \par - \par -low_pow; \par - \par - 2 \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # NIL} - -${\footnote \pard\plain \sl240 \fs20 $ NIL} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0656} - - K{\footnote \pard\plain \sl240 \fs20 K false;NIL constant;constant} - -}{\b\f2 NIL}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - - \par -\par -}{\f3 nil} {\f2 represents the truth value false in symbolic mode, and is -a synonym for 0 in algebraic mode. It cannot be used for any other -purpose, even inside procedures or } -{\f2\uldb for}{\v\f2 FOR} -{\f2 loops. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # PI} - -${\footnote \pard\plain \sl240 \fs20 $ PI} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0657} - - K{\footnote \pard\plain \sl240 \fs20 K PI constant;constant} - -}{\b\f2 PI}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - -The identifier }{\f3 pi} {\f2 is reserved for use as the circular constant. -Its value is given by 3.14159265358..., which REDUCE gives to the current -decimal precision when REDUCE is in a floating-point mode. -\par -\par -}{\f3 pi} {\f2 may be used as a looping variable in a } -{\f2\uldb for}{\v\f2 FOR} -{\f2 statement, -or as a local variable in a } -{\f2\uldb procedure}{\v\f2 PROCEDURE} -{\f2 . Its value in such cases -will be taken from the local environment. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # requirements} - -${\footnote \pard\plain \sl240 \fs20 $ requirements} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0658} - - K{\footnote \pard\plain \sl240 \fs20 K solve;requirements variable;variable} - -}{\b\f2 REQUIREMENTS}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -After an attempt to solve an inconsistent equation system -with parameters, the variable }{\f3 requirements} {\f2 contains a list -of expressions. These expressions define a set of conditions implicitly -equated with zero. Any solution to this system defines a setting for -the parameters sufficient to make the original system consistent. - \par -examples: \par -\pard \tx3420 }{\f4 \par -solve(\{x-a,x-y,y-1\},\{x,y\}); \par - \par - \{\} \par - \par - \par -requirements; \par - \par - \{a - 1\} \par - \par -\pard \sl240 }{\f2 \par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # ROOT\_MULTIPLICITIES} - -${\footnote \pard\plain \sl240 \fs20 $ ROOT_MULTIPLICITIES} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0659} - - K{\footnote \pard\plain \sl240 \fs20 K polynomial;solve;root;ROOT_MULTIPLICITIES variable;variable} - -}{\b\f2 ROOT\_MULTIPLICITIES}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par -\par - - \par -\par -The }{\f3 root_multiplicities} {\f2 variable is set to the list of the -multiplicities of the roots of an equation by the } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 operator. -\par -\par -} -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 returns its solutions in a list. The multiplicities of -each solution are put in the corresponding locations of the list -}{\f3 root_multiplicities} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # T} - -${\footnote \pard\plain \sl240 \fs20 $ T} - -+{\footnote \pard\plain \sl240 \fs20 + g3:0660} - - K{\footnote \pard\plain \sl240 \fs20 K T constant;constant} - -}{\b\f2 T}{\f2 \tab \tab \tab \tab }{\b\f2 constant}{\f2 \par -\par - -The constant }{\f3 t} {\f2 stands for the truth value true. It cannot be used -as a scalar variable in a } -{\f2\uldb block}{\v\f2 block} -{\f2 , as a looping variable in a -} -{\f2\uldb for}{\v\f2 FOR} -{\f2 statement or as an } -{\f2\uldb operator}{\v\f2 OPERATOR} -{\f2 name. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # g3} - -${\footnote \pard\plain \sl240 \fs20 $ Variables} - -+{\footnote \pard\plain \sl240 \fs20 + index:0003} -}{\b\f2 Variables}{\f2 \par }\pard \sl240 {\f2 \par } -{\f2 \tab}{\f2\uldb assumptions variable} -{\v\f2 assumptions}{\f2 \par -}{\f2 \tab}{\f2\uldb CARD\_NO variable} -{\v\f2 CARD\_NO}{\f2 \par -}{\f2 \tab}{\f2\uldb E constant} -{\v\f2 E}{\f2 \par -}{\f2 \tab}{\f2\uldb EVAL\_MODE variable} -{\v\f2 EVAL\_MODE}{\f2 \par -}{\f2 \tab}{\f2\uldb FORT\_WIDTH variable} -{\v\f2 FORT\_WIDTH}{\f2 \par -}{\f2 \tab}{\f2\uldb HIGH\_POW variable} -{\v\f2 HIGH\_POW}{\f2 \par -}{\f2 \tab}{\f2\uldb I constant} -{\v\f2 I}{\f2 \par -}{\f2 \tab}{\f2\uldb INFINITY constant} -{\v\f2 INFINITY}{\f2 \par -}{\f2 \tab}{\f2\uldb LOW\_POW variable} -{\v\f2 LOW\_POW}{\f2 \par -}{\f2 \tab}{\f2\uldb NIL constant} -{\v\f2 NIL}{\f2 \par -}{\f2 \tab}{\f2\uldb PI constant} -{\v\f2 PI}{\f2 \par -}{\f2 \tab}{\f2\uldb requirements variable} -{\v\f2 requirements}{\f2 \par -}{\f2 \tab}{\f2\uldb ROOT\_MULTIPLICITIES variable} -{\v\f2 ROOT\_MULTIPLICITIES}{\f2 \par -}{\f2 \tab}{\f2\uldb T constant} -{\v\f2 T}{\f2 \par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # semicolon} - -${\footnote \pard\plain \sl240 \fs20 $ semicolon} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0661} - - K{\footnote \pard\plain \sl240 \fs20 K semicolon command;command} - -}{\b\f2 ;}{\f2 \tab }{\b\f2 SEMICOLON}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The semicolon is a statement delimiter, indicating results are to be printed -when used in interactive mode. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -(x+1)**2; \par - \par - 2 \par - X + 2*X + 1 \par - \par - \par -df(x**2 + 1,x); \par - \par - 2*X \par - \par -\pard \sl240 }{\f2 Entering a }{\f3 Return} {\f2 without a semicolon or dollar sign results in a -prompt on the following line. A semicolon or dollar sign can be -added at this point to execute the statement. In interactive mode, a -statement that is ended with a semicolon and }{\f3 Return} {\f2 has its results -printed on the screen. -\par -\par -Inside a group statement }{\f3 <<} {\f2 ...}{\f3 >>} {\f2 -or a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 block, a -semicolon or dollar sign separates individual REDUCE statements. Since -results are not printed from a block without a specific }{\f3 return} {\f2 -statement, there is no difference between using the semicolon or dollar -sign. In a group statement, the last value produced is the value -returned by the group statement. Thus, if a semicolon or dollar sign is -placed between the last statement and the ending brackets, the group -statement returns the value 0 or nil, rather than the value of the -last statement. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # dollar} - -${\footnote \pard\plain \sl240 \fs20 $ dollar} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0662} - - K{\footnote \pard\plain \sl240 \fs20 K dollar command;command} - -}{\b\f2 $}{\f2 \tab }{\b\f2 DOLLAR}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The dollar sign is a statement delimiter, indicating results are not to be -printed when used in interactive mode. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -(x+1)**2$ \pard \sl240 }{\f2 The workspace is set to }{\f4 x^2 + 2x + 1}{\f2 - but nothing shows on the screen}{\f4 \pard \tx3420 \par - \par - \par -ws; \par - \par - 2 \par - X + 2*X + 1 \par - \par -\pard \sl240 }{\f2 -\par -\par -Entering a }{\f3 Return} {\f2 without a semicolon or dollar sign results in a -prompt on the following line. A semicolon or dollar sign can -be added at this point to execute the statement. In interactive mode, a -statement that ends with a dollar sign }{\f3 $} {\f2 and a }{\f3 Return} {\f2 is -executed, but the results not printed. -\par -\par -Inside a } -{\f2\uldb group}{\v\f2 group} -{\f2 statement }{\f3 <<} {\f2 ...}{\f3 >>} {\f2 -or a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 , a -semicolon or dollar sign separates individual REDUCE statements. Since -results are not printed from a } -{\f2\uldb block}{\v\f2 block} -{\f2 without a specific -} -{\f2\uldb return}{\v\f2 RETURN} -{\f2 \par -\par -statement, there is no difference between using the semicolon or dollar -sign. -\par -\par -In a group statement, the last value produced is the value returned by the -group statement. Thus, if a semicolon or dollar sign is placed between the -last statement and the ending brackets, the group statement returns the -value 0 or nil, rather than the value of the last statement. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # percent} - -${\footnote \pard\plain \sl240 \fs20 $ percent} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0663} - - K{\footnote \pard\plain \sl240 \fs20 K percent command;command} - -}{\b\f2 %}{\f2 \tab }{\b\f2 PERCENT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The percent sign is used to precede comments; everything from a percent -to the end of the line is ignored. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -df(x**3 + y,x);\% This is a comment \key\{Return\} \par - \par - \par - 2 \par - 3*X \par - \par - \par -int(3*x**2,x) \%This is a comment; \key\{Return\} \par -\pard \sl240 }{\f2 A prompt is given, waiting for the semicolon that was not -detected in the comment}{\f4 \pard \tx3420 \pard \sl240 }{\f2 -\par -\par -Statement delimiters }{\f3 ;} {\f2 and }{\f3 $} {\f2 are not detected between a -percent sign and the end of the line. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # dot} - -${\footnote \pard\plain \sl240 \fs20 $ dot} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0664} - - K{\footnote \pard\plain \sl240 \fs20 K list;dot operator;operator} - -}{\b\f2 .}{\f2 \tab }{\b\f2 DOT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The . (dot) infix binary operator adds a new item to the beginning of an -existing } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . In high energy physics expressions, -it can also be used -to represent the scalar product of two Lorentz four-vectors. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 .} {\f4 -\par -\par -}{\f2 \par - can be any REDUCE scalar expression, including a list; - must be a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 to avoid producing an error message. -The dot operator is right associative. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -liss := a . \{\}; \par - \par - LISS := \{A\} \par - \par - \par -liss := b . liss; \par - \par - LISS := \{B,A\} \par - \par - \par -newliss := liss . liss; \par - \par - NEWLISS := \{\{B,A\},B,A\} \par - \par - \par -firstlis := a . b . \{c\}; \par - \par - FIRSTLIS := \{A,B,C\} \par - \par - \par -secondlis := x . y . \{z\}; \par - \par - SECONDLIS := \{X,Y,Z\} \par - \par - \par -for i := 1:3 sum part(firstlis,i)*part(secondlis,i); \par - \par - \par - \par - A*X + B*Y + C*Z \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # assign} - -${\footnote \pard\plain \sl240 \fs20 $ assign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0665} - - K{\footnote \pard\plain \sl240 \fs20 K assign;assign operator;operator} - -}{\b\f2 :=}{\f2 \tab }{\b\f2 ASSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 :=} {\f2 is the assignment operator, assigning the value on the right-hand -side to the identifier or other valid expression on the left-hand side. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 :=} {\f4 -\par -\par -}{\f2 \par - is ordinarily a single identifier, though simple -expressions may be used (see Comments below). is any -valid REDUCE expression. If is a } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 -identifier, then - can be a matrix identifier (redimensioned if -necessary) which has each element set to the corresponding elements -of the identifier on the right-hand side. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := x**2 + 1; \par - \par - 2 \par - A := X + 1 \par - \par - \par -a; \par - \par - 2 \par - X + 1 \par - \par - \par -first := second := third; \par - \par - FIRST := SECOND := THIRD \par - \par - \par -first; \par - \par - THIRD \par - \par - \par -second; \par - \par - THIRD \par - \par - \par -b := for i := 1:5 product i; \par - \par - B := 120 \par - \par - \par -b; \par - \par - 120 \par - \par - \par -w + (c := x + 3) + z; \par - \par - W + X + Z + 3 \par - \par - \par -c; \par - \par - X + 3 \par - \par - \par -y + b := c; \par - \par - Y + B := C \par - \par - \par -y; \par - \par - - (B - C) \par - \par -\pard \sl240 }{\f2 The assignment operator is right associative, as shown in the second and -third examples. A string of such assignments has all but the last -item set to the value of the last item. Embedding an assignment statement -in another expression has the side effect of making the assignment, as well -as causing the given replacement in the expression. -\par -\par -Assignments of values to expressions rather than simple identifiers (such as in -the last example above) can also be done, subject to the following remarks: -\par -\par -\tab (i) -If the left-hand side is an identifier, an operator, or a power, the -substitution rule is added to the rule table. -\par -\par -\tab (ii) -If the operators }{\f3 - + /} {\f2 appear on the left-hand side, all but the first -term of the expression is moved to the right-hand side. -\par -\par -\tab (iii) -If the operator }{\f3 *} {\f2 appears on the left-hand side, any constant terms are -moved to the right-hand side, but the symbolic factors remain. -\par -\par -Assignment is valid for } -{\f2\uldb array}{\v\f2 ARRAY} -{\f2 elements, but not for entire arrays. -The assignment operator can also be used to attach functionality to operators. -\par -\par -A recursive construction such as }{\f3 a := a + b} {\f2 is allowed, but when -}{\f3 a} {\f2 is referenced again, the process of resubstitution continues -until the expression stack overflows (you get an error message). -Recursive assignments can be done safely inside controlled loop -expressions, such as } -{\f2\uldb for}{\v\f2 FOR} -{\f2 ... or } -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # equalsign} - -${\footnote \pard\plain \sl240 \fs20 $ equalsign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0666} - - K{\footnote \pard\plain \sl240 \fs20 K equalsign operator;operator} - -}{\b\f2 =}{\f2 \tab }{\b\f2 EQUALSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 =} {\f2 operator is a prefix or infix equality comparison operator. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 =} {\f4 (}{\f3 ,} {\f4 ) - or - }{\f3 =} {\f4 -\par -\par -}{\f2 \par - can be any REDUCE scalar expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 4; \par - \par - A := 4 \par - \par - \par -if =(a,10) then write "yes" else write "no"; \par - \par - \par - \par - no \par - \par - \par -b := c; \par - \par - B := C \par - \par - \par -if b = c then write "yes" else write "no"; \par - \par - \par - \par - yes \par - \par - \par -on rounded; \par - \par -if 4.0 = 4 then write "yes" else write "no"; \par - \par - \par - \par - yes \par - \par -\pard \sl240 }{\f2 This logical equality operator can only be used inside a conditional -statement, such as } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 -or } -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 . In other places the equal -sign establishes an algebraic object of type } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # replace} - -${\footnote \pard\plain \sl240 \fs20 $ replace} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0667} - - K{\footnote \pard\plain \sl240 \fs20 K replace operator;operator} - -}{\b\f2 =>}{\f2 \tab }{\b\f2 REPLACE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -\par -\par -The }{\f3 =>} {\f2 operator is a binary operator used in } -{\f2\uldb rule}{\v\f2 RULE} -{\f2 lists to -denote replacements. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -operator f; \par - \par -let f(x) => x^2; \par - \par -f(x); \par - \par - 2 \par - x \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # plussign} - -${\footnote \pard\plain \sl240 \fs20 $ plussign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0668} - - K{\footnote \pard\plain \sl240 \fs20 K plussign operator;operator} - -}{\b\f2 +}{\f2 \tab }{\b\f2 PLUSSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 +} {\f2 operator is a prefix or infix n-ary addition operator. -\par -\par - \par -syntax: \par -}{\f4 \{}{\f3 +} {\f4 \}+ -\par -\par -or }{\f3 +} {\f4 ( \{,\}+) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x**4 + 4*x**2 + 17*x + 1; \par - \par - 4 2 \par - X + 4*X + 17*X + 1 \par - \par - \par -14 + 15 + x; \par - \par - X + 29 \par - \par - \par -+(1,2,3,4,5); \par - \par - 15 \par - \par -\pard \sl240 }{\f2 }{\f3 +} {\f2 is also valid as an addition operator for } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 variables -that are of the same dimensions and for } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # minussign} - -${\footnote \pard\plain \sl240 \fs20 $ minussign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0669} - - K{\footnote \pard\plain \sl240 \fs20 K minussign operator;operator} - -}{\b\f2 -}{\f2 \tab }{\b\f2 MINUSSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 -} {\f2 operator is a prefix or infix binary subtraction operator, as well -as the unary minus operator. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 -} {\f4 -or }{\f3 -} {\f4 (,) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -15 - 4; \par - \par - 11 \par - \par - \par -x*(-5); \par - \par - - 5*X \par - \par - \par -a - b - 15; \par - \par - A - B - 15 \par - \par - \par --(a,4); \par - \par - A - 4 \par - \par -\pard \sl240 }{\f2 The subtraction operator is left associative, so that a - b - c is equivalent -to (a - b) - c, as shown in the third example. The subtraction operator is -also valid with } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 expressions of the correct dimensions -and with } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # asterisk} - -${\footnote \pard\plain \sl240 \fs20 $ asterisk} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0670} - - K{\footnote \pard\plain \sl240 \fs20 K asterisk operator;operator} - -}{\b\f2 *}{\f2 \tab }{\b\f2 ASTERISK}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 *} {\f2 operator is a prefix or infix n-ary multiplication operator. -\par -\par - \par -syntax: \par -}{\f4 \{}{\f3 *} {\f4 \}+ -\par -\par -or }{\f3 *} {\f4 ( \{,\}+) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -15*3; \par - \par - 45 \par - \par - \par -24*x*yvalue*2; \par - \par - 48*X*YVALUE \par - \par - \par -*(6,x); \par - \par - 6*X \par - \par - \par -on rounded; \par - \par -3*1.5*x*x*x; \par - \par - 3 \par - 4.5*X \par - \par - \par -off rounded; \par - \par -2x**2; \par - \par - 2 \par - 2*X \par - \par -\pard \sl240 }{\f2 REDUCE assumes you are using an implicit multiplication operator when an -identifier is preceded by a number, as shown in the last line above. Since -no valid identifiers can begin with numbers, there is no ambiguity in -making this assumption. -\par -\par -The multiplication operator is also valid with } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 expressions -of the -proper dimensions: matrices A and B -can be multiplied if -A is n x m and B is -m x p. Matrices and } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s can also be -multiplied by scalars: the -result is as if each element was multiplied by the scalar. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # slash} - -${\footnote \pard\plain \sl240 \fs20 $ slash} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0671} - - K{\footnote \pard\plain \sl240 \fs20 K slash operator;operator} - -}{\b\f2 /}{\f2 \tab }{\b\f2 SLASH}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 /} {\f2 operator is a prefix or infix binary division operator or -prefix unary } -{\f2\uldb recip}{\v\f2 RECIP} -{\f2 rocal operator. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 /} {\f4 or - }{\f3 /} {\f4 -\par -\par -or }{\f3 /} {\f4 (,) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -20/5; \par - \par - 4 \par - \par - \par -100/6; \par - \par - 50 \par - -- \par - 3 \par - \par - \par -16/2/x; \par - \par - 8 \par - - \par - X \par - \par - \par -/b; \par - \par - 1 \par - - \par - B \par - \par - \par -/(y,5); \par - \par - Y \par - - \par - 5 \par - \par - \par -on rounded; \par - \par -35/4; \par - \par - 8.75 \par - \par - \par -/20; \par - \par - 0.05 \par - \par -\pard \sl240 }{\f2 The division operator is left associative, so that }{\f3 a/b/c} {\f2 is equivalent -to }{\f3 (a/b)/c} {\f2 . The division operator is also valid with square -} -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 expressions of the same dimensions: With A and -B both n x n matrices and B -invertible, A/B is -given by A*B^-1. -Division of a matrix by a scalar is defined, with the results being the -division of each element of the matrix by the scalar. Division of a -scalar by a matrix is defined if the matrix is invertible, and has the -effect of multiplying the scalar by the inverse of the matrix. When -}{\f3 /} {\f2 is used as a reciprocal operator for a matrix, the inverse of -the matrix is returned if it exists. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # power} - -${\footnote \pard\plain \sl240 \fs20 $ power} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0672} - - K{\footnote \pard\plain \sl240 \fs20 K power operator;operator} - -}{\b\f2 **}{\f2 \tab }{\b\f2 POWER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 **} {\f2 operator is a prefix or infix binary exponentiation operator. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 **} {\f4 - or }{\f3 **} {\f4 (,) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x**15; \par - \par - 15 \par - X \par - \par - \par -x**y**z; \par - \par - Y*Z \par - X \par - \par - \par -x**(y**z); \par - \par - Z \par - Y \par - X \par - \par - \par - **(y,4); \par - \par - 4 \par - Y \par - \par - \par -on rounded; \par - \par -2**pi; \par - \par - 8.82497782708 \par - \par -\pard \sl240 }{\f2 The exponentiation operator is left associative, so that }{\f3 a**b**c} {\f2 is -equivalent to }{\f3 (a**b)**c} {\f2 , as shown in the second example. Note -that this is not }{\f3 a**(b**c)} {\f2 , which would be right associative. -\par -\par -When } -{\f2\uldb nat}{\v\f2 NAT} -{\f2 is on (the default), REDUCE output produces raised -exponents, as shown. The symbol }{\f3 ^} {\f2 , which is the upper-case 6 on -most keyboards, may be used in the place of }{\f3 **} {\f2 . -\par -\par -A square } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 may also be raised to positive and negative powers -with the exponentiation operator (negative powers require the matrix to be -invertible). Scalar expressions and } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s may be raised to -fractional and floating-point powers. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # caret} - -${\footnote \pard\plain \sl240 \fs20 $ caret} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0673} - - K{\footnote \pard\plain \sl240 \fs20 K caret operator;operator} - -}{\b\f2 ^}{\f2 \tab }{\b\f2 CARET}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 ^} {\f2 operator is a prefix or infix binary exponentiation operator. -It is equivalent to } -{\f2\uldb power}{\v\f2 power} -{\f2 or **. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 ^} {\f4 - or }{\f3 ^} {\f4 (,) -\par -\par -}{\f2 \par - may be any valid REDUCE expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -x^15; \par - \par - 15 \par - X \par - \par - \par -x^y^z; \par - \par - Y*Z \par - X \par - \par - \par -x^(y^z); \par - \par - Z \par - Y \par - X \par - \par - \par -^(y,4); \par - \par - 4 \par - Y \par - \par - \par -on rounded; \par - \par -2^pi; \par - \par - 8.82497782708 \par - \par -\pard \sl240 }{\f2 The exponentiation operator is left associative, so that }{\f3 a^b^c} {\f2 is -equivalent to }{\f3 (a^b)^c} {\f2 , as shown in the second example. Note -that this is }{\f3 a^(b^c)} {\f2 , which would be right associative. -\par -\par -When } -{\f2\uldb nat}{\v\f2 NAT} -{\f2 is on (the default), REDUCE output produces raised -exponents, as shown. -\par -\par -A square } -{\f2\uldb matrix}{\v\f2 MATRIX} -{\f2 may also be raised to positive -and negative powers with -the exponentiation operator (negative powers require the matrix to be -invertible). Scalar expressions and } -{\f2\uldb equation}{\v\f2 EQUATION} -{\f2 s -may be raised to fractional and floating-point powers. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # geqsign} - -${\footnote \pard\plain \sl240 \fs20 $ geqsign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0674} - - K{\footnote \pard\plain \sl240 \fs20 K geqsign operator;operator} - -}{\b\f2 >=}{\f2 \tab }{\b\f2 GEQSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -}{\f3 >=} {\f2 is an infix binary comparison operator, which returns true if -its first argument is greater than or equal to its second argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 >=} {\f4 -\par -\par -}{\f2 \par - must evaluate to an integer or floating-point number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -if (3 >= 2) then yes; \par - \par - yes \par - \par - \par -a := 15; \par - \par - A := 15 \par - \par - \par -if a >= 20 then big else small; \par - \par - \par - small \par - \par -\pard \sl240 }{\f2 The binary comparison operators can only be used for comparisons between -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 -or } -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 or } -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # greater} - -${\footnote \pard\plain \sl240 \fs20 $ greater} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0675} - - K{\footnote \pard\plain \sl240 \fs20 K greater operator;operator} - -}{\b\f2 >}{\f2 \tab }{\b\f2 GREATER}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 >} {\f2 is an infix binary comparison operator that returns - true if its first argument is strictly greater than its second. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 >} {\f4 -\par -\par -}{\f2 \par - must evaluate to a number, e.g., integer, rational or -floating point number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -on rounded; \par - \par -if 3.0 > 3 then write "different" else write "same"; \par - \par - \par - same \par - \par - \par -off rounded; \par - \par -a := 20; \par - \par - A := 20 \par - \par - \par -if a > 20 then write "bigger" else write "not bigger"; \par - \par - \par - not bigger \par - \par -\pard \sl240 }{\f2 The binary comparison operators can only be used for comparisons between -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 or -} -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 or } -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # leqsign} - -${\footnote \pard\plain \sl240 \fs20 $ leqsign} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0676} - - K{\footnote \pard\plain \sl240 \fs20 K leqsign operator;operator} - -}{\b\f2 <=}{\f2 \tab }{\b\f2 LEQSIGN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -}{\f3 <=} {\f2 is an infix binary comparison operator that returns - true if its first argument is less than or equal to its second argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 <=} {\f4 -\par -\par -}{\f2 \par - must evaluate to a number, e.g., integer, rational or -floating point number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 10; \par - \par - A := 10 \par - \par - \par -if a <= 10 then true; \par - \par - true \par - \par -\pard \sl240 }{\f2 The binary comparison operators can only be used for comparisons between -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 or -} -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 or } -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # less} - -${\footnote \pard\plain \sl240 \fs20 $ less} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0677} - - K{\footnote \pard\plain \sl240 \fs20 K less operator;operator} - -}{\b\f2 <}{\f2 \tab }{\b\f2 LESS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -}{\f3 <} {\f2 is an infix binary logical comparison operator that -returns true if its first argument is strictly less than its second -argument. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 <} {\f4 -\par -\par -}{\f2 \par - must evaluate to a number, e.g., integer, rational or -floating point number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -f := -3; \par - \par - F := -3 \par - \par - \par -if f < -3 then write "yes" else write "no"; \par - \par - \par - no \par - \par -\pard \sl240 }{\f2 The binary comparison operators can only be used for comparisons between -numbers or variables that evaluate to numbers. The truth values returned -by such a comparison can only be used inside programming constructs, such -as } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 -or } -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 or -} -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # tilde} - -${\footnote \pard\plain \sl240 \fs20 $ tilde} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0678} - - K{\footnote \pard\plain \sl240 \fs20 K tilde operator;operator} - -}{\b\f2 ~}{\f2 \tab }{\b\f2 TILDE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 ~} {\f2 is used as a unary prefix operator in the left-hand -sides of } -{\f2\uldb rule}{\v\f2 RULE} -{\f2 s to mark } -{\f2\uldb free variable}{\v\f2 Free_Variable} -{\f2 s. A double tilde -marks an optional } -{\f2\uldb free variable}{\v\f2 Free_Variable} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # group} - -${\footnote \pard\plain \sl240 \fs20 $ group} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0679} - - K{\footnote \pard\plain \sl240 \fs20 K group command;command} - -}{\b\f2 <<}{\f2 \tab }{\b\f2 GROUP}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The }{\f3 <<} {\f2 ...}{\f3 >>} {\f2 command is a group statement, -used to group statements -together where REDUCE expects a single statement. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 <<} {\f4 \{; }{\f3 or} {\f4 - }{\f2 \}* }{\f3 >>} {\f2 -\par -\par -\par - may be any valid REDUCE statement or expression. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 2; \par - \par - A := 2 \par - \par - \par -if a < 5 then <>; \par - \par - \par - 12 \par - \par - \par -<>; \par - \par - \par - 2 \par - C + 90*C + 202 \par - ---------------- \par - 225 \par - \par -\pard \sl240 }{\f2 The value returned from a group statement is the value of the last -individual statement executed inside it. Note that when a semicolon is -placed between the last statement and the closing brackets, 0 or - nil is returned. Group statements are often used in the -consequence portions of } -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 , -} -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 , and -} -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 -clauses. They may also be used in interactive -operation to execute several statements at one time. Statements inside -the group statement are separated by semicolons or dollar signs. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # AND} - -${\footnote \pard\plain \sl240 \fs20 $ AND} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0680} - - K{\footnote \pard\plain \sl240 \fs20 K AND operator;operator} - -}{\b\f2 AND}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 and} {\f2 binary logical operator returns true if both of its -arguments are true. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 and} {\f4 -\par -\par -}{\f2 \par - must evaluate to true or nil. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 12; \par - \par - A := 12 \par - \par - \par -if numberp a and a < 15 then write a**2 else write "no"; \par - \par - \par - \par - 144 \par - \par - \par -clear a; \par - \par -if numberp a and a < 15 then write a**2 else write "no"; \par - \par - \par - \par - no \par - \par -\pard \sl240 }{\f2 Logical operators can only be used inside conditional statements, such as -} -{\f2\uldb while}{\v\f2 WHILE} -{\f2 ...}{\f3 do} {\f2 or -} -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 . }{\f3 and} {\f2 examines each of -its arguments in order, and quits, returning nil, on finding an -argument that is not true. An error results if it is used in other -contexts. -\par -\par -}{\f3 and} {\f2 is left associative: }{\f3 x and y and z} {\f2 is equivalent to -}{\f3 (x and y) and z} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # BEGIN} - -${\footnote \pard\plain \sl240 \fs20 $ BEGIN} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0681} - - K{\footnote \pard\plain \sl240 \fs20 K BEGIN command;command} - -}{\b\f2 BEGIN}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -}{\f3 begin} {\f2 is used to start a } -{\f2\uldb block}{\v\f2 block} -{\f2 statement, which is closed with -}{\f3 end} {\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 begin} {\f4 \{}{\f3 ;} {\f4 \}* }{\f3 end} {\f4 -\par -\par -}{\f2 \par - is any valid REDUCE statement. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -begin for i := 1:3 do write i end; \par - \par - \par - 1 \par - 2 \par - 3 \par - \par - \par -begin scalar n;n:=1;b:=for i:=1:4 product(x-i);return n end; \par - \par - \par - \par - 1 \par - \par - \par -b; \par - \par - 4 3 2 \par - X - 10*X + 35*X - 50*X + 24 \par - \par -\pard \sl240 }{\f2 A }{\f3 begin} {\f2 ...}{\f3 end} {\f2 block can do actions (such as }{\f3 write} {\f2 ), but -does not -return a value unless instructed to by a } -{\f2\uldb return}{\v\f2 RETURN} -{\f2 statement, which must -be the last statement executed in the block. It is unnecessary to insert -a semicolon before the }{\f3 end} {\f2 . -\par -\par -Local variables, if any, are declared in the first statement immediately -after }{\f3 begin} {\f2 , and may be defined as }{\f3 scalar, integer,} {\f2 or -}{\f3 real} {\f2 . } -{\f2\uldb array}{\v\f2 ARRAY} -{\f2 variables declared -within a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 block -are global in every case, and } -{\f2\uldb let}{\v\f2 LET} -{\f2 statements have global -effects. A } -{\f2\uldb let}{\v\f2 LET} -{\f2 statement involving a formal parameter affects -the calling parameter that corresponds to it. } -{\f2\uldb let}{\v\f2 LET} -{\f2 statements -involving local variables make global assignments, overwriting outside -variables by the same name or creating them if they do not exist. You -can use this feature to affect global variables from procedures, but be -careful that you do not do it inadvertently. -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # block} - -${\footnote \pard\plain \sl240 \fs20 $ block} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0682} - - K{\footnote \pard\plain \sl240 \fs20 K block command;command} - -}{\b\f2 BLOCK}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -A }{\f3 block} {\f2 is a sequence of statements enclosed by -commands } -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 and } -{\f2\uldb end}{\v\f2 END} -{\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 begin} {\f4 \{}{\f3 ;} {\f4 \}* }{\f3 end} {\f4 -\par -\par -}{\f2 \par -For more details see } -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 . -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # COMMENT} - -${\footnote \pard\plain \sl240 \fs20 $ COMMENT} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0683} - - K{\footnote \pard\plain \sl240 \fs20 K COMMENT command;command} - -}{\b\f2 COMMENT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -Beginning with the word }{\f3 comment} {\f2 , all text until the next statement -terminator (}{\f3 ;} {\f2 or }{\f3 $} {\f2 ) is ignored. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -x := a**2 comment--a is the velocity of the particle;; \par - \par - \par - \par - 2 \par - X := A \par - \par -\pard \sl240 }{\f2 Note that the first semicolon ends the comment and the second one -terminates the original REDUCE statement. -\par -\par -Multiple-line comments are often needed in interactive files. The -}{\f3 comment} {\f2 command allows a normal-looking text to accompany the -REDUCE statements in the file. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # CONS} - -${\footnote \pard\plain \sl240 \fs20 $ CONS} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0684} - - K{\footnote \pard\plain \sl240 \fs20 K CONS operator;operator} - -}{\b\f2 CONS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 cons} {\f2 operator adds a new element to the beginning of a -} -{\f2\uldb list}{\v\f2 LIST} -{\f2 . Its -operation is identical to the symbol } -{\f2\uldb dot}{\v\f2 dot} -{\f2 (dot). It can be used -infix or prefix. -\par -\par - \par -syntax: \par -}{\f4 }{\f3 cons} {\f4 (,) or }{\f3 cons} {\f4 -\par -\par -}{\f2 \par - can be any REDUCE scalar expression, including a list; -must be a list. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par - \par -liss := cons(a,\{b\}); \par - \par - \{A,B\} \par - \par - \par - \par -liss := c cons liss; \par - \par - \{C,A,B\} \par - \par - \par - \par -newliss := for each y in liss collect cons(y,list x); \par - \par - \par - \par - NEWLISS := \{\{C,X\},\{A,X\},\{B,X\}\} \par - \par - \par - \par -for each y in newliss sum (first y)*(second y); \par - \par - \par - \par - X*(A + B + C) \par - \par -\pard \sl240 }{\f2 If you want to use }{\f3 cons} {\f2 to put together two elements into a new list, -you must make the second one into a list with curly brackets or the }{\f3 list} {\f2 -command. You can also start with an empty list created by }{\f3 \{\}} {\f2 . -\par -\par -The }{\f3 cons} {\f2 operator is right associative: }{\f3 a cons b cons c} {\f2 is valid -if }{\f3 c} {\f2 is a list; }{\f3 b} {\f2 need not be a list. The list produced is -}{\f3 \{a,b,c\}} {\f2 . -\par -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # END} - -${\footnote \pard\plain \sl240 \fs20 $ END} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0685} - - K{\footnote \pard\plain \sl240 \fs20 K END command;command} - -}{\b\f2 END}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -The command }{\f3 end} {\f2 has two main uses: -\par -\par -\tab (i) -as the ending of a } -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 ; and -\par -\tab (ii) -to end input from a file. -\par -\par -In a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 , there need not be a delimiter -(}{\f3 ;} {\f2 or }{\f3 $} {\f2 ) before the }{\f3 end} {\f2 , though there must be one -after it, or a right bracket matching an earlier left bracket. -\par -\par -Files to be read into REDUCE should end with }{\f3 end;} {\f2 , which must be -preceded by a semicolon (usually the last character of the previous line). -The additional semicolon avoids problems with mistakes in the files. If -you have suspended file operation by answering }{\f3 n} {\f2 to a }{\f3 pause} {\f2 -command, you are still, technically speaking, ``in" the file. Use -}{\f3 end} {\f2 to exit the file. -\par -\par -An }{\f3 end} {\f2 at the top level of a program is ignored. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # EQUATION} - -${\footnote \pard\plain \sl240 \fs20 $ EQUATION} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0686} - - K{\footnote \pard\plain \sl240 \fs20 K =;arithmetic;equal;equation;EQUATION type;type} - -}{\b\f2 EQUATION}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par -\par - - \par -\par -An }{\f3 equation} {\f2 is an expression where two algebraic expressions -are connected by the (infix) operator } -{\f2\uldb equal}{\v\f2 EQUAL} -{\f2 or by }{\f3 =} {\f2 . -For access to the components of an }{\f3 equation} {\f2 the operators -} -{\f2\uldb lhs}{\v\f2 LHS} -{\f2 , } -{\f2\uldb rhs}{\v\f2 RHS} -{\f2 or } -{\f2\uldb part}{\v\f2 PART} -{\f2 can be used. The -evaluation of the left-hand side of an }{\f3 equation} {\f2 is controlled -by the switch } -{\f2\uldb evallhseqp}{\v\f2 EVALLHSEQP} -{\f2 , while the right-hand side is -evaluated unconditionally. When an }{\f3 equation} {\f2 is part of a -logical expression, e.g. in a } -{\f2\uldb if}{\v\f2 IF} -{\f2 or } -{\f2\uldb while}{\v\f2 WHILE} -{\f2 statement, -the equation is evaluated by subtracting both sides can comparing -the result with zero. -\par -\par -Equations occur in many contexts, e.g. as arguments of the } -{\f2\uldb sub}{\v\f2 SUB} -{\f2 -operator and in the arguments and the results -of the operator } -{\f2\uldb solve}{\v\f2 SOLVE} -{\f2 . An equation can be member of a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 -and you may assign an equation to a variable. Elementary arithmetic is supported -for equations: if } -{\f2\uldb evallhseqp}{\v\f2 EVALLHSEQP} -{\f2 is on, you may add and subtract -equations, and you can combine an equation with a scalar expression by -addition, subtraction, multiplication, division and raise an equation -to a power. - \par -examples: \par -\pard \tx3420 }{\f4 \par -on evallhseqp; \par - \par -u:=x+y=1$ \par - \par -v:=2x-y=0$ \par - \par -2*u-v; \par - \par - - 3*y=-2 \par - \par - \par -ws/3; \par - \par - 2 \par - y=-- \par - 3 \par - \par -\pard \sl240 }{\f2 \par -\par -Important: the equation must occur in the leftmost term of such an expression. -For other operations, e.g. taking function values of both sides, use the -} -{\f2\uldb map}{\v\f2 MAP} -{\f2 operator. -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FIRST} - -${\footnote \pard\plain \sl240 \fs20 $ FIRST} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0687} - - K{\footnote \pard\plain \sl240 \fs20 K decomposition;list;FIRST operator;operator} - -}{\b\f2 FIRST}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - - \par -\par -The }{\f3 first} {\f2 operator returns the first element of a } -{\f2\uldb list}{\v\f2 LIST} -{\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 first} {\f4 () or }{\f3 first} {\f4 -\par -\par -}{\f2 \par - must be a non-empty list to avoid an error message. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -alist := \{a,b,c,d\}; \par - \par - ALIST := \{A,B,C,D\} \par - \par - \par -first alist; \par - \par - A \par - \par - \par -blist := \{x,y,\{ww,aa,qq\},z\}; \par - \par - BLIST := \{X,Y,\{WW,AA,QQ\},Z\} \par - \par - \par -first third blist; \par - \par - WW \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FOR} - -${\footnote \pard\plain \sl240 \fs20 $ FOR} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0688} - - K{\footnote \pard\plain \sl240 \fs20 K loop;FOR command;command} - -}{\b\f2 FOR}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -The }{\f3 for} {\f2 command is used for iterative loops. There are many -possible forms it can take. -\par -\par -\pard \tx3420 }{\f4 \par - / \ \par - / |STEP UNTIL| \ \par - |:=| || \par -FOR| | : | | \par - | \ / | \par - |EACH IN | \par - \ / \par - \par - where ::= DO|PRODUCT|SUM|COLLECT|JOIN. \par -\pard \sl240 }{\f2 can be any valid REDUCE identifier except }{\f3 t} {\f2 or -}{\f3 nil} {\f2 , , and can be any expression -that evaluates to a positive or negative integer. must be a -valid } -{\f2\uldb list}{\v\f2 LIST} -{\f2 structure. -The action taken must be one of the actions shown -above, each of which is followed by a single REDUCE expression, statement -or a } -{\f2\uldb group}{\v\f2 group} -{\f2 (}{\f3 <<} {\f2 ...}{\f3 >>} {\f2 ) or } -{\f2\uldb block}{\v\f2 block} -{\f2 -(} -{\f2\uldb begin}{\v\f2 BEGIN} -{\f2 ...} -{\f2\uldb end}{\v\f2 END} -{\f2 ) statement. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -for i := 1:10 sum i; \par - \par - \par - \par - 55 \par - \par - \par -for a := -2 step 3 until 6 product a; \par - \par - \par - \par - -8 \par - \par - \par -a := 3; \par - \par - A := 3 \par - \par - \par -for iter := 4:a do write iter; \par - \par -m := 0; \par - \par - M := 0 \par - \par - \par -for s := 10 step -1 until 3 do <>; \par - \par -m; \par - \par - 520 \par - \par - \par -for each x in \{q,r,s\} sum x**2; \par - \par - 2 2 2 \par - Q + R + S \par - \par - \par -for i := 1:4 collect 1/i; \par - \par - \par - \par - 1 1 1 \par - \{1,-,-,-\} \par - 2 3 4 \par - \par - \par -for i := 1:3 join list solve(x**2 + i*x + 1,x); \par - \par - \par - \par - SQRT(3)*I + 1 \par - \{\{X= --------------, \par - 2 \par - SQRT(3)*I - 1 \par - X= --------------\} \par - 2 \par - \{X=-1\}, \par - SQRT(5) + 3 SQRT(5) - 3 \par - \{X= - -----------,X=-----------\}\} \par - 2 2 \par - \par -\pard \sl240 }{\f2 The behavior of each of the five action words follows: -\par -\par -\pard \tx3420 }{\f4 \par - Action Word Behavior \par -Keyword Argument Type Action \par - do statement, command, group Evaluates its argument once \par - or block for each iteration of the loop, \par - not saving results \par -collect expression, statement, Evaluates its argument once for \par - command, group, block, list each iteration of the loop, \par - storing the results in a list \par - which is returned by the for \par - statement when done \par - join list or an operator which Evaluates its argument once for \par - produces a list each iteration of the loop, \par - appending the elements in each \par - individual result list onto the \par - overall result list \par -product expression, statement, Evaluates its argument once for \par - command, group or block each iteration of the loop, \par - multiplying the results together \par - and returning the overall product \par - sum expression, statement, Evaluates its argument once for \par - command, group or block each iteration of the loop, \par - adding the results together and \par - returning the overall sum \par -\pard \sl240 }{\f2 For number-driven }{\f3 for} {\f2 statements, if the ending limit is smaller -than the beginning limit (larger in the case of negative steps) the action -statement is not executed at all. The iterative variable is local to the -}{\f3 for} {\f2 statement, and does not affect the value of an identifier with -the same name. For list-driven }{\f3 for} {\f2 statements, if the list is -empty, the action statement is not executed, but no error occurs. -\par -\par -You can use nested }{\f3 for} {\f2 statements, with the inner }{\f3 for} {\f2 -statement after the action keyword. You must make sure that your inner -statement returns an expression that the outer statement can handle. -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # FOREACH} - -${\footnote \pard\plain \sl240 \fs20 $ FOREACH} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0689} - - K{\footnote \pard\plain \sl240 \fs20 K loop;FOREACH command;command} - -}{\b\f2 FOREACH}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - - \par -\par -}{\f3 foreach} {\f2 is a synonym for the }{\f3 for each} {\f2 variant of the -} -{\f2\uldb for}{\v\f2 FOR} -{\f2 construct. It is designed to iterate down a list, and an -error will occur if a list is not used. The use of }{\f3 for each} {\f2 is -preferred to }{\f3 foreach} {\f2 . -\par -\par - \par -syntax: \par -}{\f4 }{\f3 foreach} {\f4 in -\par -\par -where ::= }{\f3 do | product | sum | collect | join} {\f4 -\par -\par -}{\f2 \par - \par -examples: \par -\pard \tx3420 }{\f4 \par -foreach x in \{q,r,s\} sum x**2; \par - \par - 2 2 2 \par - Q + R + S \par - \par -\pard \sl240 }{\f2 - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GEQ} - -${\footnote \pard\plain \sl240 \fs20 $ GEQ} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0690} - - K{\footnote \pard\plain \sl240 \fs20 K GEQ operator;operator} - -}{\b\f2 GEQ}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par -\par - -The }{\f3 geq} {\f2 operator is a binary infix or prefix logical operator. It -returns true if its first argument is greater than or equal to its second -argument. As an infix operator it is identical with }{\f3 >=} {\f2 . - \par -syntax: \par -}{\f4 \par -\par -}{\f3 geq} {\f4 (,) or -}{\f3 geq} {\f4 -\par -\par -}{\f2 \par - can be any valid REDUCE expression that evaluates to a -number. -\par -\par - \par -examples: \par -\pard \tx3420 }{\f4 \par -a := 20; \par - \par - A := 20 \par - \par - \par -if geq(a,25) then write "big" else write "small"; \par - \par - \par - \par - small \par - \par - \par -if a geq 20 then write "big" else write "small"; \par - \par - \par - \par - big \par - \par - \par -if (a geq 18) then write "big" else write "small"; \par - \par - \par - \par - big \par - \par -\pard \sl240 }{\f2 Logical operators can only be used in conditional statements such as -\par -\par -} -{\f2\uldb if}{\v\f2 IF} -{\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 or -} -{\f2\uldb repeat}{\v\f2 REPEAT} -{\f2 ...}{\f3 until} {\f2 . -\par -\par -\par - - -\page - - -#{\footnote \pard\plain \sl240 \fs20 # GOTO} - -${\footnote \pard\plain \sl240 \fs20 $ GOTO} - -+{\footnote \pard\plain \sl240 \fs20 + g4:0691} - - K{\footnote \pard\plain \sl240 \fs20 K GOTO command;command} - -}{\b\f2 GOTO}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par -\par - -Inside a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } -{\f2\uldb block}{\v\f2 block} -{\f2 , }{\f3 goto} {\f2 , or -preferably, }{\f3 go to} {\f2 , transfers flow of control to a labeled statement. - \par -syntax: \par -}{\f4 \par -\par -}{\f3 go to} {\f4 or }{\f3 goto} {\f4 -\par -\par -}{\f2 \par - is of the form