Artifact ae9c6f8f08728f51cab334592026302c3adfdfdf3be4bec289b0bd04506df786:
- Executable file
r38/help/redhelp.rtf
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[f2fda60abd]
at
2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 902092) [annotate] [blame] [check-ins using] [more...]
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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. \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 <item> }{\f3 .} {\f4 <list> \par \par }{\f2 \par <item> can be any REDUCE scalar expression, including a list; <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 <restricted\_expression> }{\f3 :=} {\f4 <expression> \par \par }{\f2 \par <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 } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 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. \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 (<expression>}{\f3 ,} {\f4 <expression>) or <expression> }{\f3 =} {\f4 <expression> \par \par }{\f2 \par <expression> 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 <expression> \{}{\f3 +} {\f4 <expression>\}+ \par \par or }{\f3 +} {\f4 (<expression> \{,<expression>\}+) \par \par }{\f2 \par <expression> 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 <expression> }{\f3 -} {\f4 <expression> or }{\f3 -} {\f4 (<expression>,<expression>) \par \par }{\f2 \par <expression> 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 <expression> \{}{\f3 *} {\f4 <expression>\}+ \par \par or }{\f3 *} {\f4 (<expression> \{,<expression>\}+) \par \par }{\f2 \par <expression> 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 <expression>}{\f3 /} {\f4 <expression> or }{\f3 /} {\f4 <expression> \par \par or }{\f3 /} {\f4 (<expression>,<expression>) \par \par }{\f2 \par <expression> 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 <expression> }{\f3 **} {\f4 <expression> or }{\f3 **} {\f4 (<expression>,<expression>) \par \par }{\f2 \par <expression> 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 <expression> }{\f3 ^} {\f4 <expression> or }{\f3 ^} {\f4 (<expression>,<expression>) \par \par }{\f2 \par <expression> 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. \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 <expression> }{\f3 >=} {\f4 <expression> \par \par }{\f2 \par <expression> 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 <expression> }{\f3 >} {\f4 <expression> \par \par }{\f2 \par <expression> 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 <expression> }{\f3 <=} {\f4 <expression> \par \par }{\f2 \par <expression> 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 <expression> }{\f3 <} {\f4 <expression> \par \par }{\f2 \par <expression> 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 <statement>\{; <statement> }{\f3 or} {\f4 }{\f2 <statement>\}* }{\f3 >>} {\f2 \par \par \par <statement> 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 <<b := a + 10; write b>>; \par \par \par 12 \par \par \par <<d := c/15; f := d + 3; f**2>>; \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 <logical\_expression> }{\f3 and} {\f4 <logical\_expression> \par \par }{\f2 \par <logical\_expression> 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 <statement>\{}{\f3 ;} {\f4 <statement>\}* }{\f3 end} {\f4 \par \par }{\f2 \par <statement> 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 <statement>\{}{\f3 ;} {\f4 <statement>\}* }{\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 (<item>,<list>) or <item> }{\f3 cons} {\f4 <list> \par \par }{\f2 \par <item> can be any REDUCE scalar expression, including a list; <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 (<list>) or }{\f3 first} {\f4 <list> \par \par }{\f2 \par <list> 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 <number> UNTIL| \ \par |<var>:=<number>| |<number>| \par FOR| | : | |<action> <exprn> \par | \ / | \par |EACH <var> IN <list> | \par \ / \par \par where <action> ::= DO|PRODUCT|SUM|COLLECT|JOIN. \par \pard \sl240 }{\f2 <var> can be any valid REDUCE identifier except }{\f3 t} {\f2 or }{\f3 nil} {\f2 , <inc>, <start> and <stop> can be any expression that evaluates to a positive or negative integer. <list> 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 <<d := 10*s;m := m + d>>; \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 <variable> in <list> <action> <expression> \par \par where <action> ::= }{\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 (<expression>,<expression>) or <expression> }{\f3 geq} {\f4 <expression> \par \par }{\f2 \par <expression> 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 <labeled_statement> or }{\f3 goto} {\f4 <labeled_statement> \par \par }{\f2 \par <labeled_statement> is of the form <label> }{\f3 :} {\f2 <statement> \par \par \par examples: \par \pard \tx3420 }{\f4 \par procedure dumb(a); \par begin scalar q; \par go to lab; \par q := df(a**2 - sin(a),a); \par write q; \par lab: return a \par end; \par \pard \sl240 \par \par DUMB \par \par \par \par dumb(17); \par \par 17 \par \par \pard \sl240 }{\f2 }{\f3 go to} {\f2 can only be used inside a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } {\f2\uldb block}{\v\f2 block} {\f2 , and inside the block only statements at the top level can be labeled, not ones inside }{\f3 <<} {\f2 ...}{\f3 >>} {\f2 , } {\f2\uldb while}{\v\f2 WHILE} {\f2 ...}{\f3 do} {\f2 , etc. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # GREATERP} ${\footnote \pard\plain \sl240 \fs20 $ GREATERP} +{\footnote \pard\plain \sl240 \fs20 + g4:0692} K{\footnote \pard\plain \sl240 \fs20 K GREATERP operator;operator} }{\b\f2 GREATERP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par The }{\f3 greaterp} {\f2 logical operator returns true if its first argument is strictly greater than its second argument. As an infix operator it is identical with }{\f3 >} {\f2 . \par syntax: \par }{\f4 \par \par }{\f3 greaterp} {\f4 (<expression>,<expression>) or <expression> }{\f3 greaterp} {\f4 <expression> \par \par }{\f2 \par <expression> can be any valid REDUCE expression that evaluates to a number. \par \par \par examples: \par \pard \tx3420 }{\f4 \par \par a := 20; \par \par A := 20 \par \par \par if greaterp(a,25) then write "big" else write "small"; \par \par \par \par small \par \par \par if a greaterp 20 then write "big" else write "small"; \par \par \par \par small \par \par \par if (a greaterp 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 ...} {\f2\uldb while}{\v\f2 WHILE} {\f2 . \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # IF} ${\footnote \pard\plain \sl240 \fs20 $ IF} +{\footnote \pard\plain \sl240 \fs20 + g4:0693} K{\footnote \pard\plain \sl240 \fs20 K then;else;IF command;command} }{\b\f2 IF}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par \par The }{\f3 if} {\f2 command is a conditional statement that executes a statement if a condition is true, and optionally another statement if it is not. \par syntax: \par }{\f4 \par \par }{\f3 if} {\f4 <condition> }{\f3 then} {\f4 <statement> \tab option(}{\f3 else} {\f4 <statement>) \par \par }{\f2 \par <condition> must be a logical or comparison operator that evaluates to a } {\f2\uldb boolean value}{\v\f2 boolean_value} {\f2 . <statement> must be a single REDUCE statement or a } {\f2\uldb group}{\v\f2 group} {\f2 (}{\f3 <<} {\f2 ...}{\f3 >>} {\f2 ) or } {\f2\uldb block}{\v\f2 block} {\f2 (}{\f3 begin} {\f2 ...}{\f3 end} {\f2 ) statement. \par \par \par examples: \par \pard \tx3420 }{\f4 \par if x = 5 then a := b+c else a := d+f; \par \par \par \par D + F \par \par \par x := 9; \par \par X := 9 \par \par \par if numberp x and x<20 then y := sqrt(x) else write "illegal"; \par \par \par \par 3 \par \par \par clear x; \par \par if numberp x and x<20 then y := sqrt(x) else write "illegal"; \par \par \par \par illegal \par \par \par x := 12; \par \par X := 12 \par \par \par a := if x < 5 then 100 else 150; \par \par \par \par A := 150 \par \par \par b := u**(if x < 10 then 2); \par \par \par B := 1 \par \par \par bb := u**(if x > 10 then 2); \par \par \par 2 \par BB := U \par \par \pard \sl240 }{\f2 An }{\f3 if} {\f2 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 }{\f3 else} {\f2 clause, the value is 0 if a number is expected, and nothing otherwise. \par \par The }{\f3 else} {\f2 clause may be left out if no action is to be taken if the condition is false. \par \par The condition may be a compound conditional statement using } {\f2\uldb and}{\v\f2 AND} {\f2 or } {\f2\uldb or}{\v\f2 OR} {\f2 . If a non-conditional statement, such as a constant, is used by accident, it is assumed to have value true. \par \par Be sure to use } {\f2\uldb group}{\v\f2 group} {\f2 or } {\f2\uldb block}{\v\f2 block} {\f2 statements after }{\f3 then} {\f2 or }{\f3 else} {\f2 . \par \par The }{\f3 if} {\f2 operator is right associative. The following constructions are examples: \par \par \tab (1) \par syntax: \par }{\f4 \par \par }{\f3 if} {\f4 <condition> }{\f3 then} {\f4 }{\f3 if} {\f4 <condition> }{\f3 then} {\f4 <action> }{\f3 else} {\f4 <action> \par \par }{\f2 \par which is equivalent to \par syntax: \par }{\f4 \par \par }{\f3 if} {\f4 <condition> }{\f3 then} {\f4 (}{\f3 if} {\f4 <condition> }{\f3 then} {\f4 <action> }{\f3 else} {\f4 <action>); \par \par }{\f2 \par \tab (2) \par syntax: \par }{\f4 \par \par }{\f3 if} {\f4 <condition> }{\f3 then} {\f4 <action> }{\f3 else if} {\f4 <condition> }{\f3 then} {\f4 <action> }{\f3 else} {\f4 <action> \par \par }{\f2 \par which is equivalent to \par syntax: \par }{\f4 \par \par }{\f3 if} {\f4 <condition> }{\f3 then} {\f4 <action> }{\f3 else} {\f4 \par \par (}{\f3 if} {\f4 <condition> }{\f3 then} {\f4 <action> }{\f3 else} {\f4 <action>). \par \par }{\f2 \par \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # LIST} ${\footnote \pard\plain \sl240 \fs20 $ LIST} +{\footnote \pard\plain \sl240 \fs20 + g4:0694} K{\footnote \pard\plain \sl240 \fs20 K list;LIST operator;operator} }{\b\f2 LIST}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 list} {\f2 operator constructs a list from its arguments. \par syntax: \par }{\f4 \par \par }{\f3 list} {\f4 (<item> \{,<item>\}*) or }{\f3 list} {\f4 () to construct an empty list. \par \par }{\f2 \par <item> can be any REDUCE scalar expression, including another list. Left and right curly brackets can also be used instead of the operator }{\f3 list} {\f2 to construct a list. \par \par \par examples: \par \pard \tx3420 }{\f4 \par liss := list(c,b,c,\{xx,yy\},3x**2+7x+3,df(sin(2*x),x)); \par \par \par \par 2 \par LISS := \{C,B,C,\{XX,YY\},3*X + 7*X + 3,2*COS(2*X)\} \par \par \par length liss; \par \par 6 \par \par \par liss := \{c,b,c,\{xx,yy\},3x**2+7x+3,df(sin(2*x),x)\}; \par \par \par \par 2 \par LISS := \{C,B,C,\{XX,YY\},3*X + 7*X + 3,2*COS(2*X)\} \par \par \par emptylis := list(); \par \par EMPTYLIS := \{\} \par \par \par a . emptylis; \par \par \{A\} \par \par \pard \sl240 }{\f2 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 } {\f2\uldb part}{\v\f2 PART} {\f2 operator can be used to access elements anywhere within a list hierarchy. The } {\f2\uldb length}{\v\f2 LENGTH} {\f2 operator counts the number of top-level elements of its list argument; elements that are themselves lists still only count as one element. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # OR} ${\footnote \pard\plain \sl240 \fs20 $ OR} +{\footnote \pard\plain \sl240 \fs20 + g4:0695} K{\footnote \pard\plain \sl240 \fs20 K OR operator;operator} }{\b\f2 OR}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par The }{\f3 or} {\f2 binary logical operator returns true if either one or both of its arguments is true. \par syntax: \par }{\f4 \par \par <logical expression> }{\f3 or} {\f4 <logical expression> \par \par }{\f2 \par <logical expression> must evaluate to true or nil. \par \par \par examples: \par \pard \tx3420 }{\f4 \par a := 10; \par \par A := 10 \par \par \par if a<0 or a>140 then write "not a valid human age" else \par write "age = ",a; \par \pard \sl240 \par \par \par \par age = 10 \par \par \par a := 200; \par \par A := 200 \par \par \par if a < 0 or a > 140 then write "not a valid human age"; \par \par \par \par not a valid human age \par \par \pard \sl240 }{\f2 The }{\f3 or} {\f2 operator is left associative: }{\f3 x or y or z} {\f2 is equivalent to }{\f3 (x or y)} {\f2 }{\f3 or z} {\f2 . \par \par Logical operators can only be used in conditional expressions, such as \par \par } {\f2\uldb if}{\v\f2 IF} {\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 and } {\f2\uldb while}{\v\f2 WHILE} {\f2 ...}{\f3 do} {\f2 . }{\f3 or} {\f2 evaluates its arguments in order and quits, returning true, on finding the first true statement. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # PROCEDURE} ${\footnote \pard\plain \sl240 \fs20 $ PROCEDURE} +{\footnote \pard\plain \sl240 \fs20 + g4:0696} K{\footnote \pard\plain \sl240 \fs20 K PROCEDURE command;command} }{\b\f2 PROCEDURE}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par \par The }{\f3 procedure} {\f2 command allows you to define a mathematical operation as a function with arguments. \par syntax: \par }{\f4 \par \par \tab <option> }{\f3 procedure} {\f4 <identifier> (<arg>\{,<arg>\}+)}{\f3 ;} {\f4 <body> \par \par }{\f2 \par The <option> may be } {\f2\uldb algebraic}{\v\f2 ALGEBRAIC} {\f2 or } {\f2\uldb symbolic}{\v\f2 SYMBOLIC} {\f2 , indicating the mode under which the procedure is executed, or } {\f2\uldb real}{\v\f2 REAL} {\f2 or } {\f2\uldb integer}{\v\f2 INTEGER} {\f2 , 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, } {\f2\uldb array}{\v\f2 ARRAY} {\f2 or } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . <arg> is a formal parameter that may be any valid REDUCE identifier. <body> is a single statement (a } {\f2\uldb group}{\v\f2 group} {\f2 or } {\f2\uldb block}{\v\f2 block} {\f2 statement may be used) with the desired activities in it. \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 \par FAC \par \par \par fac(0); \par \par 1 \par \par \par fac(5); \par \par 120 \par \par \par fac(-5); \par \par ***** choose nonneg. integer only \par \par \pard \sl240 }{\f2 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 } {\f2\uldb let}{\v\f2 LET} {\f2 statements, new definitions under the same procedure name replace the previous definitions completely. \par \par 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. \par \par 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 }{\f3 procedure} {\f2 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 }{\f3 :=} {\f2 with a formal parameter on the left-hand side only changes the value of the calling parameter within the procedure. \par \par Using a } {\f2\uldb let}{\v\f2 LET} {\f2 statement inside a procedure always changes the value globally: a }{\f3 let} {\f2 with a formal parameter makes the change to the calling parameter. }{\f3 let} {\f2 statements cannot be made on local variables inside } {\f2\uldb begin}{\v\f2 BEGIN} {\f2 ...}{\f3 end} {\f2 } {\f2\uldb block}{\v\f2 block} {\f3 s} {\f2 . When } {\f2\uldb clear}{\v\f2 CLEAR} {\f2 statements are used on formal parameters, the calling variables associated with them are cleared globally too. The use of }{\f3 let} {\f2 or }{\f3 clear} {\f2 statements inside procedures should be done with extreme caution. \par \par 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. \par \par 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 } {\f2\uldb let}{\v\f2 LET} {\f2 statement on the name of the procedure (i.e., }{\f3 let noargs = noargs()} {\f2 ) after which you can call the procedure by just its name. \par \par 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. \par \par Procedures often have a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 block in them. Inside the block, local variables are declared using }{\f3 scalar} {\f2 , }{\f3 real} {\f2 or }{\f3 integer} {\f2 declarations. The declarations must be made immediately after the word }{\f3 begin} {\f2 , and if more than one type of declaration is made, they are separated by semicolons. REDUCE currently does no type checking on local variables; }{\f3 real} {\f2 and }{\f3 integer} {\f2 are treated just like }{\f3 scalar} {\f2 . 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. \par \par If a return value is desired from a procedure call, a specific } {\f2\uldb return}{\v\f2 RETURN} {\f2 command must be the last statement executed before exiting from the procedure. If no }{\f3 return} {\f2 is used, a procedure returns a zero or no value. \par \par Procedures are often written in a file using an editor, then the file is input using the command } {\f2\uldb in}{\v\f2 IN} {\f2 . 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. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # REPEAT} ${\footnote \pard\plain \sl240 \fs20 $ REPEAT} +{\footnote \pard\plain \sl240 \fs20 + g4:0697} K{\footnote \pard\plain \sl240 \fs20 K until;loop;REPEAT command;command} }{\b\f2 REPEAT}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par \par \par \par The } {\f2\uldb repeat}{\v\f2 REPEAT} {\f2 command causes repeated execution of a statement }{\f3 until} {\f2 \par \par the given condition is found to be true. The statement is always executed at least once. \par syntax: \par }{\f4 \par \par }{\f3 repeat} {\f4 <statement> }{\f3 until} {\f4 <condition> \par \par }{\f2 \par <statement> can be a single statement, } {\f2\uldb group}{\v\f2 group} {\f2 statement, or a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } {\f2\uldb block}{\v\f2 block} {\f2 . <condition> must be a logical operator that evaluates to true or nil. \par \par \par examples: \par \pard \tx3420 }{\f4 \par <<m := 4; repeat <<write 100*x*m;m := m-1>> until m = 0>>; \par \par \par \par 400*X \par 300*X \par 200*X \par 100*X \par \par \par \par <<m := -1; repeat <<write m; m := m-1>> until m <= 0>>; \par \par \par \par -1 \par \par \pard \sl240 }{\f2 }{\f3 repeat} {\f2 must always be followed by an }{\f3 until} {\f2 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 }{\f3 m = 0} {\f2 , it would never have been true since }{\f3 m} {\f2 already had value -2 when the condition was first evaluated. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # REST} ${\footnote \pard\plain \sl240 \fs20 $ REST} +{\footnote \pard\plain \sl240 \fs20 + g4:0698} K{\footnote \pard\plain \sl240 \fs20 K decomposition;list;REST operator;operator} }{\b\f2 REST}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 rest} {\f2 operator returns a } {\f2\uldb list}{\v\f2 LIST} {\f2 containing all but the first element of the list it is given. \par syntax: \par }{\f4 \par \par }{\f3 rest} {\f4 (<list>) or }{\f3 rest} {\f4 <list> \par \par \par \par }{\f2 <list> must be a non-empty list, but need not have more than one element. \par \par \par examples: \par \pard \tx3420 }{\f4 \par alist := \{a,b,c,d\}; \par \par ALIST := \{A,B,C,D\}; \par \par \par rest alist; \par \par \{B,C,D\} \par \par \par blist := \{x,y,\{aa,bb,cc\},z\}; \par \par BLIST := \{X,Y,\{AA,BB,CC\},Z\} \par \par \par second rest blist; \par \par \{AA,BB,CC\} \par \par \par clist := \{c\}; \par \par CLIST := C \par \par \par rest clist; \par \par \{\} \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # RETURN} ${\footnote \pard\plain \sl240 \fs20 $ RETURN} +{\footnote \pard\plain \sl240 \fs20 + g4:0699} K{\footnote \pard\plain \sl240 \fs20 K RETURN command;command} }{\b\f2 RETURN}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par \par The }{\f3 return} {\f2 command causes a value to be returned from inside a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } {\f2\uldb block}{\v\f2 block} {\f2 . \par \par \par syntax: \par }{\f4 }{\f3 begin} {\f4 <statements> }{\f3 return} {\f4 <(expression)> }{\f3 end} {\f4 \par \par \par \par }{\f2 <statements> can be any valid REDUCE statements. The value of <expression> is returned. \par \par \par examples: \par \pard \tx3420 }{\f4 \par begin write "yes"; return a end; \par \par yes \par A \par \par \par procedure dumb(a); \par begin if numberp(a) then return a else return 10 end; \par \pard \sl240 \par \par \par DUMB \par \par \par dumb(x); \par \par 10 \par \par \par dumb(-5); \par \par -5 \par \par \par procedure dumb2(a); \par begin c := a**2 + 2*a + 1; d := 17; c*d; return end; \par \pard \sl240 \par \par DUMB2 \par \par \par dumb2(4); \par \par c; \par \par 25 \par \par \par d; \par \par 17 \par \par \pard \sl240 }{\f2 Note in }{\f3 dumb2} {\f2 above that the assignments were made as requested, but the product }{\f3 c*d} {\f2 cannot be accessed. Changing the procedure to read }{\f3 return c*d} {\f2 would remedy this problem. \par \par The }{\f3 return} {\f2 statement is always the last statement executed before leaving the block. If }{\f3 return} {\f2 has no argument, the block is exited but no value is returned. A block statement does not need a }{\f3 return} {\f2 ; 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. \par \par The }{\f3 return} {\f2 command can be used inside }{\f3 <<} {\f2 ...}{\f3 >>} {\f2 } {\f2\uldb group}{\v\f2 group} {\f2 statements and } {\f2\uldb if}{\v\f2 IF} {\f2 ...}{\f3 then} {\f2 ...}{\f3 else} {\f2 commands that are inside }{\f3 begin} {\f2 ...}{\f3 end} {\f2 } {\f2\uldb block}{\v\f2 block} {\f2 s. It is not valid in these constructions that are not inside a }{\f3 begin} {\f2 ...}{\f3 end} {\f2 block. It is not valid inside } {\f2\uldb for}{\v\f2 FOR} {\f2 , } {\f2\uldb repeat}{\v\f2 REPEAT} {\f2 ...}{\f3 until} {\f2 or } {\f2\uldb while}{\v\f2 WHILE} {\f2 ...}{\f3 do} {\f2 loops in any construction. To force early termination from loops, the }{\f3 go to} {\f2 (} {\f2\uldb goto}{\v\f2 GOTO} {\f2 ) command must be used. When you use nested block statements, a }{\f3 return} {\f2 from an inner block exits returning a value to the next-outermost block, rather than all the way to the outside. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # REVERSE} ${\footnote \pard\plain \sl240 \fs20 $ REVERSE} +{\footnote \pard\plain \sl240 \fs20 + g4:0700} K{\footnote \pard\plain \sl240 \fs20 K list;REVERSE operator;operator} }{\b\f2 REVERSE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 reverse} {\f2 operator returns a } {\f2\uldb list}{\v\f2 LIST} {\f2 that is the reverse of the list it is given. \par syntax: \par }{\f4 \par \par }{\f3 reverse} {\f4 (<list>) or }{\f3 reverse} {\f4 <list> \par \par }{\f2 \par <list> must be a } {\f2\uldb list}{\v\f2 LIST} {\f2 . \par \par \par examples: \par \pard \tx3420 }{\f4 \par aa := \{c,b,a,\{x**2,z**3\},y\}; \par \par 2 3 \par AA := \{C,B,A,\{X ,Z \},Y\} \par \par \par reverse aa; \par \par 2 3 \par \{Y,\{X ,Z \},A,B,C\} \par \par \par reverse(q . reverse aa); \par \par 2 3 \par \{C,B,A,\{X ,Z \},Y,Q\} \par \par \pard \sl240 }{\f2 }{\f3 reverse} {\f2 and } {\f2\uldb cons}{\v\f2 CONS} {\f2 can be used together to add a new element to the end of a list (}{\f3 .} {\f2 adds its new element to the beginning). The }{\f3 reverse} {\f2 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. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # RULE} ${\footnote \pard\plain \sl240 \fs20 $ RULE} +{\footnote \pard\plain \sl240 \fs20 + g4:0701} K{\footnote \pard\plain \sl240 \fs20 K ~;rule list;rule;RULE type;type} }{\b\f2 RULE}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par \par \par \par A }{\f3 rule} {\f2 is an instruction to replace an algebraic expression or a part of an expression by another one. \par syntax: \par }{\f4 \par \par <lhs> => <rhs> or <lhs> => <rhs> }{\f3 when} {\f4 <cond> \par \par }{\f2 \par <lhs> is an algebraic expression used as search pattern and <rhs> is an algebraic expression which replaces matches of <rhs>. }{\f3 =>} {\f2 is the operator } {\f2\uldb replace}{\v\f2 replace} {\f2 . \par \par <lhs> can contain } {\f2\uldb free variable}{\v\f2 Free_Variable} {\f2 s which are symbols preceded by a tilde }{\f3 ~} {\f2 in their leftmost position in <lhs>. A double tilde marks an } {\f2\uldb optional free variable}{\v\f2 Optional_Free_Variable} {\f2 . If a rule has a }{\f3 when} {\f2 <cond> part it will fire only if the evaluation of <cond> has a result } {\f2\uldb true}{\v\f2 TRUE} {\f2 . <cond> may contain references to free variables of <lhs>. \par \par Rules can be collected in a } {\f2\uldb list}{\v\f2 LIST} {\f2 which then forms a }{\f3 rule list} {\f2 . }{\f3 Rule lists} {\f2 can be used to collect algebraic knowledge for a specific evaluation context. \par \par }{\f3 Rules} {\f2 and }{\f3 rule lists} {\f2 are globally activated and deactivated by } {\f2\uldb let}{\v\f2 LET} {\f2 , } {\f2\uldb forall}{\v\f2 FORALL} {\f2 , } {\f2\uldb clearrules}{\v\f2 CLEARRULES} {\f2 . For a single evaluation they can be locally activate by } {\f2\uldb where}{\v\f2 WHERE} {\f2 . The active rules for an operator can be visualized by } {\f2\uldb showrules}{\v\f2 SHOWRULES} {\f2 . \par \par \par examples: \par \pard \tx3420 }{\f4 \par operator f,g,h; \par \par let f(x) => x^2; \par \par f(x); \par \par 2 \par x \par \par \par g_rules:=\{g(~n,~x)=>h(n/2,x) when evenp n, \par \par g(~n,~x)=>h((1-n)/2,x) when not evenp n\}$ \par \par let g_rules; \par \par g(3,x); \par \par h(-1,x) \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # Free_Variable} ${\footnote \pard\plain \sl240 \fs20 $ Free_Variable} +{\footnote \pard\plain \sl240 \fs20 + g4:0702} K{\footnote \pard\plain \sl240 \fs20 K variable;Free Variable type;type} }{\b\f2 FREE VARIABLE}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par \par \par \par A variable preceded by a tilde is considered as }{\f3 free variable} {\f2 and stands for an arbitrary part in an algebraic form during pattern matching. Free variables occur in the left-hand sides of } {\f2\uldb rule}{\v\f2 RULE} {\f2 s, in the side relations for } {\f2\uldb compact}{\v\f2 COMPACT} {\f2 and in the first arguments of } {\f2\uldb map}{\v\f2 MAP} {\f2 and } {\f2\uldb select}{\v\f2 SELECT} {\f2 calls. See } {\f2\uldb rule}{\v\f2 RULE} {\f2 for examples. \par \par In rules also } {\f2\uldb optional free variable}{\v\f2 Optional_Free_Variable} {\f2 s may occur. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # Optional_Free_Variable} ${\footnote \pard\plain \sl240 \fs20 $ Optional_Free_Variable} +{\footnote \pard\plain \sl240 \fs20 + g4:0703} K{\footnote \pard\plain \sl240 \fs20 K variable;Optional Free Variable type;type} }{\b\f2 OPTIONAL FREE VARIABLE}{\f2 \tab \tab \tab \tab }{\b\f2 type}{\f2 \par \par \par \par A variable preceded by a double tilde is considered as }{\f3 optional free variable} {\f2 \par \par and stands for an arbitrary part part in an algebraic form during pattern matching. In contrast to ordinary } {\f2\uldb free variable}{\v\f2 Free_Variable} {\f2 s an operator pattern with an }{\f3 optional free variable} {\f2 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 \par \par term in a sum: set to 0 if missing, \par \par factor in a product: set to 1 if missing, \par \par exponent: set to 1 if missing \par \par \par examples: \par \pard \tx3420 }{\f4 \pard \sl240 }{\f2 Optional free variables are allowed only in the left-hand sides of } {\f2\uldb rule}{\v\f2 RULE} {\f2 s. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # SECOND} ${\footnote \pard\plain \sl240 \fs20 $ SECOND} +{\footnote \pard\plain \sl240 \fs20 + g4:0704} K{\footnote \pard\plain \sl240 \fs20 K decomposition;list;SECOND operator;operator} }{\b\f2 SECOND}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 second} {\f2 operator returns the second element of a list. \par syntax: \par }{\f4 \par \par }{\f3 second} {\f4 (<list>) or }{\f3 second} {\f4 <list> \par \par \par \par }{\f2 <list> must be a list with at least two elements, 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 second alist; \par \par B \par \par \par blist := \{x,\{aa,bb,cc\},z\}; \par \par BLIST := \{X,\{AA,BB,CC\},Z\} \par \par \par second second blist; \par \par BB \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # SET} ${\footnote \pard\plain \sl240 \fs20 $ SET} +{\footnote \pard\plain \sl240 \fs20 + g4:0705} K{\footnote \pard\plain \sl240 \fs20 K assign;SET operator;operator} }{\b\f2 SET}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 set} {\f2 operator is used for assignments when you want both sides of the assignment statement to be evaluated. \par syntax: \par }{\f4 \par \par }{\f3 set} {\f4 (<restricted\_expression>,<expression>) \par \par }{\f2 \par <expression> can be any REDUCE expression; <restricted\_expression> must be an identifier or an expression that evaluates to an identifier. \par \par \par examples: \par \pard \tx3420 }{\f4 \par a := y; \par \par A := Y \par \par \par set(a,sin(x^2)); \par \par 2 \par SIN(X ) \par \par \par a; \par \par 2 \par SIN(X ) \par \par \par y; \par \par 2 \par SIN(X ) \par \par \par a := b + c; \par \par A := B + C \par \par \par set(a-c,z); \par \par Z \par \par \par b; \par \par Z \par \par \pard \sl240 }{\f2 Using an } {\f2\uldb array}{\v\f2 ARRAY} {\f2 or } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 reference as the first argument to }{\f3 set} {\f2 has the result of setting the contents of the designated element to }{\f3 set} {\f2 's second argument. You should be careful to avoid unwanted side effects when you use this facility. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # SETQ} ${\footnote \pard\plain \sl240 \fs20 $ SETQ} +{\footnote \pard\plain \sl240 \fs20 + g4:0706} K{\footnote \pard\plain \sl240 \fs20 K assign;SETQ operator;operator} }{\b\f2 SETQ}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 setq} {\f2 operator is an infix or prefix binary assignment operator. It is identical to }{\f3 :=} {\f2 . \par syntax: \par }{\f4 \par \par }{\f3 setq} {\f4 (<restricted\_expression>,<expression>) or \par \par <restricted\_expression> }{\f3 setq} {\f4 <expression> \par \par }{\f2 \par <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 } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 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. \par \par \par examples: \par \pard \tx3420 }{\f4 \par setq(b,6); \par \par B := 6 \par \par \par c setq sin(x); \par \par C := SIN(X) \par \par \par w + setq(c,x+3) + z; \par \par W + X + Z + 3 \par \par \par c; \par \par X + 3 \par \par \par setq(a1 + a2,25); \par \par A1 + A2 := 25 \par \par \par a1; \par \par - (A2 - 25) \par \par \pard \sl240 }{\f2 Embedding a }{\f3 setq} {\f2 statement in an expression has the side effect of making the assignment, as shown in the third example above. \par \par Assignments are generally done for identifiers, but may be done for simple expressions as well, subject to the following remarks: \par \par \tab (i) If the left-hand side is an identifier, an operator, or a power, the 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 Be careful not to make a recursive }{\f3 setq} {\f2 assignment that is not controlled inside a loop statement. The process of resubstitution continues until you get a stack overflow message. }{\f3 setq} {\f2 can be used to attach functionality to operators, as the }{\f3 :=} {\f2 does. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # THIRD} ${\footnote \pard\plain \sl240 \fs20 $ THIRD} +{\footnote \pard\plain \sl240 \fs20 + g4:0707} K{\footnote \pard\plain \sl240 \fs20 K decomposition;list;THIRD operator;operator} }{\b\f2 THIRD}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 third} {\f2 operator returns the third item of a } {\f2\uldb list}{\v\f2 LIST} {\f2 . \par syntax: \par }{\f4 \par \par }{\f3 third} {\f4 (<list>) or }{\f3 third} {\f4 <list> \par \par \par \par }{\f2 <list> must be a list containing at least three items 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 third alist; \par \par C \par \par \par blist := \{x,\{aa,bb,cc\},y,z\}; \par \par BLIST := \{X,\{AA,BB,CC\},Y,Z\}; \par \par \par third second blist; \par \par CC \par \par \par third blist; \par \par Y \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # WHEN} ${\footnote \pard\plain \sl240 \fs20 $ WHEN} +{\footnote \pard\plain \sl240 \fs20 + g4:0708} K{\footnote \pard\plain \sl240 \fs20 K rule;WHEN operator;operator} }{\b\f2 WHEN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 when} {\f2 operator is used inside a }{\f3 rule} {\f2 to make the execution of the rule depend on a boolean condition which is evaluated at execution time. For the use see } {\f2\uldb rule}{\v\f2 RULE} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # g4} ${\footnote \pard\plain \sl240 \fs20 $ Syntax} +{\footnote \pard\plain \sl240 \fs20 + index:0004} }{\b\f2 Syntax}{\f2 \par }\pard \sl240 {\f2 \par } {\f2 \tab}{\f2\uldb semicolon command} {\v\f2 semicolon} (;){\f2 \par }{\f2 \tab}{\f2\uldb dollar command} {\v\f2 dollar} ($){\f2 \par }{\f2 \tab}{\f2\uldb percent command} {\v\f2 percent} (%){\f2 \par }{\f2 \tab}{\f2\uldb dot operator} {\v\f2 dot} (.){\f2 \par }{\f2 \tab}{\f2\uldb assign operator} {\v\f2 assign} (: =){\f2 \par }{\f2 \tab}{\f2\uldb equalsign operator} {\v\f2 equalsign} (=){\f2 \par }{\f2 \tab}{\f2\uldb replace operator} {\v\f2 replace} (= >){\f2 \par }{\f2 \tab}{\f2\uldb plussign operator} {\v\f2 plussign} (+){\f2 \par }{\f2 \tab}{\f2\uldb minussign operator} {\v\f2 minussign} (-){\f2 \par }{\f2 \tab}{\f2\uldb asterisk operator} {\v\f2 asterisk} (*){\f2 \par }{\f2 \tab}{\f2\uldb slash operator} {\v\f2 slash} (/){\f2 \par }{\f2 \tab}{\f2\uldb power operator} {\v\f2 power} (* *){\f2 \par }{\f2 \tab}{\f2\uldb caret operator} {\v\f2 caret} (^){\f2 \par }{\f2 \tab}{\f2\uldb geqsign operator} {\v\f2 geqsign} (> =){\f2 \par }{\f2 \tab}{\f2\uldb greater operator} {\v\f2 greater} (>){\f2 \par }{\f2 \tab}{\f2\uldb leqsign operator} {\v\f2 leqsign} (< =){\f2 \par }{\f2 \tab}{\f2\uldb less operator} {\v\f2 less} (<){\f2 \par }{\f2 \tab}{\f2\uldb tilde operator} {\v\f2 tilde} (~){\f2 \par }{\f2 \tab}{\f2\uldb group command} {\v\f2 group} (< <){\f2 \par }{\f2 \tab}{\f2\uldb AND operator} {\v\f2 AND}{\f2 \par }{\f2 \tab}{\f2\uldb BEGIN command} {\v\f2 BEGIN}{\f2 \par }{\f2 \tab}{\f2\uldb block command} {\v\f2 block}{\f2 \par }{\f2 \tab}{\f2\uldb COMMENT command} {\v\f2 COMMENT}{\f2 \par }{\f2 \tab}{\f2\uldb CONS operator} {\v\f2 CONS}{\f2 \par }{\f2 \tab}{\f2\uldb END command} {\v\f2 END}{\f2 \par }{\f2 \tab}{\f2\uldb EQUATION type} {\v\f2 EQUATION}{\f2 \par }{\f2 \tab}{\f2\uldb FIRST operator} {\v\f2 FIRST}{\f2 \par }{\f2 \tab}{\f2\uldb FOR command} {\v\f2 FOR}{\f2 \par }{\f2 \tab}{\f2\uldb FOREACH command} {\v\f2 FOREACH}{\f2 \par }{\f2 \tab}{\f2\uldb GEQ operator} {\v\f2 GEQ}{\f2 \par }{\f2 \tab}{\f2\uldb GOTO command} {\v\f2 GOTO}{\f2 \par }{\f2 \tab}{\f2\uldb GREATERP operator} {\v\f2 GREATERP}{\f2 \par }{\f2 \tab}{\f2\uldb IF command} {\v\f2 IF}{\f2 \par }{\f2 \tab}{\f2\uldb LIST operator} {\v\f2 LIST}{\f2 \par }{\f2 \tab}{\f2\uldb OR operator} {\v\f2 OR}{\f2 \par }{\f2 \tab}{\f2\uldb PROCEDURE command} {\v\f2 PROCEDURE}{\f2 \par }{\f2 \tab}{\f2\uldb REPEAT command} {\v\f2 REPEAT}{\f2 \par }{\f2 \tab}{\f2\uldb REST operator} {\v\f2 REST}{\f2 \par }{\f2 \tab}{\f2\uldb RETURN command} {\v\f2 RETURN}{\f2 \par }{\f2 \tab}{\f2\uldb REVERSE operator} {\v\f2 REVERSE}{\f2 \par }{\f2 \tab}{\f2\uldb RULE type} {\v\f2 RULE}{\f2 \par }{\f2 \tab}{\f2\uldb Free Variable type} {\v\f2 Free_Variable}{\f2 \par }{\f2 \tab}{\f2\uldb Optional Free Variable type} {\v\f2 Optional_Free_Variable}{\f2 \par }{\f2 \tab}{\f2\uldb SECOND operator} {\v\f2 SECOND}{\f2 \par }{\f2 \tab}{\f2\uldb SET operator} {\v\f2 SET}{\f2 \par }{\f2 \tab}{\f2\uldb SETQ operator} {\v\f2 SETQ}{\f2 \par }{\f2 \tab}{\f2\uldb THIRD operator} {\v\f2 THIRD}{\f2 \par }{\f2 \tab}{\f2\uldb WHEN operator} {\v\f2 WHEN}{\f2 \par \page #{\footnote \pard\plain \sl240 \fs20 # ARITHMETIC\_OPERATIONS} ${\footnote \pard\plain \sl240 \fs20 $ ARITHMETIC_OPERATIONS} +{\footnote \pard\plain \sl240 \fs20 + g5:0709} K{\footnote \pard\plain \sl240 \fs20 K ARITHMETIC_OPERATIONS introduction;introduction} }{\b\f2 ARITHMETIC\_OPERATIONS}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par \par 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. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # ABS} ${\footnote \pard\plain \sl240 \fs20 $ ABS} +{\footnote \pard\plain \sl240 \fs20 + g5:0710} K{\footnote \pard\plain \sl240 \fs20 K absolute value;ABS operator;operator} }{\b\f2 ABS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 abs} {\f2 operator returns the absolute value of its argument. \par \par \par syntax: \par }{\f4 }{\f3 abs} {\f4 (<expression>) \par \par }{\f2 \par <expression> can be any REDUCE scalar expression. \par \par \par examples: \par \pard \tx3420 }{\f4 \par abs(-a); \par \par ABS(A) \par \par \par abs(-5); \par \par 5 \par \par \par a := -10; \par \par A := -10 \par \par \par abs(a); \par \par 10 \par \par \par abs(-a); \par \par 10 \par \par \pard \sl240 }{\f2 If the argument has had no numeric value assigned to it, such as an identifier or polynomial, }{\f3 abs} {\f2 returns an expression involving }{\f3 abs} {\f2 of its argument, doing as much simplification of the argument as it can, such as dropping any preceding minus sign. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # ADJPREC} ${\footnote \pard\plain \sl240 \fs20 $ ADJPREC} +{\footnote \pard\plain \sl240 \fs20 + g5:0711} K{\footnote \pard\plain \sl240 \fs20 K precision;input;ADJPREC switch;switch} }{\b\f2 ADJPREC}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par \par \par \par When a real number is input, it is normally truncated to the } {\f2\uldb precision}{\v\f2 PRECISION} {\f2 in effect at the time the number is read. If it is desired to keep the full precision of all numbers input, the switch }{\f3 adjprec} {\f2 (for <adjust precision>) can be turned on. While on, }{\f3 adjprec} {\f2 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. \par \par \par examples: \par \pard \tx3420 }{\f4 \par on rounded; \par \par 1.23456789012345; \par \par 1.23456789012 \par \par \par on adjprec; \par \par 1.23456789012345; \par \par *** precision increased to 15 \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # ARG} ${\footnote \pard\plain \sl240 \fs20 $ ARG} +{\footnote \pard\plain \sl240 \fs20 + g5:0712} K{\footnote \pard\plain \sl240 \fs20 K polar angle;complex;ARG operator;operator} }{\b\f2 ARG}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par If } {\f2\uldb complex}{\v\f2 COMPLEX} {\f2 and } {\f2\uldb rounded}{\v\f2 ROUNDED} {\f2 are on, and arg evaluates to a complex number, }{\f3 arg} {\f2 returns the polar angle of arg, measured in radians. Otherwise an expression in arg is returned. \par \par \par examples: \par \pard \tx3420 }{\f4 \par arg(3+4i) \par \par ARG(3 + 4*I) \par \par \par on rounded, complex; \par \par ws; \par \par 0.927295218002 \par \par \par arg a; \par \par ARG(A) \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # CEILING} ${\footnote \pard\plain \sl240 \fs20 $ CEILING} +{\footnote \pard\plain \sl240 \fs20 + g5:0713} K{\footnote \pard\plain \sl240 \fs20 K integer;CEILING operator;operator} }{\b\f2 CEILING}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 ceiling} {\f4 (<expression>) \par \par }{\f2 \par 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 } {\f2\uldb fix}{\v\f2 FIX} {\f2 . For non-numeric arguments, the value is an expression in the original operator. \par \par \par examples: \par \pard \tx3420 }{\f4 \par ceiling 3.4; \par \par 4 \par \par \par fix 3.4; \par \par 3 \par \par \par ceiling(-5.2); \par \par -5 \par \par \par fix(-5.2); \par \par -5 \par \par \par ceiling a; \par \par CEILING(A) \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # CHOOSE} ${\footnote \pard\plain \sl240 \fs20 $ CHOOSE} +{\footnote \pard\plain \sl240 \fs20 + g5:0714} K{\footnote \pard\plain \sl240 \fs20 K CHOOSE operator;operator} }{\b\f2 CHOOSE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par }{\f3 choose} {\f2 (<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 }{\f4 m > n}{\f2 , the expression is returned unchanged. than or equal to \par examples: \par \pard \tx3420 }{\f4 \par choose(2,3); \par \par 3 \par \par \par choose(3,2); \par \par CHOOSE(3,2) \par \par \par choose(a,b); \par \par CHOOSE(A,B) \par \par \pard \sl240 }{\f2 \par \par \page #{\footnote \pard\plain \sl240 \fs20 # DEG2DMS} ${\footnote \pard\plain \sl240 \fs20 $ DEG2DMS} +{\footnote \pard\plain \sl240 \fs20 + g5:0715} K{\footnote \pard\plain \sl240 \fs20 K radians;degrees;DEG2DMS operator;operator} }{\b\f2 DEG2DMS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 deg2dms} {\f4 (<expression>) \par \par }{\f2 \par In } {\f2\uldb rounded}{\v\f2 ROUNDED} {\f2 mode, if <expression> is a real number, the operator }{\f3 deg2dms} {\f2 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. \par \par \par examples: \par \pard \tx3420 }{\f4 \par deg2dms 60; \par \par DEG2DMS(60) \par \par \par on rounded; \par \par ws; \par \par \{60,0,0\} \par \par \par deg2dms 42.4; \par \par \{42,23,60.0\} \par \par \par deg2dms a; \par \par DEG2DMS(A) \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # DEG2RAD} ${\footnote \pard\plain \sl240 \fs20 $ DEG2RAD} +{\footnote \pard\plain \sl240 \fs20 + g5:0716} K{\footnote \pard\plain \sl240 \fs20 K radians;degrees;DEG2RAD operator;operator} }{\b\f2 DEG2RAD}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 deg2rad} {\f4 (<expression>) \par \par }{\f2 \par In } {\f2\uldb rounded}{\v\f2 ROUNDED} {\f2 mode, if <expression> is a real number, the operator }{\f3 deg2rad} {\f2 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. \par \par \par examples: \par \pard \tx3420 }{\f4 \par deg2rad 60; \par \par DEG2RAD(60) \par \par \par on rounded; \par \par ws; \par \par 1.0471975512 \par \par \par deg2rad a; \par \par DEG2RAD(A) \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # DIFFERENCE} ${\footnote \pard\plain \sl240 \fs20 $ DIFFERENCE} +{\footnote \pard\plain \sl240 \fs20 + g5:0717} K{\footnote \pard\plain \sl240 \fs20 K DIFFERENCE operator;operator} }{\b\f2 DIFFERENCE}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par The }{\f3 difference} {\f2 operator may be used as either an infix or prefix binary subtraction operator. It is identical to }{\f3 -} {\f2 as a binary operator. \par \par \par syntax: \par }{\f4 }{\f3 difference} {\f4 (<expression>,<expression>) or \par \par <expression> }{\f3 difference} {\f4 <expression> \{}{\f3 difference} {\f4 <expression>\}* \par \par }{\f2 \par <expression> can be a number or any other valid REDUCE expression. Matrix expressions are allowed if they are of the same dimensions. \par \par \par examples: \par \pard \tx3420 }{\f4 \par \par difference(10,4); \par \par 6 \par \par \par \par 15 difference 5 difference 2; \par \par 8 \par \par \par \par a difference b; \par \par A - B \par \par \pard \sl240 }{\f2 The }{\f3 difference} {\f2 operator is left associative, as shown in the second example above. \par \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # DILOG} ${\footnote \pard\plain \sl240 \fs20 $ DILOG} +{\footnote \pard\plain \sl240 \fs20 + g5:0718} K{\footnote \pard\plain \sl240 \fs20 K dilogarithm function;DILOG operator;operator} }{\b\f2 DILOG}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 dilog} {\f2 operator is known to the differentiation and integration operators, but has numeric value attached only at }{\f3 dilog(0)} {\f2 . Dilog is defined by \par \par dilog(x) = -int(log(x),x)/(x-1) \par \par \par examples: \par \pard \tx3420 }{\f4 \par df(dilog(x**2),x); \par \par 2 \par 2*LOG(X )*X \par - ------------ \par 2 \par X - 1 \par \par \par \par int(dilog(x),x); \par \par DILOG(X)*X - DILOG(X) + LOG(X)*X - X \par \par \par \par dilog(0); \par \par 2 \par PI \par ---- \par 6 \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # DMS2DEG} ${\footnote \pard\plain \sl240 \fs20 $ DMS2DEG} +{\footnote \pard\plain \sl240 \fs20 + g5:0719} K{\footnote \pard\plain \sl240 \fs20 K radians;degrees;DMS2DEG operator;operator} }{\b\f2 DMS2DEG}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 dms2deg} {\f4 (<list>) \par \par }{\f2 \par In } {\f2\uldb rounded}{\v\f2 ROUNDED} {\f2 mode, if <list> is a list of three real numbers, the operator }{\f3 dms2deg} {\f2 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. \par \par \par examples: \par \pard \tx3420 }{\f4 \par dms2deg \{42,3,7\}; \par \par DMS2DEG(\{42,3,7\}) \par \par \par on rounded; \par \par ws; \par \par 42.0519444444 \par \par \par dms2deg a; \par \par DMS2DEG(A) \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # DMS2RAD} ${\footnote \pard\plain \sl240 \fs20 $ DMS2RAD} +{\footnote \pard\plain \sl240 \fs20 + g5:0720} K{\footnote \pard\plain \sl240 \fs20 K radians;degrees;DMS2RAD operator;operator} }{\b\f2 DMS2RAD}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 dms2rad} {\f4 (<list>) \par \par }{\f2 \par In } {\f2\uldb rounded}{\v\f2 ROUNDED} {\f2 mode, if <list> is a list of three real numbers, the operator }{\f3 dms2rad} {\f2 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. \par \par \par examples: \par \pard \tx3420 }{\f4 \par dms2rad \{42,3,7\}; \par \par DMS2RAD(\{42,3,7\}) \par \par \par on rounded; \par \par ws; \par \par 0.733944887421 \par \par \par dms2rad a; \par \par DMS2RAD(A) \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # FACTORIAL} ${\footnote \pard\plain \sl240 \fs20 $ FACTORIAL} +{\footnote \pard\plain \sl240 \fs20 + g5:0721} K{\footnote \pard\plain \sl240 \fs20 K gamma;FACTORIAL operator;operator} }{\b\f2 FACTORIAL}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 factorial} {\f4 (<expression>) \par \par }{\f2 \par If the argument of }{\f3 factorial} {\f2 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 } {\f2\uldb gamma}{\v\f2 GAMMA} {\f2 operator is available in the } {\f2\uldb Special Function Package}{\v\f2 Special_Function_Package} {\f2 . \par \par \par examples: \par \pard \tx3420 }{\f4 \par factorial 4; \par \par 24 \par \par \par factorial 30 ; \par \par 265252859812191058636308480000000 \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # FIX} ${\footnote \pard\plain \sl240 \fs20 $ FIX} +{\footnote \pard\plain \sl240 \fs20 + g5:0722} K{\footnote \pard\plain \sl240 \fs20 K integer;FIX operator;operator} }{\b\f2 FIX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 fix} {\f4 (<expression>) \par \par }{\f2 \par The operator }{\f3 fix} {\f2 returns the integer part of its argument, if that argument has a numerical value. For positive numbers, this is equivalent to } {\f2\uldb floor}{\v\f2 FLOOR} {\f2 , and, for negative numbers, } {\f2\uldb ceiling}{\v\f2 CEILING} {\f2 . For non-numeric arguments, the value is an expression in the original operator. \par \par \par examples: \par \pard \tx3420 }{\f4 \par fix 3.4; \par \par 3 \par \par \par floor 3.4; \par \par 3 \par \par \par ceiling 3.4; \par \par 4 \par \par \par fix(-5.2); \par \par -5 \par \par \par floor(-5.2); \par \par -6 \par \par \par ceiling(-5.2); \par \par -5 \par \par \par fix(a); \par \par FIX(A) \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # FIXP} ${\footnote \pard\plain \sl240 \fs20 $ FIXP} +{\footnote \pard\plain \sl240 \fs20 + g5:0723} K{\footnote \pard\plain \sl240 \fs20 K integer;FIXP operator;operator} }{\b\f2 FIXP}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 fixp} {\f2 logical operator returns true if its argument is an integer. \par syntax: \par }{\f4 \par \par }{\f3 fixp} {\f4 (<expression>) or }{\f3 fixp} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> can be any valid REDUCE expression, <simple\_expression> must be a single identifier or begin with a prefix operator. \par \par \par examples: \par \pard \tx3420 }{\f4 \par if fixp 1.5 then write "ok" else write "not"; \par \par \par \par not \par \par \par if fixp(a) then write "ok" else write "not"; \par \par \par \par not \par \par \par a := 15; \par \par A := 15 \par \par \par if fixp(a) then write "ok" else write "not"; \par \par \par \par ok \par \par \pard \sl240 }{\f2 Logical operators can only be used inside conditional expressions such as }{\f3 if} {\f2 ...}{\f3 then} {\f2 or }{\f3 while} {\f2 ...}{\f3 do} {\f2 . \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # FLOOR} ${\footnote \pard\plain \sl240 \fs20 $ FLOOR} +{\footnote \pard\plain \sl240 \fs20 + g5:0724} K{\footnote \pard\plain \sl240 \fs20 K integer;FLOOR operator;operator} }{\b\f2 FLOOR}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 floor} {\f4 (<expression>) \par \par }{\f2 \par 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 } {\f2\uldb fix}{\v\f2 FIX} {\f2 . For non-numeric arguments, the value is an expression in the original operator. \par \par \par examples: \par \pard \tx3420 }{\f4 \par floor 3.4; \par \par 3 \par \par \par fix 3.4; \par \par 3 \par \par \par floor(-5.2); \par \par -6 \par \par \par fix(-5.2); \par \par -5 \par \par \par floor a; \par \par FLOOR(A) \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # EXPT} ${\footnote \pard\plain \sl240 \fs20 $ EXPT} +{\footnote \pard\plain \sl240 \fs20 + g5:0725} K{\footnote \pard\plain \sl240 \fs20 K EXPT operator;operator} }{\b\f2 EXPT}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par The }{\f3 expt} {\f2 operator is both an infix and prefix binary exponentiation operator. It is identical to }{\f3 ^} {\f2 or }{\f3 **} {\f2 . \par syntax: \par }{\f4 \par \par }{\f3 expt} {\f4 (<expression>,<expression>) or <expression> }{\f3 expt} {\f4 <expression> \par \par }{\f2 \par \par examples: \par \pard \tx3420 }{\f4 \par a expt b; \par \par B \par A \par \par \par expt(a,b); \par \par B \par A \par \par \par (x+y) expt 4; \par \par 4 3 2 2 3 4 \par X + 4*X *Y + 6*X *Y + 4*X*Y + Y \par \par \pard \sl240 }{\f2 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. \par \par }{\f3 expt} {\f2 is left associative. In other words, }{\f3 a expt b expt c} {\f2 is equivalent to }{\f3 a expt (b*c)} {\f2 , not }{\f3 a expt (b expt c)} {\f2 , which would be right associative. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # GCD} ${\footnote \pard\plain \sl240 \fs20 $ GCD} +{\footnote \pard\plain \sl240 \fs20 + g5:0726} K{\footnote \pard\plain \sl240 \fs20 K polynomial;greatest common divisor;GCD operator;operator} }{\b\f2 GCD}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 gcd} {\f2 operator returns the greatest common divisor of two polynomials. \par syntax: \par }{\f4 \par \par }{\f3 gcd} {\f4 (<expression>,<expression>) \par \par }{\f2 \par <expression> must be a polynomial (or integer), otherwise an error occurs. \par \par \par examples: \par \pard \tx3420 }{\f4 \par gcd(2*x**2 - 2*y**2,4*x + 4*y); \par \par 2*(X + Y) \par \par \par gcd(sin(x),x**2 + 1); \par \par 1 \par \par \par gcd(765,68); \par \par 17 \par \par \pard \sl240 }{\f2 The operator }{\f3 gcd} {\f2 described here provides an explicit means to find the gcd of two expressions. The switch }{\f3 gcd} {\f2 described below simplifies expressions by finding and canceling gcd's at every opportunity. When the switch } {\f2\uldb ezgcd}{\v\f2 EZGCD} {\f2 is also on, gcd's are figured using the EZ GCD algorithm, which is usually faster. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # LN} ${\footnote \pard\plain \sl240 \fs20 $ LN} +{\footnote \pard\plain \sl240 \fs20 + g5:0727} K{\footnote \pard\plain \sl240 \fs20 K logarithm;LN operator;operator} }{\b\f2 LN}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 ln} {\f4 (<expression>) \par \par }{\f2 \par <expression> can be any valid scalar REDUCE expression. \par \par The }{\f3 ln} {\f2 operator returns the natural logarithm of its argument. However, unlike } {\f2\uldb log}{\v\f2 LOG} {\f2 , there are no algebraic rules associated with it; it will only evaluate when } {\f2\uldb rounded}{\v\f2 ROUNDED} {\f2 is on, and the argument is a real number. \par \par \par examples: \par \pard \tx3420 }{\f4 \par ln(x); \par \par LN(X) \par \par \par ln 4; \par \par LN(4) \par \par \par ln(e); \par \par LN(E) \par \par \par df(ln(x),x); \par \par DF(LN(X),X) \par \par \par on rounded; \par \par ln 4; \par \par 1.38629436112 \par \par \par ln e; \par \par 1 \par \par \pard \sl240 }{\f2 Because of the restricted algebraic properties of }{\f3 ln} {\f2 , users are advised to use } {\f2\uldb log}{\v\f2 LOG} {\f2 whenever possible. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # LOG} ${\footnote \pard\plain \sl240 \fs20 $ LOG} +{\footnote \pard\plain \sl240 \fs20 + g5:0728} K{\footnote \pard\plain \sl240 \fs20 K logarithm;LOG operator;operator} }{\b\f2 LOG}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The }{\f3 log} {\f2 operator returns the natural logarithm of its argument. \par syntax: \par }{\f4 \par \par }{\f3 log} {\f4 (<expression>) or }{\f3 log} {\f4 <expression> \par \par }{\f2 \par <expression> can be any valid scalar REDUCE expression. \par \par \par examples: \par \pard \tx3420 }{\f4 \par log(x); \par \par LOG(X) \par \par \par log 4; \par \par LOG(4) \par \par \par log(e); \par \par 1 \par \par \par on rounded; \par \par log 4; \par \par 1.38629436112 \par \par \pard \sl240 }{\f2 }{\f3 log} {\f2 returns a numeric value only when } {\f2\uldb rounded}{\v\f2 ROUNDED} {\f2 is on. In that case, use of a negative argument for }{\f3 log} {\f2 results in an error message. No error is given on a negative argument when REDUCE is not in that mode. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # LOGB} ${\footnote \pard\plain \sl240 \fs20 $ LOGB} +{\footnote \pard\plain \sl240 \fs20 + g5:0729} K{\footnote \pard\plain \sl240 \fs20 K logarithm;LOGB operator;operator} }{\b\f2 LOGB}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 logb} {\f4 (<expression>,<integer>) \par \par }{\f2 \par <expression> can be any valid scalar REDUCE expression. \par \par The }{\f3 logb} {\f2 operator returns the logarithm of its first argument using the second argument as base. However, unlike } {\f2\uldb log}{\v\f2 LOG} {\f2 , there are no algebraic rules associated with it; it will only evaluate when } {\f2\uldb rounded}{\v\f2 ROUNDED} {\f2 is on, and the first argument is a real number. \par \par \par examples: \par \pard \tx3420 }{\f4 \par logb(x,2); \par \par LOGB(X,2) \par \par \par logb(4,3); \par \par LOGB(4,3) \par \par \par logb(2,2); \par \par LOGB(2,2) \par \par \par df(logb(x,3),x); \par \par DF(LOGB(X,3),X) \par \par \par on rounded; \par \par logb(4,3); \par \par 1.26185950714 \par \par \par logb(2,2); \par \par 1 \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # MAX} ${\footnote \pard\plain \sl240 \fs20 $ MAX} +{\footnote \pard\plain \sl240 \fs20 + g5:0730} K{\footnote \pard\plain \sl240 \fs20 K maximum;MAX operator;operator} }{\b\f2 MAX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par The operator }{\f3 max} {\f2 is an n-ary prefix operator, which returns the largest value in its arguments. \par syntax: \par }{\f4 \par \par }{\f3 max} {\f4 (<expression>\{,<expression>\}*) \par \par \par \par }{\f2 <expression> must evaluate to a number. }{\f3 max} {\f2 of an empty list returns 0. \par \par \par examples: \par \pard \tx3420 }{\f4 \par max(4,6,10,-1); \par \par 10 \par \par \par <<a := 23;b := 2*a;c := 4**2;max(a,b,c)>>; \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 + g5:0731} 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 (<expression>\{,<expression>\}*) \par \par }{\f2 \par <expression> 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 <<a := 23;b := 2*a;c := 4**2;min(a,b,c)>>; \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 + g5:0732} 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 (<expression>) \par \par \par \par }{\f2 <expression> 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 + g5:0733} 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 (<expression>) \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 + g5:0734} 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 + g5:0735} 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 (<expression>) \par \par }{\f2 \par If }{\f3 rounded} {\f2 is on, and the argument is a real number, <norm> returns its absolute value. If }{\f3 complex} {\f2 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. \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 + g5:0736} 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(<expression1>,<expression2>) \par \par }{\f2 \par If <expression1> and <expression2> evaluate to positive integers, }{\f3 perm} {\f2 returns the number of permutations possible in selecting <expression1> objects from <expression2> 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 + g5:0737} 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 (<expression>,<expression>\{,<expression>\} *) or \par \par <expression> }{\f3 plus} {\f4 <expression> \{}{\f3 plus} {\f4 <expression>\}* \par \par }{\f2 \par <expression> 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 + g5:0738} 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 (<expression>,<expression>) or <expression> }{\f3 quotient} {\f4 <expression> or }{\f3 quotient} {\f4 (<expression>) or }{\f3 quotient} {\f4 <expression> \par \par }{\f2 \par <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. \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 + g5:0739} 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 (<expression>) \par \par }{\f2 \par In } {\f2\uldb rounded}{\v\f2 ROUNDED} {\f2 mode, if <expression> 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 + g5:0740} 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 (<expression>) \par \par }{\f2 \par In } {\f2\uldb rounded}{\v\f2 ROUNDED} {\f2 mode, if <expression> 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 + g5:0741} 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 + g5:0742} 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 (<expression>,<expression>) \par \par }{\f2 \par <expression> 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 + g5:0743} 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 (<expression>) \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 + g5:0744} 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 <integer> \par \par }{\f2 \par <integer> 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 + g5:0745} 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 <expression> \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 + g5:0746} 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 (<expression>) \par \par }{\f2 \par <expression> 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 + g5:0747} 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 <expression> }{\f3 times} {\f4 <expression> \{}{\f3 times} {\f4 <expression>\}* \par \par or }{\f3 times} {\f4 (<expression>,<expression> \{,<expression>\}*) \par \par }{\f2 \par <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. \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 + g6:0748} 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 + g6:0749} 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 <expression> }{\f3 equal} {\f4 <expression> \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 . <equal> 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 + g6:0750} 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 (<integer>) or }{\f3 evenp} {\f4 <integer> \par \par }{\f2 \par <integer> 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 + g6:0751} 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 + g6:0752} 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 (<expression>,<kernel>) or <expression> }{\f3 freeof} {\f4 <kernel> \par \par }{\f2 \par <expression> can be any valid scalar REDUCE expression, <kernel> 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 + g6:0753} 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 (<expression>,<expression>) or <expression> }{\f3 leq} {\f4 <expression> \par \par \par \par }{\f2 <expression> 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 + g6:0754} 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 (<expression>,<expression>) or <expression> }{\f3 lessp} {\f4 <expression> \par \par \par \par }{\f2 <expression> 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 + g6:0755} 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 <expression> }{\f3 member} {\f4 <list> \par \par }{\f2 \par }{\f3 member} {\f2 is an infix binary comparison operator that evaluates to } {\f2\uldb true}{\v\f2 TRUE} {\f2 if <expression> is } {\f2\uldb equal}{\v\f2 EQUAL} {\f2 to a member of the } {\f2\uldb list}{\v\f2 LIST} {\f2 <list>. \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 . <member> 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 + g6:0756} 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 <expression> }{\f3 neq} {\f4 <expression> \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 . <neq> 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 + g6:0757} 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 (<logical expression>) \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 + g6:0758} 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 (<expression>) or }{\f3 numberp} {\f4 <expression> \par \par }{\f2 \par <expression> 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 + g6:0759} 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 (<expression1>,<expression2>) \par \par \par \par }{\f2 <expression1> and <expression2> 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 + g6:0760} 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 (<expression>) or }{\f3 primep} {\f4 <simple\_expression> \par \par }{\f2 \par If <expression> evaluates to a integer, }{\f3 primep} {\f2 returns } {\f2\uldb true}{\v\f2 TRUE} {\f2 \par \par 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 } {\f2\uldb nil}{\v\f2 NIL} {\f2 otherwise. If <expression> 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 + g6:0761} 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 + g7:0762} 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 + g7:0763} 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 + 12,1\},\{X + 5,1\}\} \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 + 12,1\} \par \par \par factor2 := second results; \par \par FACTOR2 := \{X + 5,1\} \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 + g7:0764} 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 <n>, }{\f3 display} {\f2 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. \par \par \par syntax: \par }{\f4 }{\f3 display} {\f4 (<n>) or }{\f3 display} {\f4 () \par \par }{\f2 \par <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. \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 + g7:0765} 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 <package\_name>}{\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 + g7:0766} 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 + g7:0767} 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 + g7:0768} 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 + g7:0769} 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 <message> \par \par }{\f2 \par <message> 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 + g7:0770} 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 + g7:0771} 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 <identifier> \par \par }{\f2 \par <identifier> 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 + g7:0772} 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 + g7:0773} 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 <item>\{,<item>\}* \par \par }{\f2 \par <item> 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 + g8:0774} 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 (<list>,<list>) \par \par }{\f2 \par <list> 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 + g8:0775} 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 + g8:0776} 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 + g8:0777} 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 (<expression>) \par \par }{\f2 \par <expression> 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 + g8:0778} 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 (<expression>}{\f3 ,} {\f4 <variable>) \par \par }{\f2 \par <expression> 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. <variable> 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 + g8:0779} 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 (<expression>,<kernel>,<integer>) \par \par }{\f2 \par <expression> must be a polynomial, unless } {\f2\uldb ratarg}{\v\f2 RATARG} {\f2 is on which allows rational expressions. <kernel> must be a kernel, and <integer> 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 + g8:0780} 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 (<expression>) or }{\f3 conj} {\f4 <simple\_expression> \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 + g8:0781} 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 (<num>) or }{\f3 continued_fraction} {\f4 ( <num>,<size>) \par \par }{\f2 \par This operator approximates the real number <num> ( } {\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 <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. \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 + g8:0782} 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 (<expression>) or }{\f3 decompose} {\f4 <simple\_expression> \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, <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. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # DEG} ${\footnote \pard\plain \sl240 \fs20 $ DEG} +{\footnote \pard\plain \sl240 \fs20 + g8:0783} 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 (<expression>,<kernel>) \par \par }{\f2 \par <expression> 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. <variable> 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 + g8:0784} 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 (<expression>) \par \par }{\f2 \par <expression> 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 + g8:0785} 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 (<expression>}{\f3 ,} {\f4 <var> [}{\f3 ,} {\f4 <number>] \{}{\f3 ,} {\f4 <var> [ }{\f3 ,} {\f4 <number>] \}) \par \par }{\f2 \par <expression> can be any valid REDUCE algebraic expression. <var> must be a } {\f2\uldb kernel}{\v\f2 KERNEL} {\f2 , and is the differentiation variable. <number> 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 + g8:0786} 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 + g8:0787} 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 + g8:0788} 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 into a list of \{factor,power\} pairs. \par syntax: \par }{\f4 \par \par }{\f3 factorize} {\f4 (<expression>) \par \par }{\f2 \par <expression> 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 + X*Y + Y ,1\},\{X - Y,1\}\} \par \par \par fac1 := first fff; \par \par 2 2 \par FAC1 := \{\{X + X*Y + Y ,1\} \par \par \par factorize(x^15 - 1); \par \par 8 7 6 5 4 \par \{\{ X - X + X - X + X - X + 1,1\}, \par 4 3 2 \par \{X + X + X + X + 1,1\}, \par 2 \par \{X + X + 1,1\}, \par \{X - 1,1\}\} \par \par \par lastone := part(ws,length ws); \par \par LASTONE := \{X - 1,1\} \par \par \par setmod 2; \par \par 1 \par \par \par on modular; \par \par factorize(x^15 - 1); \par \par 4 3 2 \par \{\{X + X + X + X + 1,1\}, \par 4 3 \par \{X + X + 1,1\}, \par 4 \par \{X + X + 1,1\}, \par 2 \par \{ X + X + 1,1\}, \par \{X + 1,1\}\} \par \par \pard \sl240 }{\f2 The }{\f3 factorize} {\f2 command returns the factor,power pairs 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 these pairs. \par \par If the <expression> 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 + g8:0789} 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(<expression>,<expression>) \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 + g8:0790} 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 (<expression>) or }{\f3 impart} {\f4 <simple\_expression> \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 + g8:0791} 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 (<expression>,<kernel>) \par \par }{\f2 \par <expression> 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. <kernel> 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 + g8:0792} 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(<values>,<variable>,<points>) \par \par }{\f2 \par <values> and <points> are } {\f2\uldb list}{\v\f2 LIST} {\f2 s of equal length and <variable> 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(<values>)-1. The unique polynomial }{\f3 f} {\f2 is defined by the property that for corresponding elements }{\f3 v} {\f2 of <values> and }{\f3 p} {\f2 of <points> 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 + g8:0793} 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 (<expression>,<kernel>) \par \par }{\f2 \par <expression> 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. <kernel> 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 + g8:0794} 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 (<expr>) or }{\f3 length} {\f4 <expr> \par \par }{\f2 \par <expr> 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 + g8:0795} 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 (<equation>) or }{\f3 lhs} {\f4 <equation> \par \par \par \par }{\f2 <equation> 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 + g8:0796} 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 (<expr>,<var>,<limpoint>) or \par \par }{\f3 limit!+} {\f4 (<expr>,<var>,<limpoint>) or \par \par }{\f3 limit!-} {\f4 (<expr>,<var>,<limpoint>) \par \par }{\f2 \par where <expr> is an expression depending of the variable <var> (a } {\f2\uldb kernel}{\v\f2 KERNEL} {\f2 ) and <limpoint> is the limit point. If the limit depends upon the direction of approach to the <limpoint>, 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 + g8:0797} 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 (<expression>,<kernel>) \par \par }{\f2 \par <expression> 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. <kernel> 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 + g8:0798} 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 (<expression>,<kernel>) \par \par }{\f2 \par <expression> 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. <kernel> 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 + g8:0799} 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 (<expression>) \par \par \par \par }{\f2 <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 } {\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 + g8:0800} 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 (<function>,<object>) \par \par }{\f2 \par <object> is a list, a matrix or an operator expression. \par \par <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, \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 <object> 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 <var> is a variable (a <kernel> without subscript) and <rep> is an expression which contains <var>. Here }{\f3 rep} {\f2 is evaluated for each element of <object> 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 <function> 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 + g8:0801} 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 (<stem>,<leaf>) \par \par }{\f2 \par <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 } {\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 + g8:0802} 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 (<expression>) or }{\f3 nprimitive} {\f4 <simple\_expression> \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 + g8:0803} 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 (<expression>) or }{\f3 num} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> 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 + g8:0804} 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 (<expr>,<var1>,<var2>) \par \par \par \par }{\f2 <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 } {\f2\uldb equation}{\v\f2 EQUATION} {\f2 . \par \par <var1> is the name of the dependent variable, <var2> is the name of the independent variable. \par \par A differential in <expr> is expressed using the } {\f2\uldb df}{\v\f2 DF} {\f2 operator. Note that in most cases you must declare explicitly <var1> to depend of <var2> 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 + g8:0805} 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 + g8:0806} 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 (<expression,integer>\{,<integer>\}*) \par \par }{\f2 \par <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 }{\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 + g8:0807} 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(<expression>,<variable>) \par \par }{\f2 \par }{\f3 pf} {\f2 transforms <expression> into a } {\f2\uldb list}{\v\f2 LIST} {\f2 of partial fraction s with respect to the main variable, <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 + g8:0808} 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 (<expr>,<k>[,<lolim> [,<uplim> ]]) \par \par \par \par }{\f2 where <expr> is the expression to be multiplied, <k> is the control variable (a } {\f2\uldb kernel}{\v\f2 KERNEL} {\f2 ), 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. \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 + g8:0809} 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 (<expression>,<kernel>) \par \par }{\f2 \par <expression> 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. <kernel> 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 + g8:0810} 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 (<expression>) or }{\f3 repart} {\f4 <simple\_expression> \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 + g8:0811} 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 (<expression>,<expression>,<kernel>) \par \par }{\f2 \par <expression> must be a polynomial containing <kernel> ; <kernel> 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 <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: \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 + g8:0812} 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 (<equation>) or }{\f3 rhs} {\f4 <equation> \par \par }{\f2 \par <equation> 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 + g8:0813} 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 + g8:0814} 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 (<function>,<object>) \par \par }{\f2 \par <object> is a } {\f2\uldb list}{\v\f2 LIST} {\f2 . \par \par <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, \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 <object> 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 <var> is a variable (a <kernel> without subscript) and <rep> is an expression which contains <var>. Here }{\f3 rep} {\f2 is evaluated for each element of <object> 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 <function> is needed when more than one free variable occurs. The evaluation result of <function> 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 + g8:0815} 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 (<expression>) or }{\f3 showrules} {\f4 <simple\_expression> \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 + g8:0816} 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 (<expression> [ , <kernel>]) or \par \par }{\f3 solve} {\f4 (\{<expression>,...\} [ ,\{ <kernel> ,...\}] ) \par \par }{\f2 \par \par 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 } {\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 + g8:0817} 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 (<lst>,<comp>) \par \par }{\f2 \par <lst> is a } {\f2\uldb list}{\v\f2 LIST} {\f2 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 }{\f3 <} {\f2 (} {\f2\uldb lessp}{\v\f2 LESSP} {\f2 ), }{\f3 <=} {\f2 (} {\f2\uldb leq}{\v\f2 LEQ} {\f2 ) etc., or <comp> 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 <lst> in a sequence corresponding to <comp>. \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,x) then 0 else \par \par if deg(a,y)>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 + g8:0818} 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 (<expression> [,<identifier>[,<identifier> ...]]) \par \par }{\f2 \par <expression> may be any valid REDUCE scalar expression. <identifier> 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 + g8:0819} 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 (<kernel>}{\f3 =} {\f4 <expression> \{,<kernel>}{\f3 =} {\f4 <expression>\}*, <expression>) or \par \par }{\f3 sub} {\f4 (\{<kernel>}{\f3 =} {\f4 <expression>*, <kernel>}{\f3 =} {\f3 expression} {\f4 \},<expression>) \par \par }{\f2 \par <kernel> must be a } {\f2\uldb kernel}{\v\f2 KERNEL} {\f2 , <expression> 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 + g8:0820} 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 (<expr>,<k>[,<lolim> [,<uplim> ]]) \par \par \par \par }{\f2 where <expr> is the expression to be added, <k> is the control variable (a } {\f2\uldb kernel}{\v\f2 KERNEL} {\f2 ), 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. \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 + g8:0821} 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 (<number>) \par \par }{\f2 \par <number> 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 <number>}{\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 + g9:0822} 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 + g9:0823} 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 <identifier>\{}{\f3 ,} {\f4 <identifier>\}* \par \par }{\f2 \par <identifier> 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 <identifier> has not been declared an operator, the flag }{\f3 antisymmetric} {\f2 is still attached to it. When <identifier> is subsequently used as an operator, the message }{\f3 Declare} {\f2 <identifier> }{\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 + g9:0824} 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 <identifier>(<dimensions>) \{}{\f3 ,} {\f4 <identifier>(<dimensions>)\}* \par \par }{\f2 \par <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>\}*. \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 <not> 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 + g9:0825} 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 <identifier>\{,<identifier>\}+ or \par \par <let-type statement> }{\f3 clear} {\f4 <identifier> \par \par }{\f2 \par <identifier> 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. <let-type statement> 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 + g9:0826} 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 <list>\{,<list>\}+ \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. <list> 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 + g9:0827} 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 <identifier>}{\f3 =} {\f4 <substitution> \{}{\f3 ,} {\f4 <identifier>}{\f3 =} {\f4 <substitution>\}* \par \par }{\f2 \par <identifier> is any valid REDUCE identifier, <substitution> 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 + g9:0828} 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 <kernel>\{}{\f3 ,} {\f4 <kernel>\}+ \par \par }{\f2 \par <kernel> 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 + g9:0829} 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 <identifier>\{,<identifier>\}* \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 + g9:0830} 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 <kernel> \{}{\f3 ,} {\f4 <kernel>\}* \par \par }{\f2 \par <kernel> 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 Note that }{\f3 factor} {\f2 does not affect the order of its arguments. You should also use } {\f2\uldb order}{\v\f2 ORDER} {\f2 if this is important. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # FORALL} ${\footnote \pard\plain \sl240 \fs20 $ FORALL} +{\footnote \pard\plain \sl240 \fs20 + g9:0831} 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 <identifier>\{,<identifier>\}* }{\f3 let} {\f4 <let statement> \par \par or \par \par }{\f3 for all} {\f4 <identifier>\{,<identifier>\}* }{\f3 such that} {\f4 <condition> }{\f3 let} {\f4 <let statement> \par \par }{\f2 \par <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. \par \par \par examples: \par \pard \tx3420 }{\f4 \par for all x let f(x) = sin(x**2); \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 + g9:0832} 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 <identifier>\{,<identifier>\}* \par \par }{\f2 \par <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. \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 + g9:0833} 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 <identifier>\{,<identifier>\}* \par \par }{\f2 \par <identifier> 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 + g9:0834} 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 <kernel>\{}{\f3 ,} {\f4 <kernel>\}* \par \par }{\f2 \par <kernel> 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 + g9:0835} 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 <identifier> }{\f3 =} {\f4 <expression>\{,<identifier> }{\f3 =} {\f4 <expression>\}* \par \par }{\f2 \par <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. \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 <list>\{,<list>\}+ \par \par }{\f2 \par <list> 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 <vars> }{\f3 let} {\f4 <operator>(<vars>)}{\f3 =} {\f4 <block> \par \par }{\f2 \par is an alternative to the \par syntax: \par }{\f4 \par \par }{\f3 procedure} {\f4 <name>(<vars>)}{\f3 ;} {\f4 <block> \par \par }{\f2 \par 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 }{\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 + g9:0836} 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 <operator>\{}{\f3 ,} {\f4 <operator>\}* \par \par }{\f2 \par <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 } {\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 <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. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # LINELENGTH} ${\footnote \pard\plain \sl240 \fs20 $ LINELENGTH} +{\footnote \pard\plain \sl240 \fs20 + g9:0837} 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 <expression> \par \par }{\f2 \par 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. \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 + g9:0838} 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 + g9:0839} 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 <operator>\{}{\f3 ,} {\f4 <operator>\}* \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 + g9:0840} 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 <dep-kernel>\{,<kernel>\}+ \par \par \par \par }{\f2 <dep-kernel> 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 + g9:0841} 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 <expr> }{\f3 =} {\f4 <expression>\{,<expr> }{\f3 =} {\f4 <expression>\}* \par \par }{\f2 \par <expr> is generally a term involving powers, and is limited by the rules for the } {\f2\uldb let}{\v\f2 LET} {\f2 command. <expression> 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 + g9:0842} 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 <operator>\{,<operator>\}* \par \par }{\f2 \par <operator> 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 + g9:0843} 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 <identifier>\{,<identifier>\}* \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 + g9:0844} 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 <identifier>\{,<identifier>\}* \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 + g9:0845} 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 <switch>\{,<switch>\}* \par \par }{\f2 \par <switch> 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 + g9:0846} 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 <switch>\{,<switch>\}* \par \par }{\f2 \par <switch> 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 + g9:0847} 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 <identifier>\{,<identifier>\}* \par \par }{\f2 \par <identifier> 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 + g9:0848} 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 <identifier>\{,<identifier>\}* \par \par }{\f2 \par <kernel> 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 + g9:0849} 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 <operator>,<known\_operator> \par \par }{\f2 \par <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. \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 <known\_operator>. \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 + g9:0850} 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 (<integer>) or }{\f3 precision} {\f4 <integer> \par \par }{\f2 \par <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. \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 + g9:0851} 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 (<integer>) or }{\f3 print_precision} {\f4 <integer> \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 + g9:0852} 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 <identifier>\{,<identifier>\}* \par \par }{\f2 \par <identifier> 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 + g9:0853} 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 <kernel>\{,<kernel>\}+ \par \par }{\f2 \par <kernel> 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 + g9:0854} 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 <identifier>\{,<identifier>\}* \par \par }{\f2 \par <identifier> 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 + g9:0855} 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 (<m>) or }{\f3 scientific_notation} {\f4 (\{<m>,<n>\}) \par \par }{\f2 \par <m> and <n> 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 \{<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. \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 + g9:0856} 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 <identifier>\{,<identifier>\}* \par \par }{\f2 \par <identifier> 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 <REDUCE User's Manual>, and the <Standard Lisp Report>. \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 + g9:0857} 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 + g9:0858} 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 <identifier>\{,<identifier>\}* \par \par }{\f2 \par <identifier> 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 <identifier> has not been declared to be an operator, the flag }{\f3 symmetric} {\f2 is still attached to it. When <identifier> is subsequently used as an operator, the message }{\f3 Declare} {\f2 <identifier> }{\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 + g9:0859} 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 <name>\{,<name>\}* \par \par }{\f2 \par <name> 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 + g9:0860} 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 <name>\{,<name>\}* \par \par }{\f2 \par <name> 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 + g9:0861} 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 <identifier> \par \par }{\f2 \par <identifier> 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 + g9:0862} 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 <kernel> }{\f3 =} {\f4 <number> \par \par }{\f2 \par <kernel> must be a REDUCE } {\f2\uldb kernel}{\v\f2 KERNEL} {\f2 , <number> 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 + g9:0863} 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 <expression> }{\f3 where} {\f4 <kernel> }{\f3 =} {\f4 <expression> \{,<kernel> }{\f3 =} {\f4 <expression>\}* \par \par }{\f2 \par <expression> can be any REDUCE scalar expression, <kernel> 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 + g9:0864} 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 <condition> }{\f3 do} {\f4 <statement> \par \par }{\f2 \par <condition> is given by a logical operator, <statement> 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 <<write a; a := a + 1>>; \par \par \par \par 10 \par \par \par 11 \par \par 12 \par \par while a < 5 do <<write a; a := a + 1>>; \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 + g9:0865} 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 <expression> \par \par }{\f2 \par 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. }{\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 + g10:0866} 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 <filename>\{,<filename>\}* \par \par }{\f2 \par <filename> 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 + g10:0867} 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 (<number>) or }{\f3 input} {\f4 <number> \par \par \par \par }{\f2 <number> 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 + g10:0868} 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 <filename> or }{\f3 out "} {\f4 <pathname> }{\f3 "} {\f4 or }{\f3 out t} {\f4 \par \par }{\f2 \par <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. \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 + g10:0869} 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 <filename>\{,<filename>\}* \par \par }{\f2 \par <filename> 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 + g11:0870} 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 (<expression>) or }{\f3 acos} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0871} 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 (<expression>) or }{\f3 acosh} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0872} 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 (<expression>) or }{\f3 acot} {\f4 <simple\_expression> \par \par }{\f2 \par <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 }{\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 + g11:0873} 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 (<expression>) or }{\f3 acoth} {\f4 <simple\_expression> \par \par }{\f2 \par <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 }{\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 + g11:0874} 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 (<expression>) or }{\f3 acsc} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0875} 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 (<expression>) or }{\f3 acsch} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0876} 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 (<expression>) or }{\f3 asec} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0877} 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 (<expression>) or }{\f3 asech} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0878} 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 (<expression>) or }{\f3 asin} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0879} 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 (<expression>) or }{\f3 asinh} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0880} 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 (<expression>) or }{\f3 atan} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0881} 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 (<expression>) or }{\f3 atanh} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0882} 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 (<expression>,<expression>) \par \par }{\f2 \par <expression> 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 + g11:0883} 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 (<expression>) or }{\f3 cos} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> is any valid scalar REDUCE expression, <simple\_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 + g11:0884} 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 (<expression>) or }{\f3 cosh} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0885} 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 (<expression>) or }{\f3 cot} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> may be any scalar REDUCE expression. <simple\_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 + g11:0886} 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 (<expression>) or }{\f3 coth} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> may be any scalar REDUCE expression. <simple\_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 + g11:0887} 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 (<expression>) or }{\f3 csc} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> may be any scalar REDUCE expression. <simple\_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 + g11:0888} 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 (<expression>) or }{\f3 csch} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0889} 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 + g11:0890} 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 (<expression>) or }{\f3 exp} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> can be any valid REDUCE scalar expression. <simple\_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 + g11:0891} 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 (<expression>) or }{\f3 sec} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> is any valid scalar REDUCE expression, <simple\_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 + g11:0892} 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 (<expression>) or }{\f3 sech} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> is any valid scalar REDUCE expression, <simple\_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 + g11:0893} 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 (<expression>) or }{\f3 sin} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> is any valid scalar REDUCE expression, <simple\_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 + g11:0894} 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 (<expression>) or }{\f3 sinh} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g11:0895} 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 (<expression>) or }{\f3 tan} {\f4 <simple\_expression> \par \par \par \par }{\f2 <expression> is any valid scalar REDUCE expression, <simple\_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 + g11:0896} 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 (<expression>) or }{\f3 tanh} {\f4 <simple\_expression> \par \par }{\f2 \par <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. \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 + g12:0897} 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 + g12:0898} 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 + g12:0899} 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 + g12:0900} 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 + g12:0901} 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 + g12:0902} 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,...<n>), where <n> is the current modulus. With }{\f3 balanced_mod} {\f2 on, the range [-<n>/2,<n>/2], or more precisely [-floor((<n>-1)/2), ceiling((<n>-1)/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 + g12:0903} 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 + g12:0904} 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 + g12:0905} 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 + g12:0906} 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 + g12:0907} 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 ,1\}\} \par \par \par on complex; \par \par \par factorize(a**2 + b**2); \par \par \{\{A + I*B,1\},\{A - I*B,1\}\} \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 + g12:0908} 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 + g12:0909} 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 + g12:0910} 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 + g12:0911} 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 + g12:0912} 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 + g12:0913} 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 + g12:0914} 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 + g12:0915} 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 + g12:0916} 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 + g12:0917} 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 + g12:0918} 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 + g12:0919} 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 <Proceedings of the ACM>, 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 + g12:0920} 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 + g12:0921} 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 + g12:0922} 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 + g12:0923} 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 + g12:0924} 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 + g12:0925} 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 + g12:0926} 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 + g12:0927} 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 + g12:0928} 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 + g12:0929} 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 \{\{4,1\},\{X + 4,1\},\{X + 3,1\}\} \par \par \par factorize(22587); \par \par \{\{3,1\},\{7529,1\}\} \par \par \par on ifactor; \par \par factorize(4*x**2 + 28*x + 48); \par \par \{\{2,2\},\{X + 4,1\},\{X + 3,1\}\} \par \par \par factorize(22587); \par \par \{\{3,1\},\{7529,1\}\} \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 + g12:0930} 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 + g12:0931} 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 + g12:0932} 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 + g12:0933} 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 + g12:0934} 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 2 \par \{- 3*X*Y + Y + 7,1\} \par 3 \par \{2*X*Y + Y + 5,1\}, \par \{X - Y,2\}\} \par \par \par on limitedfactors; \par \par factorize a; \par \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 + 35,1\}, \par \{X - Y,2\}\} \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 + g12:0935} 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 + g12:0936} 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 + g12:0937} 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 + g12:0938} 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 <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 } {\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 + g12:0939} 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 + g12:0940} 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 + g12:0941} 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 + g12:0942} 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 + g12:0943} 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 + g12:0944} 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 + g12:0945} 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 + g12:0946} 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 + g12:0947} 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 + g12:0948} 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 + g12:0949} 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 + g12:0950} 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 + g12:0951} 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 + g12:0952} 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 + g12:0953} 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 + g12:0954} 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 + g12:0955} 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 + g12:0956} 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 + g12:0957} 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 + g12:0958} 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 + g12:0959} 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 + g12:0960} 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 + g12:0961} 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 + g12:0962} 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 + g12:0963} 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 + g12:0964} 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 + g12:0965} 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 + g12:0966} 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 + g12:0967} 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 + g12:0968} 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 + g12:0969} 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 + g12:0970} 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 + g12:0971} 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 + g12:0972} 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 + g13:0973} 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 <row> and column <column> of a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . Errors occur if <row> or <column> do not evaluate to integer expressions or if the matrix is not square. \par \par \par syntax: \par }{\f4 }{\f3 cofactor} {\f4 (<matrix\_expression>,<row>,<column>) \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 + g13:0974} 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 (<expression>) or }{\f3 det} {\f4 <expression> \par \par }{\f2 \par <expression> 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 + g13:0975} 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 ((<expr>\{,<expr>\}*) \{(<expr>\{}{\f3 ,} {\f4 <expr>\}*)\}*) \par \par }{\f2 \par <expr> 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 + g13:0976} 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 (<matrix-id>,<tag-id>) \par \par }{\f2 \par <matrix-id> must be a declared matrix of values, and <tag-id> 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 MAT(2,1) := ARBCOMPLEX(1) \par \}\} \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 + g13:0977} 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 <identifier> \tab option (<index>,<index>) \par \par \{,<identifier> \tab option (<index>,<index>)\}* \par \par }{\f2 \par <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. \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 <not> 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 + g13:0978} 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 (<matrix\_expression>) \par \par }{\f2 \par <nullspace> 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 + g13:0979} 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 (<matrix\_expression>) \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 + g13:0980} 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 <identifier> or }{\f3 tp} {\f4 (<identifier>) \par \par }{\f2 \par <identifier> 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 + g13:0981} 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 (<expression>) or }{\f3 trace} {\f4 <simple\_expression> \par \par }{\f2 \par <expression> or <simple\_expression> 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 + g14:0982} 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 + g14:0983} 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 <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. \par \par The variable sequence <var> 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 <R> 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 <exp> can be polynomials <p>, rational functions <n>/<d> or equations <lh>=<rh> 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 <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. \par \par A basis on input or output of an algorithm is coded as } {\f2\uldb list}{\v\f2 LIST} {\f2 of expressions \{<exp>,<exp>,...\} . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # Term_order} ${\footnote \pard\plain \sl240 \fs20 $ Term_order} +{\footnote \pard\plain \sl240 \fs20 + g15:0984} 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 + g15:0985} 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 (<vl>, <m>) \par \par }{\f2 \par where <vl> is a } {\f2\uldb list}{\v\f2 LIST} {\f2 of variables (} {\f2\uldb kernel}{\v\f2 KERNEL} {\f2 s) and <m> 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 (<vl>,<m>,<n>) \par \par \par \par }{\f2 where <m> 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 <n> is a positive integer. \par \par 3. weighted term order \par syntax: \par }{\f4 \par \par }{\f3 torder} {\f4 (<vl>, }{\f3 weighted} {\f4 , <n>,<n>,...); \par \par }{\f2 \par where the <n> 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 (<vl>, }{\f3 matrix} {\f4 , <m>); \par \par }{\f2 \par where <m> 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 (<vl>, }{\f3 co} {\f4 ); \par \par }{\f2 \par where <co> 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 + g15:0986} 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 (<name>,<mat>) \par \par }{\f2 \par where <name> is an identifier for the new term order and <mat> 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 <name> 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 + g15:0987} 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 + g15:0988} 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 + g15:0989} 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 + g15:0990} 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 + g15:0991} 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 + g15:0992} 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 + g15:0993} 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 + g15:0994} 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 + g15:0995} 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 + g15:0996} 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 + g16:0997} 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 (\{<exp>,<exp>,... \}) \par \par \par \par }{\f2 where <exp> 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 + g16:0998} 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 # groebner\_walk} ${\footnote \pard\plain \sl240 \fs20 $ groebner_walk} +{\footnote \pard\plain \sl240 \fs20 + g16:0999} K{\footnote \pard\plain \sl240 \fs20 K groebner_walk operator;operator} }{\b\f2 GROEBNER\_WALK}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par The operator }{\f3 groebner_walk} {\f2 computes a }{\f3 lex} {\f2 basis from a given }{\f3 graded} {\f2 (or }{\f3 weighted} {\f2 ) one. \par syntax: \par }{\f4 \par \par }{\f3 groebner_walk} {\f4 (<g>) \par \par }{\f2 \par where <g> is a }{\f3 graded} {\f2 basis (or }{\f3 weighted} {\f2 basis with a weight vector with one repeated element) of the polynomial ideal. }{\f3 Groebner_walk} {\f2 computes a sequence of monomial bases, each time lifting the full system to a complete basis. }{\f3 Groebner_walk} {\f2 should be called only in cases, where a normal }{\f3 kex} {\f2 computation would take too much computer time. \par \par The operator } {\f2\uldb torder}{\v\f2 torder} {\f2 has to be called before in order to define the variable sequence and the term order mode of <g>. \par \par The variable } {\f2\uldb gvarslast}{\v\f2 gvarslast} {\f2 is not set. \par \par Do not call }{\f3 groebner_walk} {\f2 with }{\f3 on} {\f2 } {\f2\uldb groebopt}{\v\f2 groebopt} {\f2 . \par \par }{\f3 Groebner_walk} {\f2 includes some overhead (such as e. g. computation with division). On the other hand, sometimes }{\f3 groebner_walk} {\f2 is faster than a direct }{\f3 lex} {\f2 computation. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # groebopt} ${\footnote \pard\plain \sl240 \fs20 $ groebopt} +{\footnote \pard\plain \sl240 \fs20 + g16:1000} 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 + g16:1001} 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 + g16:1002} 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 + g16:1003} 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 + g16:1004} 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 + g16:1005} 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 + g16:1006} 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 + g16:1007} 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 + g16:1008} 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 + g16:1009} 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 + g16:1010} 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 (<basis>) \par \par \par \par }{\f2 where <bas> 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 + g16:1011} 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 (<bas>) \par \par \par \par }{\f2 where <bas> 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 + g16:1012} 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 (<bas>) \par \par \par \par }{\f2 where <bas> 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 + g16:1013} 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 (<d1>,<d2>,<plist>) \par \par }{\f2 \par 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 }{\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 + g16:1014} 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 (<bas>[,<vars>][,MAXDEG=<mx>] [,NEWVARS=<nv>]) \par \par \par \par }{\f2 where <bas> is a } {\f2\uldb groebner}{\v\f2 groebner} {\f2 basis in the current term order, <mx> (optional) is a positive integer and <nvl> (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 <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. \par \par If <newvars> is a list with one element, the minimal }{\f3 univariate polynomial} {\f2 is computed. \par \par <maxdeg> 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 + g16:1015} 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 + g16:1016} 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 (<p>, \{<exp>, ... \}) \par \par \par \par }{\f2 where <p> is an expression, and \{<exp>, ... \}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 + g16:1017} 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 (\{<exp>, ...\}, <d>) \par \par \par \par }{\f2 where \{<exp>,...\} is a list of expressions or equations, <d> is a single expression or equation. \par \par }{\f3 Idealquotient} {\f2 computes the ideal quotient: ideal spanned by the expressions \{<exp>,...\} divided by the single polynomial/expression <f>. 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 + g16:1018} 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(<bas>) \par \par \par \par }{\f2 where <bas> 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 # saturation} ${\footnote \pard\plain \sl240 \fs20 $ saturation} +{\footnote \pard\plain \sl240 \fs20 + g16:1019} K{\footnote \pard\plain \sl240 \fs20 K saturation operator;operator} }{\b\f2 SATURATION}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par syntax: \par }{\f4 \par \par }{\f3 saturation} {\f4 (\{<exp>, ...\}, <p>) \par \par \par \par }{\f2 where \{<exp>,...\} is a list of expressions or equations, <p> is a single polynomial. \par \par }{\f3 Saturation} {\f2 computes 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. }{\f3 Saturation} {\f2 calls } {\f2\uldb idealquotient}{\v\f2 idealquotient} {\f2 several times until the result does not change any more. \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 groebner\_walk operator} {\v\f2 groebner\_walk}{\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 }{\f2 \tab}{\f2\uldb saturation operator} {\v\f2 saturation}{\f2 \par \page #{\footnote \pard\plain \sl240 \fs20 # groebnerf} ${\footnote \pard\plain \sl240 \fs20 $ groebnerf} +{\footnote \pard\plain \sl240 \fs20 + g17:1020} 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 (\{<exp>, ...\}[,\{\},\{<nz>, ... \}]); \par \par \par \par }{\f2 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. \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 + g17:1021} 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 + g17:1022} 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 + g17:1023} 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 + g18:1024} 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 + g18:1025} 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 + g18:1026} 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 (\{<v>=<exp>,...\}) \par \par \par \par }{\f2 where <v> are } {\f2\uldb kernel}{\v\f2 KERNEL} {\f3 s} {\f2 (simple or indexed variables), <exp> are polynomials. \par \par }{\f3 groebnert} {\f2 is functionally equivalent to a } {\f2\uldb groebner}{\v\f2 groebner} {\f2 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> \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 + g18:1027} 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 (<p>,\{<v>=<exp>...\}) \par \par }{\f2 \par where <p> is an expression, <v> are kernels (simple or indexed variables), }{\f3 exp} {\f2 are polynomials. \par \par }{\f3 preducet} {\f2 computes the remainder of <p> modulo \{<exp>,...\} 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 + g19:1028} 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 + g19:1029} 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 + g20:1030} 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 (<p>) \par \par }{\f2 \par where <p> 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 + g20:1031} 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 (<p>[,<vars>]); \par \par }{\f2 \par where <p> 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 + g20:1032} 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 (<p1>,<p2>); \par \par \par \par }{\f2 where <p1> and <p2> 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 + g21:1033} 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 + g21:1034} 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 <vector> }{\f3 .} {\f4 <vector> \par \par }{\f2 \par <vector> 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 + g21:1035} 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 (<vector-expr>,<vector-expr>,<vector-expr>, <vector-expr>) \par \par }{\f2 \par <vector-expr> 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 + g21:1036} 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 (<identifier>,<vector-expr> \{,<vector-expr>\}*) \par \par }{\f2 \par <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. \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 <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. \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, <Relativistic Quantum Mechanics,> 1964, is assumed in all operations involving gamma matrices. For an example of the use of }{\f3 g} {\f2 in a calculation, see the <REDUCE User's Manual>. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # INDEX} ${\footnote \pard\plain \sl240 \fs20 $ INDEX} +{\footnote \pard\plain \sl240 \fs20 + g21:1037} 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 <vector-id>\{,<vector-id>\}* \par \par }{\f2 \par <vector-id> 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 + g21:1038} 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 <vector-var>}{\f3 =} {\f4 <scalar-var> \{,<vector-var>}{\f3 =} {\f4 <scalar-var>\}* \par \par }{\f2 \par <vector-var> 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. <scalar-var> 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 + g21:1039} 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 <vector-var>\{,<vector-var>\}* \par \par }{\f2 \par <vector-var> 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 + g21:1040} 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 <line-id>\{,<line-id>\}* \par \par }{\f2 \par <line-id> 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 + g21:1041} 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 <identifier>\{,<identifier>\}* \par \par }{\f2 \par <identifier> 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 + g21:1042} 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 <line-id>\{,<line-id>\}* \par \par }{\f2 \par <line-id> 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 + g21:1043} 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 <dimension> \par \par }{\f2 \par <dimension> 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 + g21:1044} 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 <identifier>\{,<identifier>\}* \par \par }{\f2 \par <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. \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 <ID> }{\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 <list> \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 + g22:1045} 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 + g22:1046} 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 <var> = (<low> .. <high>) \par \par }{\f2 \par where <var> is a } {\f2\uldb kernel}{\v\f2 KERNEL} {\f2 and <low>, <high> are numbers or expression which evaluate to numbers with <low><=<high>. \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 + g22:1047} 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 + g22:1048} 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 + g22:1049} 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 (<exp>, <var>[=<val>] [,<var>[=<val>] ... [,accuracy=<a>] [,iterations=<i>]) \par \par or \par \par }{\f3 num_min} {\f4 (exp, \{ <var>[=<val>] [,<var>[=<val>] ...] \} [,accuracy=<a>] [,iterations=<i>]) \par \par }{\f2 \par where <exp> is a function expression, <var> are the variables in <exp> and <val> are the (optional) start values. For <a> and <i> 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 + g22:1050} 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 (<exp>, <var>[=<val>][,accuracy=<a>][,iterations=<i>]) \par \par or \par \par }{\f3 num_solve} {\f4 (\{<exp>,...,<exp>\}, <var>[=<val>],...,<var>[=<val>] [,accuracy=<a>][,iterations=<i>]) \par \par or \par \par }{\f3 num_solve} {\f4 (\{<exp>,...,<exp>\}, \{<var>[=<val>],...,<var>[=<val>]\} [,accuracy=<a>][,iterations=<i>]) \par \par \par \par }{\f2 where <exp> are function expressions, <var> are the variables, <val> are optional start values. For <a> and <i> 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 + g22:1051} 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. <a> 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 (<exp>,<var>=(<l> .. <u>) [,<var>=(<l> .. <u>),...] [,accuracy=<a>][,iterations=<i>]) \par \par }{\f2 \par where <exp> is the function to be integrated, <var> are the integration variables, <l> are the lower bounds, <u> 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 + g22:1052} 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 (<exp>,<depvar>=<start>, <indep>=(<from> .. <to>) [,accuracy=<a>][,iterations=<i>]) \par \par or \par \par }{\f3 num_odesolve} {\f4 (\{<exp>,<exp>,...\}, \{<depvar>=<start>,<depvar>=<start>,...\} <indep>=(<from> .. <to>) [,accuracy=<a>][,iterations=<i>]) \par \par \par \par }{\f2 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. \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 <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. \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 + g22:1053} 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 (<exp>,<var>=(<l> .. <u>) [,<var>=(<l> .. <u>) ...]) \par \par or \par \par }{\f3 bounds} {\f4 (<exp>,\{<var>=(<l> .. <u>) [,<var>=(<l> .. <u>) ...]\}) \par \par \par \par }{\f2 where <exp> is the function to be investigated, <var> are the variables of <exp>, <l> and <u> 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 + g22:1054} 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 (<fcn>,<var>=(<lo> .. <hi>),<n>) \par \par }{\f3 chebyshev_eval} {\f4 (<coeffs>,<var>=(<lo> .. <hi>), <var>=<pt>) \par \par }{\f3 chebyshev_df} {\f4 (<coeffs>,<var>=(<lo> .. <hi>)) \par \par }{\f3 chebyshev_int} {\f4 (<coeffs>,<var>=(<lo> .. <hi>)) \par \par }{\f2 \par 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 } {\f2\uldb interval}{\v\f2 Interval} {\f2 <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. \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 + g22:1055} 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 (<vals>,<basis>,<var>=<pts>) \par \par }{\f2 \par 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 }{\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 + g23:1056} 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 + g23:1057} 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 <rl> \par \par }{\f2 \par where <rl> 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 + g23:1058} 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 (<p>,<pt>) \par \par }{\f2 \par where <p> is a univariate polynomial and <pt> 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 + g23:1059} 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 (<p>) or \par \par }{\f3 realroots} {\f4 (<p>,<from>,<to>) \par \par }{\f2 \par 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 }{\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 + g23:1060} 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 (<n>) \par \par }{\f2 \par Here <n> 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 + g23:1061} 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 (<p>) \par \par }{\f2 \par where <p> 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 + g23:1062} 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 (<p>) \par \par }{\f2 \par where <p> 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 + g23:1063} 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 + g23:1064} 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 + g24:1065} 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 + g24:1066} 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 \tab }{\f3 Euler's constant } {\f2 (which can be computed as -} {\f2\uldb Psi}{\v\f2 PSI} {\f2 (1)) and \par \tab }{\f3 Khinchin's constant} {\f2 (which is defined in Khinchin's book ``Continued Fractions'') and \par \tab }{\f3 Golden_Ratio} {\f2 (which can be computed as (1 + sqrt 5)/2) and \par \tab }{\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 + g25:1067} 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 (<integer>) \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 + g25:1068} 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 (<integer>,<expression>) \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 + g25:1069} 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 (<integer>) \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 + g25:1070} 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 (<integer>,<expression>) \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 + g25:1071} 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 (<expression>) \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 + g26:1072} 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 (<order>,<argument>) \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 + g26:1073} 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 (<order>,<argument>) \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 + g26:1074} 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 (<order>,<argument>) \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 + g26:1075} 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 (<order>,<argument>) \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 + g26:1076} 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 (<order>,<argument>) \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 + g26:1077} 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 (<order>,<argument>) \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 + g26:1078} 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 (<order>,<argument>) \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 + g26:1079} 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 (<order>,<argument>) \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 + g26:1080} 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 (<parameter>,<parameter>,<argument>) \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 + g26:1081} 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 (<parameter>,<parameter>,<argument>) \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 + g26:1082} 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 (<parameter>,<parameter>,<argument>) \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 + g27:1083} 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 (<argument>) \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 + g27:1084} 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 (<argument>) \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 + g27:1085} 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 (<argument>) \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 + g27:1086} 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 (<argument>) \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 + g28:1087} 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 (<expression>,<integer>) \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 + g28:1088} 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 (<expression>,<integer>) \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 + g28:1089} 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 (<expression>,<integer>) \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 + g28:1090} 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 (<expression>,<integer>) \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 + g28:1091} 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 (<expression>,<integer>) \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 + g28:1092} 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 (<expression>,<integer>) \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 + g28:1093} 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 (<expression>,<integer>) \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 + g28:1094} 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 (<expression>,<integer>) \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 + g28:1095} 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 (<expression>,<integer>) \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 + g28:1096} 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 (<expression>,<integer>) \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 + g28:1097} 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 (<expression>,<integer>) \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 + g28:1098} 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 (<expression>,<integer>) \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 + g28:1099} 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 (<expression>,<integer>) \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 + g28:1100} 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 (<integer>,<integer>,<integer>) \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 + g28:1101} 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 (<expression>,<integer>) \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 + g28:1102} 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 (<expression>,<integer>) \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 + g28:1103} 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 (<integer>) \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 + g28:1104} 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 (<integer>) \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 + g28:1105} 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 (<expression>,<integer>) \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 (<integer>) \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 + g28:1106} 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 (<integer>,<expression>,<integer>) \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 + g28:1107} 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 (<expression>,<integer>) \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 + g29:1108} 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 (<expression>,<expression>) \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 + g29:1109} 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 (<expression>) \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 + g29:1110} 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 (<expression>,<expression>) \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 + g29:1111} 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 (<expression>) \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 + g29:1112} 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 (<integer>,<expression>) \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 + g30:1113} 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 (<order>,<expression>) \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 + g30:1114} 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 (<z>) \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 + g31:1115} 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 (<integer>,<expression>) \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 + g31:1116} 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 (<integer>,<expression>) \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 + g31:1117} 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 (<integer>,<expression>) \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 + g31:1118} 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 (<integer>,<expression>) or \par \par }{\f3 LaguerreP} {\f4 (<integer>,<expression>,<expression>) \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 + g31:1119} 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 (<integer>,<expression>) or \par \par }{\f3 LegendreP} {\f4 (<integer>,<expression>,<expression>) \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 + g31:1120} 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 (<integer>,<expression>,<expression>, <expression>) \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 + g31:1121} 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 (<integer>,<expression>,<expression>) \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 + g31:1122} 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 (<integer>,<integer>, <expression>,<expression>,<expression>,<expression>) \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 + g31:1123} 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 (<integer>,<integer>, <expression>,<expression>) \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 + g32:1124} 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 (<expression>) \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 + g32:1125} 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 (<expression>) \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 + g32:1126} 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 (<expression>) \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 + g32:1127} 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 (<expression>) \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 + g32:1128} 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 (<expression>) \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 + g32:1129} 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 (<expression>) \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 + g32:1130} 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 (<expression>) \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 + g32:1131} 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 (<expression>) \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 + g32:1132} 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 (<expression>) \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 + g32:1133} 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 (<expression>) \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 + g33:1134} 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 (<integer>,<integer>) \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 + g33:1135} 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 (<integer>,<integer>) \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 + g33:1136} 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 (<integer>,<integer>) \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 + g34:1137} 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 (<list of j1,m1>,<list of j2,m2>, <list of j3,m3>) \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 + g34:1138} 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 (<list of j1,m1>,<list of j2,m2>, <list of j3,m3>) \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 + g34:1139} 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 (<list of j1,j2,j3>,<list of l1,l2,l3>) \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 + g35:1140} 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 (<list of parameters>,<list of parameters>, <argument>) \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 + g35:1141} 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 (<list of parameters>,<list of parameters>, <argument>) \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 + g35:1142} 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 (<argument>) \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 + g35:1143} 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 (<argument>) \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 + g36:1144} 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 + g36:1145} 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 (<expression> }{\f3 ,} {\f4 <var>}{\f3 ,} {\f4 <expression>}{\f3 ,} {\f4 <number> \par \par \{}{\f3 ,} {\f4 <var>}{\f3 ,} {\f4 <expression>}{\f3 ,} {\f4 <number>\}*) \par \par }{\f2 \par <expression> can be any valid REDUCE algebraic expression. <var> must be a } {\f2\uldb kernel}{\v\f2 KERNEL} {\f2 , 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 }{\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 , <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 }{\f3 infinity} {\f2 , }{\f3 taylor} {\f2 tries to expand in a series in 1/<var>. \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 + g36:1146} 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 + g36:1147} 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 + g36:1148} 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 + g36:1149} 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 + g36:1150} 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 (<expression>) or }{\f3 taylororiginal} {\f4 <simple_expression> \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 + g36:1151} 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 + g36:1152} 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 + g36:1153} 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 (<expression>}{\f3 ,} {\f4 <var>}{\f3 ,} {\f4 <var>) \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 + g36:1154} 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 (<expression>) or }{\f3 taylorseriesp} {\f4 <simple_expression> \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 + g36:1155} 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 (<expression>) or }{\f3 taylortemplate} {\f4 <simple_expression> \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 + g36:1156} 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 (<expression>) or }{\f3 taylortostandard} {\f4 <simple_expression> \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 + g37:1157} 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 + g37:1158} 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 + g37:1159} 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 a 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 + g37:1160} 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 (<spec>,<spec>,...) \par \par }{\f2 \par where <spec> is a <function>, a <range> or an <option>. \par \par <function>: \par \par - an expression depending on one unknown (e.g. }{\f3 sin(x)} {\f2 or two unknowns (e.g. }{\f3 sin(x+y)} {\f2 , \par \par - an equation with a function on its right-hand side and a single name on its left-hand side (e.g. }{\f3 z=sin(x+y)} {\f2 where the name on the left-hand side specifies the dependent variable. \par \par - 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. }{\f3 \{sin(x),cos(x)\}} {\f2 ). \par \par - an equation with zero left or right hand side describing an implicit curve in two dimensions (e.g. }{\f3 x**3+x*y**3-9x=0} {\f2 ). \par \par - a point set: the graph can be given as point set in 2 dimensions or a } {\f2\uldb pointset}{\v\f2 Pointset} {\f2 or pointset list in 3 dimensions. \par \par <range>: \par \par 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 } {\f2\uldb interval}{\v\f2 Interval} {\f2 , e.g. }{\f3 x=( -3 .. 5)} {\f2 . \par \par 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. \par \par <option>: \par \par 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 } {\f2\uldb string}{\v\f2 STRING} {\f2 ). Available options are: \par \par } {\f2\uldb title}{\v\f2 title} {\f2 : assign a heading (default: empty) \par \par } {\f2\uldb xlabel}{\v\f2 xlabel} {\f2 : set label for the x axis \par \par } {\f2\uldb ylabel}{\v\f2 ylabel} {\f2 : set label for the y axis \par \par } {\f2\uldb zlabel}{\v\f2 zlabel} {\f2 : set label for the z axis \par \par } {\f2\uldb terminal}{\v\f2 terminal} {\f2 : select an output device \par \par } {\f2\uldb size}{\v\f2 size} {\f2 : rescale the picture \par \par } {\f2\uldb view}{\v\f2 view} {\f2 : set a viewpoint \par \par }{\f3 (no)} {\f2\uldb contour}{\v\f2 contour} {\f2 : 3d: add contour lines \par \par }{\f3 (no)} {\f2\uldb surface}{\v\f2 surface} {\f2 : 3d: draw surface (default: yes) \par \par }{\f3 (no)} {\f2\uldb hidden3d}{\v\f2 hidden3d} {\f2 : 3d: remove hidden lines (default: no) \par \par \par examples: \par \pard \tx3420 }{\f4 \par plot(cos x); \par \par plot(s=sin phi,phi=(-3 .. 3)); \par \par plot(sin phi,cos phi,phi=(-3 .. 3)); \par \par plot (cos sqrt(x**2 + y**2),x=(-3 .. 3),y=(-3 .. 3),hidden3d); \par \par plot \{\{0,0\},\{0,1\},\{1,1\},\{0,0\},\{1,0\},\{0,1\},\{0.5,1.5\},\{1,1\},\{1,0\}\}; \par \par \par \par on rounded; \par \par w:=for j:=1:200 collect \{1/j*sin j,1/j*cos j,j/200\}$ \par \par plot w; \par \pard \sl240 }{\f2 Additional control of the }{\f3 plot} {\f2 operation: } {\f2\uldb plotrefine}{\v\f2 PLOTREFINE} {\f2 , } {\f2\uldb plot_xmesh}{\v\f2 plot_xmesh} {\f2 , } {\f2\uldb plot_ymesh}{\v\f2 plot_ymesh} {\f2 , } {\f2\uldb trplot}{\v\f2 TRPLOT} {\f2 , } {\f2\uldb plotkeep}{\v\f2 PLOTKEEP} {\f2 , } {\f2\uldb show_grid}{\v\f2 SHOW_GRID} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # PLOTRESET} ${\footnote \pard\plain \sl240 \fs20 $ PLOTRESET} +{\footnote \pard\plain \sl240 \fs20 + g37:1161} K{\footnote \pard\plain \sl240 \fs20 K PLOTRESET command;command} }{\b\f2 PLOTRESET}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par \par The command }{\f3 plotreset} {\f2 closes the current GNUPLOT windows. The next call to } {\f2\uldb plot}{\v\f2 PLOT} {\f2 will create a new one. }{\f3 plotreset} {\f2 can also be used to reset the system status after technical problems. \par \par \par syntax: \par }{\f4 }{\f3 plotreset} {\f4 ; \par \par }{\f2 \par \page #{\footnote \pard\plain \sl240 \fs20 # title} ${\footnote \pard\plain \sl240 \fs20 $ title} +{\footnote \pard\plain \sl240 \fs20 + g37:1162} K{\footnote \pard\plain \sl240 \fs20 K plot;title variable;variable} }{\b\f2 TITLE}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par \par \par \par } {\f2\uldb plot}{\v\f2 PLOT} {\f2 option: Assign a title to the GNUPLOT graph. \par \par \par syntax: \par }{\f4 }{\f3 title} {\f4 = <string> \par \par }{\f2 \par \par examples: \par \pard \tx3420 }{\f4 \par title="annual revenue in 1993" \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # xlabel} ${\footnote \pard\plain \sl240 \fs20 $ xlabel} +{\footnote \pard\plain \sl240 \fs20 + g37:1163} K{\footnote \pard\plain \sl240 \fs20 K plot;xlabel variable;variable} }{\b\f2 XLABEL}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par \par \par \par } {\f2\uldb plot}{\v\f2 PLOT} {\f2 option: Assign a name to to the x axis (see } {\f2\uldb axes names}{\v\f2 Axes_names} {\f2 ). \par \par \par syntax: \par }{\f4 }{\f3 xlabel} {\f4 = <string> \par \par }{\f2 \par \par examples: \par \pard \tx3420 }{\f4 \par xlabel="month" \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # ylabel} ${\footnote \pard\plain \sl240 \fs20 $ ylabel} +{\footnote \pard\plain \sl240 \fs20 + g37:1164} K{\footnote \pard\plain \sl240 \fs20 K plot;ylabel variable;variable} }{\b\f2 YLABEL}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par \par \par \par } {\f2\uldb plot}{\v\f2 PLOT} {\f2 option: Assign a name to to the x axis (see } {\f2\uldb axes names}{\v\f2 Axes_names} {\f2 ). \par \par \par syntax: \par }{\f4 }{\f3 ylabel} {\f4 = <string> \par \par }{\f2 \par \par examples: \par \pard \tx3420 }{\f4 \par ylabel="million forint" \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # zlabel} ${\footnote \pard\plain \sl240 \fs20 $ zlabel} +{\footnote \pard\plain \sl240 \fs20 + g37:1165} K{\footnote \pard\plain \sl240 \fs20 K plot;zlabel variable;variable} }{\b\f2 ZLABEL}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par \par \par \par } {\f2\uldb plot}{\v\f2 PLOT} {\f2 option: Assign a name to to the z axis (see } {\f2\uldb axes names}{\v\f2 Axes_names} {\f2 ). \par \par \par syntax: \par }{\f4 }{\f3 zlabel} {\f4 = <string> \par \par }{\f2 \par \par examples: \par \pard \tx3420 }{\f4 \par zlabel="local weight" \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # terminal} ${\footnote \pard\plain \sl240 \fs20 $ terminal} +{\footnote \pard\plain \sl240 \fs20 + g37:1166} K{\footnote \pard\plain \sl240 \fs20 K plot;terminal variable;variable} }{\b\f2 TERMINAL}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par \par \par \par } {\f2\uldb plot}{\v\f2 PLOT} {\f2 option: Select a different output device. The possible values here depend highly on the facilities installed for your GNUPLOT software. \par \par \par syntax: \par }{\f4 }{\f3 terminal} {\f4 = <string> \par \par }{\f2 \par \par examples: \par \pard \tx3420 }{\f4 \par terminal="x11" \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # size} ${\footnote \pard\plain \sl240 \fs20 $ size} +{\footnote \pard\plain \sl240 \fs20 + g37:1167} K{\footnote \pard\plain \sl240 \fs20 K plot;size variable;variable} }{\b\f2 SIZE}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par \par \par \par } {\f2\uldb plot}{\v\f2 PLOT} {\f2 option: Rescale the graph (not the window!) in x and y direction. Default is 1.0 (no rescaling). \par \par \par syntax: \par }{\f4 }{\f3 size} {\f4 = "<sx>,<sy>" \par \par }{\f2 \par where <sx>,<sy> are floating point number not too far from 1.0. \par examples: \par \pard \tx3420 }{\f4 \par size="0.7,1" \par \pard \sl240 }{\f2 \par \par \page #{\footnote \pard\plain \sl240 \fs20 # view} ${\footnote \pard\plain \sl240 \fs20 $ view} +{\footnote \pard\plain \sl240 \fs20 + g37:1168} K{\footnote \pard\plain \sl240 \fs20 K plot;view variable;variable} }{\b\f2 VIEW}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par \par \par \par } {\f2\uldb plot}{\v\f2 PLOT} {\f2 option: Set a new viewpoint by turning the object around the x and then around the z axis (see } {\f2\uldb axes names}{\v\f2 Axes_names} {\f2 ). \par \par \par syntax: \par }{\f4 }{\f3 view} {\f4 = "<sx>,<sz>" \par \par }{\f2 \par where <sx>,<sz> are floating point number representing angles in degrees. \par examples: \par \pard \tx3420 }{\f4 \par view="30,130" \par \pard \sl240 }{\f2 \par \par \page #{\footnote \pard\plain \sl240 \fs20 # contour} ${\footnote \pard\plain \sl240 \fs20 $ contour} +{\footnote \pard\plain \sl240 \fs20 + g37:1169} K{\footnote \pard\plain \sl240 \fs20 K plot;contour switch;switch} }{\b\f2 CONTOUR}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par \par \par \par } {\f2\uldb plot}{\v\f2 PLOT} {\f2 option: If }{\f3 contour} {\f2 is member of the options for a 3d } {\f2\uldb plot}{\v\f2 PLOT} {\f2 contour lines are projected to the z=0 plane (see } {\f2\uldb axes names}{\v\f2 Axes_names} {\f2 ). The absence of contour lines can be selected explicitly by including }{\f3 nocontour} {\f2 . Default is }{\f3 nocontour} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # surface} ${\footnote \pard\plain \sl240 \fs20 $ surface} +{\footnote \pard\plain \sl240 \fs20 + g37:1170} K{\footnote \pard\plain \sl240 \fs20 K plot;surface switch;switch} }{\b\f2 SURFACE}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par \par \par \par } {\f2\uldb plot}{\v\f2 PLOT} {\f2 option: If }{\f3 surface} {\f2 is member of the options for a 3d } {\f2\uldb plot}{\v\f2 PLOT} {\f2 the surface is drawn. The absence of the surface plotting can be selected by including }{\f3 nosurface} {\f2 , e.g. if only the } {\f2\uldb contour}{\v\f2 contour} {\f2 should be visualized. Default is }{\f3 surface} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # hidden3d} ${\footnote \pard\plain \sl240 \fs20 $ hidden3d} +{\footnote \pard\plain \sl240 \fs20 + g37:1171} K{\footnote \pard\plain \sl240 \fs20 K plot;hidden3d switch;switch} }{\b\f2 HIDDEN3D}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par \par \par \par } {\f2\uldb plot}{\v\f2 PLOT} {\f2 option: If }{\f3 hidden3d} {\f2 is member of the options for a 3d } {\f2\uldb plot}{\v\f2 PLOT} {\f2 hidden lines are removed from the picture. Otherwise a surface is drawn as transparent object. Default is }{\f3 nohidden3d} {\f2 . Selecting }{\f3 hidden3d} {\f2 increases the computing time substantially. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # PLOTKEEP} ${\footnote \pard\plain \sl240 \fs20 $ PLOTKEEP} +{\footnote \pard\plain \sl240 \fs20 + g37:1172} K{\footnote \pard\plain \sl240 \fs20 K plot;PLOTKEEP switch;switch} }{\b\f2 PLOTKEEP}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par \par \par \par Normally all intermediate data sets are deleted after terminating a plot session. If the switch }{\f3 plotkeep} {\f2 is set } {\f2\uldb on}{\v\f2 ON} {\f2 , the data sets are kept for eventual post processing independent of REDUCE. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # PLOTREFINE} ${\footnote \pard\plain \sl240 \fs20 $ PLOTREFINE} +{\footnote \pard\plain \sl240 \fs20 + g37:1173} K{\footnote \pard\plain \sl240 \fs20 K plot;PLOTREFINE switch;switch} }{\b\f2 PLOTREFINE}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par \par \par \par In general } {\f2\uldb plot}{\v\f2 PLOT} {\f2 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 }{\f3 plotrefine} {\f2 which is } {\f2\uldb on}{\v\f2 ON} {\f2 by default. When you turn it } {\f2\uldb off}{\v\f2 OFF} {\f2 the functions will be evaluated only at the basic points (see } {\f2\uldb plot_xmesh}{\v\f2 plot_xmesh} {\f2 , } {\f2\uldb plot_ymesh}{\v\f2 plot_ymesh} {\f2 ). \par \par \page #{\footnote \pard\plain \sl240 \fs20 # plot_xmesh} ${\footnote \pard\plain \sl240 \fs20 $ plot_xmesh} +{\footnote \pard\plain \sl240 \fs20 + g37:1174} K{\footnote \pard\plain \sl240 \fs20 K plot;plot_xmesh variable;variable} }{\b\f2 PLOT_XMESH}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par \par \par \par The integer value of the global variable }{\f3 plot_xmesh} {\f2 defines the number of initial function evaluations in x direction (see } {\f2\uldb axes names}{\v\f2 Axes_names} {\f2 ) for } {\f2\uldb plot}{\v\f2 PLOT} {\f2 . For 2d graphs additional points will be used as long as } {\f2\uldb plotrefine}{\v\f2 PLOTREFINE} {\f2 is }{\f3 on} {\f2 . For 3d graphs this number defines also the number of mesh lines orthogonal to the x axis. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # plot_ymesh} ${\footnote \pard\plain \sl240 \fs20 $ plot_ymesh} +{\footnote \pard\plain \sl240 \fs20 + g37:1175} K{\footnote \pard\plain \sl240 \fs20 K plot;plot_ymesh variable;variable} }{\b\f2 PLOT_YMESH}{\f2 \tab \tab \tab \tab }{\b\f2 variable}{\f2 \par \par \par \par The integer value of the global variable }{\f3 plot_ymesh} {\f2 defines for 3d } {\f2\uldb plot}{\v\f2 PLOT} {\f2 calls the number of function evaluations in y direction (see } {\f2\uldb axes names}{\v\f2 Axes_names} {\f2 ) and the number of mesh lines orthogonal to the y axis. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # SHOW_GRID} ${\footnote \pard\plain \sl240 \fs20 $ SHOW_GRID} +{\footnote \pard\plain \sl240 \fs20 + g37:1176} K{\footnote \pard\plain \sl240 \fs20 K plot;SHOW_GRID switch;switch} }{\b\f2 SHOW_GRID}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par \par \par \par The grid for localizing an implicitly defined curve in } {\f2\uldb plot}{\v\f2 PLOT} {\f2 consists of triangles. These are computed initially equally distributed over the x-y plane controlled by } {\f2\uldb plot_xmesh}{\v\f2 plot_xmesh} {\f2 . The grid is refined adaptively in several levels. The final grid can be visualized by setting on the switch }{\f3 show_grid} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # TRPLOT} ${\footnote \pard\plain \sl240 \fs20 $ TRPLOT} +{\footnote \pard\plain \sl240 \fs20 + g37:1177} K{\footnote \pard\plain \sl240 \fs20 K plot;TRPLOT switch;switch} }{\b\f2 TRPLOT}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par \par \par \par 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 }{\f3 trplot} {\f2 is set on all GNUPLOT commands will be printed to the standard output additionally. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # g37} ${\footnote \pard\plain \sl240 \fs20 $ Gnuplot package} +{\footnote \pard\plain \sl240 \fs20 + index:0037} }{\b\f2 Gnuplot package}{\f2 \par }\pard \sl240 {\f2 \par } {\f2 \tab}{\f2\uldb GNUPLOT and REDUCE introduction} {\v\f2 GNUPLOT_and_REDUCE}{\f2 \par }{\f2 \tab}{\f2\uldb Axes names concept} {\v\f2 Axes_names}{\f2 \par }{\f2 \tab}{\f2\uldb Pointset type} {\v\f2 Pointset}{\f2 \par }{\f2 \tab}{\f2\uldb PLOT command} {\v\f2 PLOT}{\f2 \par }{\f2 \tab}{\f2\uldb PLOTRESET command} {\v\f2 PLOTRESET}{\f2 \par }{\f2 \tab}{\f2\uldb title variable} {\v\f2 title}{\f2 \par }{\f2 \tab}{\f2\uldb xlabel variable} {\v\f2 xlabel}{\f2 \par }{\f2 \tab}{\f2\uldb ylabel variable} {\v\f2 ylabel}{\f2 \par }{\f2 \tab}{\f2\uldb zlabel variable} {\v\f2 zlabel}{\f2 \par }{\f2 \tab}{\f2\uldb terminal variable} {\v\f2 terminal}{\f2 \par }{\f2 \tab}{\f2\uldb size variable} {\v\f2 size}{\f2 \par }{\f2 \tab}{\f2\uldb view variable} {\v\f2 view}{\f2 \par }{\f2 \tab}{\f2\uldb contour switch} {\v\f2 contour}{\f2 \par }{\f2 \tab}{\f2\uldb surface switch} {\v\f2 surface}{\f2 \par }{\f2 \tab}{\f2\uldb hidden3d switch} {\v\f2 hidden3d}{\f2 \par }{\f2 \tab}{\f2\uldb PLOTKEEP switch} {\v\f2 PLOTKEEP}{\f2 \par }{\f2 \tab}{\f2\uldb PLOTREFINE switch} {\v\f2 PLOTREFINE}{\f2 \par }{\f2 \tab}{\f2\uldb plot_xmesh variable} {\v\f2 plot_xmesh}{\f2 \par }{\f2 \tab}{\f2\uldb plot_ymesh variable} {\v\f2 plot_ymesh}{\f2 \par }{\f2 \tab}{\f2\uldb SHOW_GRID switch} {\v\f2 SHOW_GRID}{\f2 \par }{\f2 \tab}{\f2\uldb TRPLOT switch} {\v\f2 TRPLOT}{\f2 \par \page #{\footnote \pard\plain \sl240 \fs20 # Linear_Algebra_package} ${\footnote \pard\plain \sl240 \fs20 $ Linear_Algebra_package} +{\footnote \pard\plain \sl240 \fs20 + g38:1178} K{\footnote \pard\plain \sl240 \fs20 K Linear Algebra package introduction;introduction} }{\b\f2 LINEAR ALGEBRA PACKAGE}{\f2 \tab \tab \tab \tab }{\b\f2 introduction}{\f2 \par \par \par \par This section briefly describes what's available in the Linear Algebra package. \par \par Note on examples: In the examples throughout this document, the matrix A will be \pard \tx3420 }{\f4 \par [1 2 3] \par [4 5 6] \par [7 8 9]. \par \pard \sl240 }{\f2 \par \par The functions can be divided into four categories: \par \par Basic matrix handling \par \par } {\f2\uldb add_columns}{\v\f2 add_columns} {\f2 , \par \par } {\f2\uldb add_rows}{\v\f2 add_rows} {\f2 , \par \par } {\f2\uldb add_to_columns}{\v\f2 add_to_columns} {\f2 , \par \par } {\f2\uldb add_to_rows}{\v\f2 add_to_rows} {\f2 , \par \par } {\f2\uldb augment_columns}{\v\f2 augment_columns} {\f2 , \par \par } {\f2\uldb char_poly}{\v\f2 char_poly} {\f2 , \par \par } {\f2\uldb column_dim}{\v\f2 column_dim} {\f2 , \par \par } {\f2\uldb copy_into}{\v\f2 copy_into} {\f2 , \par \par } {\f2\uldb diagonal}{\v\f2 diagonal} {\f2 , \par \par } {\f2\uldb extend}{\v\f2 extend} {\f2 , \par \par } {\f2\uldb find_companion}{\v\f2 find_companion} {\f2 , \par \par } {\f2\uldb get_columns}{\v\f2 get_columns} {\f2 , \par \par } {\f2\uldb get_rows}{\v\f2 get_rows} {\f2 , \par \par } {\f2\uldb hermitian_tp}{\v\f2 hermitian_tp} {\f2 , \par \par } {\f2\uldb matrix_augment}{\v\f2 matrix_augment} {\f2 , \par \par } {\f2\uldb matrix_stack}{\v\f2 matrix_stack} {\f2 , \par \par } {\f2\uldb minor}{\v\f2 minor} {\f2 , \par \par } {\f2\uldb mult_columns}{\v\f2 mult_columns} {\f2 , \par \par } {\f2\uldb mult_rows}{\v\f2 mult_rows} {\f2 , \par \par } {\f2\uldb pivot}{\v\f2 pivot} {\f2 , \par \par } {\f2\uldb remove_columns}{\v\f2 remove_columns} {\f2 , \par \par } {\f2\uldb remove_rows}{\v\f2 remove_rows} {\f2 , \par \par } {\f2\uldb row_dim}{\v\f2 row_dim} {\f2 , \par \par } {\f2\uldb rows_pivot}{\v\f2 rows_pivot} {\f2 , \par \par } {\f2\uldb stack_rows}{\v\f2 stack_rows} {\f2 , \par \par } {\f2\uldb sub_matrix}{\v\f2 sub_matrix} {\f2 , \par \par } {\f2\uldb swap_columns}{\v\f2 swap_columns} {\f2 , \par \par } {\f2\uldb swap_entries}{\v\f2 swap_entries} {\f2 , \par \par } {\f2\uldb swap_rows}{\v\f2 swap_rows} {\f2 . \par \par Constructors -- functions that create matrices \par \par } {\f2\uldb band_matrix}{\v\f2 band_matrix} {\f2 , \par \par } {\f2\uldb block_matrix}{\v\f2 block_matrix} {\f2 , \par \par } {\f2\uldb char_matrix}{\v\f2 char_matrix} {\f2 , \par \par } {\f2\uldb coeff_matrix}{\v\f2 coeff_matrix} {\f2 , \par \par } {\f2\uldb companion}{\v\f2 companion} {\f2 , \par \par } {\f2\uldb hessian}{\v\f2 hessian} {\f2 , \par \par } {\f2\uldb hilbert}{\v\f2 hilbert} {\f2 , \par \par } {\f2\uldb jacobian}{\v\f2 jacobian} {\f2 , \par \par } {\f2\uldb jordan_block}{\v\f2 jordan_block} {\f2 , \par \par } {\f2\uldb make_identity}{\v\f2 make_identity} {\f2 , \par \par } {\f2\uldb random_matrix}{\v\f2 random_matrix} {\f2 , \par \par } {\f2\uldb toeplitz}{\v\f2 toeplitz} {\f2 , \par \par } {\f2\uldb vandermonde}{\v\f2 vandermonde} {\f2 . \par \par High level algorithms \par \par } {\f2\uldb char_poly}{\v\f2 char_poly} {\f2 , \par \par } {\f2\uldb cholesky}{\v\f2 cholesky} {\f2 , \par \par } {\f2\uldb gram_schmidt}{\v\f2 gram_schmidt} {\f2 , \par \par } {\f2\uldb lu_decom}{\v\f2 lu_decom} {\f2 , \par \par } {\f2\uldb pseudo_inverse}{\v\f2 pseudo_inverse} {\f2 , \par \par } {\f2\uldb simplex}{\v\f2 simplex} {\f2 , \par \par } {\f2\uldb svd}{\v\f2 svd} {\f2 . \par \par Normal Forms \par \par There is a separate package, NORMFORM, for computing the following matrix normal forms in REDUCE: \par \par } {\f2\uldb smithex}{\v\f2 Smithex} {\f2 , \par \par } {\f2\uldb smithex_int}{\v\f2 Smithex\_int} {\f2 , \par \par } {\f2\uldb frobenius}{\v\f2 Frobenius} {\f2 , \par \par } {\f2\uldb ratjordan}{\v\f2 Ratjordan} {\f2 , \par \par } {\f2\uldb jordansymbolic}{\v\f2 Jordansymbolic} {\f2 , \par \par } {\f2\uldb jordan}{\v\f2 Jordan} {\f2 . \par \par Predicates \par \par } {\f2\uldb matrixp}{\v\f2 matrixp} {\f2 , \par \par } {\f2\uldb squarep}{\v\f2 squarep} {\f2 , \par \par } {\f2\uldb symmetricp}{\v\f2 symmetricp} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # fast_la} ${\footnote \pard\plain \sl240 \fs20 $ fast_la} +{\footnote \pard\plain \sl240 \fs20 + g38:1179} K{\footnote \pard\plain \sl240 \fs20 K fast_la switch;switch} }{\b\f2 FAST_LA}{\f2 \tab \tab \tab \tab }{\b\f2 switch}{\f2 \par \par \par \par By turning the }{\f3 fast_la} {\f2 switch on, the speed of the following functions will be increased: \par \par } {\f2\uldb add_columns}{\v\f2 add_columns} {\f2 , \par \par } {\f2\uldb add_rows}{\v\f2 add_rows} {\f2 , \par \par } {\f2\uldb augment_columns}{\v\f2 augment_columns} {\f2 , \par \par } {\f2\uldb column_dim}{\v\f2 column_dim} {\f2 , \par \par } {\f2\uldb copy_into}{\v\f2 copy_into} {\f2 , \par \par } {\f2\uldb make_identity}{\v\f2 make_identity} {\f2 , \par \par } {\f2\uldb matrix_augment}{\v\f2 matrix_augment} {\f2 , \par \par } {\f2\uldb matrix_stack}{\v\f2 matrix_stack} {\f2 , \par \par } {\f2\uldb minor}{\v\f2 minor} {\f2 , \par \par } {\f2\uldb mult_columns}{\v\f2 mult_columns} {\f2 , \par \par } {\f2\uldb mult_rows}{\v\f2 mult_rows} {\f2 , \par \par } {\f2\uldb pivot}{\v\f2 pivot} {\f2 , \par \par } {\f2\uldb remove_columns}{\v\f2 remove_columns} {\f2 , \par \par } {\f2\uldb remove_rows}{\v\f2 remove_rows} {\f2 , \par \par } {\f2\uldb rows_pivot}{\v\f2 rows_pivot} {\f2 , \par \par } {\f2\uldb squarep}{\v\f2 squarep} {\f2 , \par \par } {\f2\uldb stack_rows}{\v\f2 stack_rows} {\f2 , \par \par } {\f2\uldb sub_matrix}{\v\f2 sub_matrix} {\f2 , \par \par } {\f2\uldb swap_columns}{\v\f2 swap_columns} {\f2 , \par \par } {\f2\uldb swap_entries}{\v\f2 swap_entries} {\f2 , \par \par } {\f2\uldb swap_rows}{\v\f2 swap_rows} {\f2 , \par \par } {\f2\uldb symmetricp}{\v\f2 symmetricp} {\f2 . \par \par 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. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # add_columns} ${\footnote \pard\plain \sl240 \fs20 $ add_columns} +{\footnote \pard\plain \sl240 \fs20 + g38:1180} K{\footnote \pard\plain \sl240 \fs20 K add_columns operator;operator} }{\b\f2 ADD_COLUMNS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par Add columns, add rows: \par syntax: \par }{\f4 \par \par }{\f3 add_columns} {\f4 (<matrix>,<c1>,<c2>,<expr>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . \par \par <c1>,<c2> :- positive integers. \par \par <expr> :- a scalar expression. \par \par The Operator }{\f3 add_columns} {\f2 replaces column <\meta\{c2>\} of <matrix> by <expr> * column(<matrix>,<c1>) + column(<matrix>,<c2>). \par \par }{\f3 add_rows} {\f2 performs the equivalent task on the rows of <matrix>. \par \par \par examples: \par \pard \tx3420 }{\f4 \par \par add_columns(A,1,2,x); \par \par [1 x + 2 3] \par [ ] \par [4 4*x + 5 6] \par [ ] \par [7 7*x + 8 9] \par \par \par \par add_rows(A,2,3,5); \par \par [1 2 3 ] \par [ ] \par [4 5 6 ] \par [ ] \par [27 33 39] \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 , } {\f2\uldb mult_columns}{\v\f2 mult_columns} {\f2 , } {\f2\uldb mult_rows}{\v\f2 mult_rows} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # add_rows} ${\footnote \pard\plain \sl240 \fs20 $ add_rows} +{\footnote \pard\plain \sl240 \fs20 + g38:1181} K{\footnote \pard\plain \sl240 \fs20 K add_rows operator;operator} }{\b\f2 ADD_ROWS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par see: } {\f2\uldb add_columns}{\v\f2 add_columns} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # add_to_columns} ${\footnote \pard\plain \sl240 \fs20 $ add_to_columns} +{\footnote \pard\plain \sl240 \fs20 + g38:1182} K{\footnote \pard\plain \sl240 \fs20 K add_to_columns operator;operator} }{\b\f2 ADD_TO_COLUMNS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par Add to columns, add to rows: \par \par \par syntax: \par }{\f4 }{\f3 add_to_columns} {\f4 (<matrix>,<column\_list>,<expr>) \par \par }{\f2 \par <matrix> :- a matrix. \par \par <column\_list> :- a positive integer or a list of positive integers. \par \par <expr> :- a scalar expression. \par \par }{\f3 add_to_columns} {\f2 adds <expr> to each column specified in <column\_list> of <matrix>. \par \par }{\f3 add_to_rows} {\f2 performs the equivalent task on the rows of <matrix>. \par \par \par examples: \par \pard \tx3420 }{\f4 \par \par add_to_columns(A,\{1,2\},10); \par \par [11 12 3] \par [ ] \par [14 15 6] \par [ ] \par [17 18 9] \par \par \par \par add_to_rows(A,2,-x) \par \par \par [ 1 2 3 ] \par [ ] \par [ - x + 4 - x + 5 - x + 6] \par [ ] \par [ 7 8 9 ] \par \par \pard \sl240 }{\f2 Related functions: } {\f2\uldb add_columns}{\v\f2 add_columns} {\f2 , } {\f2\uldb add_rows}{\v\f2 add_rows} {\f2 , } {\f2\uldb mult_rows}{\v\f2 mult_rows} {\f2 , } {\f2\uldb mult_columns}{\v\f2 mult_columns} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # add_to_rows} ${\footnote \pard\plain \sl240 \fs20 $ add_to_rows} +{\footnote \pard\plain \sl240 \fs20 + g38:1183} K{\footnote \pard\plain \sl240 \fs20 K add_to_rows operator;operator} }{\b\f2 ADD_TO_ROWS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par see: } {\f2\uldb add_to_columns}{\v\f2 add_to_columns} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # augment_columns} ${\footnote \pard\plain \sl240 \fs20 $ augment_columns} +{\footnote \pard\plain \sl240 \fs20 + g38:1184} K{\footnote \pard\plain \sl240 \fs20 K augment_columns operator;operator} }{\b\f2 AUGMENT_COLUMNS}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par Augment columns, stack rows: \par \par \par syntax: \par }{\f4 }{\f3 augment_columns} {\f4 (<matrix>,<column\_list>) \par \par }{\f2 \par <matrix> :- a matrix. \par \par <column\_list> :- either a positive integer or a list of positive integers. \par \par }{\f3 augment_columns} {\f2 gets hold of the columns of <matrix> specified in }{\f3 column_list} {\f2 and sticks them together. \par \par }{\f3 stack_rows} {\f2 performs the same task on rows of <matrix>. \par \par \par examples: \par \pard \tx3420 }{\f4 \par \par augment_columns(A,\{1,2\}) \par \par \par [1 2] \par [ ] \par [4 5] \par [ ] \par [7 8] \par \par \par \par stack_rows(A,\{1,3\}) \par \par [1 2 3] \par [ ] \par [7 8 9] \par \par \pard \sl240 }{\f2 Related functions: } {\f2\uldb get_columns}{\v\f2 get_columns} {\f2 , } {\f2\uldb get_rows}{\v\f2 get_rows} {\f2 , } {\f2\uldb sub_matrix}{\v\f2 sub_matrix} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # band_matrix} ${\footnote \pard\plain \sl240 \fs20 $ band_matrix} +{\footnote \pard\plain \sl240 \fs20 + g38:1185} K{\footnote \pard\plain \sl240 \fs20 K band_matrix operator;operator} }{\b\f2 BAND_MATRIX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 band_matrix} {\f4 (<expr\_list>,<square\_size>) \par \par }{\f2 \par <expr\_list> :- either a single scalar expression or a list of an odd number of scalar expressions. \par \par <square\_size> :- a positive integer. \par \par }{\f3 band_matrix} {\f2 creates 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. \par \par \par examples: \par \pard \tx3420 }{\f4 \par \par band_matrix(\{x,y,z\},6) \par \par [y z 0 0 0 0] \par [ ] \par [x y z 0 0 0] \par [ ] \par [0 x y z 0 0] \par [ ] \par [0 0 x y z 0] \par [ ] \par [0 0 0 x y z] \par [ ] \par [0 0 0 0 x y] \par \par \pard \sl240 }{\f2 Related functions: } {\f2\uldb diagonal}{\v\f2 diagonal} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # block_matrix} ${\footnote \pard\plain \sl240 \fs20 $ block_matrix} +{\footnote \pard\plain \sl240 \fs20 + g38:1186} K{\footnote \pard\plain \sl240 \fs20 K block_matrix operator;operator} }{\b\f2 BLOCK_MATRIX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 block_matrix} {\f4 (<r>,<c>,<matrix\_list>) \par \par }{\f2 \par <r>,<c> :- positive integers. \par \par <matrix\_list> :- a list of matrices. \par \par }{\f3 block_matrix} {\f2 creates a matrix that consists of <r> by <c> matrices filled from the <matrix\_list> row wise. \par \par \par examples: \par \pard \tx3420 }{\f4 \par B := make_identity(2); \par \par [1 0] \par b := [ ] \par [0 1] \par \par \par \par C := mat((5),(5)); \par \par [5] \par c := [ ] \par [5] \par \par \par \par D := mat((22,33),(44,55)); \par \par [22 33] \par d := [ ] \par [44 55] \par \par \par \par block_matrix(2,3,\{B,C,D,D,C,B\}); \par \par \par [1 0 5 22 33] \par [ ] \par [0 1 5 44 55] \par [ ] \par [22 33 5 1 0 ] \par [ ] \par [44 55 5 0 1 ] \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # char_matrix} ${\footnote \pard\plain \sl240 \fs20 $ char_matrix} +{\footnote \pard\plain \sl240 \fs20 + g38:1187} K{\footnote \pard\plain \sl240 \fs20 K char_matrix operator;operator} }{\b\f2 CHAR_MATRIX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 char_matrix} {\f4 (<matrix>,<lambda>) \par \par }{\f2 \par <matrix> :- a square matrix. <lambda> :- a symbol or algebraic expression. \par \par <char\_matrix> creates the characteristic matrix C of <matrix>. \par \par This is C = <lambda> * Id - A. Id is the identity matrix. \par \par \par examples: \par \pard \tx3420 }{\f4 \par \par char_matrix(A,x); \par \par [x - 1 -2 -3 ] \par [ ] \par [ -4 x - 5 -6 ] \par [ ] \par [ -7 -8 x - 9] \par \par \pard \sl240 }{\f2 Related functions: } {\f2\uldb char_poly}{\v\f2 char_poly} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # char_poly} ${\footnote \pard\plain \sl240 \fs20 $ char_poly} +{\footnote \pard\plain \sl240 \fs20 + g38:1188} K{\footnote \pard\plain \sl240 \fs20 K char_poly operator;operator} }{\b\f2 CHAR_POLY}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 char_poly} {\f4 (<matrix>,<lambda>) \par \par }{\f2 \par <matrix> :- a square matrix. \par \par <lambda> :- a symbol or algebraic expression. \par \par }{\f3 char_poly} {\f2 finds the characteristic polynomial of <matrix>. This is the determinant of <lambda> * Id - A. Id is the identity matrix. \par \par \par examples: \par \pard \tx3420 }{\f4 \par char_poly(A,x); \par \par 3 2 \par x -15*x -18*x \par \par \pard \sl240 }{\f2 Related functions: } {\f2\uldb char_matrix}{\v\f2 char_matrix} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # cholesky} ${\footnote \pard\plain \sl240 \fs20 $ cholesky} +{\footnote \pard\plain \sl240 \fs20 + g38:1189} K{\footnote \pard\plain \sl240 \fs20 K cholesky operator;operator} }{\b\f2 CHOLESKY}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 cholesky} {\f4 (<matrix>) \par \par }{\f2 \par <matrix> :- a positive definite matrix containing numeric entries. \par \par }{\f3 cholesky} {\f2 computes the cholesky decomposition of <matrix>. \par \par It returns \{L,U\} where L is a lower matrix, U is an upper matrix, A = LU, and U = }{\f4 L^T}{\f2 . \par \par \par examples: \par \pard \tx3420 }{\f4 \par F := mat((1,1,0),(1,3,1),(0,1,1)); \par \par \par [1 1 0] \par [ ] \par f := [1 3 1] \par [ ] \par [0 1 1] \par \par \par \par on rounded; \par \par cholesky(F); \par \par \{ \par [1 0 0 ] \par [ ] \par [1 1.41421356237 0 ] \par [ ] \par [0 0.707106781187 0.707106781187] \par , \par [1 1 0 ] \par [ ] \par [0 1.41421356237 0.707106781187] \par [ ] \par [0 0 0.707106781187] \par \} \par \par \pard \sl240 }{\f2 Related functions: } {\f2\uldb lu_decom}{\v\f2 lu_decom} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # coeff_matrix} ${\footnote \pard\plain \sl240 \fs20 $ coeff_matrix} +{\footnote \pard\plain \sl240 \fs20 + g38:1190} K{\footnote \pard\plain \sl240 \fs20 K coeff_matrix operator;operator} }{\b\f2 COEFF_MATRIX}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 coeff_matrix} {\f4 (\{<lineq\_list>\}) \par \par }{\f2 \par (If you are feeling lazy then the braces can be omitted.) \par \par <lineq\_list> :- linear equations. Can be of the form equation = number or just equation. \par \par }{\f3 coeff_matrix} {\f2 creates the coefficient matrix C of the linear equations. \par \par It returns \{C,X,B\} such that CX = B. \par \par \par examples: \par \pard \tx3420 }{\f4 \par \par coeff_matrix(\{x+y+4*z=10,y+x-z=20,x+y+4\}); \par \par \par \{ \par [4 1 1] \par [ ] \par [-1 1 1] \par [ ] \par [0 1 1] \par , \par [z] \par [ ] \par [y] \par [ ] \par [x] \par , \par [10] \par [ ] \par [20] \par [ ] \par [-4] \par \} \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # column_dim} ${\footnote \pard\plain \sl240 \fs20 $ column_dim} +{\footnote \pard\plain \sl240 \fs20 + g38:1191} K{\footnote \pard\plain \sl240 \fs20 K column_dim operator;operator} }{\b\f2 COLUMN_DIM}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par Column dimension, row dimension: \par \par \par syntax: \par }{\f4 }{\f3 column_dim} {\f4 (<matrix>) \par \par }{\f2 \par <matrix> :- a matrix. \par \par }{\f3 column_dim} {\f2 finds the column dimension of <matrix>. \par \par }{\f3 row_dim} {\f2 finds the row dimension of <matrix>. \par \par \par examples: \par \pard \tx3420 }{\f4 \par \par column_dim(A); \par \par 3 \par \par \par row_dim(A); \par \par 3 \par \par \pard \sl240 }{\f2 \page #{\footnote \pard\plain \sl240 \fs20 # companion} ${\footnote \pard\plain \sl240 \fs20 $ companion} +{\footnote \pard\plain \sl240 \fs20 + g38:1192} K{\footnote \pard\plain \sl240 \fs20 K companion operator;operator} }{\b\f2 COMPANION}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 companion} {\f4 (<poly>,<x>) \par \par }{\f2 \par <poly> :- a monic univariate polynomial in <x>. \par \par <x> :- the variable. \par \par }{\f3 companion} {\f2 creates the companion matrix C of <poly>. \par \par This is the square matrix of dimension n, where n is the degree of <poly> w.r.t. <x>. \par \par The entries of C are: \par \par 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. \par \par \par examples: \par \pard \tx3420 }{\f4 \par \par 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 [ ] \par [0 1 0 9 ] \par [ ] \par [0 0 1 -17] \par \par \pard \sl240 }{\f2 Related functions: } {\f2\uldb find_companion}{\v\f2 find_companion} {\f2 . \par \par \page #{\footnote \pard\plain \sl240 \fs20 # copy_into} ${\footnote \pard\plain \sl240 \fs20 $ copy_into} +{\footnote \pard\plain \sl240 \fs20 + g38:1193} K{\footnote \pard\plain \sl240 \fs20 K copy_into operator;operator} }{\b\f2 COPY_INTO}{\f2 \tab \tab \tab \tab }{\b\f2 operator}{\f2 \par \par \par \par \par syntax: \par }{\f4 }{\f3 copy_into} {\f4 (<A>,<B>,<r>,<c>) \par \par }{\f2 \par <A>,<B> :- matrices. \par \par <r>,<c> :- positive integers. \par \par }{\f3 copy_into} {\f2 copies matrix <matrix> into <B> with <matrix>(1,1) at <B>(<r>,<c>). \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 + g38:1194} 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 (\{<mat\_list>\}) \par \par }{\f2 \par (If you are feeling lazy then the braces can be omitted.) \par \par <mat\_list> :- 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 + g38:1195} 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 (<matrix>,<r>,<c>,<expr>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . \par \par <r>,<c> :- positive integers. \par \par <expr> :- algebraic expression or symbol. \par \par }{\f3 extend} {\f2 returns a copy of <matrix> that has been extended by <r> rows and <c> columns. The new entries are made equal to <expr>. \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 + g38:1196} 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 (<matrix>,<x>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . \par \par <x> :- 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 + g38:1197} 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 (<matrix>,<column\_list>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . \par \par <c> :- either a positive integer or a list of positive integers. \par \par }{\f3 get_columns} {\f2 removes the columns of <matrix> specified in <column\_list> and returns them as a list of column matrices. \par \par }{\f3 get_rows} {\f2 performs the same task on the rows of <matrix>. \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 + g38:1198} 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 + g38:1199} 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 (\{<vec\_list>\}) \par \par }{\f2 \par (If you are feeling lazy then the braces can be omitted.) \par \par <vec\_list> :- 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 + g38:1200} 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 (<matrix>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . \par \par }{\f3 hermitian_tp} {\f2 computes the hermitian transpose of <matrix>. \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 <matrix>. \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 + g38:1201} 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 (<expr>,<variable\_list>) \par \par }{\f2 \par <expr> :- a scalar expression. \par \par <variable\_list> :- either a single variable or a list of variables. \par \par }{\f3 hessian} {\f2 computes the hessian matrix of <expr> w.r.t. the variables in <variable\_list>. \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 (<expr>,<variable\_list>(i), <variable\_list>(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 + g38:1202} 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 (<square\_size>,<expr>) \par \par }{\f2 \par <square\_size> :- a positive integer. \par \par <expr> :- an algebraic expression. \par \par }{\f3 hilbert} {\f2 computes the square hilbert matrix of dimension <square\_size>. \par \par This is the symmetric matrix in which the (i,j)'th entry is 1/(i+j-<expr>). \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 + g38:1203} 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 (<expr\_list>,<variable\_list>) \par \par }{\f2 \par <expr\_list> :- either a single algebraic expression or a list of algebraic expressions. \par \par <variable\_list> :- either a single variable or a list of variables. \par \par }{\f3 jacobian} {\f2 computes the jacobian matrix of <expr\_list> w.r.t. <variable\_list>. \par \par This is a matrix whose (i,j)'th entry is } {\f2\uldb df}{\v\f2 DF} {\f2 (<expr\_list> (i),<variable\_list>(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 + g38:1204} 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 (<expr>,<square\_size>) \par \par }{\f2 \par <expr> :- an algebraic expression or symbol. \par \par <square\_size> :- a positive integer. \par \par }{\f3 jordan_block} {\f2 computes the square jordan block matrix J of dimension <square\_size>. \par \par The entries of J are: \par \par J(i,i) = <expr> 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 + g38:1205} 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 (<matrix>) \par \par }{\f2 \par <matrix> :- 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 <matrix>, 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 <matrix> during the calculation. This means that LU does not equal <matrix> 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(<matrix>,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 + g38:1206} 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 (<square\_size>) \par \par }{\f2 \par <square\_size> :- a positive integer. \par \par }{\f3 make_identity} {\f2 creates the identity matrix of dimension <square\_size>. \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 + g38:1207} 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 \{<matrix\_list>\} \par \par }{\f2 \par (If you are feeling lazy then the braces can be omitted.) \par \par <matrix\_list> :- matrices. \par \par }{\f3 matrix_augment} {\f2 sticks the matrices in <matrix\_list> together horizontally. \par \par }{\f3 matrix_stack} {\f2 sticks the matrices in <matrix\_list> 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 + g38:1208} 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 (<test\_input>) \par \par }{\f2 \par <test\_input> :- 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 + g38:1209} 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 + g38:1210} 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 (<matrix>,<r>,<c>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . <r>,<c> :- positive integers. \par \par }{\f3 minor} {\f2 computes the (<r>,<c>)'th minor of <matrix>. This is created by removing the <r>'th row and the <c>'th column from <matrix>. \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 + g38:1211} 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 (<matrix>,<column\_list>,<expr>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . \par \par <column\_list> :- a positive integer or a list of positive integers. \par \par <expr> :- an algebraic expression. \par \par }{\f3 mult_columns} {\f2 returns a copy of <matrix> in which the columns specified in <column\_list> have been multiplied by <expr>. \par \par }{\f3 mult_rows} {\f2 performs the same task on the rows of <matrix>. \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 + g38:1212} 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 + g38:1213} 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 (<matrix>,<r>,<c>) \par \par }{\f2 \par <matrix> :- a matrix. \par \par <r>,<c> :- positive integers such that <matrix>(<r>, <c>) neq 0. \par \par }{\f3 pivot} {\f2 pivots <matrix> about it's (<r>,<c>)'th entry. \par \par To do this, multiples of the <r>'th row are added to every other row in the matrix. \par \par This means that the <c>'th column will be 0 except for the (<r>,<c>)'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 + g38:1214} 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 (<matrix>) \par \par }{\f2 \par <matrix> :- 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 <matrix>. \par \par Given the singular value decomposition of <matrix>, 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 <matrix> * 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 + g38:1215} 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 (<r>,<c>,<limit>) \par \par }{\f2 \par <r>,<c>,<limit> :- positive integers. \par \par }{\f3 random_matrix} {\f2 creates an <r> by <c> 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 < <limit>. \par \par }{\f3 not_negative} {\f2 :- if on then 0 < entry < <limit>. In the imaginary case we have 0 < x,y < <limit>. \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 + g38:1216} 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 (<matrix>,<column\_list>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . <column\_list> :- either a positive integer or a list of positive integers. \par \par }{\f3 remove_columns} {\f2 removes the columns specified in <column\_list> from <matrix>. \par \par }{\f3 remove_rows} {\f2 performs the same task on the rows of <matrix>. \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 + g38:1217} 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 + g38:1218} 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 + g38:1219} 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 (<matrix>,<r>,<c>,\{<row\_list>\}) \par \par }{\f2 \par <matrix> :- a namerefmatrix. \par \par <r>,<c> :- positive integers such that <matrix>(<r>, <c>) neq 0. \par \par <row\_list> :- 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 <row\_list>. \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 + g38:1220} 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 (<max/min>,<objective function>, \{<linear inequalities>\}) \par \par }{\f2 \par <max/min> :- either max or min (signifying maximize and minimize). \par \par <objective function> :- the function you are maximizing or minimizing. \par \par <linear inequalities> :- 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 <objective function> 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 + g38:1221} 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 (<matrix>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . \par \par }{\f3 squarep} {\f2 is a predicate that returns t if the <matrix> 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 + g38:1222} 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 + g38:1223} 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 (<matrix>,<row\_list>,<column\_list>) \par \par }{\f2 \par <matrix> :- a matrix. <row\_list>, <column\_list> :- 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 <row\_list> and the columns specified in <column\_list>. \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 + g38:1224} 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 (<matrix>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 containing only numeric entries. \par \par }{\f3 svd} {\f2 computes the singular value decomposition of <matrix>. \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 <matrix>. \par \par n is the column dimension of <matrix>. \par \par The singular values of <matrix> 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 + g38:1225} 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 (<matrix>,<c1>,<c2>) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . \par \par <c1>,<c1> :- positive integers. \par \par }{\f3 swap_columns} {\f2 swaps column <c1> of <matrix> with column <c2>. \par \par }{\f3 swap_rows} {\f2 performs the same task on two rows of <matrix>. \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 + g38:1226} 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 (<matrix>,\{<r1>,<c1>\},\{<r2>, <c2>\}) \par \par }{\f2 \par <matrix> :- a } {\f2\uldb matrix}{\v\f2 MATRIX} {\f2 . \par \par <r1>,<c1>,<r2>,<c2> :- positive integers. \par \par }{\f3 swap_entries} {\f2 swaps <matrix>(<r1>,<c1>) with <matrix>(<r2>,<c2>). \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 + g38:1227} 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 + g38:1228} 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 (<matrix>) \par \par }{\f2 \par <matrix> :- 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 + g38:1229} 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 (<expr\_list>) \par \par }{\f2 \par (If you are feeling lazy then the braces can be omitted.) \par \par <expr\_list> :- list of algebraic expressions. \par \par }{\f3 toeplitz} {\f2 creates the toeplitz matrix from the <expr\_list>. \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 + g38:1230} 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 (\{<expr\_list>\}) \par \par }{\f2 \par (If you are feeling lazy then the braces can be omitted.) \par \par <expr\_list> :- list of algebraic expressions. \par \par }{\f3 vandermonde} {\f2 creates the vandermonde matrix from the <expr\_list>. \par \par This is the square matrix in which the (i,j)'th entry is <expr\_list>}{\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 + g39:1231} 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 (<matrix>,<variable>) \par \par <matrix> :- a rectangular } {\f2\uldb matrix}{\v\f2 MATRIX} {\f4 of univariate polynomials in <variable>. <variable> :- 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 + g39:1232} 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 (<matrix>) \par \par <matrix> :- 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 + g39:1233} 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 (<matrix>) \par \par <matrix> :- 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 + g39:1234} 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 (<matrix>) \par \par <matrix> :- 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 + g39:1235} 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 (<matrix>) \par \par <matrix> :- 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 + g39:1236} 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 (<matrix>) \par \par <matrix> :- 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 + g40:1237} 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 + g40:1238} 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 + g40:1239} 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 + g40:1240} 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 + g40:1241} 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 + g40:1242} 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 + g40:1243} 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 + g40:1244} 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 + g40:1245} 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 package 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 + g40:1246} 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 + g40:1247} 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 # CRACK} ${\footnote \pard\plain \sl240 \fs20 $ CRACK} +{\footnote \pard\plain \sl240 \fs20 + g40:1248} 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 + g40:1249} 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 + g40:1250} 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 + g40:1251} 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 + g40:1252} 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 + g40:1253} 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 + g40:1254} 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 + g40:1255} 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 + g40:1256} 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 + g40:1257} 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 + g40:1258} 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 + g40:1259} 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 attempts to solve 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 + g40:1260} 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 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. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # LAPLACE} ${\footnote \pard\plain \sl240 \fs20 $ LAPLACE} +{\footnote \pard\plain \sl240 \fs20 + g40:1261} 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. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # LIE} ${\footnote \pard\plain \sl240 \fs20 $ LIE} +{\footnote \pard\plain \sl240 \fs20 + g40:1262} 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 + g40:1263} 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 + g40:1264} 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 + g40:1265} 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 + g40:1266} 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 + g40:1267} 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 + g40:1268} 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 + g40:1269} 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 + g40:1270} 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 + g40:1271} 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 + g40:1272} 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 + g40:1273} 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 + g40:1274} 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 + g40:1275} 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 + g40:1276} 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 + g40:1277} 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 + g40:1278} 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 + g40:1279} 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 + g40:1280} 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 + g40:1281} 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 + g40:1282} 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 + g40:1283} 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 + g40:1284} 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 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 + g41:1285} 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 <integer> or }{\f3 ed} {\f4 \par \par }{\f2 \par }{\f3 ed} {\f2 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 }{\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<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. \par \par \tab e End the current session, causing the edited expression to be reparsed by REDUCE. \par \par \tab 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. \par \par \tab i<string>}{\f3 ESC} {\f2 Insert <string> 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<char> Replace the character at the pointer by <char>. \par \par \tab s<string>}{\f3 ESC} {\f2 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 }{\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 <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. \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). 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. \par \par 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 }{\f3 ( ) # ; ' `} {\f2 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. \par \par 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. \par \par Since the editor has no dependence on any window system, it can be used if you are running REDUCE without windows. \par \par \par \page #{\footnote \pard\plain \sl240 \fs20 # EDITDEF} ${\footnote \pard\plain \sl240 \fs20 $ EDITDEF} +{\footnote \pard\plain \sl240 \fs20 + g41:1286} K{\footnote \pard\plain \sl240 \fs20 K Top;Outmoded Operations;EDITDEF command;command} }{\b\f2 EDITDEF}{\f2 \tab \tab \tab \tab }{\b\f2 command}{\f2 \par \par \par \par The interactive editor } {\f2\uldb ed}{\v\f2 ED} {\f2 may be used to edit a user-defined procedure that has not been compiled. \par syntax: \par }{\f4 \par \par }{\f3 editdef} {\f4 (}{\f3 identifier} {\f4 ) \par \par }{\f2 \par where }{\f3 identifier} {\f2 is the name of the procedure. When }{\f3 editdef} {\f2 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 } {\f2\uldb ed}{\v\f2 ED} {\f2 commands. \par \par \page #{\footnote \pard\plain \sl240 \fs20 # g41} ${\footnote \pard\plain \sl240 \fs20 $ Outmoded Operations} +{\footnote \pard\plain \sl240 \fs20 + index:0041} }{\b\f2 Outmoded Operations}{\f2 \par }\pard \sl240 {\f2 \par } {\f2 \tab}{\f2\uldb ED command} {\v\f2 ED}{\f2 \par }{\f2 \tab}{\f2\uldb EDITDEF command} {\v\f2 EDITDEF}{\f2 \par \page #{\footnote \pard\plain \sl240 \fs20 # main_index} ${\footnote \pard\plain \sl240 \fs20 $ Top} +{\footnote \pard\plain \sl240 \fs20 + index:0001} }{\b\f2 Top}{\f2 \par }\pard \sl240 {\f2 \par } {\f2 \tab}{\f2\uldb Concepts} {\v\f2 g2}{\f2 \par }{\f2 \tab}{\f2\uldb Variables} {\v\f2 g3}{\f2 \par }{\f2 \tab}{\f2\uldb Syntax} {\v\f2 g4}{\f2 \par }{\f2 \tab}{\f2\uldb Arithmetic Operations} {\v\f2 g5}{\f2 \par }{\f2 \tab}{\f2\uldb Boolean Operators} {\v\f2 g6}{\f2 \par }{\f2 \tab}{\f2\uldb General Commands} {\v\f2 g7}{\f2 \par }{\f2 \tab}{\f2\uldb Algebraic Operators} {\v\f2 g8}{\f2 \par }{\f2 \tab}{\f2\uldb Declarations} {\v\f2 g9}{\f2 \par }{\f2 \tab}{\f2\uldb Input and Output} {\v\f2 g10}{\f2 \par }{\f2 \tab}{\f2\uldb Elementary Functions} {\v\f2 g11}{\f2 \par }{\f2 \tab}{\f2\uldb General Switches} {\v\f2 g12}{\f2 \par }{\f2 \tab}{\f2\uldb Matrix Operations} {\v\f2 g13}{\f2 \par }{\f2 \tab}{\f2\uldb Groebner package} {\v\f2 g14}{\f2 \par }{\f2 \tab}{\f2\uldb High Energy Physics} {\v\f2 g21}{\f2 \par }{\f2 \tab}{\f2\uldb Numeric Package} {\v\f2 g22}{\f2 \par }{\f2 \tab}{\f2\uldb Roots Package} {\v\f2 g23}{\f2 \par }{\f2 \tab}{\f2\uldb Special Functions} {\v\f2 g24}{\f2 \par }{\f2 \tab}{\f2\uldb Taylor series} {\v\f2 g36}{\f2 \par }{\f2 \tab}{\f2\uldb Gnuplot package} {\v\f2 g37}{\f2 \par }{\f2 \tab}{\f2\uldb Linear Algebra package} {\v\f2 g38}{\f2 \par }{\f2 \tab}{\f2\uldb Matrix Normal Forms} {\v\f2 g39}{\f2 \par }{\f2 \tab}{\f2\uldb Miscellaneous Packages} {\v\f2 g40}{\f2 \par }{\f2 \tab}{\f2\uldb Outmoded Operations} {\v\f2 g41}{\f2 \par \page }}