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"false"; "no"; "small"; "yes" (<argument>) (<coeffs>,<var>=(<lo> .. <hi>) (<exp>, (<expr>,<k>[,<lolim> [,<uplim> ]]) (<expr>,<var>,<limpoint>) or (<expression (<expression>) (<expression>) or (<expression>,<expression> (<expression>,<expression>) or <expression> (<expression>,<integer>) (<expression>,<kernel> (<integer>) (<integer>,<expression>) (<integer>,<expression>,<expression> (<integer>,<integer> (<list of j1,m1>,<list of j2,m2>, <list of j3,m3>) (<list of parameters>,<list of parameters>, <argument>) (<list>) or (<matrix>) (<matrix>,<column (<matrix>,<r>,<c> (<order>,<argument>) (<parameter>,<parameter>,<argument>) (A say). It returns (If you are feeling lazy then the braces can be omitted.) (LIST (a + b*i)*(c + d*i); (default (e.g. (i,j)'th (or other variable declaration such as (see (which (x + y)**3; (x+1)/x + x**2/sin y; ). A numeric value is not returned by ); ***** A invalid as integer ***** Matrix ***** X invalid as KERNEL , , and , its arguments are reordered to conform to the internal ordering of the system. , or , 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, , the , the printing behavior is as if , there are no algebraic rules associated with it; it will only evaluate when , where , which would be right associative. , without a semicolon between ,3*X + 7*X + 3,2*COS(2*X)} ,3x**2+7x+3,df(sin(2*x),x) - ------- ------- --------- ----------- ------------- ---------------- --------------------------------- . . Default is . For non-numeric arguments, the value is an expression in the original operator. . If . In addition, if an operator has more than one argument, no such distribution occurs, so . The . This is done by the commands . You may wish to add a check to see that its argument is properly restricted. ... .......... 0 0.......0 0.00000000000000012 0.70710678118 1 1 2 3 10*(Y + 12*Y + 54*Y + 108*Y + 81), 2 2 2 2 2 2 2 2 2 2 2.2798561599 3 3 2 4 3 2 4 3 3 4 4 3 2 2 3 4 4 3 2 5*(Y + 12*Y + 54*Y + 108*Y + 81), :- if on then ; <<a := 23;b := 2*a;c := 4**2;m <action> <column <condition> <exp> <expr <expr> <expr> :- a scalar expression. <expr> :- an algebraic expression. <expression <expression> <expression> can be any <expression> can be any REDUCE scalar expression. <expression> can be any valid REDUCE <expression> can be any valid REDUCE expression that evaluates to a number. <expression> can be any valid scalar REDUCE expression <expression> is any valid scalar REDUCE expression <expression> is expected to be a polynomial expression, not a rational expression. Rational expressions are accepted when the switch <expression> is ordinarily a polynomial. If <expression> m <expression> may be any scalar REDUCE expression, not an array, matrix or vector expression. <simple <expression> may be any scalar REDUCE expression. <expression> may be any valid REDUCE expression. <expression> must evaluate to a <expression> must evaluate to a number, e.g., integer, rational or floating point number. <filename> <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, <identifier> <identifier> can be any valid REDUCE identifier. <identifier> may be any valid REDUCE identifier <identifier> must <integer> <item> <kernel> <kernel> must be a <line-id> <list> <logical expression> <matrix <matrix> <matrix> :- a <matrix> :- a matrix. <matrix> :- a square <matrix>. <name> <number> <object> <operator> <r>,<c> :- positive integers such that <matrix>(<r>, <c>) neq 0. <r>,<c> :- positive integers. <restricted <simple <simple_expression> <square <statement> <string> <substitution> <var> <var>[=<val>] <variable <vector-var> = <string> A := A := 10 A numeric value is not returned by A variable preceded by a A(1,1) := A(2,1) := A(2,2) := A*C + A*D*I + B*C*I - B*D A*D - B*C AGM_function ALIST := ARBCOMPLEX ASIN( ATAN After Airy_Ai Airy_Bi Algebra Alternatively An explicit numeric value is not given unless the switch Any variables used inside Arithmetic Array Assign 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 n Author: Author: Eberhard Schruefer Author: Francis J. Wright Author: Herbert Melenk Authors: BERNOULLI BESSEL BLIST := Basic Groebner operators Be careful Bernoulli BernoulliP Bessel Binomial Buchberger By default, calculations are performed in the rational numbers. To extend this field the CE concept of idempotent evaluation. As an alternative, REDUCE provides two switches COS(A - B) + COS(A + B) COS(Y)*DF(Y,X Chebyshev Clebsch_Gordan Combinatorial Operators Comparison operators can only be used as conditions in conditional commands such as Computation Computing with distributive polynomials Cont? (Y or N) Declaration Declare Default ELLIPTIC Elliptic Error Euler Evaluates Expressions Extensions FACTOR Factorizing Groebner bases Field Extensions: For Fortran Fortran-compatible Fresnel Function Functions GNUPLOT GROEBNER Gamma and Related Functions Gegenbauer Groebner Groebner Bases for Modules HUGO := 1 + X + -*X + O(X ) Hankel Heaviside Herbert HermiteP However, Hypergeometric IDENT If If the If you 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 REDU Integral Integrals It has dimension n where n is the number of expressions. It returns JACOBI Jacobi Jacobi's Elliptic Functions and Elliptic Integrals JacobiAmplitude Jacobian Jacobic Jacobid Jacobin Jacobis Kredel-Weispfenning LaguerreP Legendre Linear Algebra package Lisp Logical operators can only be used in conditional expressions Logical operators can only be used in conditional statements such as Logical operators can only be used inside conditional MULTIPLICITIES Matrices Matrix Meijer Melenk Miscellaneous Modular Arithmetic: Modular operations are No simplification is done for this function. Note Numeric Package Numerical ORDER Operations Operator Otherwise PACKAGE PROCEDURE Package Pochhammer Polygamma Polynomial Procedure REDUCE REDUCE's REDUCE, REDUCE. RES := 0.785398163397 ROOT ROUND Related functions: Result Roots Package SIN(X) SIN(Y SQRT( SQRT(24*Y + 60*Y + 25) SQRT(3) STIRLING Scalar Since SixjSymbol SolidHarmonicY Special Function Package SphericalHarmonicY Stirling TAYLOR TERM Taylor The 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 The command The declaration The exponentiation operator is left associative, so that The first The following The functions The integer value of the global variable The minimal accuracy of the result values is controlled by The numeric values for the operator The operator The s The te The variable Then There These This ThreejSymbol Time: Time: 1 Time: 80 ms To change the Under normal circumstances When When an operator is declared When the When the operator When the switch WhittakerW With X X + X + X + X + X + 1 X + X + 1 X - 2*X *Y + 2*X*Y - Y X - X + X - X + X X=2*ARBINT(1)*PI + ASIN(1) + PI - 3} You You can write a procedure to attach integrals or other functions to You may attach functionality by defining You may attach further functionality by defining Zeta [ [ [ [ [ [ [ [ [ ] [ ] [ ] [ ] [ ] [ ] [ 2 2 ] [ ] [ 2 ] [ - 1 - 1 - 1 ] [ ] [ 25*y - 2*y + 1 2*(25*y - 2*y + 1) ] [ 1 i 0 ] [,accuracy=<a>][,iterations=<i>]) [----------- ----------- -----------] [---------------------- ---------------------------------] [0 [0 [0 0 0 [0 0 0 0 0] [0 1 [1 [1 2 3] [4 5 6] [7 8 9] [i + 1 i + 2 i + 3] a := a := 10; a := 15; a := [ about above above. absolute access according accuracy acosh acsc action activate actual add_columns add_rows add_to_columns add_to_rows added addition additional affect after again algebra algebraic algint algorithm algorithms alist := all allbranch allowed allows alphabetical already also alternative always ambiguity and and some simple transformations are known to the system. and stands for an arbitrary part and the and the first variable declaration. and the number zero are another answer antisymmetric any anywhere appear appear on the left-hand side, all but the first term of the expression is moved to the right-hand side. appearing appears on the left-hand side, any constant terms are moved to the right-hand side, but the symbolic factors remain. append application applied applies apply appropriate approximate approximation arbcomplex arbitrary arbvars arccosecant are are computed via the power series representation, which limits the argument range. are global, and any change made to them inside the block affects their global value. A arglength argument argument, argument. arguments arguments, arguments. arithmetic around array array, asech asinh assign assigned assignment associated associative assume assumption asymptotic attach attached attempt augment_columns automatically auxiliary available avoid axes names b := balanced_mod band_matrix based bases basic basis because become been before begin begin scalar begin scalar b; b := a + 10; WRITE A end; beginning begins behavior being between binary blist := block block. When the block is finished, the variables are removed. You may use the words block_matrix blocks that were not declared boolean boolean value both bound bounds bounds(sin x,x=(1 .. 2)); braces bracket branch but calculated calculates calculation calculations calculations. call called calling calls can 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 representat can also be used can be can be turned on. can be used cannot canonical careful case case, case. cases cases, causes certain change changed changes char_matrix char_poly character characteristic characters chebyshev_ check cholesky class clear cleared clearrules closed coeff coeff_matrix coefficient coefficients coeffn cofactor collect collected collection column column_dim columns, combination combine command command changes REDUCE's mode of operation to symbolic. When 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 command, command, group or block each iteration of the loop, command. commands comment common commutat companion comparison compiled complete complex complicated component computation compute computed computes computes the computes the Jordan normal form J of a computing concept condition conditional conditions conjugate connect cons considered consist constant constraint construc construction constructs contain containing contains contents continue continued_fraction continues contour contraction control controlled controls convert converted coordinate copy_into correct corresponding cos(a)*cos(b); cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, cos(~x)^2 => (1+cos(2*x))/2, coshval(1); could count cramer create cross-referenc current currently curve decimal declaration declaration must be made immediately after a declare declared declared inside a block is always global. declaring decompose decomposition default define defined defines defining definite definition defint( defpoly degree degrees, delimiter denominator denote depend dependencies dependency dependent depending derivative describe desired detail determin determinant df(sin(7*x + y),x); diagonal difference different differential differentiat differentiation dilog dimension dimensions direction directly directory display distribution divided division divisor documentation does does not indicate typechecking by the current REDUCE; it is only for your own information. Declaration statements must immediately follow the doing dollar domain done double during e.g. each echo editing editor effect effect. effects. efficient eigenvector either element elements eliminat else else for i := 0:n-1 product i+1; empty enclosed end end; entered entries entry equal equation equations equations. equivalent error escape establish evallhseqp evaluate evaluated evaluates evaluation even evenp eventual every exactly example example, example. examples: except exclamation execute executed executing execution existing expand_cases expanded expanding expandlogs expansion expect expensive explicit explicitly exponential exponentiation exponents expresse expression expression, expression. expression> expression> is a single identifier or begins with a prefix operator name. expression> must expression> must be a single identifier or begin with a prefix operator name. expression> must be a single identifier or begin with a prefix operator. expressions expressions. expt extend extended extract facilitate facilities factor factored factorial factoring factorization factorize factorize(4*x**2 + 28*x + 48); factorize(a**2 + b**2); factorize(x^15 - 1); factors false faster feeling field file file, file. files find find_companion finding finds first first lu * second lu; floating floating-point followed following for for all for an example. for each forall form form, form. formal format forms formula found four-vectors fraction free free variable freeof frobenius from full fullroots function function. functionality functions functions, functions. functions: further g := (exp(a) + exp(-a))/2; g(line1,a1,b1)*g(line2,b1,c1); gamma garbage general generally generat generate generated get_columns get_rows given given, given. global globally gltbasis gradlex term order gradlexgradlex term order gradlexrevgradlex term order gram_schmidt graph greater greatest groebfullreduction groebmonfac groebner groebnerf groebnert groebopt groebprereduce groebprot groebrestriction groebstat group gvarslast handle has has a limited numeric evaluation of large values of its argument. has been has no effect. have hermitian hessian high-energy higher highest hilbert homogeneous hugo := taylor(exp(x),x,0,2); hyperbolic hypergeometric ideal ideal parameters idealquotient identical identifier identifier, identifier, then <restricted identifier. identifiers identity if if fixp(a) then write "ok" else write "not"; if not (fixp(n) and n>=0) if not numberp(a) then write " if numberp a and a < 15 then write a**2 else write "no"; if numberp x and x<20 then y := sqrt(x) else write "illegal"; if you wish. Numerical values of expressions involving ifactor ignored imaginary immediately impart implementation implements implicit important in applications different from classical linear algebra. in the place of include included includes including increase indefinite independent indeterminate index indicate indicating indices individual inequalities infinite infinity infix information inhibit initial initialize initially input inside instead int integer integer, integer. integers integers. integra integral integration interactive intermediate internal interpolation interpret interval into introduction invalid inverse inversion invertible involving is is a is a list of expressions or equations. is a synonym for is also called Bessel function of the third kind. There is currently no numeric evaluation of Hankel functions. is also declared non zero by the declaration is equivalent to is evaluated for each element of <object> where the element is substituted for is known to the system. Numerical values may also be found by turning on the switch is member of the options for a 3d is off is off, regardless of the setting of is off. is on is on and its argument evaluates to a number. is on and the argument has an absolute numeric value less than or equal to 1. is on, is on, a is on, a rational expression may also be used, otherwise an error results. <kernel> must be a is on, expressions in the differentiation operator is on. is printed. If the user replies is replaced by zero unless is reserved for use as the is set is simplified towards is still attached to it. When <identifier> is subsequently used as an operator, the message iteration iterative its jacobian jordan_block jordansymbolic kernel kernel, kernel. kernels key is your delimiter for the input string. knowledge known korder lambda large last leading least left left-hand length less let level level, lex term order like limit limited limitedfactors limits line line, line. linear linelength lines liss list list, list. list> list> :- a positive integer or a list of positive integers. list> :- either a positive integer or a list of positive integers. list> :- either a single list> :- either a single variable or a list of variables. list> :- list of algebraic expressions. list>) list>. listarg lists load_package loaded local logarithm logical logical operator returns long lower lu_decom m := mat((a,b),(c,d)); made main make make_identity manipulation many mass mat mat(( mat((i+1,i+2,i+3),(4,5,2),(1,i,0)); match mateigen mathematical matrices matrices. matrix matrix, matrix. matrix_augment matrix_stack matrixp may may also be found by turning on the switch may also be raised to positive and negative powers with the exponentiation operator (negative powers require the matrix to be invertible). Scalar expressions and may be may be optionally preceded by a tilde. means member message message. messages method minimal minimum minor minus missing mode mode, mode, if <expression> is a real number, the operator mode, if <list> is a list of three real numbers, the operator mode. modular module modulus monomial more most mshell mult_columns mult_rows multiple multiplication multiplicities multiplicity multiplied multiply multivariate must n-ary name name. names natural necessary need needed negative nested new next nextprime nil nodepend non-negative non-numeric nonneg. nonzero normal normally not notation nprimitive nullspace num_odesolve num_solve number number, number. numbered numberp numberp( numbers numbers, numbers. numerator numeric numeric accuracy numerical objects occur occurs of off off exp; off rounded; off, off. often omitted omitted.) on complex; on fort; on modular; on rounded; on rounded; on, on. once one only operation operations operator operator f; operator is operator is a binary infix or prefix logical operator. It returns operator is included in the package specfn2. operator is left associative: operator is right associative operator is used operator returns operator returns the operator which implements the operator, operator. operators operators, operators. option: option: Assign a name to to the optional optional free variable options or or a or an algebraic expression with exactly one order order. ordered ordering ordinarily ordinary original orthogonal other otherwise otherwise. out output outside over package 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 parameter parameters parentheses part partial particular parts pattern pause performed performs the equivalent task on the rows of <matrix>. performs the same task performs the same task on the rows of <matrix>. pf(2/((x+1)^2*(x+2)),x); physics picture pivot place placed places plot plot_xmesh plot_ymesh plus point point. pointer points polynomial polynomial. polynomials polynomials. position positive possible power powers preceded precedence precise precision predicate preduce prefix previous previously prime print print_precision printed printed, printed. printing problem procedure procedure coshval(a); procedure fac(n); procedure. procedures process produce produced produces producing product production products program programming programs prompt properties property provide provides pseudo_inverse put quotient radians raised random random_matrix range range. ratarg rather rational ratjordan read real realroots recurrence recursive redefine rederr redimension reduce reduct reference related relation remain remainder remains remove remove_columns remove_rows removed removes repart repeat replace replacement replaces represent representation represented representing represents require required requirements reserved respect respectively. rest restricted restriction resubstitution result resultant resulting results results. return returned returned. returning returns returns a numeric value only returns an error message. reverse revgradlex term order right right-hand root root_multiplicities root_of roots round rounded rounded; rows rows_pivot rule rules s may be raised to fractional and floating-point powers. same saveas savestructr scalar scientific scientific_notation screen second see see: select semicolon separate sequence series session session, session. set setmod sets setting several share should shown showrules side sides sign significan similar simple simplif simplification simplified simultaneous sin(x) sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, sin(~x)^2 => (1-cos(2*x))/2} since single singular size> size> :- a positive integer. smithex_int solution solve solve( solve(log(sin(x+3)),x); solvesingular solving some sometimes space special specific specifically specified specifies sqrt( sqrt2**2-2; square stack_rows standard starting statement statement, statement. statement> statements statements and procedures to add further functionality to statements, statements. steps sticks the matrices in <matrix still stored string structr structure sub_matrix subexpressions subscript subsequent substitute substitution subtraction such such that sufficient sum supplie support suppressed surface swap_columns swap_entries swap_rows switch switch is switch only has effect when the switches symbol symbolic symbols symmetr symmetric syntax syntax: system system. systems taken takes tangent taylor taylor(e^(x^2+y^2),x,0,4,y,0,4); taylor(exp(x),x,0,2); taylorautoexpand taylorcombine taylorcombine(hugo + x); taylorkeeporiginal taylororiginal taylorprint taylorrevert taylorseriesp taylortemplate taylortostandard techniques term term order terminal terminat terminate terms than that the their them them. themselves then then rederr "Choose nonneg. integer only" there therefore these they third this those though three tilde time time, times to be the inverse of to carry out these operations. toeplitz together top-level torder torder_compile total trace trallfac transform transformations trgroeb trigonometric true true. truncat truth turn turned turning two type unary unchanged. under unique univariate unless unless the switch until upper use used used, used. useful user uses using usual usually valid value value, value. values values, values. vandermonde variable variable. variables variables, variables. various vector vector aa,bb,cc; vectors want warning was weight weighted weighted term order weights well when where where < where <bas> is a where <p> is a polynomial or a list of polynomials. where <p> is a univariate polynomial where <p> is an expression, where <var> is a variable (a <kernel> without subscript) and <rep> is an expression which contains <var>. Here whether which which is equivalent to while will wish with within without words workspace would write written ws; wtlevel x + x*y**3 + x**2*cos(z); x**5 + x**2 + 18 + sqrt(y); you your zero zero. zeros {,<expression>} {,<identifier>} {,<kernel> {,<vector- {1,2, {2.27985615996629, 2.27985615996629, {2x + y,4x + 2y} {3,7,15,1,292,1,1,1,2,1} {7x + 15y - z,x - y - z} {<exp>, ... {<exp>,...} {<exp>,<exp>,... {A,B,C,D} {AA,BB,CC} {Return} {X= - {X=2*ARBINT(1)*PI - ASIN(1) - 3, {a,b,c,d} {aa,bb,cc} {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2, {exp1, exp2, ... , expm} {x,y,z} {x,y} {x=sqrt(3),x= - sqrt(3)}