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REDUCE User`s Manual - REDUCE Computer Algebra System

1. ARBCOMPLEX 2 ARBCOMPLEX 2 13 4 3 TP Operator Syntax TP EXPRN matrix_expression matrix 145 This operator takes a single matrix argument and returns its transpose 13 4 4 Trace Operator Syntax TRACE EXPRN matrix_expression algebraic The operator TRACE is used to represent the trace of a square matrix 13 4 5 Matrix Cofactors Syntax COFACTOR EXPRN matrix_expression ROW integer COLUMN integer algebraic The operator COFACTOR returns the cofactor of the element in row ROW and col umn COLUMN of the matrix MATRIX Errors occur if ROW or COLUMN do not simplify to integer expressions or if MATRIX is not square 146 CHAPTER 13 MATRIX CALCULATIONS 13 4 6 NULLSPACE Operator Syntax NULLSPACE EXPRN matrix_expression list NULLSPACE calculates for a matrix A a list of linear independent vectors a basis whose linear combinations satisfy the equation Ax 0 The basis is provided in a form such that as many upper components as possible are isolated Note that withb nullspace a the expression length bis the nullity of A and that second length a length b calculates the rank of A The rank of a matrix expression can also be found more directly by the RANK operator described below Example The command nullspace mat 1 2 3 4 5 6 7 8 gives the output In addition to the REDUCE matrix form NULLSPACE accepts a
2. while mkid a b 2 gives an error The SET operator can be used to give a value to the identifiers created by MKID for example 66 CHAPTER 7 BUILT IN PREFIX OPERATORS set mkid a 3 3 will give A3 the value 2 7 8 PF Operator PF lt exp gt lt var gt transforms the expression lt exp gt into a list of partial frac tions with respect to the main variable lt var gt PF does acomplete partial fraction decomposition and as the algorithms used are fairly unsophisticated factorization and the extended Euclidean algorithm the code may be unacceptably slow in com plicated cases Example Given 2 x 1 72x x 2 in the workspace pf ws x gives the result X 2 X 1 2 X 2 X 1 If you want the denominators in factored form use off exp Thus with 2 x 1 2 x 2 inthe workspace the commands off exp pf ws x give the result 2 2 2 r X 2 X 1 2 X 1 To recombine the terms FOR EACH SUMcan be used So with the above list in the workspace for each j in ws sum j returns the result 2 2 X 2 X 1 Alternatively one can use the operations on lists to extract any desired term 7 9 SELECT OPERATOR 67 7 9 SELECT Operator The SELECT operator extracts from a list or from the arguments of an n ary operator elements corresponding to a boolean predicate It is used with the syntax SELECT U function V list Function can be
3. Label 45 Laguerre polynomials 171 LaguerreP 171 LAMBDA 179 Lambert s W 68 LAPLACE 164 LCM 108 LCOF 112 Leading coefficient 112 Legendre polynomials 151 171 LegendreP 171 LENGTH 33 50 64 103 105 143 LET 59 61 74 79 81 120 128 153 154 LHS 31 LIE 164 LIMITS 165 LINALG 165 LINEAR 78 Linear operator 78 79 82 LINELENGTH 86 LISP 177 Lisp 177 LIST 88 List 33 list 67 INDEX List operation 33 35 LISTARGP 35 LISTARGS 35 LN 57 60 LOAD 200 LOAD_PACKAGE 157 201 LOG 57 60 63 LOG10 57 60 LOGB 57 60 Lommel functions 171 Lommel1 171 Lommel 2 171 Loop 40 41 LOW_POW 100 LPOWER 113 LTERM 113 187 MACRO 181 MAINVAR 113 MAP 64 map 67 MASS 194 195 MAT 141 142 MATCH 127 MATEIGEN 144 Mathematical function 57 MATRIX 142 Matrix assignment 147 Matrix calculations 141 MAX 55 MCD 106 108 Meijer s G function 172 MIN 55 Minimum 167 MKID 65 Mode 50 Mode communication 182 MODSR 165 MODULAR 117 Modular coefficient 117 MSG 203 MSHELL 195 Multiple assignment statement 38 MULTIPLICITIES 69 INDEX NAT 96 NCPOLY 166 NERO 93 Newton s method 167 NEXTPRIME 55 NOCONVERT 116 NODEPEND 82 Non commuting operator 79 NONCOM 79 NONZERO 78 NORMFORM 166 NOSPLIT 88
4. The command BYE or alternatively QUIT stops the execution of REDUCE closes all open output files and returns you to the calling program usually the operating system Your REDUCE session is normally destroyed 6 5 SHOWTIME Command SHOWTIME prints the elapsed time since the last call of this command or on its first call since the current REDUCE session began The time is normally given in milliseconds and gives the time as measured by a system clock The operations covered by this measure are system dependent 6 6 DEFINE Command The command DEFINE allows a user to supply a new name for any identifier or replace it by any well formed expression Its argument is a list of expressions of 52 CHAPTER 6 COMMANDS AND DECLARATIONS the form lt identifier gt lt number gt lt identifier gt lt operator gt lt reserved word gt lt expression gt Example define be x yt z means that BE will be interpreted as an equal sign and X as the expression y z from then on This renaming is done at parse time and therefore takes precedence over any other replacement declared for the same identifier It stays in effect until the end of the REDUCE run The identifiers ALGEBRAIC and SYMBOLIC have properties which prevent DEFINE from being used on them To define ALG to be a synonym for ALGEBRAIC use the more complicated construction put alg newnam algebraic Chapter 7
5. Users should note however that unlike factorization this decomposition is not unique 9 9 INTERPOL operator Syntax INTERPOL lt values gt lt variable gt lt points gt where lt values gt and lt points gt are lists of equal length and lt variable gt is an algebraic expression preferably a kernel INTERPOL generates an interpolation polynomial f in the given variable of degree length lt values gt 1 The unique polynomial f is defined by the property that for corresponding elements v of lt values gt and p of lt points gt the relation f p v holds The Aitken Neville interpolation algorithm is used which guarantees a stable result even with rounded numbers and an ill conditioned problem 9 10 Obtaining Parts of Polynomials and Rationals These operators select various parts of a polynomial or rational function structure Except for the cost of rearrangement of the structure these operations take very little time to perform For those operators in this section that take a kernel VAR as their second argument an error results if the first expression is not a polynomial in VAR although the coef ficients themselves can be rational as long as they do not depend on VAR However if the switch RATARG is on denominators are not checked for dependence on VAR and are taken to be part of the coefficients 9 10 1 DEG Operator This operator is used with the syntax 112 CHAPTER 9 POLYNOMIALS AND RATIONALS
6. lt variable gt lt variable gt e g lambda x y car x cdr y is equivalent to the Lisp LAMBDA expression lambda x y cons car x cdr y The parentheses may be omitted in specifying the variable list if desired LAMBDA expressions may be used in symbolic mode in place of prefix operators or as an argument of the reserved word FUNCTION In those cases where a LAMBDA expression is used to introduce local variables to avoid recomputation a WHERE statement can also be used For example the expression lambda x y list car x cdr x car y cdr y reverse u reverse v can also be written car x cdr x car y cdr y where x reverse u y reverse v Where possible WHERE syntax is preferred to LAMBDA syntax since it is more natural 16 5 Symbolic Assignment Statements In symbolic mode if the left side of an assignment statement is a variable a SETO of the right hand side to that variable occurs If the left hand side is an expression it must be of the form of an array element otherwise an error will result For exam ple x y translates into SETQ X Y whereas a 3 3 will be valid if A has been previously declared a single dimensioned array of at least four elements 16 6 FOR EACH STATEMENT 181 16 6 FOR EACH Statement The FOR EACH form of the FOR statement designed for iteration down a list is more general in symbolic mode Its syntax is FOR EACH ID identifier IN
7. 15 58 XIDEAL Gr bner Bases for exterior algebra 15 59ZEILBERG A package for indefinite and definite summation 15 60ZTRANS Z transform package o 16 Symbolic Mode 16 1 Symbolic Infix Operators o o e 16 2 Symbolic Expressions coo ee ee 16 3 uoied Expressions gt s ca ca ec ka tae ae ae ees 16 4 Lambda Expressions s c oreca ace ee ee ee 16 5 Symbolic Assignment Statements 16 6 FOR BACH Statement 2 2 4 22 2454 44 4 2d eae ers 16 7 Symbolic Procedures gt s s c veo ea a ee 16 8 Standard Lisp Equivalent of Reduce Input 16 9 Communicating with Algebraic Mode 16 9 1 Passing Algebraic Mode Values to Symbolic Mode 16 9 2 Passing Symbolic Mode Values to Algebraic Mode 16 9 3 Complete Example lt s 224454485 4224 ace 16 9 4 Defining Procedures for Intermode Communication 16 LORIS p S852 a a ed be tbe ae he eed b TG LUPE ss cag was Sok a Sede ae ae Ga ae 17 Calculations in High Energy Physics 17 1 High Energy Physics Operators IALL Cams PUDCLOOE ka ca a a 17 1 2 G Operator for Gamma Matrices EAN Oper AA II 17 2 Vector Variables s o s coso ee errar we ww He 17 3 Additional Expression Types 4 17 3 1 Vector Expressions 0 20 ee ee 174 174 175 175 175 177 179 179 179 179 180 181 181 182 182 183 185 186 186 188 189 1
8. Documentation for this package is in plain text Author Richard Liska 15 19 FPS Automatic calculation of formal power series This package can expand a specific class of functions into their corresponding Laurent Puiseux series 15 20 GENTRAN A CODE GENERATION PACKAGE 163 Authors Wolfram Koepf and Winfried Neun 15 20 GENTRAN A code generation package GENTRAN is an automatic code GENerator and TRANslator It constructs com plete numerical programs based on sets of algorithmic specifications and symbolic expressions Formatted FORTRAN RATFOR PASCAL or C code can be gener ated through a series of interactive commands or under the control of a template processing routine Large expressions can be automatically segmented into subex pressions 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 si multaneously and to facilitate recursive invocation of the whole code generation process Author Barbara L Gates 15 21 GNUPLOT Display of functions and surfaces This package is an interface to the popular GNUPLOT package It allows you to display functions in 2D and surfaces in 3D on a variety of output devices including X terminals PC monitors and postscript and Latex printer files NOTE The GNUPLOT package may not be included in all versions of REDUCE Author Herbert Melenk 15 22 GROEBNER A Grobner basis package GROEB
9. It returns the remainder when EXPRN1 is divided by EXPRN2 This is the true remainder based on the internal ordering of the variables and not the pseudo remainder The pseudo remainder and in general pseudo division of polynomials can be calculated after loading the polydiv package Please refer to the docu mentation of this package for details Examples remainder x y x x 2x y X 3x y gt 2x Yxx2 remainder 2xx y 2 gt Y CAUTION 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 Example remainder x 2 2 x sqrt 2 gt X 2 2 load_package arnum defpoly sqrt2xx2 2 remainder x 2 2 x sqrt2 gt 0 9 7 RESULTANT Operator This is used with the syntax RESULTANT EXPRN1 polynomial EXPRN2 polynomial VAR kernel polynomial It computes the resultant of the two given polynomials with respect to the given variable the coefficients of the polynomials can be taken from any domain The result can be identified as the determinant of a Sylvester matrix but can often also be thought of informally as the result obtained when the given variable is eliminated between the two input polynomials If the two input polynomials have a non trivial GCD their resultant vanishes The switch Bezout controls the computation of the resultants It is off by default In this case a subresultant algorithm is used
10. The SUB operator gives the algebraic list of one or more equations of the form lgebraic result of replacing every occurrence of the variable VAR in the expression EXPRN1 by the expression EXPRN Specifically EXPRNL is first evaluated using all available rules Next the substitutions are made and finally the substituted expression is reevaluated When more than one variable occurs in the substitution list the subst itution is performed by recursively walking down the tree representing EXPRN1 and replacing every VAR found by the ap propriate EXPRN The EXPRN are not themselves searched for any occurrences of the various VARs The trivial case SUI EXPRN1 Examples B EXPRN1 returns the algebraic value of 119 120 CHAPTER 10 SUBSTITUTION COMMANDS sub x aty y yt1l x 2 ty 2 gt A 2xAxY 2xY 2 Y 1 and with s x aty y y 1 2 2 sub s x 2 y 2 gt A 2xAxY 2xY 2 Y 1 Note that the global assignments x a y etc do not take place EXPRN1 can be any valid algebraic expression whose type is such that a substi tution process is defined for it e g scalar expressions lists and matrices An error will occur if an expression of an invalid type for substitution occurs either in EXPRN or EXPRN1 The braces around the substitution list may also be omitted as in 2 2 sub x aty y y 1 x 2 y 2 gt A 2xAxY 2xY 2xY 1 10 2 LET Ru
11. sin x cos x sqrt x x 2 will result in the output ANS 4 x LOG X COS X X 2 4 LOG X COS X X 3 x LOG X COS X 4 LOG X SIN X X 2 4 LOG X SIN X X 3 LOG X SIN X 8 COS X X 8 COS X 8 SIN X X 8 SIN X 4 SORT X X 2 These algebraic manipulations illustrate the algebraic mode of REDUCE RE DUCE is based on Standard Lisp A symbolic mode is also available for executing Lisp statements These statements follow the syntax of Lisp e g symbolic car a Communication between the two modes is possible With this simple introduction you are now in a position to study the material in the full REDUCE manual in order to learn just how extensive the range of facilities really is If further tutorial material is desired the seven REDUCE Interactive Lessons by David R Stoutemyer are recommended These are normally distributed with the system Chapter 2 Structure of Programs A REDUCE program consists of a set of functional commands which are evaluated sequentially by the computer These commands are built up from declarations statements and expressions Such entities are composed of sequences of numbers variables operators strings reserved words and delimiters such as commas and parentheses which in turn are sequences of basic characters 2 1 The REDUCE Standard Character Set The basic characters which are used to build REDUCE symbols are the following
12. directory and examples in specfn tst and specfmor tst in the examples directory The package SPECEN is designed to provide algebraic and numerical manipula tions of several common special functions namely e Bernoulli Numbers and Euler Numbers e Stirling Numbers e Binomial Coefficients e Pochhammer notation e The Gamma function e The Psi function and its derivatives e The Riemann Zeta function e The Bessel functions J and Y of the first and second kind 172 CHAPTER 15 USER CONTRIBUTED PACKAGES The modified Bessel functions I and K e The Hankel functions H1 and H2 e The Kummer hypergeometric functions M and U e The Beta function and Struve Lommel and Whittaker functions The Airy functions e The Exponential Integral the Sine and Cosine Integrals The Hyperbolic Sine and Cosine Integrals e The Fresnel Integrals and the Error function The Dilog function Hermite Polynomials Jacobi Polynomials Legendre Polynomials Spherical and Solid Harmonics Laguerre Polynomials Chebyshev Polynomials e Gegenbauer Polynomials Euler Polynomials Bernoulli Polynomials Jacobi Elliptic Functions and Integrals 3j symbols 6j symbols and Clebsch Gordan coefficients Author Chris Cannam with contributions from Winfried Neun Herbert Melenk Victor Adamchik Francis Wright and several others 15 49 SPECFN2 Package for special special functions This package provides algebraic mani
13. Advanced Use of Rule Lists Some advanced features of the rule list mechanism make it possible to write more complicated rules than those discussed so far and in many cases to write more compact rule lists These features are e Free operators e Double slash operator e Double tilde variables A free operator in the left hand side of a pattern will match any operator with the same number of arguments The free operator is written in the same style as a variable For example the implementation of the product rule of differentiation can be written as operator diff f g prule diff f x g x x gt diff f x x x g x diff g x x x let prule diff sin z x cos Z 2 cos z diff sin z z diff cos z z xsin z The double slash operator may be used as an alternative to a single slash quo tient in order to match quotients properly E g in the example of the Gamma function above one can use 10 3 RULE LISTS 131 gammarule gamma z c gamma zz gt gamma z c gamma zz 1 zz when fixp zz z and zz z gt 0 gamma z gamma zz gt gamma z gamma zz 1 zz when fixp zz z and zz z gt 0 let gammarule gamma z gamma z 3 1 3 2 z 6xz llxz 6 The above example suffers from the fact that two rules had to be written in order to perform the required operation This can be simplified by the use of double tilde variables E
14. Built in Prefix Operators In the following subsections are descriptions of the most useful prefix operators built into REDUCE that are not defined in other sections such as substitution operators Some are fully defined internally as procedures others are more nearly abstract operators with only some of their properties known to the system In many cases an operator is described by a prototypical header line as follows Each formal parameter is given a name and followed by its allowed type The names of classes referred to in the definition are printed in lower case and param eter names in upper case If a parameter type is not commonly used it may be a specific set enclosed in brackets Operators that accept formal parame ter lists of arbitrary length have the parameter and type class enclosed in square brackets indicating that zero or more occurrences of that argument are permitted Optional parameters and their type classes are enclosed in angle brackets 7 1 Numerical Operators REDUCE includes a number of functions that are analogs of those found in most numerical systems With numerical arguments such functions return the expected result However they may also be called with non numerical arguments In such cases except where noted the system attempts to simplify the expression as far as it can In such cases a residual expression involving the original operator usually remains These operators are as follows 7 1 1 A
15. In many cases this is all that the user needs The declarations by which the user can exercise some control over the way in which the evaluation is performed are explained in other sections For example if a real floating point number is encountered during evaluation the system will normally convert it into a ratio of two integers If the user wants to use real arithmetic he can effect this by the command on rounded Other modes for coefficient arithmetic are described elsewhere If an illegal action occurs during evaluation such as division by zero or functions are called with the wrong number of arguments and so on an appropriate error message is generated 3 2 Integer Expressions These are expressions which because of the values of the constants and variables in them evaluate to whole numbers Examples 2r 37 xk 999 x 3 2 x72 6 x are obviously integer expressions 7 pik 2 ea is an integer expression when J and K have values that are integers or if not integers are such that the variables and fractions cancel out as in 3 3 BOOLEAN EXPRESSIONS 29 le PB a OS GAD 3 3 Boolean Expressions A boolean expression returns a truth value In the algebraic mode of REDUCE boolean expressions have the syntactical form lt expression gt lt relational operator gt lt expression gt or lt boolean operator gt lt arguments gt or lt boolean expression gt lt logical operator gt l
16. gt Similarly coupled substitutions such as let 1l m n n 1 4 4 would lead to the same error As a result if you try to evaluate an X L or N defined as above you will get an error such as X improperly defined in terms of itself Array and matrix elements can appear on the left hand side of a LET statement However because of their instant evaluation property it is the value of the element that is substituted for rather than the element itself E g array a 5 a 2 b let a 2 a results in B being substituted by C the assignment for a 2 does not change Finally if an error occurs in any equation in a LET statement including generalized statements involving FOR ALL and SUCH THAT the remaining rules are not evaluated 10 2 1 FOR ALL LET If a substitution for all possible values of a given argument of an operator is re quired the declaration FOR ALL may be used The syntax of such a command is FOR ALL lt variable gt lt variable gt lt LET statement gt lt terminator gt e g for all x y let h x y x y for all x let k x y x y 10 2 LET RULES 123 The first of these declarations would cause h a b to be evaluated as A B h u v u w to be V W etc If the operator symbol H is used with more or fewer argument places not two the LET would have no effect and no error would result The second declaration would cause k a y to be evaluated as a
17. 2x U 2x wWx x 14495 Ux 2x Vx wWx x 8 55 Ux 2 wxx 9 11 ux v 10 110 x u v 9 rw 495 U Vx 8x wx 2 1320 UxUV 7 xw xx3 2310 x Ux Vx 6xwWxx x44 2772 UxV xbawxx5 2310 Ux Uxx 4xwWxx 6 1320 xUx UVx x x3xwx x 7 495 x Ux Vx 2x wx 8 110 Uux v xwx 9 11 xuxw x10 vxx114 11 vxx 10xw 55 Vx x x9 wxx2 165 UVx 8xwxx 3 330 x UVA Xx Tx WXx44 462 Vx 6xwWx x5 462 xkUVx xbxwxx06 330 x Uxx4x xwx x7 165 xvVxx3xwWx x8 55 Ux 2xwxx9 11 v wx x 10 wxx11 x uxx11 11 ux 10xv 11 ux 10xw 55 x Uux 9xUVx 2 110 U x 9x Vx wW 55 ruxrr9 wxx 2 165 Ux 8xUV x r3 495 xU 8xU x 2x w 495 xUux xx 8 AUVXWxXx 2 165 U 8xW x 3 330 Uux x 1 vx 4 1320 Ux x 7 Vx 3x w 1980 xUx x Tx UVxx 2x xwxx x2 1320 Ux x x 7 VxwWx x3 330 xUux x x71 wxxr4 462 UX O XUVA RD 2310 xU r6rvx 4xw 4620 Uux x 6 xvx 3xwxx2 4620 x u RXOXVA 2xWxx 3 2310 U x 6x VxWxx4 462 xUxx x6xwxx5 462 xUxx5x UVxx6 2 772 Ux 5xrvVx x5biw t6930 xUx DxUx x4xwx 2 9240 xUxx5x V xx3xw x x3 6930 xux x5xV xx2xwWw xx4 2772 xuxx5xvxwxx5 462 xuxx5 xwxx6 330 xu xx x4xvx xx7 2310 xu xx4xvx xx6xw 6930 xu x x4 xv x x5 xW 2 11550 Ux 4x Vx 4 wx x 3 11550 Ux 4 UVx 3xwx 4 6930 Uux x 4xyxx2xwxx5 2310 xuxx4xv xw xx6 330 xuxx4xwxx7 165 xu xx3 xv xx8 1320 x Uxx3xUxx 71xw 4620 Ux 3xUxx 6 xwxx2 9240 xUxx 3x Ux xx 8 3 OUTPUT OF EXPRESSIONS 95 5xwxx3 11550 xUux 3xUv x 4x xwx x4 9240 xUxx 3xUVx 3xw xx5 4620 x u x 3xUVx 2 wx x6 ansl c it was foolish to expand this expression print 1 x
18. Both are off by default Thus to expand LOG X Y into a sum of logs one can say ON EXPANDLOGS LOG X Y and to combine this sum into a single log ON COMBINELOGS LOG X LOG Y At the present time it is possible to have both switches on at once which could lead to infinite recursion However an expression is switched from one form to the other in this case Users should not rely on this behavior since it may change in the next release The current version of REDUCE does a poor job of simplifying surds In particular expressions involving the product of variables raised to non integer powers do not usually have their powers combined internally even though they are printed as if those powers were combined For example the expression x 1 3 x 1 6 will print as SORT X but will have an internal form containing the two exponentiated terms If you now subtract sqrt x from this expression you will not get zero Instead the confusing form SORT X SORT X will result To combine such exponentiated terms the switch COMBINEEXPT should be turned on 60 CHAPTER 7 BUILT IN PREFIX OPERATORS The square root function can be input using the name SORT or the power opera tion 1 2 On output unsimplified square roots are normally represented by the operator SORT rather than a fractional power With the default system switch settings the argument of a square root is first simplif
19. DEG EXPRN polynomial VAR kernel integer It returns the leading degree of the polynomial EXPRN in the variable VAR If VAR does not occur as a variable in EXPRN 0 is returned Examples deg a b x c 2 d 22 deg a b c 2xd 2 d gt 2 deg a b c 2x d 2 Note also that if RATARG is on deg a b 3 a a gt 3 since in this case the denominator A is considered part of the coefficients of the numerator in A With RATARG off however an error would result in this case 9 10 2 DEN Operator This is used with the syntax DEN EXPRN rational polynomial It returns the denominator of the rational expression EXPRN If EXPRN is a poly nomial 1 is returned Examples den x y 2 gt Yxx2 den 100 6 gt 3 since 100 6 is first simplified to 50 3 den a 4 b 6 gt 12 den a b gt 1 9 10 3 LCOF Operator LCOF is used with the syntax LCOF EXPRN polynomial VAR kernel polynomial It returns the leading coefficient of the polynomial EXPRN in the variable VAR If VAR does not occur as a variable in EXPRN EXPRN is returned 9 10 OBTAINING PARTS OF POLYNOMIALS AND RATIONALS Examples lcof a lcof a lcof a tb c4 tb c4 t2 d 2 a F2xd 2 d Ho c4 t2 d re 9 10 4 LPOWER Operator Syntax LPOW ER LPOWER returns the leading power of does not depend on VAR 1 is returned Example
20. NOSPUR 195 ULLSPACE 146 UM 114 UM_INT 167 UM_MIN 167 UM_ODESOLVE 167 UM_SOLVE 167 umber 19 20 UMBERP 29 Numerical operator 53 Numerical precision 22 ZX222 2Z3 2Z2Z DD 78 dd operator 78 DESOLVE 167 FF 50 51 N 50 51 ONE_OF 69 OPERATOR 186 Operator 24 26 Operator precedence 25 26 RDER 86 99 RDP 29 79 rthogonal polynomials 171 RTHOVEC 168 UT 135 136 UTPUT 85 Output 91 95 Output declaration 86 O OOO 000000 PART 33 98 100 PAUSE 139 213 Percent sign 23 ERIOD 95 LOT 163 PM 168 Pochhammer 171 Pochhammer s symbol 171 Polygamma 171 Polygamma functions 171 Polynomial 103 Polynomial equations 163 PRECEDENCE 81 PRECISE 60 PRECISION 115 Prefix 53 80 81 Prefix operator 24 25 PRET 203 PRETTYPRINT 203 Prettyprinting 203 PRI 86 PRIMEP 29 PRINT_PRECISION 116 PROCEDURE 149 Procedure body 151 153 Procedure heading 150 PRODUCT 40 41 Program 23 Program structure 19 Proper statement 31 37 PS 173 PSEUDO DIVIDE 109 PSEUDO_REMAINDER 109 Psi 171 Psi function 171 Quadrature 167 QUIT 51 QUOTE 179 RANDOM 56 RANDOM_NEW_SEED 56 RANDPOLY 169 214 RANK 147 RAT 89 RATARG 100 111 RATIONAL 115 Rational coefficient 115 Rational function 103
21. Put into words the Legendre polynomial p x is the result of substituting y 0 in the nt partial derivative with respect to y of a certain fraction involving x and y then dividing that by n This verbal formula can easily be written in REDUCE procedure p n x sub y 0 df 1 y 2 2 xxy 1 1 2 y n for i 1 n product i Having input this definition the expression evaluation 2p 2 w would result in the output 2 3xW 1 152 CHAPTER 14 PROCEDURES If the desired process is best described as a series of steps then a group or com pound statement can be used 14 3 USING LET INSIDE PROCEDURES 153 Example The above Legendre polynomial example can be rewritten as a series of steps in stead of a single formula as follows procedure p n x begin scalar seed deriv top fact seed 1 y 2 2x xxy 1 1 2 deriv df seed y n top sub y 0 deriv fact for i l n product i return top fact end Procedures may also be defined recursively In other words the procedure body can include references to the procedure name itself or to other procedures that them selves reference the given procedure As an example we can define the Legendre polynomial through its standard recurrence relation procedure p n x if n lt 0 then rederr Invalid argument to P N X else if n 0 then 1 else if n 1 then x else 2 n 1 x p n 1 x n 1 p n 2 x n The operator REDERR in the above exampl
22. SIN 72 CHAPTER 7 BUILT IN PREFIX OPERATORS SORT 31 x I ATAN 3xSORT 3 COS SQRT 3 3 SORT 31 1I ATAN 3xSORT 3 X Ix SORT 3 SIN 3 SORT 31 I ATAN 3xSORT 3 COS SORT 3 3 SORT 31 I ATAN 3xSORT 3 2 COS I 3 X SQRT 3 off trigform XX 2 3 X SORT 31 3x SORT 3 SORT 3 1 2 3 2 3 SORT 31 3 SORT 3 2 SORT 3 I 2 3 1 43 0 43 2 2x SORT 31 3 SQRT 3 x6 1 6 3 2 3 7 10 SOLVE OPERATOR 73 X SORT 31 3xSORT 3 SORT 3 I 27 3 2 3 SQRT 31 3xSORT 3 2 SORT 3 I 2 3 1 73 T73 2 2x SORT 31 3 SQRT 3 x6 1 6 3 2 3 2 3 SORT 31 3xSORT 3 2 x 1 3 1 3 1 6 SORT 31 3 SQRT 3 x6 3 7 10 3 Other Options If SOLVESINGULAR is on the default setting degenerate systems such as x y 0 2x 2y 0 will be solved by introducing appropriate arbitrary constants The consistent singular equation 0 0 or equations involving functions with multi ple inverses may introduce unique new indeterminant kernels ARBCOMP LEX 3 or ARBINT 3 1 2 representing arbitrary complex or integer numbers re spectively To automatically select the principal branches do off allbranch To avoid the introduction of new indeterminant kernels do OFF ARBVARS then no equations are generated for the free variables and their or
23. and the user is not redefining his own procedure he is well advised to rename it and possibly start over because he has already redefined some internal procedure whose correct functioning may be required for his job 149 150 CHAPTER 14 PROCEDURES All required procedures should be defined at the top level since they have global scope throughout a program In particular an attempt to define a procedure within a procedure will cause an error to occur 14 1 Procedure Heading Each procedure has a heading consisting of the word PROCEDURE optionally preceded by the word ALGEBRAIC followed by the name of the procedure to be defined and followed by its formal parameters the symbols that will be used in the body of the definition to illustrate what is to be done There are three cases 1 No parameters Simply follow the procedure name with a terminator semi colon or dollar sign procedure abc When such a procedure is used in an expression or command abc with empty parentheses must be written 2 One parameter Enclose it in parentheses or just leave at least one space then follow with a terminator procedure abc x or procedure abc x 3 More than one parameter Enclose them in parentheses separated by com mas then follow with a terminator procedure abc x y Z Referring to the last example if later in some expression being evaluated the sym bols abc u pq 123 appear the operations of the
24. lt substitution operator gt where lt arithmetic operator gt x 1 17 lt construction operator gt MEMO and EQ are not used in the algebraic mode of REDUCE They are explained in the section on symbolic mode WHERE is described in the section on substitu tions In previous versions of REDUCE not was also defined as an infix operator In the present version it is a regular prefix operator and interchangeable with null For compatibility with the intermediate language used by REDUCE each special character infix operator has an alternative alphanumeric identifier associated with 1t These identifiers may be used interchangeably with the corresponding special character names on input This correspondence is as follows setq the assignment operator equal gt geq gt greaterp lt leq 26 CHAPTER 2 STRUCTURE OF PROGRAMS lt lessp plus difference if unary minus times quotient if unary recip or expt raising to a power cons Note NEO is used to mean not equal There is no special symbol provided for it The above operators are binary except NOT which is unary and and which are nary i e taking an arbitrary number of arguments In addition and may be used as unary operators e g 2 means the same as 1 2 Any other operator is parsed as a binary operator using a left association rule Thus a b c is interpreted as a b c There are two except
25. please let us know A trace of a y matrix expression involving a line identifier which has been declared NOSPUR may be later taken by making the declaration SPUR See also the CVIT package for an alternative mechanism chapter 15 12 17 5 Mass Declarations It is often necessary to put a particle on the mass shell in a calculation This can of course be accomplished with a LET command such as let p p m 2 but an alternative method is provided by two commands MASS and MSHELL MASS takes a list of equations of the form lt vector variable gt lt scalar variable gt 196 CHAPTER 17 CALCULATIONS IN HIGH ENERGY PHYSICS for example mass pl m ql mu The only effect of this command is to associate the relevant scalar variable as a mass with the corresponding vector If we now say mshell lt vector variable gt lt vector variable gt and a mass has been associated with these arguments a substitution of the form lt vector variable gt lt vector variable gt lt mass gt 2 is set up An error results if the variable has no preassigned mass 17 6 Example We give here as an example of a simple calculation in high energy physics the computation of the Compton scattering cross section as given in Bjorken and Drell Egs 7 72 through 7 74 We wish to compute the trace of a ey 22 tm Ge reverts 4 m 2 k 2m 2k p 2k p 2m j renere 2k pi 2k pi where k and kf are the fo
26. 1 The 26 letters a through z 2 The 10 decimal digits 0 through 9 3 The special characters _ lt gt lt blank gt With the exception of strings and characters preceded by an exclamation mark the case of characters is ignored depending of the underlying LISP they will all be converted internally into lower case or upper case ALPHA Alpha and alpha represent the same symbol Most implementations allow you to switch this con version off The operating instructions for a particular implementation should be consulted on this point For portability we shall limit ourselves to the standard character set in this exposition 2 2 Numbers There are several different types of numbers available in REDUCE Integers consist of a signed or unsigned sequence of decimal digits written without a decimal point 19 20 CHAPTER 2 STRUCTURE OF PROGRAMS for example 2 7 9396 32 In principle there is no practical limit on the number of digits permitted as ex act arithmetic is used in most implementations You should however check the specific instructions for your particular system implementation to make sure that this is true For example if you ask for the value of 220 you get it displayed as a number of 603 decimal digits taking up nine lines of output on an interactive display It should be borne in mind of course that computations with such long numbers can be quite slow Numbers that aren t integers are usuall
27. 135 FIRST 34 FIX 54 FIXP 29 FLOOR 55 FOR 47 FOR ALL 122 123 FOR EACH 41 42 181 FORT 93 FORT_WIDTH 95 FORTRAN 93 95 FORTUPPER 95 FPS 162 FREEOF 29 FULLROOTS 71 Function 155 G 192 Gamma 171 Gamma function 171 GCD 107 Gegenbauer polynomials 171 GegenbauerP 171 Generalized Hypergeometric functions 172 211 ENTRAN 163 NUPLOT 163 O TO 45 ROEBNER 163 Groebner 68 Group statement 39 43 G G G G Hankel functions 171 Hankell 171 Hankel 2 171 Hermite polynomials 171 HermiteP 171 High energy trace 195 High energy vector expression 191 194 HIGH_Pow 100 History 137 HYPOT 57 60 E 22 IDEALS 164 Identifier 21 IF 39 IFACTOR 105 IMPART 54 56 IN 135 Indefinite integration 62 INDEX 192 INEO 164 INFINITY 22 INFIX 81 Infix operator 24 26 INPUT 138 Input 135 Instant evaluation 50 102 122 142 144 INT 62 139 INTEGER 44 Integer 28 Integration 62 79 Interactive use 137 139 INTERPOL 111 Introduction 15 INTSTR 84 INVBASE 164 212 Jacobi Elliptic Functions and Integrals 171 Jacobi s polynomials 171 Jacobiamplitude 171 acobicn 171 acobidn 171 acobiP 171 acobisn 171 acobiZeta 171 OIN 40 Cy Cy Y Es EQ Ep Kernel 83 87 99 kernel form 84 KORDER 99 Kummer functions 171 Kummer 171 Kummeru 171
28. A ALLFAC is normally on and is on in the following examples except where other wise stated DIV Switch This switch makes the system search the denominator of an expression for simple factors that 1t divides into the numerator so that rational fractions and negative powers appear in the output With DIV on our expression would print as 2 2 1 1 Xx X Y 2xXxY 1 2xY xA 1 2 A Z DIV is normally off LIST Switch This switch causes the system to print each term in any sum on a separate line With LIST on our expression prints as 2 Xx 2 X Y xA 4xXxYxA 2 Y Z 2 A LIST is normally off NOSPLIT Switch Under normal circumstances the printing routines try to break an expression across lines at a natural point This is a fairly expensive process If you are not overly concerned about where the end of line breaks come you can speed up the printing of expressions by turning off the switch NOSPLIT This switch is normally on 8 3 OUTPUT OF EXPRESSIONS 89 RAT Switch This switch is only useful with expressions in which variables are factored with FACTOR With this mode the overall denominator of the expression is printed with each factored sub expression We assume a prior declaration factor x in the following output We first print the expression with RAT off 2 2 2xX Y Ax Y 2 Xx Y Z 2xA With RAT on the output becomes 90 CHAPTER 8 DISPLAY AND STRUCTURING OF EXPRESSION
29. If the switch Bezout is turned on the resultant is computed via the Bezout Matrix However in the latter case only polynomial coefficients are permitted 110 CHAPTER 9 POLYNOMIALS AND RATIONALS The sign conventions used by the resultant function follow those in R Loos Com puting in Algebraic Extensions in Computer Algebra Symbolic and Algebraic Computation Second Ed Edited by B Buchberger G E Collins and R Loos Springer Verlag 1983 Namely with A and B not dependent on X deg p deg q resultant p x q x x 1 resultant q p x deg p resultant a p x x a resultant a b x 1 Examples 2 resultant x rx xuty u y u Say calculation in an algebraic extension load arnum defpoly sqrt2 2 2 resultant x sqrt2 sqrt2 x 1 x gt 1 or in a modular domain setmod 17 on modular resultant 2x 1 3x 4 x gt 5 9 8 DECOMPOSE Operator The D ECOMPOS E operator takes a multivariate polynomial as argument and re turns an expression and a list of equations from which the original polynomial can be found by composition Its syntax is DECOMPOSE EXPRN polynomial list For example 9 9 INTERPOL OPERATOR 111 decompose x 8 88xx 7 2924xx 6 43912x x 5 263431xx 4 218900 x 3 65690 x 2 7700 x 234 2 2 2 gt U 35 U 234 U V 10xV V X 22xX 2 decompose u 2 v 2 2uxv 1 gt W 1 W U V
30. Report The Report also defines a function type FEXPR However its use is discouraged since it is hard to implement efficiently and most uses can be replaced by macros At the present time there are no FEXPRs in the core REDUCE system 16 8 Standard Lisp Equivalent of Reduce Input A user can obtain the Standard Lisp equivalent of his REDUCE input by turning on the switch DEFN for definition The system then prints the Lisp translation of his input but does not evaluate it Normal operation is resumed when DEFN is turned off 16 9 Communicating with Algebraic Mode One of the principal motivations for a user of the algebraic facilities of REDUCE to learn about symbolic mode is that it gives one access to a wider range of techniques than is possible in algebraic mode alone For example if a user wishes to use parts of the system defined in the basic system source code or refine their algebraic code definitions to make them more efficient then it is necessary to understand the source language in fairly complete detail Moreover it is also necessary to know a little more about the way REDUCE operates internally Basically REDUCE con siders expressions in two forms prefix form which follow the normal Lisp rules of function composition and so called canonical form which uses a completely different syntax Once these details are understood the most critical problem faced by a user is how to make expressions and procedures commu
31. This package is a careful implementation of the Gosper and Zeilberger algorithms for indefinite and definite summation of hypergeometric terms respectively Ex tensions of these algorithms are also included that are valid for ratios of products of powers factorials C function terms binomial coefficients and shifted factorials that are rational linear in their arguments Authors Gregor St lting and Wolfram Koepf 15 60 ZTRANS Z transform package This package is an implementation of the Z transform of a sequence This is the discrete analogue of the Laplace Transform Authors Wolfram Koepf and Lisa Temme 176 CHAPTER 15 USER CONTRIBUTED PACKAGES Chapter 16 Symbolic Mode At the system level REDUCE is based on a version of the programming language Lisp known as Standard Lisp which is described in J Marti Hearn A C Griss M L and Griss C Standard LISP Report SIGPLAN Notices ACM New York 14 No 10 1979 48 68 We shall assume in this section that the reader is familiar with the material in that paper This also assumes implicitly that the reader has a reasonable knowledge about Lisp in general say at the level of the LISP 1 5 Programmer s Manual McCarthy J Abrahams P W Edwards D J Hart T P and Levin M I LISP 1 5 Programmer s Manual M I T Press 1965 or any of the books mentioned at the end of this section Persons unfamiliar with this material will have some difficulty understan
32. a procedure LEADTERM which gives the leading term of an algebraic expression one could do this as follows symbolic operator leadterm Declare LEADTERM as a symbolic mode procedure to be used in algebraic mode Ad A A symbolic procedure leadterm u Define LEADTERM mk sq t2q lt numr simp x u Note that this operator has a different effect than the operator LTERM In the latter case the calculation is done with respect to the second argument of the operator In the example here we simply extract the leading term with respect to the system s choice of main variable Finally if you wish to use the algebraic evaluator on an argument in a symbolic mode definition the function REVAL can be used The one argument of REVAL must be the prefix form of an expression REVAL returns the evaluated expression as a true Lisp prefix form 188 CHAPTER 16 SYMBOLIC MODE 16 10 Rlisp 88 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 Evo lutionary Approach to Program Design and Reuse World Scientific Singapore 1993 Rlisp 88 adds to the traditional Rlisp the following facilities 1 more general versions of the looping constructs for repeat and while 2 support for a backquote construct 3 support for active comments 4 support for vectors of the form namelindex 5 suppo
33. an op erator symbol with its list of arguments filled in or an array name with the proper number of integer subscript values within the array bounds For example al b c h l m x 2xy where h is an operator k 3 5 x 2x y where k is a 2 dim array More general assignments such as a b care also allowed The effect of these is explained in Section 10 2 5 An assignment statement causes the expression on the right hand side to be evalu ated If the left hand side is a variable the value of the right hand side is assigned to that unevaluated variable If the left hand side is an operator or array expression the arguments of that operator or array are evaluated but no other simplification done The evaluated right hand side is then assigned to the resulting expression For example if A is a single dimensional array a 1 1 b assigns the value B to the array element a 2 If a semicolon is used as the terminator when an assignment is issued as a command i e not as a part of a group statement or procedure or other similar construct the left hand side symbol of the assignment statement is printed out followed by a followed by the value of the expression on the right It is also possible to write a multiple assignment statement lt expression gt lt expression gt lt expression gt In this form each lt expression gt but the last is set to the value of the last lt expression gt If a
34. argument is returned as an expression in the original operator For example 7 1 NUMERICAL OPERATORS 55 fix 5 4 gt Li fix a gt FIX A 7 1 6 FLOOR This operator returns the floor i e the greatest integer less than the given argu ment if its single argument has a numerical value A non numerical argument is returned as an expression in the original operator For example KH loor 5 4 gt 2 loor a gt FLOOR A Hh 7 1 7 IMPART This operator returns the imaginary part of an expression if that argument has an numerical value A non numerical argument is returned as an expression in the operators REPART and IMPART For example impart 1 i gt 1 impart atixb gt REPART B IMPART A 7 1 8 MAXMIN MAX and MIN can take an arbitrary number of expressions as their arguments If all arguments evaluate to numerical values the maximum or minimum of the argument list is returned If any argument is non numeric an appropriately reduced expression is returned For example max 2 3 4 5 gt 5 min 2 2 gt 2 max a 2 3 gt MAX A 3 min x gt X MAX or MIN of an empty list returns 0 7 1 9 NEXTPRIME NEXTPRIME returns the next prime greater than its integer argument using a prob abilistic algorithm A type error occurs if the value of the argument is not an inte ger For example 56 CHAPTER 7 BUILT IN PREFIX OPERATORS nextprime 5 gt 7
35. by itself on the left hand side of the equation On the other hand if the solve package could not find a solution the solution will be an equation for the unknown in terms of the operator ROOT_OF If there are several unknowns each solution will be a list of equations for the unknowns For example solve x 2 1 x gt X 1 X 1 solve x 7 x 6 x 2 1 x 6 gt X ROOT_OF X_ X_ 1 X_ TAG_1 X 1 7 10 SOLVE OPERATOR 69 solve x 3y 7 y x 1 x y gt X 1 Y 2 The TAG argument is used to uniquely identify those particular solutions Solution multiplicities are stored in the global variable ROOT MULTIPLICITIES rather than the solution list The value of this variable is a list of the multiplicities of the solutions for the last call of SOLVE For example solve x 2 2x 1 x root_multiplicities gives the results X 1 2 If you want the multiplicities explicitly displayed the switch MULTIPLICITIES can be turned on For example on multiplicities solve x 2 2x 1 x yields the result X 1 X 1 7 10 1 Handling of Undetermined Solutions When SOLVE cannot find a solution to an equation it normally returns an equation for the relevant indeterminates in terms of the operator ROOT_OF For example the expression solve cos x log x x returns the result X ROOT_OF COS X_ LOG X_ X_ TAG_1 An expression with a top level ROOT_OF operator is implicitly a list with an un know
36. files for numerical pro cessing First when the switch FORT is on the system will print expressions in a FOR TRAN notation Expressions begin in column seven If an expression extends over one line a continuation mark followed by a blank appears on subsequent cards After a certain number of lines have been produced according to the value of the variable CARD_NO a new expression is started If the expression printed arises from an assignment to a variable the variable is printed as the name of the expres sion Otherwise the expression is given the default name ANS An error occurs if identifiers or numbers are outside the bounds permitted by FORTRAN A second option is to use the WRITE command to produce other programs 94 CHAPTER 8 DISPLAY AND STRUCTURING OF EXPRESSIONS Example The following REDUCE statements on fort out Eortil write C this is a fortran program write 1 format el3 5 write u 1 23 write v 2 17 write w 5 2 x u tv w 711 write C it was foolish to expand this expression write print Ly xs write end shut forfil off fort will generate a file for 11 that contains c this is a fortran program 1 format e13 5 u 1 23 v 2 17 w 5 2 ans1 1320 xuU xx3xvxw xx7 165 xUxx3xwxx8 55 xUxx2xvxx9 495 xu RX2 XVxX 8xwW 1980 x U 2x Vxx 7xwx 2 4620 U 2x U 60xwxx3 6930 xU 2x UVx 5xwx x 4 6930 U 2x UVx 4xwxx5 4620 xU 2x Vx 3x W x 6 1980 U
37. form and is not of much interest to the general user It is normally off If the switch FAILHARD is on the algorithm will terminate with an error if the integral cannot be done in closed terms rather than return a formal integration form FATLHARD is normally off The switch NOLNR suppresses the use of the linear properties of integration in cases when the integral cannot be found in closed terms It is normally off 7 4 2 Advanced Use If a function appears in the integrand that is not one of the functions EXP ERF TAN ATAN LOG DILOG then the algorithm will make an attempt to inte grate the argument if it can differentiate it and reach a known function However the answer cannot be guaranteed in this case If a function is known to be alge braically independent of this set it can be flagged transcendental by flag trilog transcendental in which case this function will be added to the permitted field descriptors for a genuine decision procedure If this is done the user is responsible for the mathe matical correctness of his actions The standard version does not deal with algebraic extensions Thus integration of expressions involving square roots and other like things can lead to trouble A contributed package that supports integration of functions involving square roots is available however ALGINT chapter 15 1 In addition there is a definite integra tion package DEFINT chapter 15 13 7 4 3 Reference
38. g the rule list GGrule gamma z c xgamma zz gt gamma z c gamma zz 1 zz when fixp zz z and zz z gt 0 will implement the same operation in a much more compact way In general dou ble tilde variables are bound to the neutral element with respect to the operation in which they are used Pattern given Argument used Binding TZ y x z x y 0 Z x 3 Z X y 3 or z 3 y x 7 y x z x y 1 Z y x 3 Z Xx y 3 or Z 3 y x ZI Y xX z x y 1 z y x 3 Z Xx y 3 Remarks A double tilde variable as the numerator of a pattern is not allowed Also using double tilde variables may lead to recursion errors when the zero case is not handled properly let f a xk x x gt a x f x x when freeof a x 132 CHAPTER 10 SUBSTITUTION COMMANDS z Z xxxxx f z z improperly defined in terms of itself BUT let ff Ta x x gt a x ff x x when freeof a x and a neq 1 Ti Gz EZ f z z ff 3 z z 3x ff 2 2 Displaying Rules Associated with an Operator The operator SHOWRULES takes a single identifier as argument and returns in rule list form the operator rules associated with that argument For example showrules log DF LOG 7X X gt X Such rules can then be manipulated further as with any list For example rhs first ws has the value 1 Note that an operator may have other properties that cannot be displayed in such a form such as the fac
39. in this section All such coefficient modes are supported in a table driven manner so that it is straightforward to extend the range of possibilities A description of how to do this is given in R J Brad ford A C Hearn J A Padget and E Schriifer Enlarging the REDUCE Domain of Computation Proc of SYMSAC 86 ACM New York 1986 100 106 9 11 1 Rational Coefficients in Polynomials Instead of treating rational numbers as the numerator and denominator of a rational expression it is also possible to use them as polynomial coefficients directly This is accomplished by turning on the switch RATIONAL Example With RATIONAL off the input expression a 2 would be converted into a rational expression whose numerator was A and denominator 2 With RATIONAL on the same input would become a rational expression with numerator 1 2x A and denominator 1 Thus the latter can be used in operations that require polynomial input whereas the former could not 9 11 2 Real Coefficients in Polynomials The switch ROUNDED permits the use of arbitrary sized real coefficients in poly nomial expressions The actual precision of these coefficients can be set by the operator PRECISION For example precision 50 sets the precision to fifty decimal digits The default precision is system dependent and can be found by precision 0 In this mode denominators are automatically made monic and an appropriate adjustment is made to the numerator Examp
40. korder vl vnj where the Vi are kernels With this declaration the Vi are ordered internally ahead of any other kernels in the system V1 has the highest order V2 the next highest and so on A further call of KORDER replaces a previous one KORDER NIL resets the internal order to the system default Unlike the ORDER declaration that has a purely cosmetic effect on the way results are printed the use of KORDER can have a significant effect on computation time In critical cases then the user can experiment with the ordering of the variables used to determine the optimum set for a given problem 8 5 Obtaining Parts of Algebraic Expressions There are many occasions where it is desirable to obtain a specific part of an ex pression or even change such a part to another expression A number of operators are available in REDUCE for this purpose and will be described in this section In addition operators for obtaining specific parts of polynomials and rational funct ions such as a denominator are described in another section 8 5 1 COEFF Operator Syntax COEFF EXPRN polynomial VAR kernel COEFF is an operator that partitions EXPRN into its various coefficients with re spect to VAR and returns them as a list with the coefficient independent of VAR first 100 CHAPTER 8 DISPLAY AND STRUCTURING OF EXPRESSIONS Under normal circumstances an error results if EXPRN is not a polynomial in VAR although the coefficients them
41. level operator in an expression If the expression does not contain a top level operator then 1 is returned For example arglength a b c gt 3 arglength f gt 0 arglength a gt vel 102 CHAPTER 8 DISPLAY AND STRUCTURING OF EXPRESSIONS 8 5 4 Substituting for Parts of Expressions PART may also be used to substitute for a given part of an expression In this case the PART construct appears on the left hand side of an assignment statement and the expression to replace the given part on the right hand side For example with the normal settings of the REDUCE switches xx atb part xx 2 C gt A C part c d 0 A gt C D Note that xx in the above is not changed by this substitution In addition un like expressions such as array and matrix elements that have an instant evaluation property the values of part xx 2 and part c d 0 are also not changed Chapter 9 Polynomials and Rationals Many operations in computer algebra are concerned with polynomials and rational functions In this section we review some of the switches and operators available for this purpose These are in addition to those that work on general expressions such as DF and INT described elsewhere In the case of operators the arguments are first simplified before the operations are applied In addition they operate only on arguments of prescribed types and produce a type mismatch error if given arguments which
42. morning Ssign 5goldrings CAUTION Many system identifiers have such special characters in their names especially and If the user accidentally picks the name of one of them for his own purposes it may have catastrophic consequences for his REDUCE run Users are therefore advised to avoid such names Identifiers are used as variables labels and to name arrays operators and proce dures Restrictions The reserved words listed in another section may not be used as identifiers No spaces may appear within an identifier and an identifier may not extend over a line of text Hyphenation of an identifier by using a reserved character as a hyphen before an end of line character is possible in some versions of REDUCE 22 CHAPTER 2 STRUCTURE OF PROGRAMS 2 4 Variables Every variable is named by an identifier and is given a specific type The type is of no concern to the ordinary user Most variables are allowed to have the default type called scalar These can receive as values the representation of any ordinary algebraic expression In the absence of such a value they stand for themselves Reserved Variables Several variables in REDUCE have particular properties which should not be changed by the user These variables include Ei Intended to represent the base of the natural logarithms log e if it occurs in an expression is automatically replaced by 1 If ROUNDED is on E is replaced by the value of E to the cu
43. nextprime 2 gt 2 nextprime 7 gt 5 nextprime 1000000 gt 1000003 whereas nextprime a gives a type error 7 1 10 RANDOM random n returns a random number r in the range 0 lt r lt n A type error occurs if the value of the argument is not a positive integer in algebraic mode or positive number in symbolic mode For example random 5 gt 3 random 1000 gt 191 whereas random a gives a type error 7 1 11 RANDOM NEW SEED random_new_seed n reseeds the random number generator to a sequence de termined by the integer argument n It can be used to ensure that a repeat able pseudo random sequence will be delivered regardless of any previous use of RANDOM or can be called early in a run with an argument derived from something variable such as the time of day to arrange that different runs of a REDUCE pro gram will use different random sequences When a fresh copy of REDUCE is first created it is as if random_new_seed 1 has been obeyed A type error occurs if the value of the argument is not a positive integer 7 112 REPART This returns the real part of an expression 1f that argument has an numerical value A non numerical argument is returned as an expression in the operators REPART and IMPART For example repart 1 i gt 1 repart atixb gt REPART A IMPART B 7 2 MATHEMATICAL FUNCTIONS 57 7 1 13 ROUND This operator returns the rounded value i e the neares
44. one of the following forms 1 the name of an operator for a single argument the operator is evaluated once with each element of object as its single argument an algebraic expression with exactly one free variable that is a variable pre ceded by the tilde symbol The expression is evaluated for each element of object where the element is substituted for the free variable a replacement rule of the form var gt rep where var is a variable a kernel without subscript and rep is an expression that contains var Rep is evaluated for each element of object where the element is substituted for var var may be optionally preceded by a tilde The rule form for function is needed when more than one free variable occurs The result of evaluating function is interpreted as a boolean value correspond ing to the conventions of REDUCE These values are composed with the leading operator of the input expression Examples select w gt 0 1 1 2 3 3 gt 1 2 3 select evenp deg w y part xty 75 0 list gt X75 10 X 3 Y72 5xXx Y 4 s ct evenp deg w x 2x 2 3x 3 4x 4 gt 4X 4 2X 2 68 CHAPTER 7 BUILT IN PREFIX OPERATORS 7 10 SOLVE Operator SOLVE is an operator for solving one or more simultaneous algebraic equations It is used with the syntax SOLVE EXPRN algebraic VAR kernel VARLIST list of kernels fast EXPRN is of the form lt expression gt or lt expressionl1
45. printed NOTE When DEEN is on input is not evaluated 204 CHAPTER 18 REDUCE AND RLISP UTILITIES Chapter 19 Maintaining REDUCE REDUCE continues to evolve both in terms of the number of facilities available and the power of the individual facilities Corrections are made as bugs are discov ered and awkward features simplified In order to provide users with easy access to such enhancements a REDUCE network library has been established from which material can be extracted by anyone with electronic mail access to the Internet computer network In addition to miscellaneous documents source and utility files the library in cludes a bibliography of papers referencing REDUCE which contains over 800 entries Instructions on using this library are sent to all registered REDUCE users who provide a network address If you would like a more complete list of the con tents of the library send to reduce netlib rand org the single line message send index or help The current REDUCE information package can be obtained from the network library by including on a separate line send info package and a demonstra tion file by including the line send demonstration If you prefer hard copies of the information package and the bibliography are available from the REDUCE secre tary at RAND 1700 Main Street P O Box 2138 Santa Monica CA 90407 2138 reduce rand org Copies of the network library are also maintained at other ad dresses At the tim
46. procedure body will be carried out as if X had the same value as U does Y the same value as pxq does and Z the value 123 The values of X Y Z after the procedure body operations are completed are unchanged So normally are the values of U P O and of course 123 This is technically referred to as call by value The reader will have noted the word normally a few lines earlier The call by value protections can be bypassed if necessary as described elsewhere 14 2 PROCEDURE BODY 151 14 2 Procedure Body Following the delimiter that ends the procedure heading must be a single statement defining the action to be performed or the value to be delivered A terminator must follow the statement If 1t is a semicolon the name of the procedure just defined is printed It is not printed if a dollar sign is used If the result wanted is given by a formula of some kind the body is just that for mula using the variables in the procedure heading Simple Example If f x isto mean x 5 x x 6 x 7 the entire procedure definition could read procedure f x x 5 x x 6 x 7 5 Then f 10 would evaluate to 240 17 f a 6 to Ax A 1 A 1 and so on More Complicated Example Suppose we need a function p n x that for any positive integer N is the Legen dre polynomial of order n We can define this operator using the textbook formula defining these functions 1 dr 1 n dy y2 2ey 1 2 l 0 Pr
47. saving and retrieving previously evaluated expressions The simplest of these refers to the last algebraic expression simpli fied When an assignment of an algebraic expression is made or an expression is evaluated at the top level i e not inside a compound statement or procedure the results of the evaluation are automatically saved in a variable WS that we shall refer to as the workspace More precisely the expression is assigned to the variable WS that is then available for further manipulation 8 3 OUTPUT OF EXPRESSIONS 85 Example If we evaluate the expression x y 2 at the top level and next wish to differen tiate it with respect to Y we can simply say df ws y to get the desired answer If the user wishes to assign the workspace to a variable or expression for later use the SAVEAS statement can be used It has the syntax SAVEAS lt expression gt For example after the differentiation in the last example the workspace holds the expression 2xx 2x y If we wish to assign this to the variable Z we can now say saveas Z If the user wishes to save the expression in a form that allows him to use some of its variables as arbitrary parameters the FOR ALL command can be used Example for all x saveas h x with the above expression would mean that h z evaluates to 2xY 2xZ A further method for referencing more than the last expression is described in the section on interactive use of REDUCE 8 3 Outp
48. system requirements x 4xy The non existence of a formal solution is caused by a contradiction which disap pears only if the parameters of the initial system are set such that all members of the requirements list take the value zero For a linear system the set is complete a solution of the requirements list makes the initial system solvable E g in the above case a substitution x 4y makes the equation set consistent For a non linear system only one inconsistency is detected If such a system has more than one inconsistency you must reduce them one after the other The set shows you also the dependency among the parameters here one of x and y is free and a formal solution of the system can be computed by adding it to the variable list of solve The requirement set is not unique there may be other such sets A system with parameters may have a formal solution e g solve x axztl 0 b z y z x y axy b z 1X b b The difference between linear and non linear inconsistent systems is based on the algorithms which produce this information as a side effect when attempting to find a formal solution example solve x a x b y c y d x y gives a set a b c d while solve x a x b y c y d x y leads to a b 7 10 SOLVE OPERATOR 75 which is not valid for all possible values of the parameters The variable assumptions contains then a list of re
49. that two equivalent expressions need not necessarily simplify to the same form Example With EXP on the two expressions atb a 2xb and a 2 3xaxb 2xb 2 will both simplify to the latter form With EXP off they would remain unchanged unless the complete factoring ALLFAC option were in force EXP is normally on Several operators that expect a polynomial as an argument behave differently when EXP is off since there is often only one term at the top level For example with EXP off length a b c 3 ctd returns the value 1 9 2 Factorization of Polynomials REDUCE is capable of factorizing univariate and multivariate polynomials that have integer coefficients finding all factors that also have integer coefficients The package for doing this was written by Dr Arthur C Norman and Ms P Mary Ann Moore at The University of Cambridge It is described in P M A Moore and A C Norman Implementing a Polynomial Factorization and GCD Package Proc SYMSAC 81 ACM New York 1981 109 116 The easiest way to use this facility 1s to turn on the switch FACTOR which causes all expressions to be output in a factored form For example with FACTOR on the expression A 2 B 2 is returned as A B A B 9 2 FACTORIZATION OF POLYNOMIALS 105 It is also possible to factorize a given expression explicitly The operator FACTORIZE that invokes this facility is used with the syntax FACTO
50. the following example presenting one possible definition of a process to calculate the factorial of N for preassigned N will suffice Example begin scalar m m 1 l if n 0 then return m m mxn n n 1 go to 1 end 5 7 2 Labels and GO TO Statements Within a BEGIN END compound statement it is possible to label statements and transfer to them out of sequence using GO TO statements Only statements on the top level inside compound statements can be labeled not ones inside subsidiary 46 CHAPTER 5 STATEMENTS constructions like lt lt gt gt IF THEN WHILE DO etc Labels and GO TO statements have the syntax lt go to statement gt GO TO lt label gt GOTO lt label gt lt label gt lt identifier gt lt labeled statement gt lt label gt lt statement gt Note that statement names cannot be used as labels While GO TO is an unconditional transfer it is frequently used in conditional state ments such as if x gt 5 then go to abcd giving the effect of a conditional transfer Transfers using GO TOs can only occur within the block in which the GO TO is used In other words you cannot transfer from an inner block to an outer block us ing a GO TO However if a group statement occurs within a compound statement it is possible to jump out of that group statement to a point within the compound statement using a GO TO 5 7 3 RETURN St
51. the operator at a later stage In those cases where the evaluation process leaves an operator expression with non trivial arguments the form of the argument can vary depending on the state of the system at the point of evaluation Such arguments are normally produced in expanded form with no terms factored or grouped in any way For example the expression cos 2xx 2x y will normally be returned in the same form If the argument 2xx 2xy were evaluated at the top level however it would be printed as 2x X Y If it is desirable to have the arguments themselves in a similar form the switch INTSTR for internal structure if on will cause this to happen In cases where the arguments of the kernel operators may be reordered the sys tem puts them in a canonical order based on an internal intrinsic ordering of the variables However some commands allow arguments in the form of kernels and the user has no way of telling what internal order the system will assign to these arguments To resolve this difficulty we introduce the notion of a kernel form as an expression that transforms to a kernel on evaluation Examples of kernel forms are a cos xx y log sin x whereas axb a b 4 are not We see that kernel forms can usually be used as generalized variables and most algebraic properties associated with variables may also be associated with kernels 8 2 The Expression Workspace Several mechanisms are available for
52. user documentation In doing this build as with the production of a Standard Lisp form of such state ments it is important to remember that some of the commands must be instantiated during the building process For example macros must be expanded and some property list operations must happen The REDUCE sources should be consulted for further details on this To avoid excessive printout input statements should be followed by a instead of the semicolon With LOAD however the input doesn t print out regardless of which terminator is used with the command 18 3 THE STANDARD LISP CROSS REFERENCE PROGRAM 201 If you subsequently change the source files used in producing a fast loading file don t forget to repeat the above process in order to update the fast loading file correspondingly Remember also that the text which is read in during the creation of the fast load file in the compiling process described above is not stored in your REDUCE environment but only translated and output If you want to use the file just created you must then use LOAD to load the output of the fast loading file generation program When the file to be loaded contains a complete package for a given application LOAD_PACKAGE rather than LOAD should be used The syntax is the same How ever LOAD PACKAGE does some additional bookkeeping such as recording that this package has now been loaded that is required for the correct operation of the system
53. usually much easier to use then ED or EDITDEF 12 3 Interactive File Control If input is coming from an external file the system treats it as a batch processed calculation If the user desires interactive response in this case he can include the command on int in the file Likewise he can issue the command off int in the main program if he does not desire continual questioning from the system Regardless of the setting of INT input commands from a file are not kept in the system and so cannot be edited using ED However many implementations of RE DUCE provide a link to an external system editor that can be used for such editing The specific instructions for the particular implementation should be consulted for information on this Two commands are available in REDUCE for interactive use of files PAUSE may be inserted at any point in an input file When this command is encountered on input the system prints the message CONT on the user s terminal and halts If the user responds Y for yes the calculation continues from that point in the file If the user responds N for no control is returned to the terminal and the user can input further statements and commands Later on he can use the command cont to transfer control back to the point in the file following the last PAUSE encountered A top level pause from the user s terminal has no effect 140 CHAPTER 12 COMMANDS FOR INTERACTIVE USE Chapter 13 Matrix C
54. will cause h 5 to be evaluated as 0 but H of a positive integer or of an argument that is not an integer at all would not be affected Any boolean expression can follow the SUCH THAT keywords 124 CHAPTER 10 SUBSTITUTION COMMANDS 10 2 3 Removing Assignments and Substitution Rules The user may remove all assignments and substitution rules from any expression by the command CLEAR in the form CLEAR lt expression gt lt expression gt lt terminator gt e g clear x h x y Because of their instant evaluation property array and matrix elements cannot be cleared with CLEAR For example if A is an array you must say a 3 0 rather than clear a 3 to clear element a 3 On the other hand a whole array or matrix A can be cleared by the command clear a This means much more than resetting to O all the elements of A The fact that A is an array and what its dimensions are are forgotten so A can be redefined as another type of object for example an operator The more general types of LET declarations can also be deleted by using CLEAR Simply repeat the LET rule to be deleted using CLEAR in place of LET and omit ting the equal sign and right hand part The same dummy variables must be used in the FOR ALL part and the boolean expression in the SUCH THAT part must be written the same way The placing of blanks doesn t have to be identical Example The LET rule for all x such that numberp
55. x and x lt 0 let h x 0 can be erased by the command for all x such that numberp x and x lt 0 clear h x 10 2 4 Overlapping LET Rules CLEAR is not the only way to delete a LET rule A new LET rule identical to the first but with a different expression after the equal sign replaces the first 10 2 LET RULES 125 Replacements are also made in other cases where the existing rule would be in conflict with the new rule For example a rule for x 4 would replace a rule for x 5 The user should however be cautioned against having several LET rules in effect that relate to the same expression No guarantee can be given as to which rules will be applied by REDUCE or in what order It is best to CLEAR an old rule before entering a new related LET rule 10 2 5 Substitutions for General Expressions The examples of substitutions discussed in other sections have involved very sim ple rules However the substitution mechanism used in REDUCE is very general and can handle arbitrarily complicated rules without difficulty The general substitution mechanism used in REDUCE is discussed in Hearn A C REDUCE A User Oriented Interactive System for Algebraic Simplification Interactive Systems for Experimental Applied Mathematics edited by M Klerer and J Reinfelds Academic Press New York 1968 79 90 and Hearn A C The Problem of Substitution Proc 1968 Summer Institute on Symbolic Mathe matical Computation IBM Pr
56. 0 17 32 Dirac Expressions o eo qo catas 17 4 Trace Calculations 17 5 Mass Declarations 170 Example ocios dada oa a 17 7 Extensions to More Than Four Dimensions 18 REDUCE and Rlisp Utilities 18 1 The Standard Lisp Compiler 18 2 Fast Loading Code Generation Program 18 3 The Standard Lisp Cross Reference Program 18 3 1 Restrictions o e oe e doa e a a 16 3 2 Usage 2 ee a we E E A 1833 Opos ood a SS os ea eS ele 18 4 Prettyprinting Reduce Expressions 18 5 Prettyprinting Standard Lisp S Expressions 19 Maintaining REDUCE A Reserved Identifiers CONTENTS Abstract This document provides the user with a description of the algebraic programming system REDUCE The capabilities of this system include 10 11 12 expansion and ordering of polynomials and rational functions substitutions and pattern matching in a wide variety of forms automatic and user controlled simplification of expressions calculations with symbolic matrices arbitrary precision integer and real arithmetic facilities for defining new functions and extending program syntax analytic differentiation and integration factorization of polynomials facilities for the solution of a variety of algebraic equations facilities for the output of expressions in a variety of formats facilities for generating numerical programs from symbolic input Dirac matrix calculations of
57. 11 4 Complex Number Coefficients in Polynomials Although REDUCE routinely treats the square of the variable i as equivalent to 1 this is not sufficient to reduce expressions involving i to lowest terms or to factor such expressions over the complex numbers For example in the default case factorize a 2 1 gives the result Ax 2 1 1 and a 2 b 2 atixb 118 CHAPTER 9 POLYNOMIALS AND RATIONALS is not reduced further However if the switch COMP LEX is turned on full complex arithmetic is then carried out In other words the above factorization will give the result A T I 1 A la I 1 and the quotient will be reduced to A I xB The switch COMP LEX may be combined with ROUNDED to give complex real num bers the appropriate arithmetic is performed in this case Complex conjugation is used to remove complex numbers from denominators of expressions To do this if COMPLEX is off you must turn the switch RATIONALIZE on Chapter 10 Substitution Commands An important class of commands in REDUCE define substitutions for variables and expressions to be made during the evaluation of expressions Such substitutions use the prefix operator SUB various forms 10 1 SUB Operator Syntax of the command LET and rule sets SUB lt substitution_list gt EXPRN1 algebraic algebraic where lt substitution_list gt isa VAR kernel EXPRN a or a kernel that evaluates to such a list
58. 18 3 The Standard Lisp Cross Reference Program CREF is a Standard Lisp program for processing a set of Standard LISP function definitions to produce 1 A summary showing a A list of files processed b A list of entry points functions which are not called or are only called by themselves c A list of undefined functions functions called but not defined in this set of functions d A list of variables that were used non locally but not declared GLOBAL or FLUID before their use e A list of variables that were declared GLOBAL but not used as FLUIDs i e bound in a function f A list of FLUID variables that were not bound in a function so that one might consider declaring them GLOBALs g A list of all GLOBAL variables present h A list of all FLUID variables present i A list of all functions present 2 A global variable usage table showing for each non local variable a Functions in which it is used as a declared FLUID or GLOBAL b Functions in which it is used but not declared c Functions in which it is bound 202 CHAPTER 18 REDUCE AND RLISP UTILITIES d Functions in which it is changed by SETO 3 A function usage table showing for each function a Where it is defined b Functions which call this function c Functions called by it d Non local variables used The program will also check that functions are called with the correct n
59. 2 will result in the algebraic value X 12 The use of SYMBOLIC and equivalently ALGEBRAIC in this manner is the same as any operator That means that parentheses could be omitted in the above ex amples since the meaning is obvious In other cases parentheses must be used as in symbolic x a Omitting the parentheses as in symbolic x a would be wrong since it would parse as symbolic x a For convenience it is assumed that any operator whose first argument is quoted is being evaluated in symbolic mode regardless of the mode in effect at that time Thus the first example above could be equally well written car a Except where explicit limitations have been made most REDUCE algebraic con structions carry over into symbolic mode However there are some differences First expression evaluation now becomes Lisp evaluation Secondly assignment statements are handled differently as we shall discuss shortly Thirdly local vari ables and array elements are initialized to NIL rather than 0 In fact any variables not explicitly declared INTEGER are also initialized to NIL in algebraic mode but the algebraic evaluator recognizes NIL as 0 Finally function definitions follow the conventions of Standard Lisp To begin with we mention a few extensions to our basic syntax which are designed primarily if not exclusively for symbolic mode 16 1 SYMBOLIC INFIX OPERATORS 179 16 1 Symbolic
60. 2 2 2 X 2xXxY 2xXxZ Y 2xYxZ Z Let us review this simple example to learn a little more about the way that RE DUCE works First we note that REDUCE deals with variables and constants like other computer languages but that in evaluating the former a variable can stand for itself Expression evaluation normally follows the rules of high school algebra so the only surprise in the above example might be that the expression was expanded REDUCE normally expands expressions where possible collecting like terms and ordering the variables in a specific manner However expansion order ing of variables format of output and so on is under control of the user and various declarations are available to manipulate these Another characteristic of the above example is the use of lower case on input and upper case on output In fact input may be in either mode but output is usually in lower case To make the difference between input and output more distinct in this manual all expressions intended for input will be shown in lower case and output in upper case However for stylistic reasons we represent all single identifiers in the text in upper case Finally the numerical prompt can be used to reference the result in a later compu tation As a further illustration of the system features the user should try for i 1 40 product i The result in this case is the value of 40 815915283247897734345611269596115894272000000000 You can al
61. 37 5 61 gt 0 0480793490914 0 998843519372xI cos 2 31 gt 4 18962569097 9 10922789376x1I 7 3 DF OPERATOR 61 7 3 DF Operator The operator DF is used to represent partial differentiation with respect to one or more variables It is used with the syntax DF EXPRN algebraic VAR kernel lt NUM integer gt algebraic The first argument is the expression to be differentiated The remaining arguments specify the differentiation variables and the number of times they are applied The number NUM may be omitted if it is 1 For example df y x 0y 0x df y X 2 8 y x df y x1 2 X2 X3 2 0 y 0x 022023 The evaluation of df y x proceeds as follows first the values of Y and X are found Let us assume that X has no assigned value so its value is X Each term or other part of the value of Y that contains the variable X is differentiated by the standard rules If Z is another variable not X itself then its derivative with respect to X is taken to be 0 unless Z has previously been declared to DEPEND on X in which case the derivative is reported as the symbol df z x 7 3 1 Adding Differentiation Rules The LET statement can be used to introduce rules for differentiation of user defined operators Its general form is FOR ALL lt varl gt lt varn gt LET DF lt operator gt lt varlist gt lt vari gt lt expression gt where lt varlist gt lt varl gt lt var
62. ATORS Beside these identities there are a lot of simplifications for elementary funct ions defined in the REDUCE system as rulelists In order to view these the SHOWRULES operator can be used e g SHOWRULES tan tan nxarbint i pi x gt tan x when fixp n tan 7x gt trigquot sin x cos x when knowledge_about sin x tan Tx 7 k pi tan ie x k 1 gt cot when x freeof pi and abs d d 2 w k xpi w remainder k d pi tan gt tan d d k when w freeof pi and ratnump and fixp k d k and abs gt 1 a tan atan x gt x 2 df tan x x gt 1 tan x For further simplification especially of expressions involving trigonometric funct ions see the TRIGSIMP package documentation 7 2 MATHEMATICAL FUNCTIONS 59 Functions not listed above may be defined in the special functions package SPECEN The user can add further rules for the reduction of expressions involving these operators by using the LET command In many cases it is desirable to expand product arguments of logarithms or collect a sum of logarithms into a single logarithm Since these are inverse operations it is not possible to provide rules for doing both at the same time and preserve the REDUCE concept of idempotent evaluation As an alternative REDUCE provides two switches EXPANDLOGS and COMBINELOGS to carry out these operations
63. BS ABS returns the absolute value of its single argument if that argument has a nu merical value A non numerical argument is returned as an absolute value with an 53 54 CHAPTER 7 BUILT IN PREFIX OPERATORS overall numerical coefficient taken outside the absolute value operator For exam ple abs 3 4 gt 3 4 abs 2a gt 2x ABS A abs 1 gt 1 abs x gt ABS X 7 1 2 CEILING This operator returns the ceiling i e the least integer greater than the given argu ment if its single argument has a numerical value A non numerical argument is returned as an expression in the original operator For example ceiling 5 4 gt 1 ceiling a gt CEILING A 7 1 3 CONJ This returns the complex conjugate of an expression if that argument has an numer ical value A non numerical argument is returned as an expression in the operators REPART and IMPART For example conj 1 i gt 1 I conJ a ixb gt REPART A REPART B x1I IMPART A I IMPART B 7 1 4 FACTORIAL If the single argument of FACTORIAL evaluates to a non negative integer its fac torial is returned Otherwise an expression involving FACTORIAL is returned For example factorial 5 gt 120 factorial a gt FACTORIAL A 7 1 5 FIX This operator returns the fixed value i e the integer part of the given argument if its single argument has a numerical value A non numerical
64. D delete next character E end editing and reread text F lt character gt move pointer to next occurrence of character I lt string gt lt escape gt insert string in front of pointer K lt character gt delete all characters until character P print string from current pointer Q give up with error exit S lt string gt lt escape gt search for first occurrence of string posi tioning pointer just before it space or X move pointer right one character 12 3 INTERACTIVE FILE CONTROL 139 The above table can be displayed online by typing a question mark followed by a carriage return to the editor The editor prompts with an angle bracket Commands can be combined on a single line and all command sequences must be followed by a carriage return to become effective Thus to change the command x a 1 to x a 2 and cause it to be executed the following edit command sequence could be used flc2e lt return gt The interactive editor may also be used to edit a user defined procedure that has not been compiled To do this one says editdef lt id gt where lt id gt is the name of the procedure The procedure definition will then be displayed in editing mode and may then be edited and redefined on exiting from the editor Some versions of REDUCE now include input editing that uses the capabilities of modern window systems Please consult your system dependent documentation to see if this is possible Such editing techniques are
65. If both sides are to be evaluated the switch EVALLHSEOP should be turned on To facilitate the handling of equations two selectors LHS and RHS which return the left and right hand sides of a equation respectively are provided For example lhs atb c gt atb and rhs at tb c gt c 3 5 Proper Statements as Expressions Several kinds of proper statements deliver an algebraic or numerical result of some kind which can in turn be used as an expression or part of an expression For example an assignment statement itself has a value namely the value assigned So 2 x x atb is equal to 2 a b as well as having the side effect of assigning the value a b to X In context y i 2 x x a b sets X to a b and Y to 2x a b The sections on the various proper statement types indicate which of these state ments are also useful as expressions 32 CHAPTER 3 EXPRESSIONS Chapter 4 Lists A list is an object consisting of a sequence of other objects including lists them selves separated by commas and surrounded by braces Examples of lists are a b c 1 a b c d a b c d e The empty list is represented as 4 1 Operations on Lists Several operators in the system return their results as lists and a user can create new lists using braces and commas Alternatively one can use the operator LIST to construct a list An important class of operations on lists are MAP and SE
66. Infix Operators There are three binary infix operators in REDUCE intended for use in symbolic mode namely CONS EQ and MEMO The precedence of these operators was given in another section 16 2 Symbolic Expressions These consist of scalar variables and operators and follow the normal rules of the Lisp meta language Examples x car u reverse v simp u v 2 16 3 Quoted Expressions Because symbolic evaluation requires that each variable or expression has a value it is necessary to add to REDUCE the concept of a quoted expression by analogy with the Lisp QUOTE function This is provided by the single quote mark For example a represents the Lisp S expression quote a a b c represents the Lisp S expression quote a b c Note however that strings are constants and therefore evaluate to themselves in symbolic mode Thus to print the string A String one would write prin2 A String Within a quoted expression identifier syntax rules are those of REDUCE Thus A B is the list consisting of the three elements A and B whereas A B is the dotted pair of A and B 16 4 Lambda Expressions LAMBDA expressions provide the means for constructing Lisp LAMBDA expres sions in symbolic mode They may not be used in algebraic mode 180 CHAPTER 16 SYMBOLIC MODE Syntax lt LAMBDA expression gt LAMBDA lt varlist gt lt terminator gt lt statement gt where lt varlist gt
67. LECT operations For details please refer to the chapters on MAP SELECT and the FOR command See also the documentation on the ASSIST package To facilitate the use of lists a number of operators are also available for manipu lating them PART lt list gt n for example will return the nt element of a list LENGTH will return the length of a list Several operators are also defined uniquely for lists For those familiar with them these operators in fact mirror the operations defined for Lisp lists These operators are as follows 33 34 CHAPTER 4 LISTS 4 1 1 LIST The operator LIST is an alternative to the usage of curly brackets LIST accepts an arbitrary number of arguments and returns a list of its arguments This operator is useful in cases where operators have to be passed as arguments E g list a list list b c d e gt tajp b e d e 4 1 2 FIRST This operator returns the first member of a list An error occurs if the argument is not a list or the list is empty 4 1 3 SECOND SECOND returns the second member of a list An error occurs if the argument is not a list or has no second element 4 1 4 THIRD This operator returns the third member of a list An error occurs if the argument is not a list or has no third element 4 1 5 REST REST returns its argument with the first element removed An error occurs if the argument is not a list or is empty 4 1 6 Cons Operator This operator ad
68. MP and Mathematica and is based on the pattern matcher described in Kevin MclIsaac Pattern Matching Algebraic Identities SIGSAM Bulletin 19 1985 4 13 Documentation for this package is in plain text Author Kevin MclIsaac 15 38 RANDPOLY A RANDOM POLYNOMIAL GENERATOR 169 15 38 RANDPOLY A random polynomial generator 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 Author Francis J Wright 15 39 REACTEQN Support for chemical reaction equat ion systems This package allows a user to transform chemical reaction systems into ordinary differential equation systems ODE corresponding to the laws of pure mass action Documentation for this package is in plain text Author Herbert Melenk 15 40 RESET Code to reset REDUCE to its initial state This package defines a 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 There is no further documentation on this package Author John Fitch 15 41 RESIDUE A residue package This package supports the calcul
69. ND or if the switch FORCE is on without the property list flag NOEXPAND will be expanded before the definition is seen by the cross reference program so this flag can also be used to select those macros you require expanded and those you do not 18 4 Prettyprinting Reduce Expressions REDUCE includes a module for printing REDUCE syntax in a standard format This module is activated by the switch PRET which is normally off Since the system converts algebraic input into an equivalent symbolic form the printing program tries to interpret this as an algebraic expression before printing it In most cases this can be done successfully However there will be occasional instances where results are printed in symbolic mode form that bears little resem blance to the original input even though it is formally equivalent If you want to prettyprint a whole file say off output msg and hopefully only clean output will result Unlike DEFN input is also evaluated with PRET on 18 5 Prettyprinting Standard Lisp S Expressions REDUCE includes a module for printing S expressions in a standard format The Standard Lisp function for this purpose is PRETTYPRINT which takes a Lisp ex pression and prints the formatted equivalent Users can also have their REDUCE input printed in this form by use of the switch DEEN This is in fact a convenient way to convert REDUCE or Rlisp syntax into Lisp off msg will prevent warning messages from being
70. NER is a package for the computation of Gr bner Bases using the Buch berger algorithm and related methods for polynomial ideals and modules It can be used over a variety of different coefficient domains and for different variable and term orderings Gr bner Bases can be used for various purposes in commutative algebra e g for elimination of variables converting surd expressions to implicit polynomial form computation of dimensions solution of polynomial equation systems etc The package is also used internally by the SOLVE operator Authors Herbert Melenk H M Moller and Winfried Neun 164 CHAPTER 15 USER CONTRIBUTED PACKAGES 15 23 IDEALS Arithmetic for polynomial ideals This package implements the basic arithmetic for polynomial ideals by exploiting the Gr bner bases package of REDUCE In order to save computing time all inter mediate Gr bner bases are stored internally such that time consuming repetitions are inhibited Author Herbert Melenk 15 24 INEQ Support for solving inequalities This package supports the operator ineq_solve that tries to solves single inequalities and sets of coupled inequalities Author Herbert Melenk 15 25 INVBASE A package for computing involutive bases Involutive bases are a new tool for solving problems in connection with multivari ate polynomials such as solving systems of polynomial equations and analyzing polynomial ideals An involutive basis of polynomial ideal is noth
71. NOMIAL COEFFICIENT ARITHMETIC 117 9 11 3 Modular Number Coefficients in Polynomials REDUCE includes facilities for manipulating polynomials whose coefficients are computed modulo a given base To use this option two commands must be used SETMOD lt integer gt to set the prime modulus and ON MODULAR to cause the actual modular calculations to occur For example with setmod 3 and on modular the polynomial a 2xb 3 would become A 3 2xB 3 The argument of SETMOD is evaluated algebraically except that non modular in teger arithmetic is used Thus the sequence setmod 3 on modular setmod 7 will correctly set the modulus to 7 Modular numbers are by default represented by integers in the interval 0 p 1 where p is the current modulus Sometimes it is more convenient to use an equiv alent symmetric representation in the interval p 2 1 p 2 or more precisely floor p 1 2 ceiling p 1 2 especially if the modular numbers map objects that include negative quantities The switch BALANCED_MOD allows you to select the symmetric representation for output Users should note that the modular calculations are on the polynomial coefficients only It is not currently possible to reduce the exponents since no check for a prime modulus is made which would allow x 7 to be reduced to 1 mod p Note also that any division by a number not co prime with the modulus will result in the error Invalid modular division 9
72. ON LST list DO COLLECT JOIN PRODUCT SUM EXPRN S expr As in algebraic mode if the keyword IN is used iteration is on each element of the list With ON iteration is on the whole list remaining at each point in the iteration As a result we have the following equivalence between each form of FOR EACH and the various mapping functions in Lisp DO COLLECT JOIN IN MAPC MAPCAR MAPCAN ON MAP MAPLIST MAPCON Example To list each element of the list a b c for each x in abc collect list x 16 7 Symbolic Procedures All the functions described in the Standard Lisp Report are available to users in symbolic mode Additional functions may also be defined as symbolic procedures For example to define the Lisp function ASSOC the following could be used symbolic procedure assoc u v if null v then nil else if u caar v then car v else assoc u cdr v If the default mode were symbolic then SYMBOLIC could be omitted in the above definition MACROs may be defined by prefixing the keyword PROCEDURE by the word MACRO In fact ordinary functions may be defined with the keyword EXPR prefixing PROCEDURE as was used in the Standard Lisp Report For example we could define a MACRO CONSCONS by symbolic macro procedure conscons l expand cdr 1 cons 182 CHAPTER 16 SYMBOLIC MODE Another form of macro the SMACRO is also available These are described in the Standard Lisp
73. R switch for this purpose rather it is a separation command All terms involving fixed powers of the declared expressions are printed as a product of the fixed powers and a sum of the rest of the terms All expressions involving a given prefix operator may also be factored by putting the operator name in the list of factored identifiers For example factor x cos sin x causes all powers of X and SIN X and all functions of COS to be factored Note that FACTOR does not affect the order of its arguments You should also use ORDER if this is important The declaration remfac vl vnj removes the factoring flag from the ex pressions v1 through vn 8 3 3 Output Control Switches In addition to these declarations the form of the output can be modified by switch ing various output control switches using the declarations ON and OFF We shall illustrate the use of these switches by an example namely the printing of the ex pression X72 y 2 2x y xx y 2 z 2 a The relevant switches are as follows ALLFAC Switch This switch will cause the system to search the whole expression or any sub expression enclosed in parentheses for simple multiplicative factors and print them outside the parentheses Thus our expression with ALLFAC off will print as 2 E 2 2 2xX xY xA 4xX xYxA XxY XxZ 2xA and with ALLFAC on as 88 CHAPTER 8 DISPLAY AND STRUCTURING OF EXPRESSIONS 2 2 Xx 2x Xx Y xA 4xXx YxA Y Z 2
74. RATIONALIZB 118 RATPRI 90 REACTEON 169 REAL 44 Real 20 21 Real coefficient 115 REDERR 153 REDUCT 114 REMAINDER 108 REMFAC 87 REMIND 192 REPART 54 56 REPEAT 43 45 47 requirements 74 Reserved variable 22 RESET 169 RESIDUE 169 REST 34 RESULTANT 109 RETRY 137 RETURN 44 46 REVERSE 35 REVPRI 90 RHS 31 RLFI 169 Rlisp 199 RLISP88 188 ROOT_OF 68 69 ROOTS 170 ROUND 57 ROUNDALL 116 ROUNDBF 116 ROUNDED 22 28 60 93 115 RSOLVE 170 Rule lists 127 SAVEAS 85 SAVESTRUCTR 98 INDEX Saving an expression 96 SCALAR 44 Scalar 27 SCIENTIFIC_NOTATION 20 SCOPE 170 SEC 57 60 SECH 57 60 SECOND 34 SELECT 67 Selector 183 Semicolon 37 SET 38 65 SETMOD 117 SETS 171 SHARE 182 SHOWRULES 132 SHOWTIME 51 SHUT 135 136 Side effect 31 SIGN 57 Simplification 28 83 SIN 57 60 SINH 57 60 Six jSymbol 171 SMACRO 182 SolidHarmonicy 171 SOLVE 68 69 73 163 SOLVES INGULAR 73 SPDE 171 SPECEN 59 171 SPECFN2 172 Spherical and Solid Harmonics 171 SphericalHarmonicy 171 SPUR 195 SORT 57 60 Standard form 183 Standard quotient 183 Statement 37 Stirling numbers 171 Stirlingl1 171 Stirling2 171 String 23 STRUCTR 96 98 Structuring 83 INDEX Struve functions 171 StruveH 171 StruvelL 171 SUB 31 119 S
75. REDUCE User s Manual Version 3 8 Anthony C Hearn Santa Monica CA USA Email reduce rand org February 2004 Copyright 2004 Anthony C Hearn All rights reserved Registered system holders may reproduce all or any part of this publication for internal purposes provided that the source of the material is clearly acknowledged and the copyright notice is retained Contents Abstract 1 Introductory Information 2 Structure of Programs 2 1 The REDUCE Standard Character Set 22 BM a ds A A ee eh Woh eB ee A er dd o A Be OUES o a A Oe a 26 COMMEME o a a da LE OPIO a a e E e 3 Expressions S A Scalar Expressions o oo ace uce O e ee 3 2 nteger EXPIESSIO OS coco eri ar a a a 33 Boblean Expressions soc oey para eda do da a gars SA EQUIBONS e aa a AAA A ee e 3 5 Proper Statements as Expressions 4 Lists 4 1 Operations on Lists o be eee ee raso ALA LIST 64 ae pate alee bea eke es el Alea FIRST e ends ala aia Seas EE hoe ae doe eed 11 15 19 19 19 21 22 23 23 24 27 27 28 29 30 31 ALS SECOND ocres bee dE wes 414 THIRD 2 ce ee ee ew vs ALS REST o nicas 4 1 6 Cons Operator Ll APPEND o aoe 6A ewe a 4 18 REVERSE cesses ee bw eda es 4 1 9 List Arguments of Other Operators 4 1 10 Caveats and Examples 5 Statements 5 1 Assignment Statements S11 Set Statement e e eee i es 5 2 GroupStatements o o aoc c
76. REDUCT Operator s so ee ee a 114 9 11 Polynomial Coefficient Arithmetic 115 9 11 1 Rational Coefficients in Polynomials 115 9 11 2 Real Coefficients in Polynomials 115 9 11 3 Modular Number Coefficients in Polynomials 117 9 11 4 Complex Number Coefficients in Polynomials 117 10 Substitution Commands 119 101 SUB Opeaior co ceda bade ee eed RA ee 119 102 LET Bales o o e ck ek eee A oe 120 10 21 FORADL DET ooo oca beet eee eee ase 122 10 2 2 POR ALL SUCH THAT LET ee ee ee we sos 123 10 2 3 Removing Assignments and Substitution Rules 124 10 2 4 Overlapping LET Rules 124 10 2 5 Substitutions for General Expressions 125 6 CONTENTS IS Ble Ligh 024 blo ORR e eR Ae aw Aad 127 10 4 Asymptotic Commands o 133 11 File Handling Commands 135 11 1 IN Command osos rd o o A a eS 135 11 2 OUT Command ooo cc 136 11 3 SHUT Command ie ro care p ea panas o Gad 136 12 Commands for Interactive Use 137 12 1 Referencing Previous Results o 137 12 2 Interactivo Haine oeoa a a ad a 138 12 3 Interactive File Control o o oa oo oca caese c barrer 139 13 Matrix Calculations 141 13 1 MAT Operator o o ca ke RR eee eS 141 132 Matrix Variables e deur a and eg ob es 142 13 3 Matriz EXPRESSIONS o ooe aa ee a ee a we wg aa 142 13 4 Operators with Matrix Arguments 143 1341 DEV pero lt o bab ae
77. RIZE EXPRN polynomial INTEXP prime integer the optional argument of which will be described later Thus to find and display all factors of the cyclotomic polynomial x 1 one could write factorize x 105 1 The result is a list of factor exponent pairs In the above example there is no overall numerical factor in the result so the results will consist only of polynomials in x The number of such polynomials can be found by using the operator LENGTH If there is a numerical factor as in factorizing 127 12 that factor will appear as the first member of the result It will however not be factored further Prime factors of such numbers can be found using a probabilistic algorithm by turning on the switch IFACTOR For example on ifactor factorize 12x 2 12 would result in the output 2 2 3 1 X T 1 17 1X AS AI A If the first argument of FACTORIZE is an integer it will be decomposed into its prime components whether or not IFACTOR is on Note that the IFACTOR switch only affects the result of FACTORIZE It has no effect if the FACTOR switch is also on The order in which the factors occur in the result with the exception of a possi ble overall numerical coefficient which comes first can be system dependent and should not be relied on Similarly it should be noted that any pair of individ ual factors can be negated without altering their product and that REDUCE may sometimes do that The fact
78. S 2 2 X xYx Y 2 Xx Y 2 2xA RAT is normally off Next if we leave X factored and turn on both DIV and RAT the result becomes 2 1 2 X Yx Y 2 1 2xXxA Y 2 Finally with X factored RAT on and ALLFAC off we retrieve the original structure 2 2 2 X x Y 2xY Xx Y Z 2 A RATPRI Switch If the numerator and denominator of an expression can each be printed in one line the output routines will print them in a two dimensional notation with numerator and denominator on separate lines and a line of dashes in between For example a b 2 will print as Turning this switch off causes such expressions to be output in a linear form REVPRI Switch The normal ordering of terms in output is from highest to lowest power In some situations e g when a power series is output the opposite ordering is more con venient The switch REVPRI if on causes such a reverse ordering of terms For example the expression yx x 1 2 y 3 2 will normally print as 2 2 X xY 2xXxY Y 7xY 9 whereas with REVPRI on it will print as 2 2 9 7xY Y 2xXxY X xY 8 3 OUTPUT OF EXPRESSIONS 91 8 3 4 WRITE Command In simple cases no explicit output command is necessary in REDUCE since the value of any expression is automatically printed if a semicolon is used as a delim iter There are however several situations in which such a command is useful In a FOR WHILE or REPEAT statement it
79. T S STRUCTR SUB SUM THIRD TP TRACE VARNAME CARD_NO E INTERPOL F LTERM MAINVAR NULLSPACE NUM PART RANDOM_NEW_SEED RANK EVAL MODE PF PRECISION I EPS ERF FACTORIZE FIRST GCD G IMPART INT LCM LCOF LENGTH LHS LINELENGT MAT MATEIGEN MAX MIN MKI RANDOM REDERR REDUCT H D REST RESULTANT REVERSE E HOWRULES SIGN SOLVE E FORT_WIDT HIGH_POW I INFINITY K x LOW_POW NII H PI ROOTMULTIPLICITY T ADJPREC Al GINT ALLBRANCH ALLFAC BFSPACE COMBINEEXPT COMBINELOGS COMP CRAMER CREF DEFN DEMO DIV EVALLHSEQP EXP LCM LIST MULTIPLICITIES NAT PERIOD PRECISE RATIONAL RLISP88 ROUNDALL SAVESTRUCTR SOLVES FORT FULLROOTS GCD LISTARGS PRET RATIONALIZE IFACTOR MCD MODULAR COMP LEX ECHO ERRCONT EXPANDLOGS EZGCD FACTOR INT INTSTR MSG NERO NOSPLIT OUTPUT PRI RAT RATARG RATPRI REVPR ROUNDBF ROUNDED NGULAR TIME TRA TRFAC TRIGFORM TRINT BEGIN DO EXPR FEXPR NPUT LAMBDA LISP MACRO PRODUCT REPEAT SMAC WS RO SUM UNTIL WHEN WHILE Index CONS 34 3j and 6j symbols 171 ABS 53 ACOS 57 60 ACOSH 57 60 ACOT 57 60 ACOTH 57 60 ACSC 57 60 ACSCH 57 60 ADJPREC 116 Airy f
80. TO OR FORALL FOREACH GO GOTO A CLEARRULES COMMENT CONT DECOMPOS D R IF IN INDEX INFIX INPUT INTEGER KORDER LE T LINEAR LISP LISTARGP LOAD LOAD PACKAGE MATRIX MSHELL NOSPUR ODD OFF ON OPERATOR ORI REAL REMFAC REMIND RETRY RI SCALAR SETMOD SHARE SHOWTIME MASS MATC NODEPEND NONCOM NONZERO DER OUT PAUSI H G ri PRECEDENCE PRINT_PRECISION PROCEDURE QUI ETURN SAVEAS SHUT SPUR SYMBOLIC SYMMETRIC VECDIM V T ECTOR WE IGH WRITE WTLEVEL EVENP FIXP FREEOF NUMBERP ORDP PRIMEP gt gt lt lt gt 4 MEMBER MEMO EQUAL NEO EQ GEQ T x xx WHERE SETQ OR AND GREATERP LEQ LESSP PLUS DIFFERENCE MINUS TIMES QUOTIEN EXPT CONS T ABS ACOS ACOSH ACOT ACOTH ACSC ACSCH ASEC ASECH ASIN ASINH ATAN ATANH ATAN2 COS COSH COT COTH CSC CSCH EXP FACTORIAL FIX FLOOR HYPOT LN LOG LOGB LOG10 NEXTPR SECH SIN SINH SORT TAN TANH 207 ME ROUND SEC 208 Prefix Operators Reserved Variables Switches Other Reserved Ids APPENDIX A RESERVED IDENTIFIERS APPEND ARGLENGTH CEILING COEFF COEFFN COFACTOR CONJ DEG DEN DET DF DILOG E 1 REMAINDER REPART RHS SECOND SE
81. Thus y 2 1 refers to the element in the second row and first column of the matrix Y 148 CHAPTER 13 MATRIX CALCULATIONS Chapter 14 Procedures It is often useful to name a statement for repeated use in calculations with varying parameters or to define a complete evaluation procedure for an operator REDUCE offers a procedural declaration for this purpose Its general syntax is lt procedural type gt PROCEDURE lt name gt lt varlist gt lt statement gt where lt varlist gt lt variable gt lt variable gt This will be explained more fully in the following sections In the algebraic mode of REDUCE the lt procedure type gt can be omitted since the default is ALGEBRAIC Procedures of type INTEGER or REAL may also be used In the former case the system checks that the value of the procedure is an integer At present such checking is not done for a real procedure although this will change in the future when a more complete type checking mechanism is installed Users should therefore only use these types when appropriate An empty variable list may also be omitted All user defined procedures are automatically declared to be operators In order to allow users relatively easy access to the whole REDUCE source pro gram system procedures are not protected against user redefinition If a procedure is redefined a message xxx lt procedure name gt REDEFINED is printed If this occurs
82. ULATIONS is a scalar expression whose value is the determinant of the matrix OQ roo BHO Determinant expressions have the instant evaluation property In other words the statement let det mat a b c d 2 sets the value of the determinant to 2 and does not set up a rule for the determinant itself 13 4 2 MATEIGEN Operator Syntax MATEIGEN EXPRN matrix_expression 1ID list MATE IGEN calculates the eigenvalue equation and the corresponding eigenvectors of a matrix using the variable ID to denote the eigenvalue A square free decom position of the characteristic polynomial is carried out The result is a list of lists of 3 elements where the first element is a square free factor of the characteristic polynomial the second its multiplicity and the third the corresponding eigenvector as an n by 1 matrix If the square free decomposition was successful the product of the first elements in the lists is the minimal polynomial In the case of degener acy several eigenvectors can exist for the same eigenvalue which manifests itself in the appearance of more than one arbitrary variable in the eigenvector To extract the various parts of the result use the operations defined on lists Example The command mateigen mat 2 1 1 0 1 1 1 1 1 eta gives the output TETA 1 2 ARBCOMP LEX 1 ARBCOMP LEX 1 13 4 OPERATORS WITH MATRIX ARGUMENTS hy ETA D 1
83. X 118 Complex coefficient 117 Compound statement 43 45 Conditional statement 39 40 CONJ 54 Constructor 183 CONT 139 COS 57 60 COSH 57 60 COT 57 60 COTH 57 60 CRACK 160 CRAMER 68 143 CREF 201 202 Cross reference 201 CSC 57 60 CSCH 57 60 CVIT 161 Declaration 49 DECOMPOSE 110 DEFINE 51 52 DEF INT 161 DEFN 182 203 DEG 111 INDEX Degree 112 DEMO 50 DEN 103 112 DEPEND 82 depend 76 DESIR 161 DET 83 143 DF 61 62 DFPART 161 Differentiation 61 62 82 Digamma 171 Digamma function 171 DILOG 57 63 Dilog 171 Dilogarithm function 171 Dirac y matrix 192 DISPLAY 138 Display 83 Displaying structure 96 DIV 88 115 DO 40 42 Dollar sign 37 Dot product 191 DUMMY 162 E 22 ECHO 135 ED 137 138 EDITDEF 139 Ei 57 Elliptick 171 EllipticF 171 EllipticTheta 171 END 51 EPS 193 Equation 30 31 ERF 63 ERRCONT 137 Euler 171 Euler numbers 171 Euler polynomials 171 EulerP 171 EVAL_MODE 177 EVALLHSEOP 31 INDEX EVEN 78 Even operator 78 EVENP 29 EXCALC 162 Exclamation mark 19 EXP 57 60 63 104 107 EXPAND CASES 69 EXPANDLOGS 59 EXPR 181 Expression 27 EZGCD 107 FACTOR 87 104 105 FACTORIAL 54 154 Factorization 104 FACTORIZE 105 Fast loading of code 200 FEXPR 182 FIDE 162 File handling
84. X and the second 0 Thus the first rule will be instantiated as a substitution for X and the second will result in an error 10 3 RULE LISTS 127 The order in which a list of rules is applied is not easily understandable without a detailed knowledge of the system simplification protocol It is also possible for this order to change from release to release as improved substitution techniques are implemented Users should therefore assume that the order of application of rules is arbitrary and program accordingly After a substitution has been made the expression being evaluated is reexamined in case a new allowed substitution has been generated This process is continued until no more substitutions can be made As mentioned elsewhere when a substitution expression appears in a product the substitution is made if that expression divides the product For example the rule let a 2xc 3x2 would cause a 2 cx to be replaced by 3 Z X and a 2xc 2 by 3 Z C If the substitution is desired only when the substitution expression appears in a product with the explicit powers supplied in the rule the command MATCH should be used instead For example match a 2xc 3x2 would cause a 2x cxx to be replaced by 3 2Z X but a 2 c 2 would not be replaced MATCH can also be used with the FOR ALL constructions described above To remove substitution rules of the type discussed in this section the CLEAR com mand can be used combin
85. alculations A very powerful feature of REDUCE is the ease with which matrix calculations can be performed To extend our syntax to this class of calculations we need to add another prefix operator MAT and a further variable and expression type as follows 13 1 MAT Operator This prefix operator is used to represent n x m matrices MAT has n arguments in terpreted as rows of the matrix each of which is a list of m expressions representing elements in that row For example the matrix a b c d e f would be written as mat a b c d e f x y becomes mat x y The inside parentheses are required to distinguish it from the single row matrix 57 that would be written as mat x y Note that the single column matrix 141 142 CHAPTER 13 MATRIX CALCULATIONS 13 2 Matrix Variables An identifier may be declared a matrix variable by the declaration MATRIX The size of the matrix may be declared explicitly in the matrix declaration or by default in assigning such a variable to a matrix expression For example matrix x 2 1 y 3 4 2Z declares X to be a2 x 1 column matrix Y to be a 3 x 4 matrix and Z a matrix whose size is to be declared later Matrix declarations can appear anywhere in a program Once a symbol is declared to name a matrix it can not also be used to name an array operator or a procedure or used as an ordinary variable It can however be redeclared to be a matrix and its size may be
86. alse expression is found The rest are not evaluated Thus numberp x and numberp y and x gt y does not attempt to make the x gt y comparison unless X and Y are both verified to be numbers Similarly evaluation of a sequence of boolean expressions connected by OR stops as soon as a true expression is found NB Ina boolean expression and in a place where a boolean expression is expected the algebraic value O is interpreted as false while all other algebraic values are converted to true So in algebraic mode a procedure can be written for direct usage in boolean expressions returning say or 0 as its value as in procedure polynomialp u x if den u 1 and deg u x gt 1 then 1 else 0 One can then use this in a boolean construct such as if polynomialp q z and not polynomialp a y then In addition any procedure that does not have a defined return value for example a block without a RETURN statement in it has the boolean value false 3 4 Equations Equations are a particular type of expression with the syntax lt expression gt lt expression gt 3 5 PROPER STATEMENTS AS EXPRESSIONS 31 In addition to their role as boolean expressions they can also be used as arguments to several operators e g SOLVE and can be returned as values Under normal circumstances the right hand side of the equation is evaluated but not the left hand side This also applies to any substitutions made by the SUB operator
87. arbitrary Several statements can be on one line or one statement can be freely broken onto several lines If the program is run interactively statements ending with or are not processed until an end of line character is encountered This character can vary from system to system but is normally the Return key on an ASCII terminal Specific systems may also use additional keys as statement terminators If a statement is a proper statement the appropriate action takes place Depending on the nature of the proper statement some result or response may or may not be printed out and the response may or may not depend on the terminator used If a statement is an expression it is evaluated If the terminator is a semicolon the result is printed If the terminator is a dollar sign the result is not printed Because it is not usually possible to know in advance how large an expression will be no explicit format statements are offered to the user However a variety of output declarations are available so that the output can be produced in different forms These output declarations are explained in Section 8 3 3 The following sub sections describe the types of proper statements in REDUCE 37 38 CHAPTER 5 STATEMENTS 5 1 Assignment Statements These statements have the syntax lt assignment statement gt lt expression gt lt expression gt The lt expression gt on the left side is normally the name of a variable
88. atements The value corresponding to a BEGIN END compound statement such as a procedure body is normally 0 NIL in symbolic mode By executing a RETURN statement in the compound statement a different value can be returned After a RETURN statement is executed no further statements within the compound state ment are executed Examples return xty return m return Note that parentheses are not required around the x y although they are permitted The last example is equivalent to return 0 or return nil depending on whether the block is used as part of an expression or not Since RETURN actually moves up only one block level in a sense the casual user is not expected to understand we tabulate some cautions concerning its use 1 RETURN can be used on the top level inside the compound statement i e as one of the statements bracketed together by the BEGIN END 5 7 COMPOUND STATEMENTS 47 2 RETURN can be used within a top level lt lt gt gt construction within the compound statement In this case the RETURN transfers control out of both the group statement and the compound statement 3 RETURN can be used within an IF THEN ELSE on the top level within the compound statement NOTE At present there is no construct provided to permit early termination of a FOR WHILE or REPEAT statement In particular the use of RETURN in such cases results in a syntax error For e
89. ation it is not practical to terminate the calculation at such points and require resubmission of the job so the system makes the most obvious guess of the user s intentions and continues the calculation There is also a difference in the handling of errors In the former case the computa tion can continue since the user has the opportunity to correct the mistake In batch mode the error may lead to consequent erroneous and possibly time consuming computations So in the default case no further evaluation occurs although the remainder of the input is checked for syntax errors A message Continuing with parsing only informs the user that this is happening On the other hand the switch ERRCONT if on will cause the system to continue evaluating expressions after such errors occur When a syntactical error occurs the place where the system detected the error is marked with three dollar signs In interactive mode the user can then use ED to correct the error or retype the command When a non syntactical error occurs in interactive mode the command being evaluated at the time the last error occurred is saved and may later be reevaluated by the command RETRY 12 1 Referencing Previous Results It is often useful to be able to reference results of previous computations during a REDUCE session For this purpose REDUCE maintains a history of all interactive 137 138 CHAPTER 12 COMMANDS FOR INTERACTIVE USE inputs and the r
90. ation of residues of arbitrary expressions Author Wolfram Koepf 15 42 RLFI REDUCE LaTeX formula interface This package adds I4T amp X syntax to REDUCE Text generated by REDUCE in this mode can be directly used in IATRX source documents Various mathematical con structions are supported by the interface including subscripts superscripts font 170 CHAPTER 15 USER CONTRIBUTED PACKAGES changing Greek letters divide bars integral and sum signs derivatives and so on Author Richard Liska 15 43 ROOTS A REDUCE root finding package This root finding package can be used to find some or all of the roots of a univariate polynomial with real or complex coefficients to the accuracy specified by the user It is designed so that it can be used as an independent package or it may be called from SOLVE if ROUNDED is on For example the evaluation of on rounded complex solve x 3 x 5 x yields the result X 1 51598 X 0 75799 1 65035 1 X 0 75799 1 65035x1 This package loads automatically Author Stanley L Kameny 15 44 RSOLVE Rational integer polynomial solvers This package provides operators that compute the exact rational zeros of a single univariate polynomial using fast modular methods The algorithm used is that described by R Loos 1983 Computing rational zeros of integral polynomials by p adic expansion SIAM J Computing 12 286 293 Author Francis J Wright 15 45 SCOPE REDUCE sourc
91. aw Hill New York 1965 Similarly an arbitrary component P may be represented by p u If contraction over components of vectors is required then the declaration INDEX must be used Thus index u declares U as an index and the simplification of p u q u would result in P Q The metric tensor g may be represented by u v If contraction over U and V is required then they should be declared as indices Errors occur if indices are not properly matched in expressions If a user later wishes to remove the index property from specific vectors he can do it with the declaration REMIND Thus remind vl vn removes the index flags from the variables V1 through Vn However these variables remain vectors in the system 17 1 2 G Operator for Gamma Matrices Syntax G ID identifier EXPRN vector_expression gamma_matrix_ expression G is an n ary operator used to denote a product of y matrices contracted with Lorentz four vectors Gamma matrices are associated with fermion lines in a Feyn man diagram If more than one such line occurs then a different set of y matrices operating in independent spin spaces is required to represent each line To facil itate this the first argument of G is a line identification identifier not a number used to distinguish different lines Thus g 11 p g 12 q denotes the product of y p associated with a fermion line identified as L1 and y q associated with another lin
92. cannot be interpreted in the required mode with the current switch settings For example if an argument is required to be a kernel and a 2 is used with no other rules for A an error A 2 invalid as kernel will result With the exception of those that select various parts of a polynomial or rational function these operations have potentially significant effects on the space and time associated with a given calculation The user should therefore experiment with their use in a given calculation in order to determine the optimum set for a given problem One such operation provided by the system is an operator LENGTH which returns the number of top level terms in the numerator of its argument For example length a b c 3 c d has the value 10 To get the number of terms in the denominator one would first select the denominator by the operator DEN and then call LENGTH as in length den atbtc 3 ct d 103 104 CHAPTER 9 POLYNOMIALS AND RATIONALS Other operations currently supported the relevant switches and operators and the required argument and value modes of the latter follow 9 1 Controlling the Expansion of Expressions The switch EXP controls the expansion of expressions If it is off no expansion of powers or products of expressions occurs Users should note however that in this case results come out in a normal but not necessarily canonical form This means that zero expressions simplify to zero but
93. cations it is desirable to load previously prepared REDUCE files into the system or to write output on other files REDUCE offers four commands for this purpose namely IN OUT SHUT LOAD and LOAD_PACKAGE The first three operators are described here LOAD and LOAD_PACKAGE are discussed in Section 18 2 11 1 IN Command This command takes a list of file names as argument and directs the system to input each file that should contain REDUCE statements and commands into the system File names can either be an identifier or a string The explicit format of these will be system dependent and in many cases site dependent The explicit instructions for the implementation being used should therefore be consulted for further details For example in f1 ggg rr s will first load file F1 then ggg rr s When a semicolon is used as the terminator of the IN statement the statements in the file are echoed on the terminal or written on the current output file If is used as the terminator the input is not shown Echoing of all or part of the input file can be prevented even if a semicolon was used by placing an off echo command in the input file Files to be read using IN should end with END Note the two semicolons First of all this is protection against obscure difficulties the user will have if there are by mistake more BEGINs than ENDs on the file Secondly it triggers some file control book keeping which may improve sy
94. cessor currently likes to transfer prefix forms from command to command but doesn t like to re convert standard forms which represent polynomials and standard quotients back to a true Lisp prefix form for the expression which would result in excessive com putation So SQ is used to tell the algebraic processor that it is dealing with a prefix form which is really a standard quotient and the second argument T or NIL tells it whether it needs further processing essentially an already simplified flag So to get the true standard quotient form in symbolic mode one needs CADR of the variable E g Zz cadr x would store in Z the standard quotient form for a b 2 Once you have this expression you can now manipulate it as you wish To facilitate this a standard set of selectors and constructors are available for getting at parts of the form Those presently defined are as follows 184 TC TDEG TPOW CHAPTER 16 SYMBOLIC MODE REDUCE Selectors denominator of standard quotient leading coefficient of polynomial leading degree of polynomial leading power of polynomial leading term of polynomial main variable of polynomial numerator of standard quotient degree of a power reductum of polynomial coefficient of a term degree of a term power of a term REDUCE Constructors add a term to a polynomial divide two polynomials to get quotient multiply power by coefficient to produc
95. changed at that time Note however that matrices once declared are global in scope and so can then be referenced anywhere in the program In other words a declaration within a block or a procedure does not limit the scope of the matrix to that block nor does the matrix go away on exiting the block use CLEAR instead for this purpose An element of a matrix is referred to in the expected manner thus x 1 1 gives the first element of the matrix X defined above References to elements of a matrix whose size has not yet been declared leads to an error All elements of a matrix whose size is declared are initialized to 0 As a result a matrix element has an instant evaluation property and cannot stand for itself If this is required then an operator should be used to name the matrix elements as in matrix m operator x m mat x 1 1 x 1 2 13 3 Matrix Expressions These follow the normal rules of matrix algebra as defined by the following syntax lt matrix expression gt MAT lt matrix description gt lt matrix variable gt lt scalar expression gt lt matrix expression gt lt matrix expression gt lt matrix expression gt lt matrix expression gt lt integer gt Aa lt matrix expression gt lt matrix expression gt a ai lt matrix expression gt lt matrix expression gt Sums and products of matrix expressions must be of compatible size otherwise an error will result during their evaluation Similarl
96. d replaced by a space following the operator name if the operator is unary and the argument is a single symbol or begins with a prefix operator name cos y means cos y cos y parentheses necessary log cos y means log cos y log cos a b means log cos a b but cos axb means cos a b cos y is erroneous treated as a variable cos minus the variable y 2 7 OPERATORS 25 A unary prefix operator has a precedence higher than any infix operator including unary infix operators In other words REDUCE will always interpret cos y 3as cos y 3ratherthanas cos y 3 Infix operators may also be used in a prefix format on input e g a b c On output however such expressions will always be printed in infix form i e a b c for this example A number of prefix operators are built into the system with predefined properties Users may also add new operators and define their rules for simplification The built in operators are described in another section Built In Infix Operators The following infix operators are built into the system They are all defined inter nally as procedures lt infix operator gt where orland member memq neqleq Ale ES These operators may be further divided into the following subclasses lt assignment operator gt 22S lt logical operator gt Orland member memq lt relational operator gt negleql gt gt lt lt
97. d to an array name returns a list of its dimensions All array elements are initialized to O at declaration time In other words an array element has an instant evaluation property and cannot stand for itself If this is required then an operator should be used instead Array declarations can appear anywhere in a program Once a symbol is declared to name an array it can not also be used as a variable or to name an operator or a procedure It can however be re declared to be an array and its size may be changed at that time An array name can also continue to be used as a parameter in a procedure or a local variable in a compound statement although this use is not recommended since it can lead to user confusion over the type of the variable Arrays once declared are global in scope and so can then be referenced anywhere in the program In other words unlike arrays in most other languages a declara tion within a block or a procedure does not limit the scope of the array to that block nor does the array go away on exiting the block use CLEAR instead for this purpose 6 2 Mode Handling Declarations The ON and OFF declarations are available to the user for controlling various sys tem options Each option is represented by a switch name ON and OFF take a list of switch names as argument and turn them on and off respectively e g on time causes the system to print a message after each command giving the elapsed CPU time
98. d to be so terminated Remember also to begin each line of a multi line comment with a sign 24 CHAPTER 2 STRUCTURE OF PROGRAMS 2 7 Operators Operators in REDUCE are specified by name and type There are two types in fix and prefix Operators can be purely abstract just symbols with no properties they can have values assigned using or simple LET declarations for specific arguments they can have properties declared for some collection of arguments using more general LET declarations or they can be fully defined usually by a procedure declaration Infix operators have a definite precedence with respect to one another and normally occur between their arguments For example a b spaces optional x lt y and y z spaces required where shown Spaces can be freely inserted between operators and variables or operators and operators They are required only where operator names are spelled out with let ters such as the AND in the example and must be unambiguously separated from another such or from a variable like Y Wherever one space can be used so can any larger number Prefix operators occur to the left of their arguments which are written as a list enclosed in parentheses and separated by commas as with normal mathematical functions e g cos u df x 2 x q v w Unmatched parentheses incorrect groupings of infix operators and the like natu rally lead to syntax errors The parentheses can be omitte
99. des an alternative method for computing traces of Dirac gamma matrices based on an algorithm by Cvitanovich that treats gamma matrices as 3 symbols Authors V Ilyin A Kryukov A Rodionov A Taranov 15 13 DEFINT A definite integration interface 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 Authors Kerry Gaskell Stanley M Kameny Winfried Neun 15 14 DESIR Differential linear homogeneous equation solutions in the neighborhood of irregular and reg ular singular points This package enables the basis of formal solutions to be computed for an ordinary homogeneous differential equation with polynomial coefficients over Q of any or der in the neighborhood of zero regular or irregular singular point or ordinary point Documentation for this package is in plain text Authors C Dicrescenzo F Richard Jung E Tournier 15 15 DFPART Derivatives of generic functions 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 Author Herbert Melenk 162 CHAPTER 15 USER CONTRIBUTED PACKAGES 15 16 DUMMY Canonical form of expressions with dummy variables This package allows a user to find the canonical form of expressions involvin
100. df f mu WHEE Ei PS miL gl f x1xdf g eps x2x df g sig x3xdf g mu write g i gl f f1 g g1 end A portion of the output to illustrate the printout from the WRITE command is as follows lt prior output gt 8 3 OUTPUT OF EXPRESSIONS 93 F 4 MUx 3x EPS 15 SIG MU G 4 6 SIG MU 2 F 5 15 SIG MUx 3 EPS 7 SIG MU 2 G 5 MU 9 EPS 45 SIG MU lt more output gt 8 3 5 Suppression of Zeros It is sometimes annoying to have zero assignments i e assignments of the form lt expression gt 0 printed especially in printing large arrays with many zero elements The output from such assignments can be suppressed by turning on the switch NERO 8 3 6 FORTRAN Style Output Of Expressions It is naturally possible to evaluate expressions numerically in REDUCE by giving all variables and sub expressions numerical values However as we pointed out elsewhere the user must declare real arithmetical operation by turning on the switch ROUNDED However it should be remembered that arithmetic in REDUCE is not particularly fast since results are interpreted rather than evaluated in a compiled form The user with a large amount of numerical computation after all necessary algebraic manipulations have been performed is therefore well advised to perform these calculations in a FORTRAN or similar system For this purpose REDUCE offers facilities for users to produce FORTRAN compatible
101. ding this section Although REDUCE is designed primarily for algebraic calculations its source lan guage is general enough to allow for a full range of Lisp like symbolic calculations To achieve this generality however 1t is necessary to provide the user with two modes of evaluation namely an algebraic mode and a symbolic mode To enter symbolic mode the user types symbolic or 1isp and to return to algebraic mode one types algebraic Evaluations proceed differently in each mode so the user is advised to check what mode he is in if a puzzling error arises He can find his mode by typing eval_mode The current mode will then be printed as ALGEBRAIC or SYMBOLIC Expression evaluation may proceed in either mode at any level of a calculation provided the results are passed from mode to mode in a compatible manner One simply prefixes the relevant expression by the appropriate mode If the mode name prefixes an expression at the top level it will then be handled as if the global system mode had been changed for the scope of that particular calculation For example if the current mode is ALGEBRAIC then the commands 177 178 CHAPTER 16 SYMBOLIC MODE symbolic car a x y will cause the first expression to be evaluated and printed in symbolic mode and the second in algebraic mode Only the second evaluation will thus affect the expression workspace On the other hand the statement x symbolic car 1
102. dos ee ee ee od 62 TAL PUORS oe cda aa ne hada eee la 63 742 Advanced Use dk a el ead we a a ew a a 63 PEA References cok aah ee ee A a 63 LENGTH Opersiot gt lt cs erca ee he Gee She ae ears 64 MAP Operator ce ks cs a e RA 64 MIRID Operatot coo paca be eb be a 65 PE Operator ica and ela al ah ah wh Sea a 66 SELECTUPBSIMOE coca Mew eae a ee ew 67 SOLVE Operator o os cac eee eee eee 68 7 10 1 Handling of Undetermined Solutions 69 7 10 2 Solutions of Equations Involving Cubics and Quartics 71 FIDA Other Options o os oa 04 beak bee ee ado 73 CONTENTS 7 10 4 Parameters and Variable Dependency 74 7 11 Eyen and Odd Operators ne a eaer ca a ee eS 78 7 12 Linear Operators oc osa ce nan ded e kai ae T aaa 78 7 13 Non Commuting Operators oaoa a 79 7 14 Symmetric and Antisymmetric Operators 80 7 15 Declaring New Prefix Operators 80 7 16 Declaring New Infix Operators noaoae 81 7 17 Creating Removing Variable Dependency 82 Display and Structuring of Expressions 83 A A 83 8 2 The Expression Workspace o o 84 8 5 Output of Expressions lt o qa ca co cnc oras 85 8 3 1 LINELENGTH Operator o csa ee ee 86 8 3 2 Output Declarations lt o oa ca ca o o esa 86 8 3 3 Output Control Switches 87 234 WRITE Command s o e s es s ceda 91 8 3 5 Suppression of Zeros ea ecese esteru 93 8 3 6 FORTRAN Style Outp
103. ds conses an expression to the front of a list For example a b c gt a b c 4 1 7 APPEND This operator appends its first argument to its second to form a new list Examples append a b c d gt a b c d append a b c qd gt a b c d 4 1 OPERATIONS ON LISTS 35 4 1 8 REVERSE The operator REVERSE returns its argument with the elements in the reverse or der It only applies to the top level list not any lower level lists that may occur Examples are reverse a b c gt c b a reverse a b c d gt d a b c 4 1 9 List Arguments of Other Operators If an operator other than those specifically defined for lists is given a single argu ment 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 For example the result of evaluating log a b c is the expression LOG A LOG B LOG C There are two ways to inhibit this operator distribution Firstly the switch LISTARGS if on will globally inhibit such distribution Secondly one can in hibit this distribution for a specific operator by the declaration LISTARGP For example with the declaration Listargp log log a b c would evaluate to LOG A B C If an operator has more than one argument no such distribution occurs 4 1 10 Caveats and Examples Some of the natural list operations such as member or delete are available on
104. e code optimization pack age SCOPE is a package for the production of an optimized form of a set of expres sions It applies an heuristic search for common sub expressions to almost any set of proper REDUCE assignment statements The output is obtained as a sequence of assignment statements GENTRAN is used to facilitate expression output Author J A van Hulzen 15 46 SETS A BASIC SET THEORY PACKAGE 171 15 46 SETS A basic set theory package The SETS package provides algebraic mode support for set operations on lists re garded as sets or representing explicit sets and on implicit sets represented by identifiers Author Francis J Wright 15 47 SPDE Finding symmetry groups of PDE s The package SPDE provides a set of functions which may be used to determine the symmetry group of Lie or point symmetries of a given system of partial dif ferential 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 Author Fritz Schwarz 15 48 SPECFN Package for special functions This special function package is separated into two portions to make it easier to handle The packages are called SPECFN and SPECEN2 The first one is more general in nature whereas the second is devoted to special special functions Doc umentation for the first package can be found in the file specfn tex in the doc
105. e identified as L2 and where p and q are Lorentz 17 2 VECTOR VARIABLES 193 four vectors A product of y matrices associated with the same line may be written in a contracted form Thus gll pl p2 p3 G 11 pl g 117 2 9 11 p3 The vector A is reserved in arguments of G to denote the special y matrix y Thus 7 associated with the line L g 1 a g l p a y p xy associated with the line L y associated with the line L may be written as g 1 u with U flagged as an index if contraction over U is required The notation of Bjorken and Drell is assumed in all operations involving y matri ces 17 1 3 EPS Operator Syntax EPS EXPRN1 vector_expression EXPRN4 vector_exp vector_exp The operator EPS has four arguments and is used only to denote the completely antisymmetric tensor of order 4 and its contraction with Lorentz four vectors Thus 1 ifi j k lis an even permutation of 0 1 2 3 ijkl 1 if an odd permutation O otherwise A contraction of the form e u Py qu May be written as eps i j p q with I and J flagged as indices and so on 17 2 Vector Variables Apart from the line identification identifier in the G operator all other arguments of the operators in this section are vectors Variables used as such must be declared so by the type declaration VECTOR for example vector pl p2 194 CHAPTER 17 CALCULATIONS IN HIGH ENERGY PHYSICS declares P1 and P2 t
106. e of writing reduce netlib can nl and reduce netlibQ pi cc u tokyo ac jp may also be used instead of reduce netlib rand org A World Wide Web REDUCE server with URL http www rrz uni koeln de REDUCE is also supported In addition to general information about REDUCE this server has pointers to the network library the demonstration versions examples of RE DUCE programming a set of manuals and the REDUCE online help system Finally there is a REDUCE electronic forum accessible from the same networks This enables REDUCE users to raise questions and discuss ideas concerning the 205 206 CHAPTER 19 MAINTAINING REDUCE use and development of REDUCE with other users Additions and changes to the network library and new releases of REDUCE are also announced in this forum Any user with appropriate electronic mail access is encouraged to register for mem bership in this forum To do so send a message requesting inclusion to reduce forum request rand org Appendix A Reserved Identifiers We list here all identifiers that are normally reserved in REDUCE including names of commands operators and switches initially in the system Excluded are words that are reserved in specific implementations of the system Commands Boolean Operators Infix Operators Numerical Operators LGEBRAIC ANTISYMMETRIC ARR EPEND DISPLAY ED AY BYE CLEAR T DEFINE G EDITDEF END EVEN FAC
107. e provides for a simple error exit from an algebraic procedure and also a block It can take a string as argument It should be noted however that all the above definitions of p n x are quite inefficient if extensive use is to be made of such polynomials since each call ef fectively recomputes all lower order polynomials It would be better to store these expressions in an array and then use say the recurrence relation to compute only those polynomials that have not already been derived We leave it as an exercise for the reader to write such a definition 14 3 Using LET Inside Procedures By using LET instead of an assignment in the procedure body it is possible to bypass the call by value protection If X is a formal parameter or local variable of the procedure i e is in the heading or in a local declaration and LET is used instead of to make an assignment to X e g let x 123 154 CHAPTER 14 PROCEDURES then it is the variable that is the value of X that is changed This effect also occurs with local variables defined in a block If the value of X is not a variable but a more general expression then it is that expression that is used on the left hand side of the LET statement For example if X had the value pq itisasif let pxq 123 had been executed 14 4 LET Rules as Procedures The LET statement offers an alternative syntax and semantics for procedure defi nition In place of procedure abc x y z lt proc
108. e term raise a variable to a power For example to find the numerator of the standard quotient above one could say numr zZ or to find the leading term of the numerator lt numr Zz Conversion between various data structures is facilitated by the use of a set of functions defined for this purpose Those currently implemented include 16 9 COMMUNICATING WITH ALGEBRAIC MODE 185 A2F convert an algebraic expression to a standard form If result is rational an error results A2K converts an algebraic expression to a kernel If this is not possible an error results IxF2A converts a standard form to an algebraic expression xF2Q convert a standard form to a standard quotient xK2F convert a kernel to a standard form 1xK20 convert a kernel to a standard quotient xP2F convert a standard power to a standard form xP2Q convert a standard power to a standard quotient xQ2F convert a standard quotient to a standard form If the quotient de nominator is not 1 an error results Q2K convert a standard quotient to a kernel If this is not possible an error results 1xT2F convert a standard term to a standard form xT2Q convert a standard term to a standard quotient 16 9 2 Passing Symbolic Mode Values to Algebraic Mode In order to pass the value of a shared variable from symbolic mode to algebraic mode the only thing to do is make sure that the value in symbolic mode is a prefix expression E g one uses ex
109. e value initially Example Suppose we want to add up a series of terms generated one by one until we reach a term which is less than 1 1000 in value For our simple example let us suppose the first term equals 1 and each term is obtained from the one before by taking one third of it and adding one third its square We would write 5 6 REPEAT UNTIL 43 x 0 term 1 while num term 1 1000 gt 0 do lt lt ex extterm term term term 2 3 gt gt ex As long as TERM is greater than or equal to gt 1 1000 it will be added to EX and the next TERM calculated As soon as TERM becomes less than 1 1000 the WHILE test fails and the TERM will not be added 5 6 REPEAT UNTIL REPEAT UNTILis very similar in purpose to WHILE DO Its syntax is REPEAT lt statement gt UNTIL lt boolean expression gt PASCAL users note Only a single statement usually a group statement is allowed between the REPEAT and the UNTIL There are two essential differences 1 The test is performed after the controlled statement or group of statements 1s executed so the controlled statement is always executed at least once 2 The test is a test for when to stop rather than when to continue so its polar ity is the opposite of that in WHILE DO As an example we rewrite the example from the WHILE DO section x 0 term 1 repeat lt lt ex
110. ed if necessary with the same FOR ALL clause with which the rule was defined for example for all x clear log e x e log x cos wxt theta x Note however that the arbitrary variable names in this case must be the same as those used in defining the substitution 10 3 Rule Lists Rule lists offer an alternative approach to defining substitutions that is different from either SUB or LET In fact they provide the best features of both since they have all the capabilities of LET but the rules can also be applied locally as is pos sible with SUB In time they will be used more and more in REDUCE However since they are relatively new much of the REDUCE code you see uses the older constructs 128 CHAPTER 10 SUBSTITUTION COMMANDS A rule list is a list of rules that have the syntax lt expression gt gt lt expression gt WHEN lt boolean expression gt For example cos x cos y gt cos x y cos x y 2 cos nx pi gt 1 n when remainder n 2 0 The tilde preceding a variable marks that variable as free for that rule much as a variable in a FOR ALL clause in a LET statement The first occurrence of that variable in each relevant rule must be so marked on input otherwise inconsistent results can occur For example the rule list cos x x cos y gt cos x y cos x y 2 cos x 72 gt 1 cos 2x 2 designed to replace products of cosines would not be correct since the sec
111. ed or the operator can be given a definition as a procedure If the user forgets to declare an identifier as an operator the system will prompt the user to do so in interactive mode or do it automatically in non interactive mode A diagnostic message will also be printed if an identifier is declared OPERATOR more than once Operators once declared are global in scope and so can then be referenced any where in the program In other words a declaration within a block or a procedure does not limit the scope of the operator to that block nor does the operator go away on exiting the block use CLEAR instead for this purpose 7 16 Declaring New Infix Operators Users can add new infix operators by using the declarations INF IX and PRECEDENCE For example infix mm precedence mm The declaration infix mm would allow one to use the symbol MM as an infix operator a mm b instead of mm a b The declaration precedence mm says that MM should be inserted into the infix operator precedence list just after the operator This gives it higher prece dence than and lower precedence than Thus a bmmc d means a b mmc qd while a k bnmmcxd means ax b mm c gt d Both infix and prefix operators have no transformation properties unless LET state ments or procedure declarations are used to assign a meaning 82 CHAPTER 7 BUILT IN PREFIX OPERATORS We should note here that infix operators s
112. edure body gt one can write for all x y z let abc x y z lt procedure body gt There are several differences to note If the procedure body contains an assignment to one of the formal parameters e g x tS 123 in the PROCEDURE case it is a variable holding a copy of the first actual argument that is changed The actual argument is not changed In the LET case the actual argument is changed Thus if ABC is defined using LET and abc u v w is evaluated the value of U changes to 123 That is the LET form of definition allows the user to bypass the protections that are enforced by the call by value conventions of standard PROCEDURE definitions Example We take our earlier FACTORIAL procedure and write it as a LET state ment for all n let factorial n begin scalar m s m l s n 11 if s 0 then return m m M s s s l go to ll end 14 4 LET RULES AS PROCEDURES 155 The reader will notice that we introduced a new local variable S and set it equal to N The original form of the procedure contained the statement n n 1 If the user asked for the value of factorial 5 then N would correspond to not just have the value of 5 and REDUCE would object to trying to execute the statement 5 5 1 If POR is a procedure with no parameters procedure pqr lt procedure body gt it can be written as a LET statement quite simply let pqr lt procedure body gt To call procedure POR if defined i
113. ee e aS 143 13 4 2 MATEIGEN Operator 4 144 13 4 3 TP Operator c e ee ee ee 145 1344 Trace Operator ns coc ore a ee Bae Se 145 1345 Marit COSES conri 145 13 4 6 NULLSPACE Operator lt o lt lt lt 146 134 7 RANK Operator 20 2 e 147 13 5 Matrix Assignments 0 0 6 recaer ee a 147 13 6 Evaluating Matrix Elements 147 14 Procedures 149 14 1 Procedure Heading gt o oo oo co es Be ed a oc acs madra 150 14 2 Procedure Body rs a edoae ales go ge de oes 151 14 3 Using LET Inside Procedures os o scacco e 153 CONTENTS 14 4 LET Rules as Procedures 15 User Contributed Packages 15 1 ALGINT Integration of square roots 15 2 APPLYSYM Infinitesimal symmetries of differential equations 15 3 ARNUM An algebraic number package 15 4 ASSIST Useful utilities for various applications 15 5 AVECTOR A vector algebra and calculus package 15 6 BOOLEAN A package for boolean algebra 15 7 CALI A package for computational commutative algebra 15 8 CAMAL Calculations in celestial mechanics 15 9 CHANGEVR Change of Independent Variable s in DEs 15 10COMPACT Package for compacting expressions 15 11CRACK Solving overdetermined systems of PDEs or ODEs 15 12CVIT Fast calculation of Dirac gamma matrix traces 15 13DEFINT A definite integration i
114. efined by Lie bracket commutators The package uses the REDUCE noncom mechanism for elementary polynomial arithmetic the commutator rules are automatically computed from the Lie brackets Authors Herbert Melenk and Joachim Apel 15 32 NORMFORM Computation of matrix normal forms This package contains routines for computing the following normal forms of ma trices e smithex_int e smithex e frobenius e ratjordan e jordansymbolic e jordan Author Matt Rebbeck 15 33 NUMERIC SOLVING NUMERICAL PROBLEMS 167 15 33 NUMERIC Solving numerical problems This package implements basic algorithms of numerical analysis These include e solution of algebraic equations by Newton s method num_solve sin x cos y x y 1 x 1 y 2 solution of ordinary differential equations num_odesolve df y x y y 1 x 0 1 iterations 5 bounds of a function over an interval bounds sin x x x 1 2 e minimizing a function Fletcher Reeves steepest descent num_min sin x x 5 x Chebyshev curve fitting chebyshev_fit sin x x x 1 3 5 e numerical quadrature num_int sin x x 0 pi Author Herbert Melenk 15 34 ODESOLVE Ordinary differential equations solver The ODESOLVE package is a solver for ordinary differential equations At the present time it has very limited capabilities It can handle only a single scalar equation presented as an algebraic expression or equation and it can solve only first o
115. el algebraic This will return correctly the indefinite integral for expressions comprising poly nomials log functions exponential functions and tan and atan The arbitrary con stant is not represented If the integral cannot be done in closed terms it returns a formal integral for the answer in one of two ways 1 It returns the input INT unchanged 2 It returns an expression involving INTs of some other functions sometimes more complicated than the original one unfortunately Rational functions can be integrated when the denominator is factorizable by the program In addition it will attempt to integrate expressions involving error funct ions dilogarithms and other trigonometric expressions In these cases it might not always succeed in finding the solution even if one exists Examples int log x x gt X LOG X 1 int e x x gt ExxX The program checks that the second argument is a variable and gives an error 1f it 1s not 7 4 INT OPERATOR 63 Note If the int operator is called with 4 arguments REDUCE will implicitly call the definite integration package DEFINT and this package will interpret the third and fourth arguments as the lower and upper limit of integration respectively For details consult the documentation on the DEFINT package 7 4 1 Options The switch TRINT when on will trace the operation of the algorithm It produces a great deal of output in a somewhat illegible
116. en fixp 2xx gamma x Here rule by rule cases of known or definitely uncomputable values are sorted out e g the rule leading to the error expression will be applied for negative integers only since the positive integers are caught by the preceding rule and the last rule will apply for negative odd multiples of 1 2 only Alternatively the first rule could have been written as gamma 1 2 gt sqrt pi 2 but then the case x 1 2 should be excluded in the WHEN part of the last rule explicitly because a rule without free variables cannot take precedence over the other rules 10 4 Asymptotic Commands In expansions of polynomials involving variables that are known to be small it is often desirable to throw away all powers of these variables beyond a certain point 134 CHAPTER 10 SUBSTITUTION COMMANDS to avoid unnecessary computation The command LET may be used to do this For example if only powers of X up to x 7 are needed the command let x 8 0 will cause the system to delete all powers of X higher than 7 CAUTION This particular simplification works differently from most substitu tion mechanisms in REDUCE in that it is applied during polynomial manipulation rather than to the whole evaluated expression Thus with the above rule in effect x 10 x 5 would give the result zero since the numerator would simplify to zero Similarly x 20 x 10 would give a Zero divisor error message since bot
117. en outside This means that F must have at least two arguments In addition the second argument must be an identifier or more generally a kernel not an expression Example If F were declared linear then 5 f axx 5 bxx c x gt F X X xA F X X B F 1 X C More precisely not only will the variable and its powers remain within the scope of the F operator but so will any variable and its powers that had been declared to DEPEND on the prescribed variable and so would any expression that contains that variable or a dependent variable on any level e g cos sin x To declare operators F and G to be linear operators use linear f 9 7 13 NON COMMUTING OPERATORS 79 The analysis is done of the first argument with respect to the second any other arguments are ignored It uses the following rules of evaluation 0 gt 0 f y x gt F Y X E y z x gt F Y X F Z X f y z x gt ZxF Y X if Z does not depend on X f y z x gt F Y X Z if Z does not depend on X To summarize Y depends on the indeterminate X in the above if either of the following hold 1 Y is an expression that contains X at any level as a variable e g cos sin x 2 Any variable in the expression Y has been declared dependent on X by use of the declaration DEPEND The use of such linear operators can be seen in the paper Fox J A and A C Hearn Analytic Computation of Some Integrals in Fourth Order Q
118. end If the arguments of a WRITE statement include an expression that requires con tinuation records the output will need editing since the output routine prints the arguments of WRITE sequentially and the continuation mechanism therefore gen erates its auxiliary variables after the preceding expression has been printed Finally since there is no direct analog of list in FORTRAN a comment line of the form C xxxxx invalid fortran construct list not printed will be printed if you try to print a list with FORT on FORTRAN Output Options There are a number of methods available to change the default format of the FOR TRAN output The breakup of the expression into subparts is such that the number of continuation lines produced is less than a given number This number can be modified by the assignment card_no lt number gt where lt numbe r gt is the total number of cards allowed in a statement The default value of CARD_NO is 20 The width of the output expression is also adjustable by the assignment fort_width lt integer gt which sets the total width of a given line to lt integer gt The initial FORTRAN output width is 70 REDUCE automatically inserts a decimal point after each isolated integer coeffi cient in a FORTRAN expression so that for example 4 becomes 4 To prevent this set the PERIOD mode switch to OFF FORTRAN output is normally produced in lower case If upper case is desired the s
119. epresentation for the expression and subsequent ele ments are the sub expression relations Thus with SAVESTRUCTR on STRUCTR WS in the above example would return 3 2 ANS3 ANS3 ANS2 D ANS2 ANS1 C ANS1 A B The PART operator can be used to retrieve the required parts of the expression For example to get the value of ANS2 in the above one could say part ws 3 2 If FORT is on then the results are printed in the reverse order the algorithm in fact guaranteeing that no sub expression will be referenced before it is defined The second optional argument ID2 may also be used in this case to name the actual expression or expressions in the case of a matrix argument Example Let us suppose that M a 2 by 1 matrix contains the elements a b 2 c 3 dand a b c d respectively and that V has been declared to be an array With EXP off and FORT on the statement structr 2xm v k will result in the output 8 4 CHANGING THE INTERNAL ORDER OF VARIABLES 99 K 1 1 2 V 3 K 2 1 2 V 1 V 4 8 4 Changing the Internal Order of Variables The internal ordering of variables more specifically kernels can have a significant effect on the space and time associated with a calculation In its default state RE DUCE uses a specific order for this which may vary between sessions However it is possible for the user to change this internal order by means of the declaration KORDER The syntax for this is
120. escription of this case is given in Section 10 2 5 The rule pertaining to w 3 will apply to any power of W greater than or equal to the third Note especially the last example let z 10 0 This declaration means in effect ignore the tenth or any higher power of Z Such declarations when appropriate often speed up a computation to a considerable degree See Section 10 4 for more details Any new operators occurring in such LET rules will be automatically declared OPERATOR by the system if the rules are being read from a file If they are being entered interactively the system will ask DECLARE OPERATOR Answer Y or N and hit Return In each of these examples substitutions are only made for the explicit expressions given i e none of the variables may be considered arbitrary in any sense For example the command let h u v u v will cause h u v to evaluate to U V but will not affect h u z or H with any arguments other than precisely the symbols U V These simple LET rules are on the same logical level as assignments made with the operator An assignment x p q cancels a rule let x y 2 made earlier and vice versa CAUTION A recursive rule such as 122 CHAPTER 10 SUBSTITUTION COMMANDS let x x t 1 is erroneous since any subsequent evaluation of X would lead to a non terminating chain of substitutions xo gt Kot UL gt Cx 1 1 gt Ce 4 1 1
121. esults of all interactive computations during a given session These results are referenced by the command number that REDUCE prints automatically in interactive mode To use an input expression in a new computation one writes input n where n is the command number To use an output expression one writes WS n WS references the previous command E g if command number 1 was INT X 1 X and the result of command number 7 was X 1 then 2x1input 1 ws 7 2 would give the result 1 whereas 2xws 1 ws 7 2 would yield the same result but without a recomputation of the integral The operator DISPLAY is available to display previous inputs If its argument is a positive integer n say then the previous n inputs are displayed If its argu ment is ALL or in fact any non numerical expression then all previous inputs are displayed 12 2 Interactive Editing It is possible when working interactively to edit any REDUCE input that comes from the user s terminal and also some user defined procedure definitions At the top level one can access any previous command string by the command ed n where n is the desired command number as prompted by the system in interactive mode ED i e no argument accesses the previous command After ED has been called you can now edit the displayed string using a string editor with the following commands B move pointer to beginning C lt character gt replace next character by character
122. f the answer The operator STRUCTR that takes a single expression as argument will do this for you Its syntax is STRUCTR EXPRN algebraic ID1 identifier ID2 identifier The structure is printed effectively as a tree in which the subparts are laid out with auxiliary names If the optional 1D1 is absent the auxiliary names are prefixed by the root ANS This root may be changed by the operator VARNAME If the optional ID1 is present and is an array name the subparts are named as elements of that array otherwise 1D1 is used as the root prefix The second optional argument ID2 is explained later 8 3 OUTPUT OF EXPRESSIONS 97 The EXPRN can be either a scalar or a matrix expression Use of any other will result in an error Example Let us suppose that the workspace contains A B 2 C 7 3 D Then the input STRUCTR WS will with EXP off result in the output 98 CHAPTER 8 DISPLAY AND STRUCTURING OF EXPRESSIONS ANS3 where 3 ANS3 ANS2 D 2 ANS2 ANS1 ENE ANS1 A B The workspace remains unchanged after this operation since STRUCTR in the de fault situation returns no value if STRUCTR is used as a sub expression its value is taken to be 0 In addition the sub expressions are normally only displayed and not retained If you wish to access the sub expressions with their displayed names the switch SAVESTRUCTR should be turned on In this case STRUCTR returns a list whose first element is a r
123. f the rational expression EXPRN If EXPRN is a polyno mial that polynomial is returned Examples num x y 2 gt X num 100 6 gt 50 num a 4 b 6 gt 3xA 2xB num a b gt A B 9 10 8 REDUCT Operator Syntax REDUCT EXPRN polynomial VAR kernel polynomial Returns the reductum of EXPRN with respect to VAR i e the part of EXPRN left after the leading term is removed If EXPRN does not depend on the variable VAR 0 is returned Examples reduct a b x c 2xd a gt Bx C 2D reduct a b x c 2xd d gt Cx A B reduct a b x c 2xd e gt 0 9 11 POLYNOMIAL COEFFICIENT ARITHMETIC 115 Compatibility Note In some earlier versions of REDUCE REDUCT returned EXPRN if it did not depend on VAR In the present version EXPRN is always equal to LTERM EXPRN VAR REDUCT EXPRN VAR 9 11 Polynomial Coefficient Arithmetic REDUCE allows for a variety of numerical domains for the numerical coefficients of polynomials used in calculations The default mode is integer arithmetic al though the possibility of using real coefficients has been discussed elsewhere Ra tional coefficients have also been available by using integer coefficients in both the numerator and denominator of an expression using the ON DIV option to print the coefficients as rationals However REDUCE includes several other coefficient opt ions in its basic version which we shall describe
124. form procedures of the CAMAL package for celestial mechanics Author John P Fitch 160 CHAPTER 15 USER CONTRIBUTED PACKAGES 15 9 CHANGEVR Change of Independent Variable s in DEs This package provides facilities for changing the independent variables in a differ ential equation It is basically the application of the chain rule Author G Ucoluk 15 10 COMPACT Package for compacting expressions COMPACT is a package of functions for the reduction of a polynomial in the pres ence of side relations COMPACT applies the side relations to the polynomial so that an equivalent expression results with as few terms as possible For example the evaluation of compact sx l1 sin x 2 cx l cos x 2 sin x 2 cos x72 cos x 2 sin x 2 1 yields the result 2 2 SIN X C COS X S 1 Author Anthony C Hearn 15 11 CRACK Solving overdetermined systems of PDEs or ODEs CRACK is a package for solving overdetermined systems of partial or ordinary differential equations PDEs ODEs Examples of programs which make use of CRACK finding symmetries of ODEs PDEs first integrals an equivalent La grangian or a differential factorization of ODEs are included The application of symmetries is also possible by using the APPLYSYM package Authors Andreas Brand Thomas Wolf 15 12 CVIT FAST CALCULATION OF DIRAC GAMMA MATRIX TRACES 161 15 12 CVIT Fast calculation of Dirac gamma matrix traces This package provi
125. g 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 Author Alain Dresse 15 17 EXCALC A differential geometry package EXCALC is designed for easy use by all who are familiar with the calculus of Mod ern 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 Author Eberhard Schriifer 15 18 FIDE Finite difference method for partial differ ential equations 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 RE DUCE and the numerical programming language FORTRAN are used in the pre sented 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
126. g simplifications of trigonometric or hyperbolic expressions with many options the second for factoriz ing them and the third for finding the greatest common divisor of two trigonometric or hyperbolic polynomials Author Wolfram Koepf 15 56 WU Wu algorithm for polynomial systems 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 Author Russell Bradford 15 57 XCOLOR Color factor in some field theories This package calculates the color factor in non abelian gauge field theories using an algorithm due to Cvitanovich Documentation for this package is in plain text Author A Kryukov 15 58 XIDEAL GROBNER BASES FOR EXTERIOR ALGEBRA 175 15 58 XIDEAL Grobner Bases for exterior algebra XIDEAL constructs Grobner bases for solving the left ideal membership problem Gr bner left ideal bases or GLIBs For graded ideals where each form is homo geneous in degree the distinction between left and right ideals vanishes Further more if the generating forms are all homogeneous then the Gr bner bases for the non graded and graded ideals are identical In this case XIDEAL is able to save time by truncating the Gr bner basis at some maximum degree if desired Author David Hartley 15 59 ZEILBERG A package for indefinite and definite summation
127. gs print exactly as quoted The WRITE command itself however does not return a value The print line is closed at the end of a WRITE command evaluation Therefore the command WRITE specifying nothing to be printed except the empty string causes a line to be skipped Examples 1 If Ais X 5 Bis itself C is 123 Mis an array and Q 3 then write m q a b c THANK YOU will set M 3 to x 5 and print 92 CHAPTER 8 DISPLAY AND STRUCTURING OF EXPRESSIONS M Q X 5 B 123 THANK YOU The blanks between the 5 and B and the 3 and T come from the blanks in the quoted strings To print a table of the squares of the integers from 1 to 20 for i 1 20 do write i 172 To print a table of the squares of the integers from 1 to 20 and at the same time store them in positions 1 to 20 of an array A for i 1 20 do lt lt a i i72 write i a i gt gt This will give us two columns of numbers If we had used for i 1 20 do write i a i 1 2 we would also get A i repeated on each line The following more complete example calculates the famous f and g se ries first reported in Sconzo P LeSchack A R and Tobey R Symbolic Computation of f and g Series by Computer Astronomical Journal 70 May 1965 xl sig mut 2 eps x2 eps 2xsig 2 x3 3 muxsig f 1 g 0 for i 1 step 1 until 10 do begin fl muxg x1x xdf f eps x2x df f sig x3x
128. gt IF lt boolean expression gt THEN lt statement gt ELSE lt statement gt The boolean expression is evaluated If this is true the first lt statement gt is executed If it is false the second is Examples if x 5 then a b c else d e f if x 5 and numberp y then lt lt ff ql a b c gt gt else lt lt ff q2 d e f gt gt Note the use of the group statement Conditional statements associate to the right i e 40 CHAPTER 5 STATEMENTS IF lt a gt THEN lt b gt ELSE IF lt c gt THEN lt d gt ELSE lt e gt is equivalent to IF lt a gt THEN lt b gt ELSE IF lt c gt THEN lt d gt ELSE lt e gt In addition the construction IF lt a gt THEN IF lt b gt THEN lt c gt ELSE lt d gt parses as IF lt a gt THEN IF lt b gt THEN lt c gt ELSE lt d gt If the value of the conditional statement is of primary interest it is often called a conditional expression instead Its value is the value of whichever statement was executed If the executed statement has no value the conditional expression has no value or the value 0 depending on how it is used Examples a if x lt 5 then 123 else 456 b u v if numberp z then 10 z else 1 w If the value is of no concern the ELSE clause may be omitted if no action is required in the false case if x 5 then a btc Note As explained in Section 3 3 a if a scalar or numerical e
129. gt lt expression2 gt Each expression is an algebraic equation or is the difference of the two sides of the equation The second argument is either a kernel or a list of kernels represent ing the unknowns in the system This argument may be omitted if the number of distinct non constant top level kernels equals the number of unknowns in which case these kernels are presumed to be the unknowns For one equation SOLVE recursively uses factorization and decomposition to gether with the known inverses of LOG SIN COS ACOS ASIN and linear quadratic cubic quartic or binomial factors Solutions of equations built with exponentials or logarithms are often expressed in terms of Lambert s W function This function is partially implemented in the special functions package Linear equations are solved by the multi step elimination method due to Bareiss unless the switch CRAMER is on in which case Cramer s method is used The Bareiss method is usually more efficient unless the system is large and dense Non linear equations are solved using the Groebner basis package Users should note that this can be quite a time consuming process Examples solve log sin x 3 5 8 x solve axlog sin x 3 5 b sin x 3 solve a x ty 3 y 2 x y SOLVE returns a list of solutions If there is one unknown each solution is an equation for the unknown If a complete solution was found the unknown will appear
130. h numerator and denominator would first simplify to zero The method just described is not adequate when expressions involve several vari ables having different degrees of smallness In this case it is necessary to supply an asymptotic weight to each variable and count up the total weight of each product in an expanded expression before deciding whether to keep the term or not There are two associated commands in the system to permit this type of asymptotic con straint The command WEIGHT takes a list of equations of the form lt kernel form gt lt number gt where lt number gt must be a positive integer not just evaluate to a positive inte ger This command assigns the weight lt number gt to the relevant kernel form A check is then made in all algebraic evaluations to see if the total weight of the term is greater than the weight level assigned to the calculation If it is the term is deleted To compute the total weight of a product the individual weights of each kernel form are multiplied by their corresponding powers and then added The weight level of the system is initially set to 1 The user may change this setting by the command wtlevel lt number gt which sets lt number gt as the new weight level of the system lt number gt must evaluate to a positive integer WTLEVEL will also allow NIL as an argument in which case the current weight level is returned Chapter 11 File Handling Commands In many appli
131. he elements of the list a b c since the JOIN operation joins the individual lists into one list for each x in a b c join x72 The control variable used in the FOR statement is actually a new variable not related to the variable of the same name outside the FOR statement In other words executing a statement for i doesn t change the system s assumption that i 1 Furthermore in algebraic mode the value of the control variable is substituted in lt exprn gt only if it occurs explicitly in that expression It will not replace a variable of the same name in the value of that expression For example b a for a 1 2 do write b prints A twice not 1 followed by 2 5 5 WHILE DO The FOR DO feature allows easy coding of a repeated operation in which the number of repetitions is known in advance If the criterion for repetition is more complicated WHILE DO can often be used Its syntax is WHILE lt boolean expression gt DO lt statement gt E The WHILE DO controls the single statement following DO If several state ments are to be repeated as is almost always the case they must be grouped using the lt lt gt gt or BEGIN END as in the example below The WHILE condition is tested each time before the action following the DO is attempted If the condition is false to begin with the action is not performed at all Make sure that what is to be tested has an appropriat
132. hmetic is also employed where applicable This package defines a LIMIT operator called with the syntax LIMIT EXPRN algebraic VAR kernel LIMPOINT algebraic algebraic For example limit x sin 1 x x infinity gt 1 limit sin x x 2 x 0 gt INFINITY Direction dependent limit operators LIMIT and LIMIT are also defined This package loads automatically Author Stanley L Kameny 15 29 LINALG Linear algebra package This package provides a selection of functions that are useful in the world of linear algebra Author Matt Rebbeck 15 30 MODSR Modular solve and roots This package supports solve M_SOLVE and roots M_ROOTS operators for modular polynomials and modular polynomial systems The moduli need not be 166 CHAPTER 15 USER CONTRIBUTED PACKAGES primes M_SOLVE requires a modulus to be set M_ROOTS takes the modulus as a second argument For example on modular setmod 8 m_solve 2x 4 gt X 2 X 6 m_solve x 2 y 3 3 gt X 0 Y 5 X 2 Y 1 X 4 Y 5 X 6 Y 1 m_solve x 2 x 2 y 3 3 gt X 2 Y 1 off modular m_roots x 2 1 8 gt ST m_roots x 3 x 7 gt 0 1 6 There is no further documentation for this package Author Herbert Melenk 15 31 NCPOLY Non commutative polynomial ideals This package allows the user to set up automatically a consistent environment for computing in an algebra where the non commutativity is d
133. ial dependency among the variables can be declared using the depend statement solve then rearranges the variable sequence but keeps any variable ahead of those on which it depends 7 10 SOLVE OPERATOR TTI on varopt s a 3t b b 2 c solve s a b c 3 6 a arbcomplex 1 b a c a depend a c depend b c solve s a b c c arbcomplex 2 6 a root_of a_ c a_ 3 oE Here solve is forced to put c after a and after b but there is no obstacle to inter changing a and b 78 CHAPTER 7 BUILT IN PREFIX OPERATORS 7 11 Even and Odd Operators An operator can be declared to be even or odd in its first argument by the declara tions EVEN and ODD respectively Expressions involving an operator declared in this manner are transformed if the first argument contains a minus sign Any other arguments are not affected In addition if say F is declared odd then f 0 is replaced by zero unless F is also declared non zero by the declaration NONZERO For example the declarations even f1 odd 2 mean that a F1 A 2 a gt F2 A f a b 2 FI A B 2 0 gt 0 To inhibit the last transformation say nonzero f2 7 12 Linear Operators An operator can be declared to be linear in its first argument over powers of its second argument If an operator F is so declared F of any sum is broken up into sums of Fs and any factors that are not powers of the variable are tak
134. ied and any divisors of the expression that are perfect squares taken outside the square root argument The remaining expression is left under the square root Thus the expression sqrt 8a72xb becomes 2xaxsqrt 2xb Note that such simplifications can cause trouble if A is eventually given a value that is a negative number If it is important that the positive property of the square root and higher even roots always be preserved the switch PRECISE should be set on the default value This causes any non numerical factors taken out of surds to be represented by their absolute value form With PRECISE on then the above example would become 2xabs a xsqrt 2xb The statement that REDUCE knows very little about these functions applies only in the mathematically exact off rounded mode If ROUNDED is on any of the functions ACOS ACOSH ACOT ACOTH ACSC ACSCH ASEC ASECH ASIN ASINH ATAN ATANH ATAN2 COS COSH COT COTH CSC CSCH EXP HYPOT LN LOG LOGB LOG10 SEC SECH SIN SINH SORT TAN TANH when given a numerical argument has its value calculated to the current degree of floating point precision In addition real non integer valued powers of numbers will also be evaluated If the COMPLEX switch is turned on in addition to ROUNDED these funct ions will also calculate a real or complex result again to the current degree of floating point precision if given complex arguments For example with on rounded complex 2
135. iginal names are used to express the solution forms To suppress solutions of consistent singular equations do OFF SOLVESINGULAR To incorporate additional inverse functions do for example put sinh inverse asinh put asinh inverse sinh together with any desired simplification rules such as for all x let sinh asinh x x asinh sinh x x For completeness functions with non unique inverses should be treated as SIN and COS are in the SOLVE module source Arguments of ASIN and ACOS are not checked to ensure that the absolute value of the real part does not exceed 1 and arguments of LOG are not checked to ensure 74 CHAPTER 7 BUILT IN PREFIX OPERATORS that the absolute value of the imaginary part does not exceed 7 but checks perhaps involving user response for non numerical arguments could be introduced using LET statements for these operators 7 10 4 Parameters and Variable Dependency E The proper design of a variable sequence supplied as a second argument to SOLV is important for the structure of the solution of an equation system Any unknown in the system not in this list is considered totally free E g the call solve x 2 z z 2 y z produces an empty list as a result because there is no function z z x y which fulfills both equations for arbitrary x and y values In such a case the share variable requirements displays a set of restrictions for the parameters of the
136. ing but a special form of a redundant Grobner basis The construction of involutive bases reduces the problem of solving polynomial systems to simple linear algebra Authors A Yu Zharkov and Yu A Blinkov 15 26 LAPLACE Laplace transforms This package can calculate ordinary and inverse Laplace transforms of expressions Documentation is in plain text Authors C Kazasov M Spiridonova V Tomov 15 27 LIE Functions for the classification of real n dimensional Lie algebras LIE is a package of functions for the classification of real n dimensional Lie al gebras It consists of two modules liendmcl1 and lie1234 With the help of the 15 28 LIMITS A PACKAGE FOR FINDING LIMITS 165 functions in the liendmel module real n dimensional Lie algebras L with a derived algebra LU of dimension 1 can be classified Authors Carsten and Franziska Sch bel 15 28 LIMITS A package for finding limits 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 lan 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 expan sion around that point L H6pital s rule is used in critical cases with preprocessing of co oo forms and reformatting of product forms in order to be able to apply P H6pital s rule A limited amount of bounded arit
137. interest to high energy physicists 11 12 CONTENTS Acknowledgment The production of this version of the manual has been the result of the contribu tions of a large number of individuals who have taken the time and effort to suggest improvements to previous versions and to draft new sections Particular thanks are due to Gerry Rayna who provided a draft rewrite of most of the first half of the manual Other people who have made significant contributions have included John Fitch Martin Griss Stan Kameny Jed Marti Herbert Melenk Don Morri son Arthur Norman Eberhard Schriifer Larry Seward and Walter Tietze Finally Richard Hitt produced a TEX version of the REDUCE 3 3 manual which has been a useful guide for the production of the IATEX version of this manual 13 14 CONTENTS Chapter 1 Introductory Information REDUCE is a system for carrying out algebraic operations accurately no matter how complicated the expressions become It can manipulate polynomials in a va riety of forms both expanding and factoring them and extract various parts of them as required REDUCE can also do differentiation and integration but we shall only show trivial examples of this in this introduction Other topics not con sidered include the use of arrays the definition of procedures and operators the specific routines for high energy physics calculations the use of files to eliminate repetitious typing and for saving results and the ed
138. internal canonical form This form which bears little resemblance to the original expression is described in detail in Hearn A C REDUCE 2 A System and Lan guage for Algebraic Manipulation Proc of the Second Symposium on Symbolic and Algebraic Manipulation ACM New York 1971 128 133 The evaluation function may transform its arguments in one of two alternative ways First it may convert the expression into other operators in the system leav ing no functions of the original operator for further manipulation This is in a sense true of the evaluation functions associated with the operators x and for ex ample because the canonical form does not include these operators explicitly It is also true of an operator such as the determinant operator DET because the rel evant evaluation function calculates the appropriate determinant and the operator DET no longer appears On the other hand the evaluation process may leave some residual functions of the relevant operator For example with the operator COS a residual expression like COS X may remain after evaluation unless a rule for the reduction of cosines into exponentials for example were introduced These residual functions of an operator are termed kernels and are stored uniquely like 83 84 CHAPTER 8 DISPLAY AND STRUCTURING OF EXPRESSIONS variables Subsequently the kernel is carried through the calculation as a variable unless transformations are introduced for
139. ion gt is a binary operator that causes a result to be built up and returned by FOR In those cases the loop is initialized to a default value 0 for SUM 1 for PRODUCT and an empty list for the other actions The test for the end condition is made before any action is taken As in Pascal if the variable is out of range in the assignment case or the lt list gt is empty inthe FOR EACH case lt exprn gt is not evaluated at all Examples 1 If A B have been declared to be arrays the following stores 5 through 10 in A 5 through A 10 and at the same time stores the cubes in the B array for i 5 step 1 until 10 do lt lt a i 1 2 b i 173 gt gt 2 Asa convenience the common construction STEP 1 UNTIL may be abbreviated to a colon Thus instead of the above we could write for i 5 10 do lt lt a i 1i72 b i 173 gt gt 3 The following sets C to the sum of the squares of 1 3 5 7 9 and D to the expression xx x 1 x 2 x x 3 x x 4 e t for J 1 step 2 until 9 sum 3 2 d for k 0 step 1 until 4 product x k 4 The following forms a list of the squares of the elements of the list la b ep 42 CHAPTER 5 STATEMENTS for each x in a b c collect x2 5 The following forms a list of the listed squares of the elements of the list a b c ie A 2 B 2 C 2 for each x in a b c collect x72 6 The following also forms a list of the squares of t
140. ions to this rule and are right associa tive Example a b c is interpreted as a b c Unlike ALGOL and PASCAL is left associative In other words a b c is interpreted as a b c The operators lt lt gt gt can only be used for making comparisons between numbers No meaning is currently assigned to this kind of comparison between general expressions Parentheses may be used to specify the order of combination If parentheses are omitted then this order is by the ordering of the precedence list defined by the right hand side of the lt infix operator gt table at the beginning of this section from lowest to highest In other words WHERE has the lowest precedence and the dot operator the highest Chapter 3 Expressions REDUCE expressions may be of several types and consist of sequences of num bers variables operators left and right parentheses and commas The most com mon types are as follows 3 1 Scalar Expressions Using the arithmetic operations power and parentheses scalar ex pressions are composed from numbers ordinary scalar variables identifiers ar ray names with subscripts operator or procedure names with arguments and state ment expressions Examples x x73 2key 2 z 2 df x z p 2 m 2 1 2 log y m a 5 b i q The symbol may be used as an alternative to the caret symbol for forming powers particularly in those systems
141. ist gt lt rule gt lt rule list gt so the first example above could also be written cos a cos b c where cos x cos y gt cos x y cos x y 2 The effect of this construct is that the rule list s in the WHERE clause only apply to the expression on the left of WHERE They have no effect outside the expression In particular they do not affect previously defined WHERE clauses or LET statements For example the sequence let a 2 a where a gt 4 a would result in the output Although WHERE has a precedence less than any other infix operator it still binds higher than keywords such as ELSE THEN DO REPEAT and so on Thus the expression if a 2 then 3 else a 2 where a 3 130 CHAPTER 10 SUBSTITUTION COMMANDS will parse as if a 2 then 3 els a 2 where a 3 WHERE may be used to introduce auxiliary variables in symbolic mode expres sions as described in Section 16 4 However the symbolic mode use has different semantics so expressions do not carry from one mode to the other Compatibility Note In order to provide compatibility with older versions of rule lists released through the Network Library it is currently possible to use an equal sign interchangeably with the replacement sign gt in rules and LET statements However since this will change in future versions the replacement sign is prefer able in rules and the equal sign in non rule based LET statements
142. iting of the input text Also not considered in any detail in this introduction are the many options that are available for varying computational procedures output forms number systems used and so on REDUCE is designed to be an interactive system so that the user can input an al gebraic expression and see its value before moving on to the next calculation For those systems that do not support interactive use or for those calculations espe cially long ones for which a standard script can be defined REDUCE can also be used in batch mode In this case a sequence of commands can be given to RE DUCE and results obtained without any user interaction during the computation In this introduction we shall limit ourselves to the interactive use of REDUCE since this illustrates most completely the capabilities of the system When RE DUCE is called it begins by printing a banner message like REDUCE 3 8 15 Jul 2003 where the version number and the system release date will change from time to time It then prompts the user for input by d 15 16 CHAPTER 1 INTRODUCTORY INFORMATION You can now type a REDUCE statement terminated by a semicolon to indicate the end of the expression for example xt yt z 2 This expression would normally be followed by another character a Return on an ASCII keyboard to wake up the system which would then input the expres sion evaluate it and return the result
143. later resetting of the output line if needed 8 3 2 Output Declarations We now describe a number of switches and declarations that are available for con trolling output formats It should be noted however that the transformation of large expressions to produce these varied output formats can take a lot of comput ing time and space If a user wishes to speed up the printing of the output in such cases he can turn off the switch PRI If this is done then output is produced in one fixed format which basically reflects the internal form of the expression and none of the options below apply PRI is normally on With PRI on the output declarations and switches available are as follows ORDER Declaration The declaration ORDER may be used to order variables on output The syntax is order vl vn where the vi are kernels Thus order X y Z orders X ahead of Y Y ahead of Z and all three ahead of other variables not given an order order nil resets the output order to the system default The order of variables may be changed by further calls of ORDER but then the reordered variables would have an order lower than those in earlier ORDER calls Thus order X y Z order y X 8 3 OUTPUT OF EXPRESSIONS 87 would order Z ahead of Y and X The default ordering is usually alphabetic FACTOR Declaration This declaration takes a list of identifiers or kernels as argument FACTOR is not a factoring command use FACTORIZE or the FACTO
144. le With ROUNDED on the input expression a 2 would be converted into a rational expression whose numerator is 0 5 A and denominator 1 Internally REDUCE uses floating point numbers up to the precision supported by 116 CHAPTER 9 POLYNOMIALS AND RATIONALS the underlying machine hardware and so called bigfloats for higher precision or whenever necessary to represent numbers whose value cannot be represented in floating point The internal precision is two decimal digits greater than the external precision to guard against roundoff inaccuracies Bigfloats represent the fraction and exponent parts of a floating point number by means of arbitrary precision integers which is a more precise representation in many cases than the machine floating point arithmetic but not as efficient If a case arises where use of the machine arithmetic leads to problems a user can force REDUCE to use the bigfloat representation at all precisions by turning on the switch ROUNDBF In rare cases this switch is turned on by the system and the user informed by the message ROUNDBF turned on to increase accuracy Rounded numbers are normally printed to the specified precision However if the user wishes to print such numbers with less precision the printing precision can be set by the command PRINT_PRECISION For example print precision 5 will cause such numbers to be printed with five digits maximum Under normal circumstances when ROUNDED is on REDUCE c
145. les Unlike substitutions introduced via SUB LET rules are global in scope and stay in effect until replaced or CLEARed The simplest use of the LET statement is in the form LET lt substitution list gt where lt substitution list gt isa list of rules separated by commas each of the form lt variable gt lt expression gt or lt prefix operator gt lt argument gt lt argument gt lt expression gt or lt argument gt lt infix operator gt lt argument gt lt expression gt For example let x gt y 2 h u v gt u v 10 2 LET RULES 121 cos pi 3 gt 1 2 axb gt c l m gt n w 3 gt 2xz 3 z 10 gt 0 The list brackets can be left out if preferred The above rules could also have been entered as seven separate LET statements After such LET rules have been input X will always be evaluated as the square of Y and so on This is so even if at the time the LET rule was input the variable Y had a value other than Y In contrast the assignment x y 2 will set X equal to the square of the current value of Y which could be quite different The rule let a b c means that whenever A and B are both factors in an ex pression their product will be replaced by C For example a 5 b 7xw would be replaced by c 5 b 2 w The rule for 1 m will not only replace all occurrences of 1 m by N but will also normally replace L by n m but not M by n 1 A more complete d
146. lways better than the basic algorithm often by orders of magnitude We therefore strongly advise users to use the EZGCD switch where they have the resources available for supporting the package For a description of the EZGCD algorithm see J Moses and D Y Y Yun The EZ GCD Algorithm Proc ACM 1973 ACM New York 1973 159 166 NOTE This package shares code with the factorizer so a certain amount of trace information can be produced using the factorizer trace switches 9 3 1 Determining the GCD of Two Polynomials This operator used with the syntax GCD EXPRN1 polynomial EXPRN2 polynomial polynomial returns the greatest common divisor of the two polynomials EXPRN1 and EXPRN2 Examples gcd x72 2xx 1 x 2 3xx 2 gt X 1 gcd 2 x 2 2 y 2 4xxt4xy gt 2xX 2xY gcd x 2 y 2 x y gt 1 108 CHAPTER 9 POLYNOMIALS AND RATIONALS 9 4 Working with Least Common Multiples Greatest common divisor calculations can often become expensive if extensive work with large rational expressions is required However in many cases the only significant cancellations arise from the fact that there are often common factors in the various denominators which are combined when two rationals are added Since these denominators tend to be smaller and more regular in structure than the numerators considerable savings in both time and space can occur if a full GCD check is made when the denominators are combined and on
147. ly a partial check when numerators are constructed In other words the true least common multiple of the denominators is computed at each step The switch LCM is available for this purpose and is normally on In addition the operator LCM used with the syntax LCM EXPRN1 polynomial EXPRN2 polynomial polynomial returns the least common multiple of the two polynomials EXPRN1 and EXPRN2 Examples lom x 2 2 x 1 x72 3 x 2 gt Xxx3 4xXxx2 5 X 2 lcm 2xx 2 2xy 2 4xx 4xy gt 4x Xxx2 Yxx2 9 5 Controlling Use of Common Denominators When two rational functions are added REDUCE normally produces an expression over a common denominator However 1f the user does not want denominators combined he or she can turn off the switch MCD which controls this process The latter switch is particularly useful if no greatest common divisor calculations are desired or excessive differentiation of rational functions is required CAUTION With MCD off results are not guaranteed to come out in either normal or canonical form In other words an expression equivalent to zero may in fact not be simplified to zero This option is therefore most useful for avoiding expression swell during intermediate parts of a calculation MCD is normally on 9 6 REMAINDER Operator This operator is used with the syntax 9 7 RESULTANT OPERATOR 109 REMAINDER EXPRN1 polynomial EXPRN2 polynomial polynomial
148. ly after loading the package ASSIST Please note that a non list as second argument to CONS a dotted pair in LISP terms is not allowed and causes an invalid as list error a 17 4 xxxxx 17 4 invalid as list Also the initialization of a scalar variable is not the empty list one has to set list type variables explicitly as in the following example load_package assist procedure lotto n m begin scalar list_1_n luckies hit list_1on 36 CHAPTER 4 LISTS luckies for k 1 n do list_1n k list_ 1 n for k 1 m do lt lt hit part list_1_n random n k 1 1 list_1 n delete hit list_1 _ n luckies hit luckies gt gt return luckies end In Germany try lotto 49 6 Another example Find all coefficients of a multivariate polynomial with respect to a list of variables procedure allcoeffs q lis q polynomial lis list of vars allcoeffs list q lis procedure allcoeffs1 q lis if lis then q else allcoeffs1 foreach qq in q join coeff qq first lis rest lis Chapter 5 Statements A statement is any combination of reserved words and expressions and has the syntax lt statement gt lt expression gt lt proper statement gt A REDUCE program consists of a series of commands which are statements fol lowed by a terminator lt terminator gt The division of the program into lines is
149. may be desired to output something each time the statement within the loop construct is repeated It may be desired for a procedure to output intermediate results or other information while it is running It may be desired to have results labeled in special ways especially if the output is directed to a file or device other than the terminal The WRITE command consists of the word WRITE followed by one or more items separated by commas and followed by a terminator There are three kinds of items that can be used 1 Expressions including variables and constants The expression is evalu ated and the result is printed out 2 Assignments The expression on the right side of the operator is evalu ated and is assigned to the variable on the left then the symbol on the left is printed followed by a followed by the value of the expression on the right almost exactly the way an assignment followed by a semicolon prints out normally The difference is that if the WRITE is in a FOR statement and the left hand side of the assignment is an array position or something similar containing the variable of the FOR iteration then the value of that variable is inserted in the printout 3 Arbitrary strings of characters preceded and followed by double quote marks e g string The items specified by a single WRITE statement print side by side on one line The line is broken automatically if it is too long Strin
150. ments reordered to conform to the internal order used by REDUCE The user can change this order for kernels by the command KORDER For example u x v 1 2 would become u v 2 1 x since numbers are ordered in decreasing order and expressions are ordered in decreasing order of complexity Similarly the declaration ANTISYMMETRIC declares an operator antisymmetric For example antisymmetric l m means that any expression involving the top level operators L or M will have its arguments reordered to conform to the internal order of the system and the sign of the expression changed if there are an odd number of argument interchanges necessary to bring about the new order For example 1 x m 1 2 would become 1 m 2 1 x since one inter change occurs with each operator An expression like 1 x x would also be replaced by 0 7 15 Declaring New Prefix Operators The user may add new prefix operators to the system by using the declaration OPERATOR For example operator h gl arctan 7 16 DECLARING NEW INFIX OPERATORS 81 adds the prefix operators H G1 and ARCTAN to the system This allows symbols like h w h x y z gl ptq arctan u v to be used in expressions but no meaning or properties of the operator are implied The same operator symbol can be used equally well as a 0 1 2 3 etc place operator To give a meaning to an operator symbol or express some of its properties LET statements can be us
151. n gt and lt var1 gt lt varn gt are the dummy variable arguments of lt operator gt An analogous form applies to infix operators Examples for all x let df tan x x 1 tan x 2 This is how the tan differentiation rule appears in the REDUCE source for all x y let df f x y x 2 f x y Af x y y xxf x y 62 CHAPTER 7 BUILT IN PREFIX OPERATORS Notice that all dummy arguments of the relevant operator must be declared arbi trary by the FOR ALL command and that rules may be supplied for operators with any number of arguments If no differentiation rule appears for an argument in an operator the differentiation routines will return as result an expression in terms of DF For example if the rule for the differentiation with respect to the second argument of F is not supplied the evaluation of df f x z z would leave this expression unchanged No DEPEND declaration is needed here since f x z obviously depends on Z Once such a rule has been defined for a given operator any future differentiation rules for that operator must be defined with the same number of arguments for that operator otherwise we get the error message Incompatible DF rule argument length for lt operator gt 7 4 INT Operator INT is an operator in REDUCE for indefinite integration using a combination of the Risch Norman algorithm and pattern matching It is used with the syntax INT EXPRN algebraic VAR kern
152. n number of elements since we don t always know how many solutions an equation has If a substitution is made into such an expression closed form solu tions can emerge If this occurs the ROOT_OF construct is replaced by an operator ONE_OF At this point it is of course possible to transform the result of the original SOLVE operator expression into a standard SOLVE solution To effect this the operator EXPAND_CASES can be used 70 CHAPTER 7 BUILT IN PREFIX OPERATORS The following example shows the use of these facilities 7 10 SOLVE OPERATOR 71 solve axx 3 axx 2 x 4 x 3 4xx 2 4 x 2 3 X ROOT_OF A X__ X_ 4 X_ 4 X_ TAG_2 X 1 sub a 1 ws X ONE_OF 2 1 2 TAG_2 X 1 expand_cases ws X 2 X 1 X 2 X 1 7 10 2 Solutions of Equations Involving Cubics and Quartics Since roots of cubics and quartics can often be very messy a switch FULLROOTS is available that when off the default will prevent the production of a result in closed form The ROOT_OF construct will be used in this case instead In constructing the solutions of cubics and quartics trigonometrical forms are used where appropriate This option is under the control of a switch TRIGFORM which is normally on The following example illustrates the use of these facilities let xx solve x 3 x 1 x XX 3 X ROOT_OF X_ X_ 1 X_ on fullroots XX SQRT 31 I ATAN 3 SQRT 3 X Ix SORT 3
153. n the latter form the empty parentheses would not be used use POR not POR where a call on the procedure is needed The two notations for a procedure with no arguments can be combined POR can be defined in the standard PROCEDURE form Then a LET statement let pqr pqr would allow a user to use POR instead of POR in calling the procedure A feature available with LET defined procedures and not with procedures defined in the standard way is the possibility of defining partial functions for all x such that numberp x let uvw x lt procedure body gt Now UVW of an integer would be calculated as prescribed by the procedure body while UVW of a general argument such as Z or p q assuming these evaluate to themselves would simply stay uvw z or uvw p q as the case may be 156 CHAPTER 14 PROCEDURES Chapter 15 User Contributed Packages The complete REDUCE system includes a number of packages contributed by users that are provided as a service to the user community Questions regarding these packages should be directed to their individual authors All such packages have been precompiled as part of the installation process How ever many must be specifically loaded before they can be used Those that are loaded automatically are so noted in their description You should also consult the user notes for your particular implementation for further information on whether this is necessary If it is the relevant command i
154. nicate between symbolic and algebraic mode The purpose of this section is to teach a user the basic principles for this If one wants to evaluate an expression in algebraic mode and then use that ex pression in symbolic mode calculations or vice versa the easiest way to do this is to assign a variable to that expression whose value is easily obtainable in both modes To facilitate this a declaration SHARE is available SHARE takes a list of identifiers as argument and marks these variables as having recognizable values in both modes The declaration may be used in either mode E g share x y says that X and Y will receive values to be used in both modes 16 9 COMMUNICATING WITH ALGEBRAIC MODE 183 If a SHARE declaration is made for a variable with a previously assigned algebraic value that value is also made available in symbolic mode 16 9 1 Passing Algebraic Mode Values to Symbolic Mode If one wishes to work with parts of an algebraic mode expression in symbolic mode one simply makes an assignment of a shared variable to the relevant expres sion in algebraic mode For example if one wishes to work with a b 2 one would say in algebraic mode x t atb 2 assuming that X was declared shared as above If we now change to symbolic mode and say Xx its value will be printed as a prefix form with the syntax SQ lt standard quotient gt T This particular format reflects the fact that the algebraic mode pro
155. nt The reader therefore needs some familiarity with the way that expressions are represented in prefix form in REDUCE to use these operators effectively Furthermore it is assumed that PRI is ON at that point in the calcula tion The reason for this is that with PRI off an expression is printed by walking the tree representing the expression internally To save space it is never actually transformed into the equivalent prefix expression as occurs when PRI is on How ever the operations on polynomials described elsewhere can be equally well used in this case to obtain the relevant parts The evaluation proceeds recursively down the integer expression list In other words PART lt expression gt lt integerl gt lt integer2 gt gt PART PART lt expression gt lt integerl gt lt integer2 gt and so on and PART lt expression gt gt lt expression gt INTEXP can be any expression that evaluates to an integer If the integer is pos itive then that term of the expression is found If the integer is 0 the operator is returned Finally if the integer is negative the counting is from the tail of the expression rather than the head For example if the expression a b is printed as A B 1 e the ordering of the variables is alphabetical then part a b 2 gt B part at tb 1 gt B and part atb 0 gt LUS An operator ARGLENGTH is available to determine the number of arguments of the top
156. nterface 15 14DESIR Differential linear homogeneous equation solutions in the neighborhood of irregular and regular singular points 15 15DFPART Derivatives of generic functions 15 16DUMMY Canonical form of expressions with dummy variables 15 17EXCALC A differential geometry package 15 18FIDE Finite difference method for partial differential equations 15 19FPS Automatic calculation of formal power series 15 20GENTRAN A code generation package 15 21GNUPLOT Display of functions and surfaces 15 22GROEBNER A Gr bner basis package 15 23IDEALS Arithmetic for polynomial ideals 15 24INEQ Support for solving inequalities 15 25INVBASE A package for computing involutive bases 15 26LAPLACE Laplace transforms o CONTENTS 15 27LIE Functions for the classification of real n dimensional Lie al a s ck Samet Raa hoes BA Se ba bd Ga eH 164 15 28LIMITS A package for finding limits 165 15 29LINALG Linear algebra package 165 15 30MODSR Modular solve and roots 165 15 31 NCPOLY Non commutative polynomial ideals 166 15 32NORMFORM Computation of matrix normal forms 166 15 33NUMERIC Solving numerical problems 167 15 340DESOLVE Ordinary differential equations solver 167 15 35ORTHOVEC Manipulation
157. o 24 44 ea esea 5 3 Conditional Statements 34 FOR Statements ooo a 53 WHILE DO 2 6 sie mssoacs os eo is 530 REPEAT s UMTIL 200004 0 eae ee eS 5 7 Compound Statements 5 7 1 Compound Statements with GO TO 5 7 2 Labels and GO TO Statements 5 73 RETURN Statements 6 Commands and Declarations 6 1 Array Declarations 6 2 Mode Handling Declarations Oo BND ald a BA Sad Oe ES 64 BYE Command oc s s co be ee ea es 6 5 SHOWTIME Command 6 6 DEFINECommand 7 Built in Prefix Operators 7 1 Numerical Operators CONTENTS CONTENTS 3 T2 73 7 4 q 7 6 tat 7 8 7 9 7 10 ALL ABSen at wie ed dada da ds ira we ee 53 AR CELING A 54 ALI CONI o cc eae ae be EWES EERE SS 54 LA PACTORIAL cours OG de bed ears 54 Ol PIR ian bee ee ae hak Ee ee Be ee 54 Fl FLOOR oc a edad be ee hace eRe EE 55 Sela IMPART cocoa a df ale ae Gay a ele 3 55 TAE MAXMIN oi a Rat a 55 71 9 NEXTPRIME ooo 2 55 PLIO RANDOM oda discs ee ha a a os ees 56 7 1 11 RANDOM_NEW_SEED 56 FL 12 REPART occo cia ebb eee eee eee 56 Pcie ROUND coc a been d baa ogee wank waded 57 FLAW BUG a s eh a Be a dk oe we we a 57 Mathematical Functions o o e 57 DP Operate o Sana ea ha ae Rede PS ba Bad os ears 61 7 3 1 Adding Differentiation Rules 61 INT Operator oo eco coca tace
158. o be vectors Variables declared as indices or given a mass are automatically declared vector by these declarations 17 3 Additional Expression Types Two additional expression types are necessary for high energy calculations namely 17 3 1 Vector Expressions These follow the normal rules of vector combination Thus the product of a scalar or numerical expression and a vector expression is a vector as are the sum and difference of vector expressions If these rules are not followed error messages are printed Furthermore if the system finds an undeclared variable where it expects a vector variable it will ask the user in interactive mode whether to make that variable a vector or not In batch mode the declaration will be made automatically and the user informed of this by a message Examples Assuming P and Q have been declared vectors the following are vector expressions P 2xq 3 2xxxyxp p ax q 3xq q whereas px q and p q are not 17 3 2 Dirac Expressions These denote those expressions which involve y matrices A y matrix is implicitly a4 x 4 matrix and so the product sum and difference of such expressions or the product of a scalar and Dirac expression is again a Dirac expression There are no Dirac variables in the system so whenever a scalar variable appears in a Dirac expression without an associated y matrix expression an implicit unit 4 by 4 matrix is assumed For example g 1 p m denotes g 1 p m l
159. o defined are always binary a mm b mm c means a mm b mm c 7 17 Creating Removing Variable Dependency There are several facilities in REDUCE such as the differentiation operator and the linear operator facility that can utilize knowledge of the dependency between various variables or kernels Such dependency may be expressed by the command DEPEND This takes an arbitrary number of arguments and sets up a dependency of the first argument on the remaining arguments For example depend X y Z says that X is dependent on both Y and Z depend z cos x y says that Z is dependent on COS X and Y Dependencies introduced by DEPEND can be removed by NODEPEND The argu ments of this are the same as for DEPEND For example given the above depen dencies nodepend z cos x says that Z is no longer dependent on COS X although it remains dependent on Ye Chapter 8 Display and Structuring of Expressions In this section we consider a variety of commands and operators that permit the user to obtain various parts of algebraic expressions and also display their structure in a variety of forms Also presented are some additional concepts in the REDUCE design that help the user gain a better understanding of the structure of the system 8 1 Kernels REDUCE is designed so that each operator in the system has an evaluation or simplification function associated with it that transforms the expression into an
160. of scalars and vectors 168 15 36PHYSOP Operator calculus in quantum theory 168 15 37PM A REDUCE pattern matcher 168 15 38RANDPOLY A random polynomial generator 169 15 39REACTEQN Support for chemical reaction equation systems 169 15 40RESET Code to reset REDUCE to its initial state 169 15 41RESIDUE A residue package o cacera 169 15 42RLFI REDUCE LaTeX formula interface 169 15 43ROOTS A REDUCE root finding package 170 15 44RSOLVE Rational integer polynomial solvers 170 15 45SCOPE REDUCE source code optimization package 170 15 46SETS A basic set theory package oo 171 15 47SPDE Finding symmetry groups of PDE s 171 15 48SPECEN Package for special functions 171 15 49SPECFN2 Package for special special functions 172 15 50SUM A package for series summation 173 15 51SYMMETRY Operations on symmetric matrices 173 15 52TAYLOR Manipulation of Taylor series 173 15 53TPS A truncated power series package 173 13 54 7TRE TeX REDUCE interface lt lt ceras 174 15 55TRIGSIMP Simplification and factorization of trigonometric and hyperbolic functions soora oo e 00004 174 CONTENTS 15 56WU Wu algorithm for polynomial systems 15 57XCOLOR Color factor in some field theories
161. ogramming Laboratory Report FSC 69 0312 1969 For the reasons given in these references REDUCE does not attempt to imple ment a general pattern matching algorithm However the present system uses far more sophisticated techniques than those discussed in the above papers It is now possible for the rules appearing in arguments of LET to have the form lt substitution expression gt lt expression gt where any rule to which a sensible meaning can be assigned is permitted However this meaning can vary according to the form of lt substitution expression gt The semantic rules associated with the application of the sub stitution are completely consistent but somewhat complicated by the pragmatic need to perform such substitutions as efficiently as possible The following rules explain how the majority of the cases are handled To begin with the lt substitution expression gt is first partly simplified by collecting like terms and putting identifiers and kernels in the system order However no substitutions are performed on any part of the expression with the exception of expressions with the instant evaluation property such as array and matrix elements whose actual values are used It should also be noted that the system order used is not changeable by the user even with the KORDER command Specific cases are then handled as follows 1 If the resulting simplified rule has a left hand side that is an identifier an expression wi
162. ond rule would only apply to the explicit argument X Later occurrences in the same rule may also be marked but this is optional internally all such rules are stored with each relevant variable explicitly marked The optional WHEN clause allows constraints to be placed on the application of the rule much as the SUCH THAT clause in a LET statement A rule list may be named for example trigl cos x x cos y gt cos x y cos x y 2 cos x x sin y gt sin x y sin x y 2 sin x sin y gt cos x y cos x y 2 cos x 2 gt 1 cos 2xx 2 sin x 7 2 gt l cos 2 x 2 Such named rule lists may be inspected as needed E g the command trigl would cause the above list to be printed Rule lists may be used in two ways They can be globally instantiated by means of the command LET For example let trigl would cause the above list of rules to be globally active from then on until cancelled by the command CLEARRULES as in clearrules trigl 10 3 RULE LISTS 129 CLEARRULES has the syntax CLEARRULES lt rule list gt lt name of rule list gt The second way to use rule lists is to invoke them locally by means of a WHERE clause For example cos a cos b c where cos x x cos y gt cos x y cos x y 2 or cos a sin b where trigrules The syntax of an expression with a WHERE clause is lt expression gt WHERE lt rule gt lt rule l
163. one after another scalar a b c scalar z In place of SCALAR one can also use the declarations INTEGER or REAL In the present version of REDUCE variables declared INTEGER are expected to have only integer values and are initialized to 0 REAL variables on the other hand are currently treated as algebraic mode SCALARs CAUTION INTEGER REAL and SCALAR declarations can only be given imme diately after a BEGIN An error will result if they are used after other statements in a block including ARRAY and OPERATOR declarations which are global in scope or outside the top most block e g at the top level All variables declared SCALAR are automatically initialized to zero in algebraic mode NIL in symbolic mode Any symbols not declared as local variables in a block refer to the variables of the same name in the current calling environment In particular if they are not so declared at a higher level e g in a surrounding block or as parameters in a calling procedure their values can be permanently changed Following the SCALAR declaration s if any write the statements to be executed one after the other separated by delimiters e g or it doesn t matter which However from a stylistic point of view is preferred The last statement in the body just before END need not have a terminator since the BEGIN END are in a sense brackets confining the block statements The last statement mus
164. onverts the number 1 0 to the integer 1 If this is not desired the switch NOCONVERT can be turned on Numbers that are stored internally as bigfloats are normally printed with a space between every five digits to improve readability If this feature is not required it can be suppressed by turning off the switch BF SPACE 1 Further information on the bigfloat arithmetic may be found in T Sasaki Man ual for Arbitrary Precision Real Arithmetic System in REDUCE Department of Computer Science University of Utah Technical Note No TR 8 1979 When a real number is input it is normally truncated to the precision in effect at the time the number is read If it is desired to keep the full precision of all numbers input the switch ADJPREC for adjust precision can be turned on While on ADJPREC will automatically increase the precision when necessary to match that of any integer or real input and a message printed to inform the user of the precision increase When ROUNDED is on rational numbers are normally converted to rounded rep resentation However if a user wishes to keep such numbers in a rational form until used in an operation that returns a real number the switch ROUNDALL can be turned off This switch is normally on Results from rounded calculations are returned in rounded form with two excep tions if the result is recognized as 0 or 1 to the current precision the integer result is returned 9 11 POLY
165. orizer works by first reducing multivariate problems to univariate ones and then solving the univariate ones modulo small primes It normally selects both evaluation points and primes using a random number generator that should lead to different detailed behavior each time any particular problem is tackled If for some reason it is known that a certain probably univariate factorization can be performed effectively with a known prime P say this value of P can be handed to FACTORIZE as a second argument An error will occur if a non prime is provided to FACTORIZE in this manner It is also an error to specify a prime that divides the discriminant of the polynomial being factored but users should note that this condition is not checked by the program so this capability should be used with Pusey 106 CHAPTER 9 POLYNOMIALS AND RATIONALS care Factorization can be performed over a number of polynomial coefficient domains in addition to integers The particular description of the relevant domain should be consulted to see if factorization is supported For example the following state ments will factorize zt 1 modulo 7 setmod 7 on modular factorize x 4 1 The factorization module is provided with a trace facility that may be useful as a way of monitoring progress on large problems and of satisfying curiosity about the internal workings of the package The most simple use of this is enabled by issuing the REDUCE command on t
166. our discussion so far we have assumed that we are working in the normal four dimensions of QED calculations However in most cases the programs will also work in an arbitrary number of dimensions The command vecdim lt expression gt sets the appropriate dimension The dimension can be symbolic as well as numer ical Users should note however that the EPS operator and the y5 symbol A are not properly defined in other than four dimensions and will lead to an error if used 198 CHAPTER 17 CALCULATIONS IN HIGH ENERGY PHYSICS Chapter 18 REDUCE and Rlisp Utilities REDUCE and its associated support language system Rlisp include a number of utilities which have proved useful for program development over the years The following are supported in most of the implementations of REDUCE currently available 18 1 The Standard Lisp Compiler Many versions of REDUCE include a Standard Lisp compiler that is automatically loaded on demand You should check your system specific user guide to make sure you have such a compiler To make the compiler active the switch COMP should be turned on Any further definitions input after this will be compiled automatically If the compiler used is a derivative version of the original Griss Hearn compiler M L Griss and A C Hearn A Portable LISP Compiler SOFTWARE Practice and Experience 11 1981 541 605 there are other switches that might also be used in this regard However these addi
167. output in IN readable form Each expression printed will end with a out abcd output to new file linelength 72 for systems with fixed input line length XYZI XyZ will output XYZ followed by the value of XYZ write end standard for ending files for IN shut abcd save ABCD return to terminal output on nat restore usual output form 11 3 SHUT Command This command takes a list of names of files that have been previously opened via an OUT statement and closes them Most systems require this action by the user before he ends the REDUCE job if not sooner otherwise the output may be lost If a file is shut and a further OUT command issued for the same file the file is erased before the new output is written If it is the current output file that is shut output will switch to the terminal At tempts to shut files that have not been opened by OUT or an input file will lead to errors Chapter 12 Commands for Interactive Use REDUCE is designed as an interactive system but naturally it can also operate in a batch processing or background mode by taking its input command by command from the relevant input stream There is a basic difference however between in teractive and batch use of the system In the former case whenever the system discovers an ambiguity at some point in a calculation such as a forgotten type assignment for instance it asks the user for the correct interpretation In batch oper
168. ple The following is the complete code for doing the above steps The end result will be that the square of the leading term of a b is calculated share x y declare X and Y as shared x a b 2 store atb 2 in X symbolic transfer to symbolic mode Z cadr x store a true standard quotient in Z lt numr z print the leading term of the numerator of Z y mk xsq t2q lt numr z store the prefix form of this leading term in Y algebraic return to algebraic mode y 2 evaluate square of the leading term of a b 2 16 9 4 Defining Procedures for Intermode Communication If one wishes to define a procedure in symbolic mode for use as an operator in alge braic mode it is necessary to declare this fact to the system by using the declaration OPERATOR in symbolic mode Thus symbolic operator leadterm would declare the procedure LEADTERM as an algebraic operator This declaration must be made in symbolic mode as the effect in algebraic mode is different The value of such a procedure must be a prefix form 16 9 COMMUNICATING WITH ALGEBRAIC MODE 187 The algebraic processor will pass arguments to such procedures in prefix form Therefore if you want to work with the arguments as standard quotients you must first convert them to that form by using the function SIMP This function takes a prefix form as argument and returns the evaluated standard quotient For example if you want to define
169. pt plus a b 2 for atb 2 or the format sq lt standard quotient gt t as described above However if you have been working with parts of a standard form they will probably not be in this form In that case you can do the following 1 If it is a standard quotient call PREPSO on it This takes a standard quo tient as argument and returns a prefix expression Alternatively you can call MK SQ on it which returns a prefix form like SQ lt standard quotient gt T and avoids translation of the expression into a true prefix form 2 If it is a standard form call PREPF on it This takes a standard form as argument and returns the equivalent prefix expression Alternatively you can convert it to a standard quotient and then call MK SQ 3 If it is a part of a standard form you must usually first build up a standard form out of it and then go to step 2 The conversion functions described 186 CHAPTER 16 SYMBOLIC MODE earlier may be used for this purpose For example a If Z is an expression which is a term T2F Z is a standard form b If Z is a standard power P2F Z is a standard form c If Z is a variable you can pass it direct to algebraic mode For example to pass the leading term of a b 2 back to algebraic mode one could say y mk lx sq t2q lt numr z where Y has been declared shared as above If you now go back to algebraic mode you can work with Y in the usual way 16 9 3 Complete Exam
170. pulations of generalized hypergeometric functions and Meijer s G function Generalized hypergeometric functions are sim plified towards special functions and Meijer s G function is simplified towards spe cial functions or generalized hypergeometric functions Author Victor Adamchik with major updates by Winfried Neun 15 50 SUM A PACKAGE FOR SERIES SUMMATION 173 15 50 SUM A package for series summation This package implements the Gosper algorithm for the summation of series It defines operators SUM and PROD The operator SUM returns the indefinite or defi nite summation of a given expression and PROD returns the product of the given expression This package loads automatically Author Fujio Kako 15 51 SYMMETRY Operations on symmetric matrices This package computes symmetry adapted bases and block diagonal forms of ma trices 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 Author Karin Gatermann 15 52 TAYLOR Manipulation of Taylor series This package carries out the Taylor expansion of an expression 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 Author Rainer Sch pf 15 53 TPS A truncated power se
171. rder equations of simple types linear equations with constant coefficients and Euler equations These solvable types are exactly those for which Lie symmetry techniques give no useful information For example the evaluation of depend y x odesolve df y Xx Xx x 2 ex X Y X 5 168 CHAPTER 15 USER CONTRIBUTED PACKAGES yields the result X 3 34E 3x ARBCONST 1 X Y Main Author Malcolm A H MacCallum Other contributors Francis Wright Alan Barnes 15 35 ORTHOVEC Manipulation of scalars and vectors ORTHOVEC is a collection of REDUCE procedures and operations which provide a simple to use environment for the manipulation of scalars and vectors Opera tions include addition subtraction dot and cross products division modulus div grad curl laplacian differentiation integration and Taylor expansion Author James W Eastwood 15 36 PHYSOP Operator calculus in quantum theory This package has been designed to meet the requirements of theoretical physicists looking for a computer algebra tool to perform complicated calculations in quan tum theory with expressions containing operators These operations consist mainly of the calculation of commutators between operator expressions and in the evalua tions of operator matrix elements in some abstract space Author Mathias Warns 15 37 PM A REDUCE pattern matcher PM is a general pattern matcher similar in style to those found in systems such as S
172. rfac Following this all calls to the factorizer will generate informative messages reporting on such things as the reduction of multivariate to univariate cases the choice of a prime and the reconstruction of full factors from their images Further levels of detail in the trace are intended mainly for system tuners and for the investigation of suspected bugs For example TRALLFAC gives tracing information at all levels of detail The switch that can be set by on timings makes it possible for one who is familiar with the algo rithms used to determine what part of the factorization code is consuming the most resources on overview reduces the amount of detail presented in other forms of trace Other forms of trace output are enabled by directives of the form symbolic set trace factor lt number gt lt filename gt where useful numbers are 1 2 3 and 100 101 This facility is intended to make it possible to discover in fairly great detail what just some small part of the code has been doing the numbers refer mainly to depths of recursion when the factorizer calls itself and to the split between its work forming and factorizing images and reconstructing full factors from these If NIL is used in place of a filename the trace output requested is directed to the standard output stream After use of this trace facility the generated trace files should be closed by calling symbolic close trace files NOTE Using the facto
173. ries package This package implements formal Laurent series expansions in one variable using the domain mechanism of REDUCE This means that power series objects can be added multiplied differentiated etc like other first class objects in the system A lazy evaluation scheme is used and thus terms of the series are not evaluated until they are required for printing or for use in calculating terms in other power series The series are extendible giving the user the impression that the full infinite series is being manipulated The errors that can sometimes occur using series that are truncated at some fixed depth for example when a term in the required series depends on terms of an intermediate series beyond the truncation depth are thus avoided 174 CHAPTER 15 USER CONTRIBUTED PACKAGES Authors Alan Barnes and Julian Padget 15 54 TRI TeX REDUCE interface This package provides facilities written in REDUCE Lisp for typesetting RE DUCE formulas using TEX The TEX REDUCE Interface incorporates three levels of TRXoutput without line breaking with line breaking and with line breaking plus indentation Author Werner Antweiler 15 55 TRIGSIMP Simplification and factorization of trigono metric and hyperbolic functions TRIGSIMP is a useful tool for all kinds of trigonometric and hyperbolic simpli fication and factorization There are three procedures included in TRIGSIMP trigsimp trigfactorize and triggcd The first is for findin
174. rizer with MCD off will result in an error 9 3 Cancellation of Common Factors Facilities are available in REDUCE for cancelling common factors in the numer ators and denominators of expressions at the option of the user The system will 9 3 CANCELLATION OF COMMON FACTORS 107 perform this greatest common divisor computation if the switch GCD is on GCD is normally off A check is automatically made however for common variable and numerical prod ucts in the numerators and denominators of expressions and the appropriate can cellations made When GCD is on and EXP is off a check is made for square free factors in an expression This includes separating out and independently checking the content of a given polynomial where appropriate For an explanation of these terms see Anthony C Hearn Non Modular Computation of Polynomial GCDs Using Trial Division Proc EUROSAM 79 published as Lecture Notes on Comp Science Springer Verlag Berlin No 72 1979 227 239 Example With EXP off and GCD on the polynomial ax cta d b c b d would be returned as A B C D Under normal circumstances GCDs are computed using an algorithm described in the above paper It is also possible in REDUCE to compute GCDs using an al ternative algorithm called the EZGCD Algorithm which uses modular arithmetic The switch EZGCD if on in addition to GCD makes this happen In non trivial cases the EZGCD algorithm is almost a
175. rrent degree of floating point precision I Intended to represent the square root of 1 i 2 is replaced by 1 and appropriately for higher powers of I This applies only to the symbol I used on the top level not as a formal parameter in a procedure a local variable nor in the context for i INFINITY Intended to represent oo in limit and power series calculations for example Note however that the current system does not do proper arithmetic on oo For example infinity infinity is 2xinfinity NIL In REDUCE algebraic mode only taken as a synonym for zero Therefore NIL cannot be used as a variable PI Intended to represent the circular constant With ROUNDED on it is replaced by the value of 7 to the current degree of floating point precision T Should not be used as a formal parameter or local variable in pro cedures since conflict arises with the symbolic mode meaning of T as true Other reserved variables such as LOW_POW described in other sections are listed in Appendix A Using these reserved variables inappropriately will lead to errors There are also internal variables used by REDUCE that have similar restrictions These usually have an asterisk in their names so it is unlikely a casual user would 2 5 STRINGS 23 use one An example of such a variable is K used in the asymptotic command package Certain words are reserved in REDUCE They may only be used in the manner intended A list of
176. rt for simple structures 6 support for records In addition is a letter in Rlisp 88 In other words A B is an identifier not the difference of the identifiers A and B If the latter construct is required it is necessary to put spaces around the character For compatibility between the two versions of Rlisp we recommend this convention be used in all symbolic mode programs To use Rlisp 88 type on rlisp88 This switches to symbolic mode with the Rlisp 88 syntax and extensions While in this environment it is impossible to switch to algebraic mode or prefix expressions by algebraic However symb olic mode programs written in Rlisp 88 may be run in algebraic mode provided the rlisp88 package has been loaded We also expect that many of the extensions de fined in Rlisp 88 will migrate to the basic Rlisp over time To return to traditional Rlisp or to switch to algebraic mode say off rlisp88 16 11 REFERENCES 189 16 11 References There are a number of useful books which can give you further information about LISP Here is a selection Allen J R The Anatomy of LISP McGraw Hill New York 1978 McCarthy J P W Abrahams J Edwards T P Hart and M I Levin LISP 1 5 Programmer s Manual M I T Press 1965 Touretzky D S LISP A Gentle Introduction to Symbolic Computation Harper amp Row New York 1984 Winston P H and Horn B K P LISP Addison We
177. s lpower a b lpower a b lpower a b 9 10 5 LTERM Operator Syntax LTERM EXPRN polyn c 2xd 2 a c 2xd 2 d gt gt gt gt c 2xd e 113 Cxx2 4xCxD 4xDxx2 4x A B AxC 2xA D Bx xC 2x BxD EXPRN polynomial VAR kernel polyno mial Dxx2 EXPRN with respect to VAR If I EXPRN omial VAR kernel polynomial LTERM returns the leading term of EXPRN with respect to VAR If EXPRN does not depend on VAR EXPRN is returned Examples Compatibility Note In some earlier versions of REDUCE LT the EXPRN did not depend on VAR In the present version EXPRN VAR REDUCT 1 to LTERM lterm a lterm a lterm a 2xd 2 a 2x d 2 d 2xd e 9 10 6 MAINVAR Operator Syntax gt Ax Cxx2 4x CxD 4xDxx2 AxXDxx2x A4 HB 2 BxD AxC 2xAxD EXPRN VAR BxC ERM returned 0 if EXPRN is always equal 114 CHAPTER 9 POLYNOMIALS AND RATIONALS MAINVAR EXPRN polynomial expression Returns the main variable based on the internal polynomial representation of EXPRN If EXPRN is a domain element 0 is returned Examples Assuming A has higher kernel order than B C or D mainvar atb c t2 d 2 gt A mainvar 2 gt 0 9 10 7 NUM Operator Syntax NUM EXPRN rational polynomial Returns the numerator o
178. s A C Norman amp P M A Moore Implementing the New Risch Algorithm Proc 4th International Symposium on Advanced Comp Methods in Theor Phys CNRS Marseilles 1977 64 CHAPTER 7 BUILT IN PREFIX OPERATORS S J Harrington A New Symbolic Integration System in Reduce Comp Journ 22 1979 2 A C Norman amp J H Davenport Symbolic Integration The Dust Settles Proc EUROSAM 79 Lecture Notes in Computer Science 72 Springer Verlag Berlin Heidelberg New York 1979 398 407 7 5 LENGTH Operator LENGTH is a generic operator for finding the length of various objects in the sys tem The meaning depends on the type of the object In particular the length of an algebraic expression is the number of additive top level terms its expanded representation Examples length a b gt 2 length 2 gt 1 Other objects that support a length operator include arrays lists and matrices The explicit meaning in these cases is included in the description of these objects 7 6 MAP Operator The MAP operator applies a uniform evaluation pattern to all members of a com posite structure a matrix a list or the arguments of an operator expression The evaluation pattern can be a unary procedure an operator or an algebraic expression with one free variable It is used with the syntax MAP U function V object Here object is a list a matrix or an operator expression Function can be one of
179. s LOAD_PACKAGE which takes a list of one or more package names as argument for example load_package algint although this syntax may vary from implementation to implementation Nearly all these packages come with separate documentation and test files except those noted here that have no additional documentation which is included along with the source of the package in the REDUCE system distribution These items should be studied for any additional details on the use of a particular package The packages available in the current release of REDUCE are as follows 15 1 ALGINT 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 where the answer exists in that class of functions It is an implemen tation of the work described in J H Davenport On the Integration of Algebraic Functions LNCS 102 Springer Verlag 1981 Both this and the source code 157 158 CHAPTER 15 USER CONTRIBUTED PACKAGES should be consulted for a more detailed description of this work Once the ALGINT package has been loaded using LOAD_PACKAGE one enters an expression for integration as with the regular integrator for example int sqrt x sgrt x 2 1 x x If one later wishes to integrate expressions without using the facilities of this pack age the switch ALGINT should be turned off This is
180. s input a matrix given as a list of lists that is interpreted as a row matrix If that form of input is chosen the vectors in the result will be represented by lists as well This addi tional input syntax facilitates the use of NULLSPACE in applications different from classical linear algebra 13 5 MATRIX ASSIGNMENTS 147 13 4 7 RANK Operator Syntax RANK EXPRN matrix_expression integer RANK calculates the rank of its argument that like NULLSPACE can either be a standard matrix expression or a list of lists that can be interpreted either as a row matrix or a set of equations Example rank mat a b Cc r d e f returns the value 2 13 5 Matrix Assignments Matrix expressions may appear in the right hand side of assignment statements If the left hand side of the assignment which must be a variable has not already been declared a matrix it is declared by default to the size of the right hand side The variable is then set to the value of the right hand side Such an assignment may be used very conveniently to find the solution of a set of linear equations For example to find the solution of the following set of equations allxx 1 al2xx 2 yl a21lxx 1 a22xx 2 y2 we simply write x 1 mat all al2 a21 a22 mat y1 y2 13 6 Evaluating Matrix Elements Once an element of a matrix has been assigned it may be referred to in standard array element notation
181. selves can be rational as long as they do not depend on VAR However if the switch RATARG is on denominators are not checked for dependence on VAR and are taken to be part of the coefficients Example coeff y 2 z 3 z y returns the result 2 Z 0 3 2Z 0 3 0 1 2 whereas coeff y 2 z 3 y y gives an error if RATARG is off and the result 3 2 Z Y 0 3 Z2 Y 0 3 Z Y 0 1 Y if RATARG is on The length of the result of COEFF is the highest power of VAR encountered plus 1 In the above examples it is 7 In addition the variable HIGH POW is set to the highest non zero power found in EXPRN during the evaluation and LOW_POW to the lowest non zero power or zero if there is a constant term If EXPRN is a constant then HIGH_POW and LOW_POW are both set to zero 8 5 2 COEFFN Operator The COEFEN operator is designed to give the user a particular coefficient of a vari able in a polynomial as opposed to COEFF that returns all coefficients COEFFN is used with the syntax COEFFN EXPRN polynomial VAR kernel N integer It returns the nt coefficient of VAR in the polynomial EXPRN 8 5 3 PART Operator Syntax 8 5 OBTAINING PARTS OF ALGEBRAIC EXPRESSIONS 101 PART EXPRN algebraic INTEXP integer This operator works on the form of the expression as printed or as it would have been printed at that point in the calculation bearing in mind all the relevant switch settings at that poi
182. semicolon is used as a terminator each expression except the last is printed followed by a ending with the value of the last expression 5 1 1 Set Statement In some cases it is desirable to perform an assignment in which both the left and right hand sides of an assignment are evaluated In this case the SET statement can be used with the syntax SET lt expression gt lt expression gt 5 2 GROUP STATEMENTS 39 For example the statements j 23 set mkid a j x assigns the value X to A23 5 2 Group Statements The group statement is a construct used where REDUCE expects a single state ment but a series of actions needs to be performed It is formed by enclosing one or more statements of any kind between the symbols lt lt and gt gt separated by semicolons or dollar signs it doesn t matter which The statements are executed one after another Examples will be given in the sections on IF and other types of statements in which the lt lt gt gt construct is useful If the last statement in the enclosed group has a value then that is also the value of the group statement Care must be taken not to have a semicolon or dollar sign after the last grouped statement if the value of the group is relevant such an extra terminator causes the group to have the value NIL or zero 5 3 Conditional Statements The conditional statement has the following syntax lt conditional statement
183. since the last command or since TIME was last turned off or the session be gan Another useful switch with interactive use is DEMO which causes the system to pause after each command in a file with the exception of comments until a Return is typed on the terminal This enables a user to set up a demonstration file and step through it command by command As with most declarations arguments to ON and OFF may be strung together sep arated by commas For example off time demo will turn off both the time messages and the demonstration switch We note here that while most ON and OFF commands are obeyed almost instanta neously some trigger time consuming actions such as reading in necessary mod 6 3 END 51 ules from secondary storage A diagnostic message is printed if ON or OFF are used with a switch that is not known to the system For example 1f you misspell DEMO and type on demq you will get the message xxxxx DEMO not defined as switch 6 3 END The identifier END has two separate uses 1 Its use in a BEGIN END bracket has been discussed in connection with compound statements 2 Files to be read using IN should end with an extra END command The reason for this is explained in the section on the IN command This use of END does not allow an immediately preceding END such as the END of a procedure definition so we advise using END there 6 4 BYE Command
184. sley 1981 190 CHAPTER 16 SYMBOLIC MODE Chapter 17 Calculations in High Energy Physics A set of REDUCE commands is provided for users interested in symbolic calcula tions in high energy physics Several extensions to our basic syntax are necessary however to allow for the different data structures encountered 17 1 High Energy Physics Operators We begin by introducing three new operators required in these calculations 17 1 1 Cons Operator Syntax EXPRN1 vector_expression EXPRN2 vector_expression algebraic The binary operator which is normally used to denote the addition of an element to the front of a list can also be used in algebraic mode to denote the scalar product of two Lorentz four vectors For this to happen the second argument must be recognizable as a vector expression at the time of evaluation With this meaning this operator is often referred to as the dot operator In the present system the index handling routines all assume that Lorentz four vectors are used but these routines could be rewritten to handle other cases Components of vectors can be represented by including representations of unit vec tors in the system Thus if EO represents the unit vector 1 0 0 0 p eo represents the zeroth component of the four vector P Our metric and notation fol 191 192 CHAPTER 17 CALCULATIONS IN HIGH ENERGY PHYSICS lows Bjorken and Drell Relativistic Quantum Mechanics McGr
185. so get the same result by saying factorial 40 Since we want exact results in algebraic calculations it is essential that integer arithmetic be performed to arbitrary precision as in the above example Further 17 more the FOR statement in the above is illustrative of a whole range of combining forms that REDUCE supports for the convenience of the user Among the many options in REDUCE is the use of other number systems such as multiple precision floating point with any specified number of digits of use if roundoff in say the 100 digit is all that can be tolerated In many cases it is necessary to use the results of one calculation in succeeding calculations One way to do this is via an assignment for a variable such as u x y z 72 If we now use U in later calculations the value of the right hand side of the above will be used The results of a given calculation are also saved in the variable WS for WorkSpace so this can be used in the next calculation for further processing For example the expression df ws x following the previous evaluation will calculate the derivative of x y z 2 with respect to X Alternatively int ws y would calculate the integral of the same expression with respect to y REDUCE is also capable of handling symbolic matrices For example matrix m 2 2 declares m to be a two by two matrix and m mat a b c d glves its elements values Expressions that incl
186. stem efficiency If END is omitted an error message End of file read will occur 135 136 CHAPTER 11 FILE HANDLING COMMANDS 11 2 OUT Command This command takes a single file name as argument and directs output to that file from then on until another OUT changes the output file or SHUT closes it Output can go to only one file at a time although many can be open If the file has previously been used for output during the current job and not SHUT the new output is appended to the end of the file Any existing file is erased before its first use for output in a job or if it had been SHUT before the new OUT To output on the terminal without closing the output file the reserved file name T for terminal may be used For example out ofile will direct output to the file OF ILE and out t will direct output to the user s terminal The output sent to the file will be in the same form that it would have on the terminal In particular x 2 would appear on two lines an X on the lower line and a 2 on the line above If the purpose of the output file is to save results to be read in later this is not an appropriate form We first must turn off the NAT switch that specifies that output should be in standard mathematical notation Example To create a file ABCD from which it will be possible to read using IN the value of the expression XYZ off echos needed if your input is from a file off nats
187. strictions the solutions are valid only as long as none of these expressions vanishes Any zero of one of them represents a special case that is not covered by the formal solution In the above case the value is 76 CHAPTER 7 BUILT IN PREFIX OPERATORS assumptions b which excludes formally the case b 0 obviously this special parameter value makes the system singular The set of assumptions is complete for both linear and non linear systems SOLVE rearranges the variable sequence to reduce the expected computing time This behavior is controlled by the switch varopt which is on by default If it is turned off the supplied variable sequence is used or the system kernel ordering is taken if the variable list is omitted The effect is demonstrated by an example s y 3 3x 0 x 2 y 2 1 solve s y x 6 2 y root_of y_ 9 y_ 9 y_ off varopt solve s y x 6 4 2 x root_of x_ 3xx_ 12 x_ 1 x_ 4 2 xx x 2xx 10 y In the first case solve forms the solution as a set of pairs y x y because the degree of x is higher such a rearrangement makes the internal computation of the Gr bner basis generally faster For the second case the explicitly given variable sequence is used such that the solution has now the form 2 y z Controlling the variable sequence is especially important if the system has one or more free variables As an alternative to turning off varopt a part
188. t 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 CAUTION The unsigned part of any number may not begin with a decimal point as this causes confusion with the CONS operator i e NOT ALLOWED 5 23 12 use0 5 0 23 0 12 instead 2 3 IDENTIFIERS 21 2 3 Identifiers Identifiers in REDUCE consist of one or more alphanumeric characters i e alpha betic letters or decimal digits the first of which must be alphabetic The maximum number of characters allowed is implementation dependent although twenty four is permitted in most implementations In addition the underscore character _ is considered a letter if it is within an identifier For example a az pl q23p a_very_long_variable are all identifiers whereas a is not A sequence of alphanumeric characters in which the first is a digit is interpreted as a product For example 2ab3c is interpreted as 2xab3c There is one exception to this If the first letter after a digit is E the system will try to interpret that part of the sequence as a real number which may fail in some cases For example 2E12 is the real number 2 0 x 102 2e3c is 2000 0 C and 2ebc gives an error Special characters such as and blank may be used in identifiers too even as the first character but each must be preceded by an exclamation mark in input For example light years d lx lxn good
189. t also be the command RETURN followed by the variable or expression whose value is to be the value returned by the procedure Ifthe RETURN is omitted or nothing is written after the word RETURN the procedure will have no value or the value zero depending on how it is used and NIL in symbolic 5 7 COMPOUND STATEMENTS 45 1 mode Remember to put a terminator after the END Example Given a previously assigned integer value for N the following block will compute the Legendre polynomial of degree N in the variable X begin scalar seed deriv top fact seed 1 y 2 2x xx y 1 1 2 deriv df seed y n top sub y 0 deriv fact for i l n product i return top fact end 5 7 1 Compound Statements with GO TO It is possible to have more complicated structures inside the BEGIN END brackets than indicated in the previous example That the individual lines of the program need not be assignment statements but could be almost any other kind of statement or command needs no explanation For example conditional state ments and WHILE and REPEAT constructions have an obvious role in defining more intricate blocks If these structured constructs don t suffice it is possible to use labels and GO TOs within a compound statement and also to use RETURN in places within the block other than just before the END The following subsections discuss these matters in detail For many readers
190. t boolean expression gt Parentheses can also be used to control the precedence of expressions In addition to the logical and relational operators defined earlier as infix operators the following boolean operators are also defined EVENP U determines if the number U is even or not FIXP U determines if the expression U is integer or not FREEOF U V determines if the expression U does not contain the kernel V anywhere in its structure NUMBERP U determines if U is a number or not ORDP U V determines if U is ordered ahead of V by some canonical ordering based on the expression structure and an internal ordering of identifiers PRIMEP U true if U is a prime object i e any object other than 0 and plus or minus 1 which is only exactly divisible by itself or a unit Examples 3 lt 1 30 CHAPTER 3 EXPRESSIONS x gt 0 or x 2 numberp x fixp x and evenp x numberp x and x neq 0 Boolean expressions can only appear directly within IF FOR WHILE and UNTIL statements as described in other sections Such expressions cannot be used in place of ordinary algebraic expressions or assigned to a variable NB For those familiar with symbolic mode the meaning of some of these oper ators is different in that mode For example NUMBERP is true only for integers and reals in symbolic mode When two or more boolean expressions are combined with AND they are evaluated one by one until a f
191. t integer of its single argu ment if that argument has a numerical value A non numeric argument is returned as an expression in the original operator For example round 5 4 round a 1 ROUND A gt 7 1 14 SIGN SIGN tries to evaluate the sign of its argument If this is possible SIGN returns one of 1 0 or 1 Otherwise the result is the original form or a simplified variant For example L SIGN B sign 5 gt sign a 2xb gt Note that even powers of formal expressions are assumed to be positive only as long as the switch COMPLEX is off 7 2 Mathematical Functions REDUCE knows that the following represent mathematical functions that can take arbitrary scalar expressions as their single argument ACOS ACOSH ACOT ACOTH ACSC ACSCH AS ATAN ATANH ATAN2 COS COSH COT COTH EC ASECH ASIN ASIN CSC CSCH DILOG EI H EXP HYPOT LN LOG LOG B LOG10 SEC SECH SIN SINH SORT TAN TANH B has two where LOG is the natural logarithm and equivalent to LN and LOGI arguments of which the second is the logarithmic base The derivatives of all these functions are also known to the system REDUCE knows various elementary identities and properties of these functions For example cos x cos x Sin x sin x cos nxpi 1 n sin n pi 0 log e 1 e ixpi 2 i log 1 0 e ixpi 1 log e x x e 3 i xpi 2 i 58 CHAPTER 7 BUILT IN PREFIX OPER
192. t it is an odd function or has a definition defined as a procedure Order of Application of Rules If rules have overlapping domains their order of application is important In gen eral it is very difficult to specify this order precisely so that it is best to assume 10 4 ASYMPTOTIC COMMANDS 133 that the order is arbitrary However if only one operator is involved the order of application of the rules for this operator can be determined from the following 1 Rules containing at least one free variable apply before all rules without free variables 2 Rules activated in the most recent LET command are applied first 3 LET with several entries generate the same order of application as a corre sponding sequence of commands with one rule or rule set each 4 Within a rule set the rules containing at least one free variable are applied in their given order In other words the first member of the list is applied first 5 Consistent with the first item any rule in a rule list that contains no free variables is applied after all rules containing free variables Example The following rule set enables the computation of exact values of the Gamma function operator gamma gamma_error gamma_rules gamma x gt sqrt pi 2 when x 1 2 gamma n gt factorial n 1 when fixp n and n gt 0 gamma n gamma x gt gamma_error n when fixp n gt x 1 gamma x 1 when fixp 2 x and x gt 1 gt gamma x 1 x wh
193. t unit 4 by 4 matrix gt Multiplication of Dirac expressions as for matrix expressions is of course non commutative 17 4 TRACE CALCULATIONS 195 17 4 Trace Calculations When a Dirac expression is evaluated the system computes one quarter of the trace of each y matrix product in the expansion of the expression One quarter of each trace is taken in order to avoid confusion between the trace of the scalar M say and M representing M lt unit 4 by 4 matrix gt Contraction over indices occurring in such expressions is also performed If an unmatched index is found in such an expression an error occurs The algorithms used for trace calculations are the best available at the time this system was produced For example in addition to the algorithm developed by Chisholm for contracting indices in products of traces REDUCE uses the elegant algorithm of Kahane for contracting indices in y matrix products These algorithms are described in Chisholm J S R Il Nuovo Cimento X 30 426 1963 and Kahane J Journal Math Phys 9 1732 1968 It is possible to prevent the trace calculation over any line identifier by the declara tion NOSPUR For example nospur 11 12 will mean that no traces are taken of y matrix terms involving the line numbers L1 and L2 However in some calculations involving more than one line a catastrophic error This NOSPUR option not implemented can occur for the reason stated If you encounter this error
194. th a top level algebraic operator or a power then the rule is added without further change to the appropriate table 126 CHAPTER 10 SUBSTITUTION COMMANDS 2 If the operator appears at the top level of the simplified left hand side then any constant arguments in that expression are moved to the right hand side of the rule The remaining left hand side is then added to the appropriate table For example let 2xxx y 3 becomes let xx y 3 2 so that xx y is added to the product substitution table and when this rule is applied the expression xx y becomes 3 2 but X or Y by themselves are not replaced 3 If the operators or appear at the top level of the simplified left hand side all but the first term is moved to the right hand side of the rule Thus the rules let l m n x 2 y a b c become let l n m x 2 y a ctb One problem that can occur in this case is that if a quantified expression is moved to the right hand side a given free variable might no longer appear on the left hand side resulting in an error because of the unmatched free variable E g for all x y let f x f y xx y would become for all x y let x x y f y which no longer has Y on both sides The fact that array and matrix elements are evaluated in the left hand side of rules can lead to confusion at times Consider for example the statements array a 5 let xta 2 3 let a 3 4 The left hand side of the first rule will become
195. that do not support a caret symbol Statement expressions usually in parentheses can also form part of a scalar ex pression as in the example w c xty z When the algebraic value of an expression is needed REDUCE determines it start ing with the algebraic values of the parts roughly as follows 27 28 CHAPTER 3 EXPRESSIONS Variables and operator symbols with an argument list have the algebraic values they were last assigned or if never assigned stand for themselves However array elements have the algebraic values they were last assigned or if never assigned are taken to be 0 Procedures are evaluated with the values of their actual parameters In evaluating expressions the standard rules of algebra are applied Unfortunately this algebraic evaluation of an expression is not as unambiguous as is numerical evaluation This process is generally referred to as simplification in the sense that the evaluation usually but not always produces a simplified form for the expression There are many options available to the user for carrying out such simplification If the user doesn t specify any method the default method is used The default evaluation of an expression involves expansion of the expression and collection of like terms ordering of the terms evaluation of derivatives and other functions and substitution for any expressions which have values assigned or declared see assignments and LET statements
196. the following 1 the name of an operator for a single argument the operator is evaluated once with each element of object as its single argument 2 an algebraic expression with exactly one free variable that is a variable pre ceded by the tilde symbol The expression is evaluated for each element of object where the element is substituted for the free variable 7 7 MKID OPERATOR 65 3 a replacement rule of the form var gt rep where var is a variable a kernel without a subscript and rep is an expression that contains var Rep is evaluated for each element of object where the element is substituted for var Var may be optionally preceded by a tilde The rule form for function is needed when more than one free variable occurs Examples map abs 1 2 a a gt 1 2 ABS A ABS A map int w x mat x72 x 5 x74 x 5 gt 3 6 x x 3 6 5 6 x x 5 6 map wx6 x 2 3 y 3 2 1 gt 2xX 2 3x Y 3 2 You can use MAP in nested expressions However you cannot apply MAP to a non composed object e g an identifier or a number 7 7 MKID Operator In many applications it is useful to create a set of identifiers for naming objects in a consistent manner In most cases it is sufficient to create such names from two components The operator MKID is provided for this purpose Its syntax is MKID U id V id non negative integer id for example mkid a 3 mkid apple s gt A3 gt APPLES
197. these is given in the section Reserved Identifiers There are of course an impossibly large number of such names to keep in mind The reader may therefore want to make himself a copy of the list deleting the names he doesn t think he is likely to use by mistake 2 5 Strings Strings are used in WRITE statements in other output statements such as error messages and to name files A string consists of any number of characters en closed in double quotes For example A String Lower case characters within a string are not converted to upper case The string represents the empty string A double quote may be included in a string by preceding it by another double quote Thus a hb is the string a b and is the string 2 6 Comments Text can be included in program listings for the convenience of human readers in such a way that REDUCE pays no attention to it There are two ways to do this 1 Everything from the word COMMENT to the next statement terminator nor mally or is ignored Such comments can be placed anywhere a blank could properly appear Note that END and gt gt are not treated as COMMENT delimiters 2 Everything from the symbol to the end of the line on which it appears is ignored Such comments can be placed as the last part of any line Statement terminators have no special meaning in such comments Remember to put a semicolon before the if the earlier part of the line is intende
198. tional switches are not supported in all compilers They are as follows 199 200 CHAPTER 18 REDUCE AND RLISP UTILITIES PLAP _ If ON causes the printing of the portable macros produced by the compiler PGWD _ If ON causes the printing of the actual assembly language instruc tions generated from the macros PWRDS If ON causes a statistic message of the form lt function gt COMPILED lt words gt WORDS lt words gt LEFT to be printed The first number is the number of words of binary program space the compiled function took and the second number the number of words left unused in binary program space 18 2 Fast Loading Code Generation Program In most versions of REDUCE it is possible to take any set of Lisp Rlisp or RE DUCE commands and build a fast loading version of them In Rlisp or REDUCE one does the following faslout lt filename gt lt commands or IN statements gt faslend To load such a file one uses the command LOAD e g load foo or load foo bah This process produces a fast loading version of the original file In some imple mentations this means another file is created with the same name but a different extension For example in PSL based systems the extension is b for binary In CSL based systems however this process adds the fast loading code to a single file in which all such code is stored Particular functions are provided by CSL for managing this file and described in the CSL
199. tor algebra and calculus package This package provides REDUCE with the ability to perform vector algebra using the same notation as scalar algebra The basic algebraic operations are supported as are differentiation and integration of vectors with respect to scalar variables cross product and dot product component manipulation and application of scalar functions e g cosine to a vector to yield a vector result Author David Harper 15 6 BOOLEAN A package for boolean algebra This package supports the computation with boolean expressions in the proposi tional calculus The data objects are composed from algebraic expressions con nected by the infix boolean operators and or implies equiv and the unary prefix operator not Boolean allows you to simplify expressions built from these oper ators and to test properties like equivalence subset property etc Author Herbert Melenk 15 7 CALI A package for computational commutative algebra This package contains algorithms for computations in commutative algebra closely related to the Gr bner algorithm for ideals and modules Its heart is a new imple mentation of the Gr bner algorithm that also allows for the computation of syzy gies This implementation is also applicable to submodules of free modules with generators represented as rows of a matrix Author Hans Gert Gr be 15 8 CAMAL Calculations in celestial mechanics This packages implements in REDUCE the Fourier trans
200. turned on automatically when the package is loaded The switches supported by the standard integrator e g TRINT are also sup ported by this package In addition the switch TRA if on will give further tracing information about the specific functioning of the algebraic integrator There is no additional documentation for this package Author James H Davenport 15 2 APPLYSYM Infinitesimal symmetries of differen tial equations This package provides programs APPLYSYM QUASILINPDE and DETRAFO for applying infinitesimal symmetries of differential equations the generalization of special solutions and the calculation of symmetry and similarity variables Author Thomas Wolf 15 3 ARNUM An algebraic number package This package provides facilities for handling algebraic numbers as polynomial co efficients in REDUCE calculations It includes facilities for introducing indetermi nates to represent algebraic numbers for calculating splitting fields and for factor ing and finding greatest common divisors in such domains Author Eberhard Schriifer 15 4 ASSIST Useful utilities for various applications 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 Author Hubert Caprasse 15 5 AVECTOR A VECTOR ALGEBRA AND CALCULUS PACKAGE 159 15 5 AVECTOR A vec
201. uantum Electrodynam ics Journ Comp Phys 14 1974 301 317 which contains a complete listing of a program for definite integration of some expressions that arise in fourth order quantum electrodynamics 7 13 Non Commuting Operators An operator can be declared to be non commutative under multiplication by the declaration NONCOM Example After the declaration noncom u v the expressions u x u y u y xu x andu x xv y v y xu x will remain unchanged on simplification and in particular will not simplify to zero Note that it is the operator U and V in the above example and not the variable that has the non commutative property The LET statement may be used to introduce rules of evaluation for such operators In particular the boolean operator ORDP is useful for introducing an ordering on such expressions Example The rule 80 CHAPTER 7 BUILT IN PREFIX OPERATORS for all x y such that x neq y and ordp x y let u x u y u y u x comm x y would introduce the commutator of u x and u y for all X and Y Note that since ordp x x is true the equality check is necessary in the degenerate case to avoid a circular loop in the rule 7 14 Symmetric and Antisymmetric Operators An operator can be declared to be symmetric with respect to its arguments by the declaration SYMMETRIC For example symmetric u v means that any expression involving the top level operators U or V will have its argu
202. ubstitution 119 SUCH THAT 123 SuM 40 41 173 Switch 50 51 SYMBOLIC 177 Symbolic mode 177 178 182 183 Symbolic procedure 181 SYMMETRIC 80 SYMMETRY 173 T 22 TAN 57 60 63 TANH 57 60 TAYLOR 173 Terminator 37 THIRD 34 ThreejSymbol 171 TIME 50 TP 145 TPS 173 TRA 158 TRACE 145 TRFAC 106 TRI 174 T T a RIGFORM 71 RIGSIMP 58 174 RINT 158 un UNTIL 40 User packages 157 Variable 22 Variable elimination 163 VARNAME 96 varopt 76 VECDIM 197 VECTOR 193 WEIGHT 134 WHEN 128 WHERE 129 215 WHILE 42 44 45 47 Whittaker functions 171 WhittakerM 171 Whittakerw 171 Workspace 84 WRITE 91 WS 17 138 WILEVEL 134 wu 174 XCOLOR 174 XIDEAL 175 ZEILBERG 175 Zeta 171 Zeta function Riemann s 171 ZTRANS 175
203. ude M and make algebraic sense may now be evaluated such as 1 m to give the inverse 2xm uxm 2 to give us another matrix and det m to give us the determinant of M REDUCE has a wide range of substitution capabilities The system knows about elementary functions but does not automatically invoke many of their well known properties For example products of trigonometrical functions are not converted automatically into multiple angle expressions but if the user wants this he can say for example sin a b cos a b sin a b cos a b 18 CHAPTER 1 INTRODUCTORY INFORMATION where cos x cos y cos x y cos x y 2 cos x sin y sin x y sin x y 2 sin x x sin y cos x y cos xty 2 where the tilde in front of the variables X and Y indicates that the rules apply for all values of those variables The result of this calculation is COS 2xA SIN 2xB See also the user contributed packages ASSIST chapter 15 4 CAMAL chap ter 15 8 and TRIGSIMP chapter 15 55 Another very commonly used capability of the system and an illustration of one of the many output modes of REDUCE is the ability to output results in a FORTRAN compatible form Such results can then be used in a FORTRAN based numerical calculation This is particularly useful as a way of generating algebraic formulas to be used as the basis of extensive numerical calculations For example the statements on Tort df log x
204. umber of arguments and print a diagnostic message otherwise The output is alphabetized on the first seven characters of each function name 18 3 1 Restrictions Algebraic procedures in REDUCE are treated as if they were symbolic so that algebraic constructs will actually appear as calls to symbolic functions such as AEVAL 18 3 2 Usage To invoke the cross reference program the switch CREF is used on cref causes the cref program to load and the cross referencing process to begin After all the required definitions are loaded off cref will cause the cross reference listing to be produced For example if you wish to cross reference all functions in the file tst red and produce the cross reference listing in the file tst crf the following sequence can be used out Cst cri on cref in tst red off cref shut tst crf To process more than one file more IN statements may be added before the call of off cref or the IN statement changed to include a list of files 18 3 3 Options Functions with the flag NOLIST will not be examined or output Initially all Standard Lisp functions are so flagged In fact they are kept on a list NOLIST x 18 4 PRETTYPRINTING REDUCE EXPRESSIONS 203 so if you wish to see references to all functions then CREF should be first loaded with the command load cref and this variable then set to NIL It should also be remembered that any macros with the property list flag EXPA
205. unctions 171 Airy Ai 171 Airy Aiprime 171 Airy Bi 171 Airy Biprime 171 GEBRAIC 177 Igebraic mode 177 182 183 LGINT 157 158 LBRANCH 73 LFAC 87 90 NTISYMMETRIC 80 D 34 PPLYSYM 158 RBVARS 73 RGLENGTH 101 RNUM 158 RRAY 49 ASEC 57 60 ASECH 57 60 ASIN 57 60 ASINH 57 60 Assignment 38 41 45 180 183 ASSIST 158 assumptions 75 Asymptotic command 121 133 gt gt ty FU E Z Dopp op D Pp p ps ATAN 57 60 63 ATAN2 57 60 ATANH 57 60 AVECTOR 159 BALANCED_MOD 117 BEGIN END 44 46 Bernoulli 171 Bernoulli numbers 171 Bessel functions 171 BessellI 171 BesselJ 171 BesselK 171 BesselY 171 Beta 171 Beta function 171 Bezout 109 BESPACE 116 Binomial 171 Binomial coefficients 171 Block 44 46 BOOLEAN 159 Boolean 29 BOUNDS 167 BYE 51 CALI 159 Call by value 150 153 CAMAL 159 Canonical form 83 CARD_NO 93 CEILING 54 CHANGEVR 160 Character set 19 Chebyshev fit 167 Chebyshev polynomials 171 209 210 ChebyshevT 171 ChebyshevJ 171 CLEAR 124 127 CLEARRULES 128 Clebsch Gordan coefficients 171 Clebsch_Gordan 171 COEFF 99 Coefficient 115 117 COEFEN 100 COFACTOR 145 COLLECT 40 COMBINEEXPT 59 COMBINELOGS 59 Command 49 Command terminator 135 COMMENT 23 COMP 199 COMPACT 160 Compiler 199 COMPLE
206. ur momenta of incoming and outgoing photons with polarization vectors e and e and laboratory energies k and k respectively and p pf are incident and final electron four momenta IN Omitting therefore an overall factor es we need to find one quarter of the trace of i a yey ey ki yeyey kf l Haas F winters 2k pi 2k pi A straightforward REDUCE program for this with appropriate substitutions using P1 for p PF for py KI for k and KF for kp is on div this gives output in same form as Bjorken and Drell mass ki 0 kf 0 pl m pf m vector e ep o if e is used as a vector it loses its scalar identity as 17 7 EXTENSIONS TO MORE THAN FOUR DIMENSIONS 197 o the base of natural logarithms mshell ki kf pl pf let pl e 0 pl ep 0 pl pf m 2 ki kf pl ki m k pl kf mxkp pf e kf e pf ep ki ep pf ki mx xkp pf kf mxk ki e 0 ki kf mx k kp kf ep 0 e 1 ep ep 1 for all p let gp p g 1 p m comment this is just to save us a lot of writing gp pf g 1 ep e ki 2 ki p1 g 1 e ep k 2 kf p1 x gp pl g 1 ki e ep 2 ki pl g 1 kf ep e 2 kf pl1 write The Compton cxn is ws We use P 1 instead of P I in the above to avoid confusion with the reserved variable PI This program will print the following result 1 1 2 The Compton cxn is 1 2 K KP 1 2x xK KP 2xE EP 17 7 Extensions to More Than Four Dimensions In
207. ut Of Expressions 93 8 3 7 Saving Expressions for Later Use as Input 96 8 3 8 Displaying Expression Structure 96 8 4 Changing the Internal Order of Variables 99 8 5 Obtaining Parts of Algebraic Expressions 99 85 1 COBRE Operator 2c a aoi a ae 99 8 5 2 LOBPEN Operator se cco e esee e A a 100 8 5 3 PART Operator ie s s e che ed he a ee a 100 8 5 4 Substituting for Parts of Expressions 102 Polynomials and Rationals 103 9 1 Controlling the Expansion of Expressions 104 9 2 Factorization of Polynomials 104 CONTENTS 5 9 3 Cancellation of Common Factors o 106 9 3 1 Determining the GCD of Two Polynomials 107 9 4 Working with Least Common Multiples 108 9 5 Controlling Use of Common Denominators 108 9 6 REMAINDER Operator o 108 9 7 RESULTANT Operator o o 109 9 8 DECOMPOSE Operator 24 2 ta as a eee 110 99 INTERPOL operator oo a a ae 111 9 10 Obtaining Parts of Polynomials and Rationals 111 9 10 1 DEG QR ceso card a aa os ears 111 910 2 DEN Operator o sacres ede ode ed ee ae e 112 S103 LCOF Operat r cido ce be eee 112 9 10 4 LPOWER Operator 113 9 10 5 LTERM Operator ec ee wk oe we we ee 113 9 10 6 MAINVAR Operator o o e 113 910 7 NUM Operon eres es risas o a ears 114 9 10 8
208. ut of Expressions A considerable degree of flexibility is available in REDUCE in the printing of expressions generated during calculations No explicit format statements are sup plied as these are in most cases of little use in algebraic calculations where the size of output or its composition is not generally known in advance Instead REDUCE provides a series of mode options to the user that should enable him to produce his output in a comprehensible and possibly pleasing form The most extreme option offered is to suppress the output entirely from any top level evaluation This is accomplished by turning off the switch OUTPUT which is normally on It is useful for limiting output when loading large files or producing clean output from the prettyprint programs In most circumstances however we wish to view the output so we need to know 86 CHAPTER 8 DISPLAY AND STRUCTURING OF EXPRESSIONS how to format it appropriately As we mentioned earlier an algebraic expression is normally printed in an expanded form filling the whole output line with terms Certain output declarations however can be used to affect this format To begin with we look at an operator for changing the length of the output line 8 3 1 LINELENGTH Operator This operator is used with the syntax LINELENGTH NUM integer integer and sets the output line length to the integer NUM It returns the previous output line length so that it can be stored for
209. witch FORTUPPER should be turned on Finally the default name ANS assigned to an unnamed expression and its subparts 96 CHAPTER 8 DISPLAY AND STRUCTURING OF EXPRESSIONS can be changed by the operator VARNAME This takes a single identifier as argu ment which then replaces ANS as the expression name The value of VARNAME is its argument Further facilities for the production of FORTRAN and other language output are provided by the SCOPE and GENTRAN packagesdescribed in chapters 15 20 and 15 45 8 3 7 Saving Expressions for Later Use as Input It is often useful to save an expression on an external file for use later as input in further calculations The commands for opening and closing output files are explained elsewhere However we see in the examples on output of expressions that the standard natural method of printing expressions is not compatible with the input syntax So to print the expression in an input compatible form we must inhibit this natural style by turning off the switch NAT If this is done a dollar sign will also be printed at the end of the expression Example The following sequence of commands off nat out out x y z 2 write end shut out on nat will generate a file out that contains X Yxx2 2xYxZ Zxx25 ENDS 8 3 8 Displaying Expression Structure In those cases where the final result has a complicated form it is often convenient to display the skeletal structure o
210. xample begin scalar y y for 1 0 99 do if a i x then return b i will lead to an error 48 CHAPTER 5 STATEMENTS Chapter 6 Commands and Declarations A command is an order to the system to do something Some commands cause visible results such as calling for input or output others usually called declara tions set options define properties of variables or define procedures Commands are formally defined as a statement followed by a terminator lt command gt lt statement gt lt terminator gt lt terminator gt Some REDUCE commands and declarations are described in the following sub sections 6 1 Array Declarations Array declarations in REDUCE are similar to FORTRAN dimension statements For example array a 10 b 2 3 4 Array indices each range from 0 to the value declared An element of an array is referred to in standard FORTRAN notation e g A 2 We can also use an expression for defining an array bound provided the value of the expression is a positive integer For example if X has the value 10 and Y the value 7 then array c 5x x y isthe same as array c 57 If an array is referenced by an index outside its range an error occurs If the array is to be one dimensional and the bound a number or a variable not a more general expression the parentheses may be omitted 49 50 CHAPTER 6 COMMANDS AND DECLARATIONS array a 10 c 57 The operator LENGTH applie
211. xpression is used in place of the boolean expression for example a variable is written there the true alternative is followed unless the expression has the value 0 5 4 FOR Statements The FOR statement is used to define a variety of program loops Its general syntax is as follows var number pacu var Nb is STEP Meade NEEE number FOR action exprn where action do product sum collect join 5 4 FOR STATEMENTS 41 The assignment form of the FOR statement defines an iteration over the indicated numerical range If expressions that do not evaluate to numbers are used in the designated places an error will result The FOR EACH form of the FOR statement is designed to iterate down a list Again an error will occur if a list is not used The action DO means that lt exprn gt is simply evaluated and no value kept the statement returning 0 in this case or no value at the top level COLLECT means that the results of evaluating lt exprn gt each time are linked together to make a list and JOIN means that the values of lt exprn gt are themselves lists that are joined to make one list similar to CONC in Lisp Finally PRODUCT and SUM form the respective combined value out of the values of lt exprn gt In all cases lt exprn gt is evaluated algebraically within the scope of the current value of lt var gt If lt action gt is DO then nothing else happens In other cases lt act
212. xtterm term term term 2 3 gt gt until num term 1 1000 lt 0 ex In this case the answer will be the same as before because in neither case is a term added to EX which is less than 1 1000 5 7 Compound Statements Often the desired process can best or only be described as a series of steps to be carried out one after the other In many cases this can be achieved by use of the group statement However each step often provides some intermediate result until at the end we have the final result wanted Alternatively iterations on the steps are 44 CHAPTER 5 STATEMENTS needed that are not possible with constructs such as WHILE or REPEAT statements In such cases the steps of the process must be enclosed between the words BEGIN and END forming what is technically called a block or compound statement Such a compound statement can in fact be used wherever a group statement appears The converse is not true BEGIN END can be used in ways that lt lt gt gt cannot If intermediate results must be formed local variables must be provided in which to store them Local means that their values are deleted as soon as the block s operations are complete and there is no conflict with variables outside the block that happen to have the same name Local variables are created by a SCALAR declaration immediately after the BEGIN scalar a b c Z If more convenient several SCALAR declarations can be given
213. y but would have no effect on k a z since the rule didn t say FOR ALL Y Where we used X and Y in the examples any variables could have been used This use of a variable doesn t affect the value it may have outside the LET statement However you should remember what variables you actually used If you want to delete the rule subsequently you must use the same variables in the CLEAR command It is possible to use more complicated expressions as a template for a LET state ment as explained in the section on substitutions for general expressions In nearly all cases the rule will be accepted and a consistent application made by the sys tem However if there is a sole constant or a sole free variable on the left hand side of a rule e g let 2 30r for all x let x 2 then the system is unable to handle the rule and the error message Substitution for not allowed will be issued Any variable listed in the FOR ALL part will have its symbol preceded by an equal sign X in the above example will appear as X An error will also occur if a variable in the FOR ALL part is not properly matched on both sides of the LET equation 10 2 2 FOR ALL SUCH THAT LET If a substitution is desired for more than a single value of a variable in an operator or other expression but not all values a conditional form of the FOR ALL LET declaration can be used Example for all x such that numberp x and x lt 0 let h x 0
214. y only square matrices may be raised to a power A negative power is computed as the inverse of the matrix raised 13 4 OPERATORS WITH MATRIX ARGUMENTS 143 to the corresponding positive power a b is interpreted as axb 1 Examples Assuming X and Y have been declared as matrices the following are matrix ex pressions y y 2 x 3xy 2 xx y mat 1 a b c 2 The computation of the quotient of two matrices normally uses a two step elimina tion method due to Bareiss An alternative method using Cramer s method is also available This is usually less efficient than the Bareiss method unless the matrices are large and dense although we have no solid statistics on this as yet To use Cramer s method instead the switch CRAMER should be turned on 13 4 Operators with Matrix Arguments The operator LENGTH applied to a matrix returns a list of the number of rows and columns in the matrix Other operators useful in matrix calculations are defined in the following subsections Attention is also drawn to the LINALG chapter 15 29 and NORMFORM chapter 15 32 packages 13 4 1 DET Operator Syntax DET EXPRN matrix_expression algebraic The operator DET is used to represent the determinant of a square matrix expres sion E g det y 2 is a scalar expression whose value is the determinant of the square of the matrix Y and det mat a b c d e f g h 3 144 CHAPTER 13 MATRIX CALC
215. y represented as the quotient of two integers in lowest terms that is as rational numbers In essentially all versions of REDUCE it is also possible but not always desirable to ask REDUCE to work with floating point approximations to numbers again to any precision Such numbers are called real They can be input in two ways 1 as a signed or unsigned sequence of any number of decimal digits with an embedded or trailing decimal point 2 as in 1 followed by a decimal exponent which is written as the letter E followed by a signed or unsigned integer e g 32 32 0 0 32E2 and 320 E 1 are all representations of 32 The declaration SCIENTIFIC_NOTATION controls the output format of floating point numbers At the default settings any number with five or less digits before the decimal point is printed in a fixed point notation e g 12345 6 Numbers with more than five digits are printed in scientific notation e g 1 234567E 5 Sim ilarly by default any number with eleven or more zeros after the decimal point is printed in scientific notation To change these defaults SCILENTIFIC_NOTATION can be used in one of two ways SCIENTIFIC_NOTATION m where mis a posi tive integer sets the printing format so that 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 SCIENTIFIC_NOTATION m n with m and n both positive integers sets the format so tha

REDUCE User`s Manual - REDUCE Computer Algebra System

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