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changed: - \documentclass{article} \normalsize\baselineskip=12pt \parskip=0pt \parindent=10pt \tolerance=5000 \pretolerance=5000 \frenchspacing \hangindent=10pt \skip\footins=18pt \global\textwidth 31pc \global\textheight 47pc \headsep 12pt \oddsidemargin 0pt \evensidemargin 0pt % \renewcommand{\textfraction}{.1} \renewcommand{\floatpagefraction}{.75} % \catcode`@=11 \def\ps@plain{\let\@mkboth\@gobbletwo% \let\@oddhead\@empty\def\@oddfoot{\sysdetails} \let\@evenhead\@empty\let\@evenfoot\@oddfoot} \def\ps@empty{\let\@mkboth\@gobbletwo% \let\@oddhead\@empty\def\@oddfoot{\sysdetails} \let\@evenhead\@empty\let\@evenfoot\@oddfoot} \catcode`@=12 % \def\sysdetails{\parbox{\textwidth}{% Based on material originally published in {\sl Computer Algebra Systems: A Practical Guide\/} edited by Michael J.\ Wester, John Wiley \& Sons, Chichester, United Kingdom, ISBN 0-471-98353-5, xvi+436 pages, 1999.}} % \pagestyle{plain} \begin{document} % % \nth{n} produces 1^{st}, 2^{nd}, 3^{rd}, 4^{th}, etc. % \def\nth#1{$#1^{\rm \ifcase#1 th\or st\or nd\or rd\else th\fi}$} % % Abbreviations % \newcommand{\Axiom}{{\sf Axiom}} \newcommand{\Derive}{{\sf Derive}} \newcommand{\DoCon}{{\sf DoCon}} \newcommand{\GAP}{{\sf GAP}} \newcommand{\Gmp}{{\sf Gmp}} \newcommand{\Macsyma}{{\sf Macsyma}} \newcommand{\Magnus}{{\sf Magnus}} \newcommand{\Maxima}{{\sf Maxima}} \newcommand{\Maple}{{\sf Maple}} \newcommand{\Mathematica}{{\sf Mathematica}} \newcommand{\MuPAD}{{\sf MuPAD}} \newcommand{\Octave}{{\sf Octave}} \newcommand{\Pari}{{\sf Pari}} \newcommand{\Reduce}{{\sf Reduce}} \newcommand{\Scilab}{{\sf Scilab}} \newcommand{\Sumit}{{\sf Sumit}} \newcommand{\Yacas}{{\sf Yacas}} \section{Introduction} The following is a collection of synonyms for various operations in the computer algebra systems \Axiom, \Derive, \GAP, \Gmp, \DoCon, \Macsyma, \Magnus, \Maxima, \Maple, \Mathematica, \MuPAD, \Octave, \Pari, \Reduce, \Scilab, \Sumit\ and \Yacas. This collection does not attempt to be comprehensive, but hopefully it will be useful in giving an indication of how to translate between the syntaxes used by the different systems in many common situations. Note that a blank entry means either (a) that there may be an exact translation of a particular operation for the indicated system, but we don't know what it is or (b) there is no exact translation but it may still be possible to work around this lack with a related functionality. While commercial systems are not provided on this CD the intent of the Rosetta effort is to make it possible for experienced Computer Algebra users to experiment with other systems. Thus the commands for commercial systems are included to allow users of those systems to translate. Some of these systems are special purpose and do not support a lot of the functionality of the more general purpose systems. Where they do support an interpreter the commands are provided. Originally written by Michael Wester. Modified for Rosetta by Timothy Daly, Alexander Hulpke (GAP). \section{System availability} \begin{tabular}{l|lll} System & \rm{License} & \rm{Status (May 2002)} & \rm{Web Location} \\ \hline \Axiom & BSD & available & http://www.aldor.org \\ \Axiom & open source & pending & http://wiki.axiom-developer.org \\ \Derive & commercial & available & http://www.mathware.com \\ \DoCon & open source & available & http://www.haskell.org/docon \\ \GAP & GPL & Rosetta & http://www.gap-system.org/~gap \\ \Gmp & GPL & Rosetta & http://www.swox.com/gmp \\ \Macsyma & commercial & dead & See Maxima \\ \Magnus & GPL & Rosetta & http://zebra.sci.ccny.cuny.edu/web \\ \Maxima & GPL & Rosetta & http://maxima.sourceforge.net \\ \Maple & commercial & available & http://www.maplesoft.com \\ \Mathematica & commercial & available & http://www.wolfram.com \\ \MuPAD & commercial & available & http://www.mupad.de \\ \Octave & GPL & Rosetta & http://www.octave.org \\ \Pari & GPL & Rosetta & http://www.parigp-home.de \\ \Reduce & commercial & available & http://www.zib.de/Symbolik/reduce \\ \Scilab & Scilab & available & http://www-rocq.inria.fr/scilab \\ \Sumit & & available & http://www-sop.inria.fr/cafe/soft-e.html \\ \Yacas & GPL & available & http://yacas.sourceforge.net \\ \end{tabular} \\[10pt] \\ \begin{tabular}{l|ll} System & \rm{Type} & \rm{Interpreted or Compiled}\\ \hline \Axiom & General Purpose & both \\ \Derive & General Purpose & \\ \DoCon & General Purpose & Interpreted in Haskell \\ \GAP & Group Theory & \\ \Gmp & arb. prec. arithmetic & \\ \Macsyma & General Purpose & \\ \Magnus & Infinite Group Theory & \\ \Maxima & General Purpose & \\ \Maple & General Purpose & \\ \Mathematica & General Purpose & \\ \MuPAD & General Purpose & \\ \Octave & Numerical Computing & \\ \Pari & Number Theory & \\ \Reduce & General Purpose & \\ \Scilab & General Purpose & \\ \Sumit & Functional Equations & \\ \Yacas & General Purpose & \\ \end{tabular} \\[10pt] \section{Programming and Miscellaneous} \begingroup \newcommand{\OR}{{\em or }} \newcommand{\fnm}{\footnotemark} \newcommand{\h}[1]{{\rm #1}} \newcommand{\m}[2]{\multicolumn{#1}{l}{#2}} \newcommand{\q}{\quad} \newcommand{\st}{\small\tt} \parindent=0pt \hfuzz=1pt \begin{tt} \begin{tabular}{l|ll} & \m{2}{\rm Unix/Microsoft user initialization file} \\ \hline \Axiom & \~{}/axiom.input & \\ \GAP & \~{}/.gaprc & GAP.RC \\ \Gmp & & \\ \DoCon & & \\ \Derive & & derive.ini \\ \Macsyma & \~{}/macsyma-init.macsyma & mac-init.mac \\ \Magnus & & \\ \Maxima & \~{}/macsyma-init.macsyma & mac-init.mac \\ \Maple & \~{}/.mapleinit & maplev5.ini \\ \Mathematica & \~{}/init.m & init.m \\ \MuPAD & \~{}/.mupadinit & $\backslash$mupad$\backslash$bin$\backslash$userinit.mu \\ \Octave & & \\ \Pari & & \\ \Reduce & \~{}/.reducerc & reduce.rc \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Describe {\em keyword}} & \h{Find keywords containing {\em pattern}} \\ \hline \Axiom & & )what operations pattern \\ \Derive & & \\ \DoCon & & \\ \GAP & ?keyword & ??keyword\\ \Gmp & & \\ \Macsyma & describe("keyword")\$ & apropos("pattern"); \\ \Magnus & & \\ \Maxima & describe("keyword")\$ & apropos("pattern"); \\ \Maple & ?keyword & ?pattern\,\fnm \\ \Mathematica & ?keyword & ?*pattern* \\ \MuPAD & ?keyword & ?*pattern* \\ \Octave & help -i keyword & \\ \Pari & & \\ \Reduce & & \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \footnotetext{Only if the pattern is not a keyword and then the matches are simplistic.} \begin{tabular}{l|l@{ }llll} & & & \h{Prev.} & \h{Case} & \h{Variables} \\ & \h{Comment} & \h{Line continuation} & \h{expr.} & \h{sensitive} & \h{assumed} \\ \hline \Axiom & -- comment & input \_<CR>input & \% & Yes & real \\ \Derive & "comment" & input \~{}<CR>input & & No & real \\ \DoCon & & & & & \\ \GAP & \# comment & input$\backslash$<CR>input&last&Yes&no assumption\\ \Gmp & & & & & \\ \Macsyma & /* comment */ & input<CR>input; & \% & No & real \\ \Magnus & & & & & \\ \Maxima & /* comment */ & input<CR>input; & \% & No & real \\ \Maple & \# comment & input<CR>input; & \% & Yes & complex \\ \Mathematica & (* comment *) & input<CR>input & \% & Yes & complex \\ \MuPAD & \# comment \# & input<CR>input; & \% & Yes & complex \\ \Octave & \#\# & & & Yes & \\ \Pari & & & & & \\ \Reduce & \% comment & input<CR>input; & ws & No & complex \\ \Scilab & & & & & \\ \Sumit & & & & & \\ \Yacas & & & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|lll} & \h{Load a file} & \h{Time a command} & \h{Quit} \\ \hline \Axiom & )read "file" )quiet & )set messages time on & )quit \\ \Derive & [Transfer Load Derive] & & [Quit] \\ \DoCon & & & \\ \GAP & Read("file"); & time; \h{(also see {\tt Runtime();})}&quit;\\ \Gmp & & & \\ \Macsyma & load("file")\$ & showtime: all\$ & quit(); \\ \Magnus & & & \\ \Maxima & load("file")\$ & showtime: all\$ & quit(); \\ \Maple & read("file"): & readlib(showtime): on; & quit \\ \Mathematica & @<< file & Timing[command] & Quit[] \\ \MuPAD & read("file"): & time(command); & quit \\ \Octave & load file & tic(); cmd ; toc() & quit \OR\ exit\\ \Pari & & & \\ \Reduce & in "file"\$ & on time; & quit; \\ \Scilab & & & quit \\ \Sumit & & & \\ \Yacas & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|lll} & \h{Display} & \h{Suppress} & \\ & \h{output} & \h{output} & \h{Substitution: $f(x, y) \rightarrow f(z, w)$} \\ \hline \Axiom & input & input; & subst(f(x, y), [x = z, y = w]) \\ \Derive & input & var:= input & [Manage Substitute] \\ \DoCon & & & \\ \GAP & input; & input;; & Value(f,[x,y],[z,w]);\fnm \\ \Gmp & & & \\ \Macsyma & input; & input\$ & subst([x = z, y = w], f(x, y)); \\ \Magnus & & & \\ \Maxima & input; & input\$ & subst([x = z, y = w], f(x, y)); \\ \Maple & input; & input: & subs(\{x = z, y = w\}, f(x, y)); \\ \Mathematica & input & input; & f[x, y] /. \{x -> z, y -> w\} \\ \MuPAD & input; & input: & subs(f(x, y), [x = z, y = w]); \\ \Octave & input & input; & \\ \Pari & & & \\ \Reduce & input; & input\$ & sub(\{x = z, y = w\}, f(x, y)); \\ \Scilab & & & \\ \Sumit & & & \\ \Yacas & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|lll} & \h{Set} & \h{List} & \h{Matrix} \\ \hline \Axiom & set [1, 2] & [1, 2] & matrix(@[[1, 2],[3, 4]]) \\ \Derive & \{1, 2\} & [1, 2] & @[[1,2], [3,4]] \\ \DoCon & & & \\ \GAP & Set([1,2]) & [1, 2] & @[[1,2], [3,4]]\fnm \\ \Gmp & & & \\ \Macsyma & [1, 2] & [1, 2] & matrix([1, 2], [3, 4]) \\ \Magnus & & & \\ \Maxima & [1, 2] & [1, 2] & matrix([1, 2], [3, 4]) \\ \Maple & \{1, 2\} & [1, 2] & matrix(@[[1, 2], [3, 4]]) \\ \Mathematica & \{1, 2\} & \{1, 2\} & \{\{1, 2\}, \{3, 4\}\} \\ \MuPAD & \{1, 2\} & [1, 2] & export(Dom): \q export(linalg): \\ & & & matrix:= ExpressionField(normal)): \\ & & & matrix(@[[1, 2], [3, 4]]) \\ \Octave & & & \\ \Pari & & & \\ \Reduce & \{1, 2\} & \{1, 2\} & mat((1, 2), (3, 4)) \\ \Scilab & & list(1,2) & A=[1,2;3,4]\\ \Sumit & & & \\ \Yacas & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|llll} & \h{Equation} & \h{List element} & \h{Matrix element} & \h{Length of a list} \\ \hline \Axiom & x = 0 & l . 2 & m(2, 3) & \#l \\ \Derive & x = 0 & l SUB 2 & m SUB 2 SUB 3 & DIMENSION(l) \\ \DoCon & & & & \\ \GAP & x=0 & l[2] & m[2][3] & Length(l) \\ \Gmp & & & & \\ \Macsyma & x = 0 & l[2] & m[2, 3] & length(l) \\ \Magnus & & & & \\ \Maxima & x = 0 & l[2] & m[2, 3] & length(l) \\ \Maple & x = 0 & l[2] & m[2, 3] & nops(l) \\ \Mathematica & x == 0 & l@[[2]] & m@[[2, 3]] & Length[l] \\ \MuPAD & x = 0 & l[2] & m[2, 3] & nops(l) \\ \Octave & & & & \\ \Pari & & & & \\ \Reduce & x = 0 & part(l, 2) & m(2, 3) & length(l) \\ \Scilab & & l(2) & & \\ \Sumit & & & & \\ \Yacas & & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|lll} & \m{2}{\rm Prepend/append an element to a list} & \h{Append two lists} \\ \hline \Axiom & cons(e, l) & concat(l, e) & append(l1, l2) \\ \Derive & APPEND([e], l) & APPEND(l, [e]) & APPEND(l1, l2) \\ \DoCon & & & \\ \GAP & Concatenation([e],l) & Add(l,e) & Append(l1, l2) \\ \Gmp & & & \\ \Macsyma & cons(e, l) & endcons(e, l) & append(l1, l2) \\ \Magnus & & & \\ \Maxima & cons(e, l) & endcons(e, l) & append(l1, l2) \\ \Maple & [e, op(l)] & [op(l), e] & [op(l1), op(l2)] \\ \Mathematica & Prepend[l, e] & Append[l, e] & Join[l1, l2] \\ \MuPAD & [e, op(l)] & append(l, e) & l1 . l2 \\ \Octave & & & \\ \Pari & & & \\ \Reduce & e . l & append(l, {e}) & append(l1, l2) \\ \Scilab & & & \\ \Sumit & & & \\ \Yacas & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Matrix column dimension} & \h{Convert a list into a column vector} \\ \hline \Axiom & ncols(m) & transpose(matrix([l])) \\ \Derive & DIMENSION(m SUB 1) & [l]\`{} \\ \DoCon & & \\ \GAP & Length(mat[1]) & \h{objects are identical} \\ \Gmp & & \\ \Macsyma & mat\_\,ncols(m) & transpose(matrix(l)) \\ \Magnus & & \\ \Maxima & mat\_\,ncols(m) & transpose(matrix(l)) \\ \Maple & linalg[coldim](m) & linalg[transpose](matrix([l])) \\ \Mathematica & Dimensions[m]@[[2]] & Transpose[\{l\}] \\ \MuPAD & linalg::ncols(m) & transpose(matrix([l]))\,\fnm \\ \Octave & & \\ \Pari & & \\ \Reduce & load\_\,package(linalg)\$ & matrix v(length(l), 1)\$ \\ & column\_dim(m) & for i:=1:length(l) do \\ & & \q\q v(i, 1):= part(l, i) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \footnotetext{See the definition of {\tt matrix} above.} \begin{tabular}{l|l} & \h{Convert a column vector into a list} \\ \hline \Axiom & [v(i, 1) for i in 1..nrows(v)] \\ \Derive & v\`{} SUB 1 \\ \DoCon & \\ \GAP & \h{objects are identical} \\ \Gmp & \\ \Macsyma & part(transpose(v), 1) \\ \Magnus & \\ \Maxima & part(transpose(v), 1) \\ \Maple & op(convert(linalg[transpose](v), listlist)) \\ \Mathematica & Flatten[v] \\ \MuPAD & [op(v)] \\ \Octave & \\ \Pari & \\ \Reduce & load\_\,package(linalg)\$ \\ & for i:=1:row\_\,dim(v) collect(v(i, 1)) \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|lllllll} & \h{True} & \h{False} & \h{And} & \h{Or} & \h{Not} & \h{Equal} & \h{Not equal} \\ \hline \Axiom & true & false & and & or & not & = & \~{}= \\ \Derive & TRUE & FALSE & AND & OR & NOT & = & /= \\ \DoCon & & & & & & & \\ \GAP & true & false\fnm & and & or & not & = & <> \\ \Gmp & & & & & & & \\ \Macsyma & true & false & and & or & not & = & \# \\ \Magnus & & & & & & & \\ \Maxima & true & false & and & or & not & = & \# \\ \Maple & true & false & and & or & not & = & <> \\ \Mathematica & True & False & \&\& & || & ! & == & != \\ \MuPAD & true & false & and & or & not & = & <> \\ \Octave & & & & & & & \\ \Pari & & & & & & & \\ \Reduce & t & nil & and & or & not & = & neq \\ \Scilab & \%t & \%f & & & & & \\ \Sumit & & & & & & & \\ \Yacas & & & & & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{If+then+else statements} & \h{Strings (concatenated)} \\ \hline \Axiom & if \_ then \_ else if \_ then \_ else \_ & concat(["x", "y"]) \\ \Derive & IF(\_, \_, IF(\_, \_, \_)) & "xy" \\ \DoCon & & \\ \GAP & if \_ then \_ elif \_ then \_ else \_ fi & Concatenation("x","y")\\ \Gmp & & \\ \Macsyma & if \_ then \_ else if \_ then \_ else \_ & concat("x", "y") \\ \Magnus & & \\ \Maxima & if \_ then \_ else if \_ then \_ else \_ & concat("x", "y") \\ \Maple & if \_ then \_ elif \_ then \_ else \_ fi & "x" . "y" \\ \Mathematica & If[\_, \_, If[\_, \_, \_]] & "x" <> "y" \\ \MuPAD & if \_ then \_ elif \_ then \_ else \_ & "x" . "y" \\ & \q\q end\_if & \\ \Octave & & \\ \Pari & & \\ \Reduce & if \_ then \_ else if \_ then \_ else \_ & "xy" \OR\ mkid(x, y)\\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Simple loop and Block} & \h{Generate the list $[1, 2, \ldots, n]$} \\ \hline \Axiom & for i in 1..n repeat ( x; y ) & [f(i) for i in 1..n] \\ \Derive & VECTOR([x, y], i, 1, n) & VECTOR(f(i), i, 1, n) \\ \DoCon & & \\ \GAP & for i in [1..n] do \_ od; & [1..n] {\rm or} [1,2..n]\\ \Gmp & & \\ \Macsyma & for i:1 thru n do (x, y); & makelist(f(i), i, 1, n); \\ \Magnus & & \\ \Maxima & for i:1 thru n do (x, y); & makelist(f(i), i, 1, n); \\ \Maple & for i from 1 to n do x; y od; & [f(i) \$ i = 1..n]; \\ \Mathematica & Do[x; y, \{i, 1, n\}] & Table[f[i], \{i, 1, n\}] \\ \MuPAD & for i from 1 to n do x; y & [f(i) \$ i = 1..n]; \\ & \q\q end\_for; & \\ \Octave & & \\ \Pari & & \\ \Reduce & for i:=1:n do @<<x; y>>; & for i:=1:n collect f(i); \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Complex loop iterating on a list} \\ \hline \Axiom & for x in [2, 3, 5] while x**2 < 10 repeat output(x) \\ \Derive & \\ \DoCon & \\ \GAP & for x in [2, 3, 5] do while x\^{}2<10 do Print(x);od;od; \\ \Gmp & \\ \Macsyma & for x in [2, 3, 5] while x\^{}2 < 10 do print(x)\$ \\ \Magnus & \\ \Maxima & for x in [2, 3, 5] while x\^{}2 < 10 do print(x)\$ \\ \Maple & for x in [2, 3, 5] while x\^{}2 < 10 do print(x) od: \\ \Mathematica & For[l = \{2, 3, 5\}, l != \{\} \&\& l@[[1]]\^{}2 < 10, \\ & \q l = Rest[l], Print[l@[[1]]] ] \\ \MuPAD & for x in [2, 3, 5] do if x\^{}2 < 10 then print(x) end\_if \\ & \q end\_for: \\ \Octave & \\ \Pari & \\ \Reduce & for each x in \{2, 3, 5\} do if x\^{}2 < 10 then write(x)\$ \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|lll} & \h{\small Assignment} & \h{Function definition} & \h{Clear vars and funs} \\ \hline \Axiom & y:= f(x) & f(x, y) == x*y & {\small\tt )clear properties y f} \\ \Derive & y:= f(x) & f(x, y):= x*y & y:= f:= \\ \DoCon & & & \\ \GAP & y:= f(x); & f:=function(x, y) return x*y; end; & \h{There are no symbolic variables}\\ \Gmp & & & \\ \Macsyma & y: f(x); & f(x, y):= x*y; & remvalue(y)\$ \\ & & & remfunction(f)\$ \\ \Magnus & & & \\ \Maxima & y: f(x); & f(x, y):= x*y; & remvalue(y)\$ \\ & & & remfunction(f)\$ \\ \Maple & y:= f(x); & f:= proc(x, y) x*y end; & y:= 'y': f:= 'f': \\ \Mathematica & y = f[x] & f[x\_, y\_\,]:= x*y & Clear[y, f] \\ \MuPAD & y:= f(x); & f:= proc(x, y) & y:= NIL: f:= NIL: \\ & & \q\q begin x*y end\_\,proc; & \\ \Octave & & & \\ \Pari & & & \\ \Reduce & y:= f(x); & procedure f(x, y); x*y; & clear y, f; \\ \Scilab & & & \\ \Sumit & & & \\ \Yacas & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Function definition with a local variable} \\ \hline \Axiom & f(x) == (local n; n:= 2; n*x) \\ \Derive & \\ \DoCon & \\ \GAP & f:=function(x) local n; n:=2;return n*x; end; \\ \Gmp & \\ \Macsyma & f(x):= block([n], n: 2, n*x); \\ \Magnus & \\ \Maxima & f(x):= block([n], n: 2, n*x); \\ \Maple & f:= proc(x) local n; n:= 2; n*x end; \\ \Mathematica & f[x\_\,]:= Module[\{n\}, n = 2; n*x] \\ \MuPAD & f:= proc(x) local n; begin n:= 2; n*x end\_\,proc; \\ \Octave & \\ \Pari & \\ \Reduce & procedure f(x); begin scalar n; n:= 2; return(n*x) end; \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Return unevaluated symbol} & \h{Define a function from an expression} \\ \hline \Axiom & e:= x*y;\q 'e & function(e, f, x, y) \\ \Derive & e:= x*y\q 'e & f(x, y):== e \\ \DoCon & & \\ \GAP & \h{No unevaluated symbols}\fnm&\\ \Gmp & & \\ \Macsyma & e: x*y\$\q 'e; & define(f(x, y), e); \\ \Magnus & & \\ \Maxima & e: x*y\$\q 'e; & define(f(x, y), e); \\ \Maple & e:= x*y:\q 'e'; & f:= unapply(e, x, y); \\ \Mathematica & e = x*y;\q HoldForm[e] & f[x\_, y\_\,] = e \\ \MuPAD & e:= x*y:\q hold(e); & f:= hold(func)(e, x, y); \\ \Octave & & \\ \Pari & & \\ \Reduce & e:= x*y\$ & for all x, y let f(x, y):= e; \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \footnotetext{Variables can be assigned to generators of a suitable free object, for example {\tt x:=X(Rationals,"x");} or {\tt f:=FreeGroup(2);x:=f.1;}.} \addtocounter{footnote}{-1}% \begin{tabular}{l|ll} & \h{Fun.\ of an indefinite number of args} & \h{Apply ``+'' to sum a list} \\ \hline \Axiom & & reduce(+, [1, 2]) \\ \Derive & LST l:= l & \\ \DoCon & & \\ \GAP & lst:=function(args) \_ end; & Sum([1,2])\\ \Gmp & & \\ \Macsyma & lst([l]):= l; & apply("+", [1, 2]) \\ \Magnus & & \\ \Maxima & lst([l]):= l; & apply("+", [1, 2]) \\ \Maple & lst:=proc() [args[1..nargs]] end; & convert([1, 2], \`{}+\`{}) \\ \Mathematica & lst[l\_\,\_\,\_\,]:= \{l\} & Apply[Plus, \{1, 2\}] \\ \MuPAD & lst:= proc(l) begin [args()] & \_\,plus(op([1, 2])) \\ & \q\q\q end\_\,proc; & \\ \Octave & & \\ \Pari & & \\ \Reduce & & xapply(+, \{1, 2\})\,\fnm \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \footnotetext{\tt procedure xapply(f, lst); lisp(f . cdr(lst))\$} \addtocounter{footnote}{-1}% \begin{tabular}{l|ll} & \h{Apply a fun.\ to a} & \\ & \h{list of its args} & \h{Map an anonymous function onto a list} \\ \hline \Axiom & reduce(f, l) & map(x +-> x + y, [1, 2]) \\ \Derive & & x:= [1, 2] \\ & & VECTOR(x SUB i + y, i, 1, DIMENSION(x)) \\ \DoCon & & \\ \GAP & List(l,f) & List([1,2],x->x+y) \\ \Gmp & & \\ \Macsyma & apply(f, l) & map(lambda([x], x + y), [1, 2]) \\ \Magnus & & \\ \Maxima & apply(f, l) & map(lambda([x], x + y), [1, 2]) \\ \Maple & f(op(l)) & map(x -> x + y, [1, 2]) \\ \Mathematica & Apply[f, l] & Map[\# + y \&, \{1, 2\}] \\ \MuPAD & f(op(l)) & map([1, 2], func(x + y, x)) \\ \Octave & & \\ \Pari & & \\ \Reduce & xapply(f, l)\,\fnm & for each x in \{1, 2\} collect x + y \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Pattern matching: $f(3 y) + f(z y) \rightarrow 3 f(y) + f(z y)$} \\ \hline \Axiom & f:= operator('f); \\ & ( rule f((n | integer?(n)) * x) == n*f(x) )( \_ \\ & \q\q f(3*y) + f(z*y)) \\ \Derive & \\ \DoCon & \\ \GAP & \\ \Gmp & \\ \Macsyma & matchdeclare(n, integerp, x, true)\$ \\ & defrule(fnx, f(n*x), n*f(x))\$ \\ & apply1(f(3*y) + f(z*y), fnx); \\ \Magnus & \\ \Maxima & matchdeclare(n, integerp, x, true)\$ \\ & defrule(fnx, f(n*x), n*f(x))\$ \\ & apply1(f(3*y) + f(z*y), fnx); \\ \Maple & map(proc(q) local m; \\ & \q\q\q if match(q = f(n*y), y, 'm') and \\ & \q\q\q\q\q type(rhs(op(m)), integer) then \\ & \q\q\q\q subs(m, n * f(y)) else q fi \\ & \q\q end, \\ & \q\q f(3*y) + f(z*y)); \\ \Mathematica & f[3*y] + f[z*y] /. f[n\_Integer * x\_\,] -> n*f[x] \\ \MuPAD & d:= domain("match"): \q d::FREEVARIABLE:= TRUE: \\ & n:= new(d, "n", func(testtype(m, DOM\_INT), m)): \\ & x:= new(d, "x", TRUE): \\ & map(f(3*y) + f(z*y), \\ & \q\q proc(q) local m; begin m:= match(q, f(n*x)); \\ & \q\q\q if m = FAIL then q \\ & \q\q\q else subs(hold("n" * f("x")), m) end\_if \\ & \q\q end\_\,proc); \\ \Octave & \\ \Pari & \\ \Reduce & operator f; \\ & f(3*y) + f(z*y) \\ & \q\q where \{f(\~{}n * \~{}x) => n*f(x) when fixp(n)\}; \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Define a new infix operator and then use it} \\ \hline \Axiom & \\ \Derive & \\ \DoCon & \\ \GAP &\h{One can overload existing infix operators for ones own purposes}\\ \Gmp & \\ \Macsyma & infix("\~{}")\$ \q "\~{}"(x, y):= sqrt(x\^{}2 + y\^{}2)\$ \q 3 \~{} 4; \\ \Magnus & \\ \Maxima & infix("\~{}")\$ \q "\~{}"(x, y):= sqrt(x\^{}2 + y\^{}2)\$ \q 3 \~{} 4; \\ \Maple & \`{}\&\~{}\`{}:= (x, y) -> sqrt(x\^{}2 + y\^{}2): \q 3 \&\~{} 4; \\ \Mathematica & x\_ $\backslash$[Tilde] y\_:= Sqrt[x\^{}2 + y\^{}2]; \q 3 $\backslash$[Tilde] 4 \\ \MuPAD & tilde:= proc(x, y) begin sqrt(x\^{}2 + y\^{}2) end\_\,proc: \\ & \q 3 \&tilde 4; \\ \Octave & \\ \Pari & \\ \Reduce & infix |\$ \q procedure |(x, y); sqrt(x\^{}2 + y\^{}2)\$ \q 3 | 4; \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|lll} & \h{Main expression} & & \\ & \h{operator} & \h{\nth{1} operand} & \h{List of expression operands} \\ \hline \Axiom\fnm & & kernels(e) . 1 & kernels(e) \\ \Derive & & & {\em various}\fnm \\ \DoCon & & & \\ \GAP &\m{3}{\rm There are no formal unevaluated expressions}\\ \Gmp & & & \\ \Macsyma & part(e, 0) & part(e, 1) & args(e) \\ \Magnus & & & \\ \Maxima & part(e, 0) & part(e, 1) & args(e) \\ \Maple & op(0, e) & op(1, e) & [op(e)] \\ \Mathematica & Head[e] & e@[[1]] & ReplacePart[e, List, 0] \\ \MuPAD & op(e, 0) & op(e, 1) & [op(e)] \\ \Octave & & & \\ \Pari & & & \\ \Reduce & part(e, 0) & part(e, 1) & for i:=1:arglength(e) \\ & & & \q\q collect part(e, i) \\ \Scilab & & & \\ \Sumit & & & \\ \Yacas & & & \\ \end{tabular} \\[10pt] \addtocounter{footnote}{-1}% \footnotetext{The following commands work only on expressions that consist of a single level (e.g., $x + y + z$ but not $a/b + c/d$).} \addtocounter{footnote}{-1}% \footnotetext{{\tt TERMS}, {\tt FACTORS}, {\tt NUMERATOR}, {\tt LHS}, etc.} \begin{tabular}{l|l} & \h{Print text and results} \\ \hline \Axiom & output(concat(["sin(", string(0), ") = ", \\ & \q string(sin(0))])); \\ \Derive & "sin(0)" = sin(0) \\ \DoCon & \\ \GAP & Print("There is no sin, but factors(10)= ",Factors(10), "$\backslash$n")\\ \Gmp & \\ \Macsyma & print("sin(", 0, ") =", sin(0))\$ \\ \Magnus & \\ \Maxima & print("sin(", 0, ") =", sin(0))\$ \\ \Maple & printf("sin(\%a) = \%a$\backslash$n", 0, sin(0)): \\ \Mathematica & Print[StringForm["sin(\`{}\`{}) = \`{}\`{}", 0, Sin[0]]]; \\ \MuPAD & print(Unquoted, "sin(".0.")" = sin(0)): \\ \Octave & \\ \Pari & \\ \Reduce & write("sin(", 0, ") = ", sin(0))\$ \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Generate FORTRAN} & \h{Generate \TeX/\LaTeX} \\ \hline \Axiom & outputAsFortran(e) & outputAsTex(e) \\ \Derive & [Transfer Save Fortran] & \\ \DoCon & & \\ \GAP &&Print(LaTeX(e));\\ \Gmp & & \\ \Macsyma & fortran(e)\$ \OR gentran(eval(e))\$ & tex(e); \\ \Magnus & & \\ \Maxima & fortran(e)\$ \OR gentran(eval(e))\$ & tex(e); \\ \Maple & fortran([e]); & latex(e); \\ \Mathematica & FortranForm[e] & TexForm[e] \\ \MuPAD & generate::fortran(e); & generate::TeX(e); \\ \Octave & & \\ \Pari & & \\ \Reduce & on fort; \q e; \q off fort; \OR & load\_\,package(tri)\$ \\ & load\_\,package(gentran)\$ gentran e; & on TeX; e; off TeX; \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Import two space separated columns of integers from {\tt file}} \\ \hline \Axiom & \\ \Derive & [Transfer Load daTa] ({\rm from} file.dat) \\ \DoCon & \\ \GAP & \\ \Gmp & \\ \Macsyma & xy: read\_num\_data\_to\_matrix("file", nrows, 2)\$ \\ \Magnus & \\ \Maxima & xy: read\_num\_data\_to\_matrix("file", nrows, 2)\$ \\ \Maple & xy:= readdata("file", integer, 2): \\ \Mathematica & xy = ReadList["file", Number, RecordLists -> True] \\ \MuPAD & \\ \Octave & \\ \Pari & \\ \Reduce & \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Export two space separated columns of integers to {\tt file}\fnm} \\ \hline \Axiom & )set output algebra "file" \q ({\rm creates} file.spout) \\ & for i in 1..n repeat output( \_ \\ & \q concat([string(xy(i, 1)), " ", string(xy(i, 2))]) ) \\ & )set output algebra console \\ \Derive & xy [Transfer Print Expressions File]\q({\rm creates} file.prt)\\ \DoCon & \\ \GAP & PrintTo("file");for i in [1..n] do\\ &\q AppendTo("file",xy[i][1]," ",xy[i][2],"$\backslash$n");od;\\ \Gmp & \\ \Macsyma & writefile("file")\$ \q for i:1 thru n do \\ & \q print(xy[i, 1], xy[i, 2])\$ \q closefile()\$ \\ \Magnus & \\ \Maxima & writefile("file")\$ \q for i:1 thru n do \\ & \q print(xy[i, 1], xy[i, 2])\$ \q closefile()\$ \\ \Maple & writedata("file", xy); \\ \Mathematica & outfile = OpenWrite["file"]; \\ & Do[WriteString[outfile, \\ & \q xy@[[i, 1]], " ", xy@[[i, 2]], "$\backslash$n"], \{i, 1, n\}] \\ & Close[outfile]; \\ \MuPAD & fprint(Unquoted, Text, "file", \\ & \q ("$\backslash$n", xy[i, 1], xy[i, 2]) \$ i = 1..n): \\ \Octave & \\ \Pari & \\ \Reduce & out "file"; \q for i:=1:n do \\ & \q write(xy(i, 1), " ", xy(i, 2)); \q shut "file"; \\ \Scilab & \\ \Sumit & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \footnotetext{Some editing of {\tt file} will be necessary for all systems but \Maple\ and \Mathematica.} \section{Mathematics and Graphics} {\rm Since {\GAP} aims at discrete mathematics, it does not provide much of the calculus functionality listed in the following section.} \begin{tabular}{l|llllll} & $e$ & $\pi$ & $i$ & $+\infty$ & $\sqrt{2}$ & $2^{1/3}$ \\ \hline \Axiom & \%e & \%pi & \%i & \%plusInfinity & sqrt(2) & 2**(1/3) \\ \Derive & \#e & pi & \#i & inf & SQRT(2) & 2\^{}(1/3) \\ \DoCon & & & & & & \\ \GAP & & & E(4) & infinity & ER(2)\fnm &\\ \Gmp & & & & & & \\ \Macsyma & \%e & \%pi & \%i & inf & sqrt(2) & 2\^{}(1/3) \\ \Magnus & & & & & & \\ \Maxima & \%e & \%pi & \%i & inf & sqrt(2) & 2\^{}(1/3) \\ \Maple & exp(1) & Pi & I & infinity & sqrt(2) & 2\^{}(1/3) \\ \Mathematica & E & Pi & I & Infinity & Sqrt[2] & 2\^{}(1/3) \\ \MuPAD & E & PI & I & infinity & sqrt(2) & 2\^{}(1/3) \\ \Octave & & & & & & \\ \Pari & & & & & & \\ \Reduce & e & pi & i & infinity & sqrt(2) & 2\^{}(1/3) \\ \Scilab & & & & & & \\ \Sumit & & & & & & \\ \Yacas & & & & & & \\ \end{tabular} \\[10pt] \footnotetext{{\tt ER} represents special cyclotomic numbers and is not a root function.} \addtocounter{footnote}{-1}% \begin{tabular}{l|llll} & \h{Euler's constant} & \h{Natural log} & \h{Arctangent} & $n!$ \\ \hline \Axiom & & log(x) & atan(x) & factorial(n) \\ \Derive & euler\_\,gamma & LOG(x) & ATAN(x) & n! \\ \DoCon & & & & \\ \GAP & & LogInt(x,base) && Factorial(n) \\ \Gmp & & & & \\ \Macsyma & \%gamma & log(x) & atan(x) & n! \\ \Magnus & & & & \\ \Maxima & \%gamma & log(x) & atan(x) & n! \\ \Maple & gamma & log(x) & arctan(x) & n! \\ \Mathematica & EulerGamma & Log[x] & ArcTan[x] & n! \\ \MuPAD & EULER & ln(x) & atan(x) & n! \\ \Octave & & & & \\ \Pari & & & & \\ \Reduce & Euler\_\,Gamma & log(x) & atan(x) & factorial(n) \\ \Scilab & & & & \\ \Sumit & & & & \\ \Yacas & & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Legendre polynomial} & \h{Chebyshev poly.\ of the \nth{1} kind} \\ \hline \Axiom & legendreP(n, x) & chebyshevT(n, x) \\ \Derive & LEGENDRE\_\,P(n, x) & CHEBYCHEV\_\,T(n, x) \\ \DoCon & & \\ \GAP & & \\ \Gmp & & \\ \Macsyma & legendre\_\,p(n, x) & chebyshev\_\,t(n, x) \\ \Magnus & & \\ \Maxima & legendre\_\,p(n, x) & chebyshev\_\,t(n, x) \\ \Maple & orthopoly[P](n, x) & orthopoly[T](n, x) \\ \Mathematica & LegendreP[n, x] & ChebyshevT[n, x] \\ \MuPAD & orthpoly::legendre(n, x) & orthpoly::chebyshev1(n, x) \\ \Octave & & \\ \Pari & & \\ \Reduce & LegendreP(n, x) & ChebyshevT(n, x) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Fibonacci number} & \h{Elliptic integral of the \nth{1} kind} \\ \hline \Axiom & fibonacci(n) & \\ \Derive & FIBONACCI(n) & ELLIPTIC\_\,E(phi, k\^{}2) \\ \DoCon & & \\ \GAP & Fibonacci(n) & \\ \Gmp & & \\ \Macsyma & fib(n) & elliptic\_\,e(phi, k\^{}2) \\ \Magnus & & \\ \Maxima & fib(n) & elliptic\_\,e(phi, k\^{}2) \\ \Maple & combinat[fibonacci](n) & EllipticE(sin(phi), k) \\ \Mathematica & Fibonacci[n] & EllipticE[phi, k\^{}2] \\ \MuPAD & numlib::fibonacci(n) & \\ \Octave & & \\ \Pari & & \\ \Reduce & & EllipticE(phi, k\^{}2) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|llll} & $\Gamma(x)$ & $\psi(x)$ & \h{Cosine integral} & \h{Bessel fun.\ (\nth{1})} \\ \hline \Axiom & Gamma(x) & psi(x) & real(Ei(\%i*x)) & besselJ(n, x) \\ \Derive & GAMMA(x) & PSI(x) & CI(x) & BESSEL\_\,J(n, x) \\ \DoCon & & & & \\ \GAP & & & & \\ \Gmp & & & & \\ \Macsyma & gamma(x) & psi[0](x) & cos\_\,int(x) & bessel\_j[n](x) \\ \Magnus & & & & \\ \Maxima & gamma(x) & psi[0](x) & cos\_\,int(x) & bessel\_j[n](x) \\ \Maple & GAMMA(x) & Psi(x) & Ci(x) & BesselJ(n, x) \\ \Mathematica & Gamma[x] & PolyGamma[x] & CosIntegral[x] & BesselJ[n, x] \\ \MuPAD & gamma(x) & psi(x) & & besselJ(n, x) \\ \Octave & & & & \\ \Pari & & & & \\ \Reduce & Gamma(x) & Psi(x) & Ci(x) & BesselJ(n, x) \\ \Scilab & & & & \\ \Sumit & & & & \\ \Yacas & & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|lll} & \h{Hypergeometric fun.\ ${}_2F_1(a, b; c; x)$} & \h{Dirac delta} & \h{Unit step fun.} \\ \hline \Axiom & & & \\ \Derive & GAUSS(a, b, c, x) & & STEP(x) \\ \DoCon & & & \\ \GAP & & & \\ \Gmp & & & \\ \Macsyma & hgfred([a, b], [c], x) & delta(x) & unit\_\,step(x) \\ \Magnus & & & \\ \Maxima & hgfred([a, b], [c], x) & delta(x) & unit\_\,step(x) \\ \Maple & hypergeom([a, b], [c], x) & Dirac(x) & Heaviside(x) \\ \Mathematica & HypergeometricPFQ[\{a,b\},\{c\},x] & \m{2}{@<< Calculus\`{}DiracDelta\`{}} \\ \MuPAD & & dirac(x) & heaviside(x) \\ \Octave & & & \\ \Pari & & & \\ \Reduce & hypergeometric(\{a, b\}, \{c\}, x) & & \\ \Scilab & & & \\ \Sumit & & & \\ \Yacas & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Define $|x|$ via a piecewise function} \\ \hline \Axiom & \\ \Derive & a(x):= -x*CHI(-inf, x, 0) + x*CHI(0, x, inf) \\ \DoCon & \\ \GAP & \\ \Gmp & \\ \Macsyma & a(x):= -x*unit\_\,step(-x) + x*unit\_\,step(x)\$ \\ \Magnus & \\ \Maxima & a(x):= -x*unit\_\,step(-x) + x*unit\_\,step(x)\$ \\ \Maple & a:= x -> piecewise(x < 0, -x, x): \\ \Mathematica & @<< Calculus\`{}DiracDelta\`{} \\ & a[x\_]:= -x*UnitStep[-x] + x*UnitStep[x] \\ \MuPAD & a:= proc(x) begin -x*heaviside(-x) + x*heaviside(x) \\ & \q\q end\_\,proc: \\ \Octave & \\ \Pari & \\ \Reduce & \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Assume $x$ is real} & \h{Remove that assumption} \\ \hline \Axiom & & \\ \Derive & x :epsilon Real & x:= \\ \DoCon & & \\ \GAP & & \\ \Gmp & & \\ \Macsyma & declare(x, real)\$ & remove(x, real)\$ \\ \Magnus & & \\ \Maxima & declare(x, real)\$ & remove(x, real)\$ \\ \Maple & assume(x, real); & x:= 'x': \\ \Mathematica & x/: Im[x] = 0; & Clear[x] \\ \MuPAD & assume(x, Type::RealNum): & unassume(x, Type::RealNum): \\ \Octave & & \\ \Pari & & \\ \Reduce & & \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Assume $0 < x \le 1$} & \h{Remove that assumption} \\ \hline \Axiom & & \\ \Derive & x :epsilon (0, 1] & x:= \\ \DoCon & & \\ \GAP & & \\ \Gmp & & \\ \Macsyma & assume(x > 0, x <= 1)\$ & forget(x > 0, x <= 1)\$ \\ \Magnus & & \\ \Maxima & assume(x > 0, x <= 1)\$ & forget(x > 0, x <= 1)\$ \\ \Maple & assume(x > 0); & x:= 'x': \\ & additionally(x <= 1); & \\ \Mathematica & Assumptions -> 0 < x <= 1\,\fnm & \\ \MuPAD & assume(x > 0): assume(x <= 1): & unassume(x): \\ \Octave & & \\ \Pari & & \\ \Reduce & & \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \footnotetext{This is an option for {\tt Integrate}.} \begin{tabular}{l|l} & \h{Basic simplification of an expression $e$} \\ \hline \Axiom & simplify(e) \OR\ normalize(e) \OR\ complexNormalize(e) \\ \Derive & e \\ \DoCon & \\ \GAP & e \\ \Gmp & \\ \Macsyma & ratsimp(e) \OR\ radcan(e) \\ \Magnus & \\ \Maxima & ratsimp(e) \OR\ radcan(e) \\ \Maple & simplify(e) \\ \Mathematica & Simplify[e] \OR\ FullSimplify[e] \\ \MuPAD & simplify(e) \OR\ normal(e) \\ \Octave & \\ \Pari & \\ \Reduce & e \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Use an unknown function} & \h{Numerically evaluate an expr.} \\ \hline \Axiom & f:= operator('f); \q f(x) & exp(1) :: Complex Float \\ \Derive & f(x):= & Precision:= Approximate \\ & f(x) & APPROX(EXP(1)) \\ & & Precision:= Exact \\ \DoCon & & \\ \GAP & & EvalF(123/456)\\ \Gmp & & \\ \Macsyma & f(x) & sfloat(exp(1)); \\ \Magnus & & \\ \Maxima & f(x) & sfloat(exp(1)); \\ \Maple & f(x) & evalf(exp(1)); \\ \Mathematica & f[x] & N[Exp[1]] \\ \MuPAD & f(x) & float(exp(1)); \\ \Octave & & \\ \Pari & & \\ \Reduce & operator f; \q f(x) & on rounded; \q exp(1); \\ & & off rounded; \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & $ n \bmod m$ & \h{Solve $e \equiv 0 \bmod m$ for $x$} \\ \hline \Axiom & rem(n, m) & solve(e = 0 :: PrimeField(m), x) \\ \Derive & MOD(n, m) & SOLVE\_\,MOD(e = 0, x, m) \\ \DoCon & & \\ \GAP & n mod m & \h{solve using finite fields}\\ \Gmp & & \\ \Macsyma & mod(n, m) & modulus: m\$ \q solve(e = 0, x) \\ \Magnus & & \\ \Maxima & mod(n, m) & modulus: m\$ \q solve(e = 0, x) \\ \Maple & n mod m & msolve(e = 0, m) \\ \Mathematica & Mod[n, m] & Solve[\{e == 0, Modulus == m\}, x] \\ \MuPAD & n mod m & solve(poly(e = 0, [x], IntMod(m)), x) \\ \Octave & & \\ \Pari & & \\ \Reduce & on modular; & load\_\,package(modsr)\$ \q on modular; \\ & setmod m\$ \q n & setmod m\$ \q m\_solve(e = 0, x) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Put over common denominator} & \h{Expand into separate fractions} \\ \hline \Axiom & a/b + c/d & (a*d + b*c)/(b*d) :: \_ \\ & & \q MPOLY([a], FRAC POLY INT) \\ \Derive & FACTOR(a/b + c/d, Trivial) & EXPAND((a*d + b*c)/(b*d)) \\ \DoCon & & \\ \GAP & a/b+c/d &\\ \Gmp & & \\ \Macsyma & xthru(a/b + c/d) & expand((a*d + b*c)/(b*d)) \\ \Magnus & & \\ \Maxima & xthru(a/b + c/d) & expand((a*d + b*c)/(b*d)) \\ \Maple & normal(a/b + c/d) & expand((a*d + b*c)/(b*d)) \\ \Mathematica & Together[a/b + c/d] & Apart[(a*d + b*c)/(b*d)] \\ \MuPAD & normal(a/b + c/d) & expand((a*d + b*c)/(b*d)) \\ \Octave & & \\ \Pari & & \\ \Reduce & a/b + c/d & on div; (a*d + b*c)/(b*d) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Manipulate the root of a polynomial} \\ \hline \Axiom & a:= rootOf(x**2 - 2); \q a**2 \\ \Derive & \\ \DoCon & \\ \GAP & x:=X(Rationals,"x");\\ &\q a:=RootOfDefiningPolynomial(AlgebraicExtension(Rationals,x\^{}2-2)); a\^{}2\\ \Gmp & \\ \Macsyma & algebraic:true\$ \q tellrat(a\^{}2 - 2)\$ \q rat(a\^{}2); \\ \Magnus & \\ \Maxima & algebraic:true\$ \q tellrat(a\^{}2 - 2)\$ \q rat(a\^{}2); \\ \Maple & a:= RootOf(x\^{}2 - 2): \q simplify(a\^{}2); \\ \Mathematica & a = Root[\#\^{}2 - 2 \&, 2] \q a\^{}2 \\ \MuPAD & \\ \Octave & \\ \Pari & \\ \Reduce & load\_\,package(arnum)\$ \q defpoly(a\^{}2 - 2); \q a\^{}2; \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Noncommutative multiplication} & \h{Solve a pair of equations} \\ \hline \Axiom & & solve([eqn1, eqn2], [x, y]) \\ \Derive & x :epsilon Nonscalar & SOLVE([eqn1, eqn2], [x, y]) \\ & y :epsilon Nonscalar & \\ & x . y & \\ \DoCon & & \\ \GAP &*&\\ \Gmp & & \\ \Macsyma & x . y & solve([eqn1, eqn2], [x, y]) \\ \Magnus & & \\ \Maxima & x . y & solve([eqn1, eqn2], [x, y]) \\ \Maple & x \&* y & solve(\{eqn1, eqn2\}, \{x, y\}) \\ \Mathematica & x ** y & Solve[\{eqn1, eqn2\}, \{x, y\}] \\ \MuPAD & & solve(\{eqn1, eqn2\}, \{x, y\}) \\ \Octave & & \\ \Pari & & \\ \Reduce & operator x, y; & solve(\{eqn1, eqn2\}, \{x, y\}) \\ & noncom x, y; & \\ & x() * y() & \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \m{2}{\rm Decrease/increase angles in trigonometric functions} \\ \hline \Axiom & \m{2}{simplify(normalize(sin(2*x)))} \\ \Derive & Trigonometry:= Expand & Trigonometry:= Collect \\ & sin(2*x) & 2*sin(x)*cos(x) \\ \DoCon & & \\ \GAP & & \\ \Gmp & & \\ \Macsyma & trigexpand(sin(2*x)) & trigreduce(2*sin(x)*cos(x)) \\ \Magnus & & \\ \Maxima & trigexpand(sin(2*x)) & trigreduce(2*sin(x)*cos(x)) \\ \Maple & expand(sin(2*x)) & combine(2*sin(x)*cos(x)) \\ \Mathematica & TrigExpand[Sin[2*x]] & TrigReduce[2*Sin[x]*Cos[x]] \\ \MuPAD & expand(sin(2*x)) & combine(2*sin(x)*cos(x), sincos) \\ \Octave & & \\ \Pari & & \\ \Reduce & load\_\,package(assist)\$ \\ & trigexpand(sin(2*x)) & trigreduce(2*sin(x)*cos(x)) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Gr\"obner basis} \\ \hline \Axiom & groebner([p1, p2, ...]) \\ \Derive & \\ \DoCon & \\ \GAP & \\ \Gmp & \\ \Macsyma & grobner([p1, p2, ...]) \\ \Magnus & \\ \Maxima & grobner([p1, p2, ...]) \\ \Maple & Groebner[gbasis]([p1, p2, ...], plex(x1, x2, ...)) \\ \Mathematica & GroebnerBasis[\{p1, p2, ...\}, \{x1, x2, ...\}] \\ \MuPAD & groebner::gbasis([p1, p2, ...]) \\ \Octave & \\ \Pari & \\ \Reduce & load\_\,package(groebner)\$ \q groebner(\{p1, p2, ...\}) \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Factorization of $e$ over $i = \sqrt{-1}$} \\ \hline \Axiom & factor(e, [rootOf(i**2 + 1)]) \\ \Derive & FACTOR(e, Complex) \\ \DoCon & \\ \GAP & Factors(GaussianIntegers,e)\\ \Gmp & \\ \Macsyma & gfactor(e); \OR\ factor(e, i\^{}2 + 1); \\ \Magnus & \\ \Maxima & gfactor(e); \OR\ factor(e, i\^{}2 + 1); \\ \Maple & factor(e, I); \\ \Mathematica & Factor[e, Extension -> I] \\ \MuPAD & QI:= Dom::AlgebraicExtension(Dom::Rational, i\^{}2 + 1); \\ & QI::name:= "QI": \q Factor(poly(e, QI)); \\ \Octave & \\ \Pari & \\ \Reduce & on complex, factor; \q e; \q off complex, factor; \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Real part} & \h{Convert a complex expr.\ to rectangular form} \\ \hline \Axiom & real(f(z)) & complexForm(f(z)) \\ \Derive & RE(f(z)) & f(z) \\ \DoCon & & \\ \GAP & (f(z)+GaloisCyc(f(z),-1))/2&\\ \Gmp & & \\ \Macsyma & realpart(f(z)) & rectform(f(z)) \\ \Magnus & & \\ \Maxima & realpart(f(z)) & rectform(f(z)) \\ \Maple & Re(f(z)) & evalc(f(z)) \\ \Mathematica & Re[f[z]] & ComplexExpand[f[z]] \\ \MuPAD & Re(f(z)) & rectform(f(z)) \\ \Octave & & \\ \Pari & & \\ \Reduce & repart(f(z)) & repart(f(z)) + i*impart(f(z)) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|lll} & \h{Matrix addition} & \h{Matrix multiplication} & \h{Matrix transpose} \\ \hline \Axiom & A + B & A * B & transpose(A) \\ \Derive & A + B & A . B & A\`{} \\ \DoCon & & & \\ \GAP & A + B & A * B & TransposedMat(A)\\ \Gmp & & & \\ \Macsyma & A + B & A . B & transpose(A) \\ \Magnus & & & \\ \Maxima & A + B & A . B & transpose(A) \\ \Maple & evalm(A + B) & evalm(A \&* B) & linalg[transpose](A) \\ \Mathematica & A + B & A . B & Transpose[A] \\ \MuPAD & A + B & A * B & transpose(A) \\ \Octave & & & \\ \Pari & & & \\ \Reduce & A + B & A * B & tp(A) \\ \Scilab & & & \\ \Sumit & & & \\ \Yacas & & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Solve the matrix equation $A x = b$} \\ \hline \Axiom & solve(A, transpose(b)) . 1 . particular :: Matrix \_\_\_ \\ \Derive & \\ \DoCon & \\ \GAP & SolutionMat(TransposedMat(A),b)\\ \Gmp & \\ \Macsyma & xx: genvector('x, mat\_nrows(b))\$ \\ & x: part(matlinsolve(A . xx = b, xx), 1, 2) \\ \Magnus & \\ \Maxima & xx: genvector('x, mat\_nrows(b))\$ \\ & x: part(matlinsolve(A . xx = b, xx), 1, 2) \\ \Maple & x:= linalg[linsolve](A, b) \\ \Mathematica & x = LinearSolve[A, b] \\ \MuPAD & \\ \Octave & \\ \Pari & \\ \Reduce & \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Sum: $\sum_{i = 1}^n f(i)$} & \h{Product: $\prod_{i = 1}^n f(i)$} \\ \hline \Axiom & sum(f(i), i = 1..n) & product(f(i), i = 1..n) \\ \Derive & SUM(f(i), i, 1, n) & PRODUCT(f(i), i, 1, n) \\ \DoCon & & \\ \GAP & Sum([1..n],f) & Product([1..n],f)\\ \Gmp & & \\ \Macsyma & closedform( & closedform( \\ & \q sum(f(i), i, 1, n)) & \q product(f(i), i, 1, n)) \\ \Magnus & & \\ \Maxima & closedform( & closedform( \\ & \q sum(f(i), i, 1, n)) & \q product(f(i), i, 1, n)) \\ \Maple & sum(f(i), i = 1..n) & product(f(i), i = 1..n) \\ \Mathematica & Sum[f[i], \{i, 1, n\}] & Product[f[i], \{i, 1, n\}] \\ \MuPAD & sum(f(i), i = 1..n) & product(f(i), i = 1..n) \\ \Octave & & \\ \Pari & & \\ \Reduce & sum(f(i), i, 1, n) & prod(f(i), i, 1, n) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Limit: $\lim_{x \rightarrow 0-} f(x)$} & \h{Taylor/Laurent/etc.\ series} \\ \hline \Axiom & limit(f(x), x = 0, "left") & series(f(x), x = 0, 3) \\ \Derive & LIM(f(x), x, 0, -1) & TAYLOR(f(x), x, 0, 3) \\ \DoCon & & \\ \GAP & & \\ \Gmp & & \\ \Macsyma & limit(f(x), x, 0, minus) & taylor(f(x), x, 0, 3) \\ \Magnus & & \\ \Maxima & limit(f(x), x, 0, minus) & taylor(f(x), x, 0, 3) \\ \Maple & limit(f(x), x = 0, left) & series(f(x), x = 0, 4) \\ \Mathematica & Limit[f[x], x->0, Direction->1] & Series[f[x],\{x, 0, 3\}] \\ \MuPAD & limit(f(x), x = 0, Left) & series(f(x), x = 0, 4) \\ \Octave & & \\ \Pari & & \\ \Reduce & limit!-(f(x), x, 0) & taylor(f(x), x, 0, 3) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Differentiate: $\frac{d^3 f(x, y)}{dx \, dy^2}$} & \h{Integrate: $\int_0^1 f(x) \, dx$} \\ \hline \Axiom & D(f(x, y), [x, y], [1, 2]) & integrate(f(x), x = 0..1) \\ \Derive & DIF(DIF(f(x, y), x), y, 2) & INT(f(x), x, 0, 1) \\ \DoCon & & \\ \GAP & & \\ \Gmp & & \\ \Macsyma & diff(f(x, y), x, 1, y, 2) & integrate(f(x), x, 0, 1) \\ \Magnus & & \\ \Maxima & diff(f(x, y), x, 1, y, 2) & integrate(f(x), x, 0, 1) \\ \Maple & diff(f(x, y), x, y\$2) & int(f(x), x = 0..1) \\ \Mathematica & D[f[x, y], x, \{y, 2\}] & Integrate[f[x], \{x, 0, 1\}] \\ \MuPAD & diff(f(x, y), x, y\$2) & int(f(x), x = 0..1) \\ \Octave & & \\ \Pari & & \\ \Reduce & df(f(x, y), x, y, 2) & int(f(x), x, 0, 1) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|ll} & \h{Laplace transform} & \h{Inverse Laplace transform} \\ \hline \Axiom & laplace(e, t, s) & inverseLaplace(e, s, t) \\ \Derive & LAPLACE(e, t, s) & \\ \DoCon & & \\ \GAP & & \\ \Gmp & & \\ \Macsyma & laplace(e, t, s) & ilt(e, s, t) \\ \Magnus & & \\ \Maxima & laplace(e, t, s) & ilt(e, s, t) \\ \Maple & inttrans[laplace](e,t,s) & inttrans[invlaplace](e,s,t) \\ \Mathematica & \m{2}{\q @<< Calculus\`{}LaplaceTransform\`{}} \\ & LaplaceTransform[e, t, s] & {\st InverseLaplaceTransform[e,s,t]} \\ \MuPAD & transform::laplace(e,t,s) & transform::ilaplace(e, s, t) \\ \Octave & & \\ \Pari & & \\ \Reduce & \m{2}{\q load\_\,package(laplace)\$ \q load\_\,package(defint)\$} \\ & laplace(e, t, s) & invlap(e, t, s) \\ \Scilab & & \\ \Sumit & & \\ \Yacas & & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Solve an ODE (with the initial condition $y'(0) = 1$)} \\ \hline \Axiom & solve(eqn, y, x) \\ \Derive & APPLY\_\,IC(RHS(ODE(eqn, x, y, y\_)), [x, 0], [y, 1]) \\ \DoCon & \\ \GAP & \\ \Gmp & \\ \Macsyma & ode\_ibc(ode(eqn, y(x), x), x = 0, diff(y(x), x) = 1) \\ \Magnus & \\ \Maxima & ode\_ibc(ode(eqn, y(x), x), x = 0, diff(y(x), x) = 1) \\ \Maple & dsolve(\{eqn, D(y)(0) = 1\}, y(x)) \\ \Mathematica & DSolve[\{eqn, y'[0] == 1\}, y[x], x] \\ \MuPAD & solve(ode(\{eqn, D(y)(0) = 1\}, y(x))) \\ \Octave & \\ \Pari & \\ \Reduce & odesolve(eqn, y(x), x) \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Define the differential operator $L = D_x + I$ and apply it to $\sin x$} \\ \hline \Axiom & DD : LODO(Expression Integer, e +-> D(e, x)) := D(); \\ & L:= DD + 1; \q L(sin(x)) \\ \Derive & \\ \DoCon & \\ \GAP & \\ \Gmp & \\ \Macsyma & load(opalg)\$ \q L: (diffop(x) - 1)\$ \q L(sin(x)); \\ \Magnus & \\ \Maxima & load(opalg)\$ \q L: (diffop(x) - 1)\$ \q L(sin(x)); \\ \Maple & id:= x -> x: \q L:= (D + id): \q L(sin)(x); \\ \Mathematica & L = D[\#, x]\& + Identity; \q Through[L[Sin[x]]] \\ \MuPAD & L:= (D + id): \q L(sin)(x); \\ \Octave & \\ \Pari & \\ \Reduce & \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{2D plot of two separate curves overlayed} \\ \hline \Axiom & draw(x, x = 0..1); \q draw(acsch(x), x = 0..1); \\ \Derive & [Plot Overlay] \\ \DoCon & \\ \GAP & \\ \Gmp & \\ \Macsyma & plot(x, x, 0, 1)\$ \q plot(acsch(x), x, 0, 1)\$ \\ \Magnus & \\ \Maxima & plot(x, x, 0, 1)\$ \q plot(acsch(x), x, 0, 1)\$ \\ \Maple & plot(\{x, arccsch(x)\}, x = 0..1): \\ \Mathematica & Plot[\{x, ArcCsch[x]\}, \{x, 0, 1\}]; \\ \MuPAD & plotfunc(x, acsch(x), x = 0..1): \\ \Octave & \\ \Pari & \\ \Reduce & load\_\,package(gnuplot)\$ \q plot(y = x, x = (0 .. 1))\$ \\ & plot(y = acsch(x), x = (0 .. 1))\$ \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] \begin{tabular}{l|l} & \h{Simple 3D plotting} \\ \hline \Axiom & draw(abs(x*y), x = 0..1, y = 0..1); \\ \Derive & [Plot Overlay] \\ \DoCon & \\ \GAP & \\ \Gmp & \\ \Macsyma & plot3d(abs(x*y), x, 0, 1, y, 0, 1)\$ \\ \Magnus & \\ \Maxima & plot3d(abs(x*y), x, 0, 1, y, 0, 1)\$ \\ \Maple & plot3d(abs(x*y), x = 0..1, y = 0..1): \\ \Mathematica & Plot3D[Abs[x*y], \{x, 0, 1\}, \{y, 0, 1\}]; \\ \MuPAD & plotfunc(abs(x*y), x = 0..1, y = 0..1): \\ \Octave & \\ \Pari & \\ \Reduce & load\_\,package(gnuplot)\$ \\ & plot(z = abs(x*y), x = (0 .. 1), y = (0 .. 1))\$ \\ \Scilab & \\ \Sumit & \\ \Yacas & \\ \end{tabular} \\[10pt] %\begin{tabular}{l|l} %& \h{} \\ %\hline %\Axiom & \\ %\Derive & \\ %\DoCon & \\ %\GAP & \\ %\Gmp & \\ %\Macsyma & \\ %\Magnus & \\ %\Maxima & \\ %\Maple & \\ %\Mathematica & \\ %\MuPAD & \\ %\Octave & \\ %\Pari & \\ %\Reduce & \\ %\Scilab & \\ %\Sumit & \\ %\Yacas & \\ %\end{tabular} \\[10pt] \end{tt} \endgroup \end{document}
