# Interface to Maple¶

AUTHORS:

• William Stein (2005): maple interface
• Gregg Musiker (2006-02-02): tutorial
• William Stein (2006-03-05): added tab completion, e.g., maple.[tab], and help, e.g, maple.sin?.

You must have the optional commercial Maple interpreter installed and available as the command maple in your PATH in order to use this interface. You do not have to install any optional Sage packages.

Type maple.[tab] for a list of all the functions available from your Maple install. Type maple.[tab]? for Maple’s help about a given function. Type maple(...) to create a new Maple object, and maple.eval(...) to run a string using Maple (and get the result back as a string).

EXAMPLES:

sage: maple('3 * 5')                                 # optional - maple
15
sage: maple.eval('ifactor(2005)')                    # optional - maple
'(5)*(401)'
sage: maple.ifactor(2005)                            # optional - maple
(5)*(401)
sage: maple.fsolve('x^2=cos(x)+4', 'x=0..5')         # optional - maple
1.914020619
sage: maple.factor('x^5 - y^5')                      # optional - maple
(x-y)*(x^4+x^3*y+x^2*y^2+x*y^3+y^4)


If the string “error” (case insensitive) occurs in the output of anything from Maple, a RuntimeError exception is raised.

## Tutorial¶

AUTHORS:

• Gregg Musiker (2006-02-02): initial version.

This tutorial is based on the Maple Tutorial for number theory from http://www.math.mun.ca/~drideout/m3370/numtheory.html.

There are several ways to use the Maple Interface in Sage. We will discuss two of those ways in this tutorial.

1. If you have a maple expression such as

factor( (x^5-1));


We can write that in sage as

sage: maple('factor(x^5-1)')                 # optional - maple
(x-1)*(x^4+x^3+x^2+x+1)


Notice, there is no need to use a semicolon.

2. Since Sage is written in Python, we can also import maple commands and write our scripts in a Pythonic way. For example, factor() is a maple command, so we can also factor in Sage using

sage: maple('(x^5-1)').factor()              # optional - maple
(x-1)*(x^4+x^3+x^2+x+1)


where expression.command() means the same thing as command(expression) in Maple. We will use this second type of syntax whenever possible, resorting to the first when needed.

sage: maple('(x^12-1)/(x-1)').simplify()     # optional - maple
x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1


The normal command will always reduce a rational function to the lowest terms. The factor command will factor a polynomial with rational coefficients into irreducible factors over the ring of integers. So for example,

sage: maple('(x^12-1)').factor( )           # optional - maple
(x-1)*(x+1)*(x^2+x+1)*(x^2-x+1)*(x^2+1)*(x^4-x^2+1)

sage: maple('(x^28-1)').factor( )           # optional - maple
(x-1)*(x^6+x^5+x^4+x^3+x^2+x+1)*(x+1)*(1-x+x^2-x^3+x^4-x^5+x^6)*(x^2+1)*(x^12-x^10+x^8-x^6+x^4-x^2+1)


Another important feature of maple is its online help. We can access this through sage as well. After reading the description of the command, you can press q to immediately get back to your original prompt.

Incidentally you can always get into a maple console by the command

sage: maple.console()          # not tested
sage: !maple                   # not tested


Note that the above two commands are slightly different, and the first is preferred.

For example, for help on the maple command fibonacci, we type

sage: maple.help('fibonacci')  # not tested, since it uses a pager


We see there are two choices. Type

sage: maple.help('combinat, fibonacci')   # not tested, since it uses a pager


We now see how the Maple command fibonacci works under the combinatorics package. Try typing in

sage: maple.fibonacci(10)                # optional - maple
fibonacci(10)


You will get fibonacci(10) as output since Maple has not loaded the combinatorics package yet. To rectify this type

sage: maple('combinat[fibonacci]')(10)     # optional - maple
55


If you want to load the combinatorics package for future calculations, in Sage this can be done as

sage: maple.with_package('combinat')       # optional - maple


or

sage: maple.load('combinat')               # optional - maple


Now if we type maple.fibonacci(10), we get the correct output:

sage: maple.fibonacci(10)                  # optional - maple
55


Some common maple packages include combinat, linalg, and numtheory. To produce the first 19 Fibonacci numbers, use the sequence command.

sage: maple('seq(fibonacci(i),i=1..19)')     # optional - maple
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,
4181


Two other useful Maple commands are ifactor and isprime. For example

sage: maple.isprime(maple.fibonacci(27))     # optional - maple
false
sage: maple.ifactor(maple.fibonacci(27))     # optional - maple
(2)*(17)*(53)*(109)


Note that the isprime function that is included with Sage (which uses PARI) is better than the Maple one (it is faster and gives a provably correct answer, whereas Maple is sometimes wrong).

sage: alpha = maple('(1+sqrt(5))/2')         # optional - maple
sage: beta = maple('(1-sqrt(5))/2')          # optional - maple
sage: f19  = alpha^19 - beta^19/maple('sqrt(5)')      # optional - maple
sage: f19                                             # optional - maple
(1/2+1/2*5^(1/2))^19-1/5*(1/2-1/2*5^(1/2))^19*5^(1/2)
sage: f19.simplify()                # somewhat randomly ordered output; optional - maple
6765+5778/5*5^(1/2)


Let’s say we want to write a maple program now that squares a number if it is positive and cubes it if it is negative. In maple, that would look like

mysqcu := proc(x)
if x > 0 then x^2;
else x^3; fi;
end;


In Sage, we write

sage: mysqcu = maple('proc(x) if x > 0 then x^2 else x^3 fi end')    # optional - maple
sage: mysqcu(5)                                                      # optional - maple
25
sage: mysqcu(-5)                                                     # optional - maple
-125


More complicated programs should be put in a separate file and loaded.

class sage.interfaces.maple.Maple(maxread=100, script_subdirectory=None, server=None, server_tmpdir=None, logfile=None)

Interface to the Maple interpreter.

Type maple.[tab] for a list of all the functions available from your Maple install. Type maple.[tab]? for Maple’s help about a given function. Type maple(...) to create a new Maple object, and maple.eval(...) to run a string using Maple (and get the result back as a string).

clear(var)

Clear the variable named var.

To clear a Maple variable, you must assign ‘itself’ to itself. In Maple ‘expr’ prevents expr to be evaluated.

EXAMPLES:

sage: maple.set('xx', '2')  # optional - maple
sage: maple.get('xx')       # optional - maple
'2'
sage: maple.clear('xx')     # optional - maple
sage: maple.get('xx')       # optional - maple
'xx'

completions(s)

Return all commands that complete the command starting with the string s. This is like typing s[Ctrl-T] in the maple interpreter.

EXAMPLES:

sage: c = maple.completions('di')  # optional - maple
sage: 'divide' in c                # optional - maple
True

console()

Spawn a new Maple command-line session.

EXAMPLES:

sage: maple.console() # not tested
|^/|     Maple 11 (IBM INTEL LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2007
<____ ____>  Waterloo Maple Inc.
|       Type ? for help.
>

cputime(t=None)

Returns the amount of CPU time that the Maple session has used. If t is not None, then it returns the difference between the current CPU time and t.

EXAMPLES:

sage: t = maple.cputime() # optional - maple
sage: t                   # random; optional - maple
0.02
sage: x = maple('x')      # optional - maple
sage: maple.diff(x^2, x)  # optional - maple
2*x
sage: maple.cputime(t)    # random; optional - maple
0.0

expect()

Returns the pexpect object for this Maple session.

EXAMPLES:

sage: m = Maple()          # optional - maple
sage: m.expect() is None   # optional - maple
True
sage: m._start()           # optional - maple
sage: m.expect()           # optional - maple
<pexpect.spawn instance at 0x...>
sage: m.quit()             # optional - maple

get(var)

Get the value of the variable var.

EXAMPLES:

sage: maple.set('xx', '2') # optional - maple
sage: maple.get('xx')      # optional - maple
'2'

help(str)

This is the same as typing ”?str” in the Maple console.

INPUT:

• str - a string to search for in the maple help system

EXAMPLES:

sage: maple.help('digamma') #not tested
Psi - the Digamma and Polygamma functions
...


Make a package of Maple procedures available in the interpreter.

INPUT:

• package - string

EXAMPLES: Some functions are unknown to Maple until you use with to include the appropriate package.

sage: maple.quit()   # reset maple; optional -- maple
sage: maple('partition(10)')              # optional - maple
partition(10)
sage: maple('bell(10)')                   # optional - maple
bell(10)
sage: maple.with_package('combinat')      # optional - maple
sage: maple('partition(10)')              # optional - maple
[[1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 2], [1, 1, 1, 1, 1, 1, 2, 2], [1, 1, 1, 1, 2, 2, 2], [1, 1, 2, 2, 2, 2], [2, 2, 2, 2, 2], [1, 1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 1, 2, 3], [1, 1, 1, 2, 2, 3], [1, 2, 2, 2, 3], [1, 1, 1, 1, 3, 3], [1, 1, 2, 3, 3], [2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 1, 1, 1, 1, 4], [1, 1, 1, 1, 2, 4], [1, 1, 2, 2, 4], [2, 2, 2, 4], [1, 1, 1, 3, 4], [1, 2, 3, 4], [3, 3, 4], [1, 1, 4, 4], [2, 4, 4], [1, 1, 1, 1, 1, 5], [1, 1, 1, 2, 5], [1, 2, 2, 5], [1, 1, 3, 5], [2, 3, 5], [1, 4, 5], [5, 5], [1, 1, 1, 1, 6], [1, 1, 2, 6], [2, 2, 6], [1, 3, 6], [4, 6], [1, 1, 1, 7], [1, 2, 7], [3, 7], [1, 1, 8], [2, 8], [1, 9], [10]]
sage: maple('bell(10)')                   # optional - maple
115975
sage: maple('fibonacci(10)')              # optional - maple
55

set(var, value)

Set the variable var to the given value.

EXAMPLES:

sage: maple.set('xx', '2') # optional - maple
sage: maple.get('xx')      # optional - maple
'2'

source(s)

Display the Maple source (if possible) about s. This is the same as returning the output produced by the following Maple commands:

interface(verboseproc=2): print(s)

INPUT:

• s - a string representing the function whose source code you want

EXAMPLES:

sage: maple.source('curry')  #not tested
p -> subs('_X' = args[2 .. nargs], () -> p(_X, args))

trait_names(verbose=True, use_disk_cache=True)

Returns a list of all the commands defined in Maple and optionally (per default) store them to disk.

EXAMPLES:

sage: c = maple.trait_names(use_disk_cache=False, verbose=False) # optional - maple
sage: len(c) > 100  # optional - maple
True
sage: 'dilog' in c  # optional - maple
True

with_package(package)

Make a package of Maple procedures available in the interpreter.

INPUT:

• package - string

EXAMPLES: Some functions are unknown to Maple until you use with to include the appropriate package.

sage: maple.quit()   # reset maple; optional -- maple
sage: maple('partition(10)')              # optional - maple
partition(10)
sage: maple('bell(10)')                   # optional - maple
bell(10)
sage: maple.with_package('combinat')      # optional - maple
sage: maple('partition(10)')              # optional - maple
[[1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 2], [1, 1, 1, 1, 1, 1, 2, 2], [1, 1, 1, 1, 2, 2, 2], [1, 1, 2, 2, 2, 2], [2, 2, 2, 2, 2], [1, 1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 1, 2, 3], [1, 1, 1, 2, 2, 3], [1, 2, 2, 2, 3], [1, 1, 1, 1, 3, 3], [1, 1, 2, 3, 3], [2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 1, 1, 1, 1, 4], [1, 1, 1, 1, 2, 4], [1, 1, 2, 2, 4], [2, 2, 2, 4], [1, 1, 1, 3, 4], [1, 2, 3, 4], [3, 3, 4], [1, 1, 4, 4], [2, 4, 4], [1, 1, 1, 1, 1, 5], [1, 1, 1, 2, 5], [1, 2, 2, 5], [1, 1, 3, 5], [2, 3, 5], [1, 4, 5], [5, 5], [1, 1, 1, 1, 6], [1, 1, 2, 6], [2, 2, 6], [1, 3, 6], [4, 6], [1, 1, 1, 7], [1, 2, 7], [3, 7], [1, 1, 8], [2, 8], [1, 9], [10]]
sage: maple('bell(10)')                   # optional - maple
115975
sage: maple('fibonacci(10)')              # optional - maple
55

class sage.interfaces.maple.MapleElement(parent, value, is_name=False, name=None)
trait_names()

EXAMPLES:

sage: a = maple(2) # optional - maple
sage: 'sin' in a.trait_names() # optional - maple
True

class sage.interfaces.maple.MapleFunction(parent, name)
class sage.interfaces.maple.MapleFunctionElement(obj, name)
sage.interfaces.maple.maple_console()

Spawn a new Maple command-line session.

EXAMPLES:

sage: maple_console() #not tested
|^/|     Maple 11 (IBM INTEL LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2007
<____ ____>  Waterloo Maple Inc.
|       Type ? for help.
>


Returns the maple object created in sage.interfaces.maple.

EXAMPLES:

sage: from sage.interfaces.maple import reduce_load_Maple
Maple