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

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

Python

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

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

   Python

   >>> from sage.all import *
   >>> 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

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

   Python

   >>> from sage.all import *
   >>> 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

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

   Python

   >>> from sage.all import *
   >>> maple('(x^12-1)/(x-1)').simplify()     # optional - maple
   (x+1)*(x^2+1)*(x^2+x+1)*(x^2-x+1)*(x^4-x^2+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

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)

Python

>>> from sage.all import *
>>> 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

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

Python

>>> from sage.all import *
>>> maple('(x^28-1)').factor( )           # optional - maple
(x-1)*(x^6+x^5+x^4+x^3+x^2+x+1)*(x+1)*(x^6-x^5+x^4-x^3+x^2-x+1)*(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

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

Python

>>> from sage.all import *
>>> maple.console()          # not tested
>>> !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

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

Python

>>> from sage.all import *
>>> maple.help('fibonacci')  # not tested, since it uses a pager

We see there are two choices. Type

Sage

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

Python

>>> from sage.all import *
>>> 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

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

Python

>>> from sage.all import *
>>> maple.fibonacci(Integer(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

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

Python

>>> from sage.all import *
>>> maple('combinat[fibonacci]')(Integer(10))     # optional - maple
55

instead.

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

Sage

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

Python

>>> from sage.all import *
>>> maple.with_package('combinat')       # optional - maple

or

Sage

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

Python

>>> from sage.all import *
>>> maple.load('combinat')               # optional - maple

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

Sage

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

Python

>>> from sage.all import *
>>> maple.fibonacci(Integer(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

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

Python

>>> from sage.all import *
>>> 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

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

Python

>>> from sage.all import *
>>> maple.isprime(maple.fibonacci(Integer(27)))     # optional - maple
false
>>> maple.ifactor(maple.fibonacci(Integer(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

sage: # optional - maple
sage: alpha = maple('(1+sqrt(5))/2')
sage: beta = maple('(1-sqrt(5))/2')
sage: f19  = alpha^19 - beta^19/maple('sqrt(5)')
sage: f19
(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
6765+5778/5*5^(1/2)

Python

>>> from sage.all import *
>>> # optional - maple
>>> alpha = maple('(1+sqrt(5))/2')
>>> beta = maple('(1-sqrt(5))/2')
>>> f19  = alpha**Integer(19) - beta**Integer(19)/maple('sqrt(5)')
>>> f19
(1/2+1/2*5^(1/2))^19-1/5*(1/2-1/2*5^(1/2))^19*5^(1/2)
>>> f19.simplify()          # somewhat randomly ordered output
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

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

Python

>>> from sage.all import *
>>> mysqcu = maple('proc(x) if x > 0 then x^2 else x^3 fi end')    # optional - maple
>>> mysqcu(Integer(5))                                                      # optional - maple
25
>>> mysqcu(-Integer(5))                                                     # optional - maple
-125

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

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

Bases: ExtraTabCompletion, Expect

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

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

Python

>>> from sage.all import *
>>> # optional - maple
>>> maple.set('xx', '2')
>>> maple.get('xx')
'2'
>>> maple.clear('xx')
>>> maple.get('xx')
'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

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

Python

>>> from sage.all import *
>>> c = maple.completions('di')  # optional - maple
>>> 'divide' in c                # optional - maple
True

console()

Spawn a new Maple command-line session.

EXAMPLES:

Sage

sage: maple.console()  # not tested
    |\^/|     Maple 2019 (X86 64 LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2019
 \  MAPLE  /  All rights reserved. Maple is a trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.

Python

>>> from sage.all import *
>>> maple.console()  # not tested
    |\^/|     Maple 2019 (X86 64 LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2019
 \  MAPLE  /  All rights reserved. Maple is a trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.

cputime(t=None)

Return 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

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

Python

>>> from sage.all import *
>>> # optional - maple
>>> t = maple.cputime()
>>> t                   # random
0.02
>>> x = maple('x')
>>> maple.diff(x**Integer(2), x)
2*x
>>> maple.cputime(t)    # random
0.0

expect()

Return the pexpect object for this Maple session.

EXAMPLES:

Sage

sage: # optional - maple
sage: m = Maple()
sage: m.expect() is None
True
sage: m._start()
sage: m.expect()
Maple with PID ...
sage: m.quit()

Python

>>> from sage.all import *
>>> # optional - maple
>>> m = Maple()
>>> m.expect() is None
True
>>> m._start()
>>> m.expect()
Maple with PID ...
>>> m.quit()

get(var)

Get the value of the variable var.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> maple.set('xx', '2') # optional - maple
>>> maple.get('xx')      # optional - maple
'2'

help(string)

Display Maple help about string.

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

INPUT:

• string – a string to search for in the maple help system

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> maple.help('Psi')  # not tested
Psi - the Digamma and Polygamma functions
...

load(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

sage: # optional - maple
sage: maple.quit()   # reset maple
sage: maple('partition(10)')
partition(10)
sage: maple('bell(10)')
bell(10)
sage: maple.with_package('combinat')
sage: maple('partition(10)')
[[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)')
115975
sage: maple('fibonacci(10)')
55

Python

>>> from sage.all import *
>>> # optional - maple
>>> maple.quit()   # reset maple
>>> maple('partition(10)')
partition(10)
>>> maple('bell(10)')
bell(10)
>>> maple.with_package('combinat')
>>> maple('partition(10)')
[[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]]
>>> maple('bell(10)')
115975
>>> maple('fibonacci(10)')
55

set(var, value)

Set the variable var to the given value.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> maple.set('xx', '2') # optional - maple
>>> 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

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

Python

>>> from sage.all import *
>>> maple.source('curry')  # not tested
... -> subs('_X' = _passed[2 .. _npassed],() -> ...(_X, _passed))

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

sage: # optional - maple
sage: maple.quit()   # reset maple
sage: maple('partition(10)')
partition(10)
sage: maple('bell(10)')
bell(10)
sage: maple.with_package('combinat')
sage: maple('partition(10)')
[[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)')
115975
sage: maple('fibonacci(10)')
55

Python

>>> from sage.all import *
>>> # optional - maple
>>> maple.quit()   # reset maple
>>> maple('partition(10)')
partition(10)
>>> maple('bell(10)')
bell(10)
>>> maple.with_package('combinat')
>>> maple('partition(10)')
[[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]]
>>> maple('bell(10)')
115975
>>> maple('fibonacci(10)')
55

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

Bases: ExtraTabCompletion, ExpectElement

op(i=None)

Return the i-th operand of this expression.

INPUT:

• i – an integer or None

EXAMPLES:

Sage

sage: V = maple(vector(QQ,[4,5,6]))      # optional - maple
sage: V.op(1)                            # optional - maple
3
sage: V.op(2)                            # optional - maple
{1 = 4, 2 = 5, 3 = 6}

Python

>>> from sage.all import *
>>> V = maple(vector(QQ,[Integer(4),Integer(5),Integer(6)]))      # optional - maple
>>> V.op(Integer(1))                            # optional - maple
3
>>> V.op(Integer(2))                            # optional - maple
{1 = 4, 2 = 5, 3 = 6}

class sage.interfaces.maple.MapleFunction(parent, name)

Bases: ExpectFunction

class sage.interfaces.maple.MapleFunctionElement(obj, name)

Bases: FunctionElement

sage.interfaces.maple.maple_console()

Spawn a new Maple command-line session.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> maple_console() #not tested
    |^/|     Maple 11 (IBM INTEL LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2007
 \  MAPLE  /  All rights reserved. Maple is a trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.
>

sage.interfaces.maple.reduce_load_Maple()

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

EXAMPLES:

Sage

sage: from sage.interfaces.maple import reduce_load_Maple
sage: reduce_load_Maple()
Maple

Python

>>> from sage.all import *
>>> from sage.interfaces.maple import reduce_load_Maple
>>> reduce_load_Maple()
Maple