Interface to Mathematica#

The Mathematica interface will only work if Mathematica is installed on your computer with a command line interface that runs when you give the math command. The interface lets you send certain Sage objects to Mathematica, run Mathematica functions, import certain Mathematica expressions to Sage, or any combination of the above. The Sage command:

Sage

sage: print(mathematica._install_hints())
...

Python

>>> from sage.all import *
>>> print(mathematica._install_hints())
...

prints more information on Mathematica installation.

To send a Sage object sobj to Mathematica, call mathematica(sobj). This exports the Sage object to Mathematica and returns a new Sage object wrapping the Mathematica expression/variable, so that you can use the Mathematica variable from within Sage. You can then call Mathematica functions on the new object; for example:

Sage

sage: mobj = mathematica(x^2-1)             # optional - mathematica
sage: mobj.Factor()                         # optional - mathematica
(-1 + x)*(1 + x)

Python

>>> from sage.all import *
>>> mobj = mathematica(x**Integer(2)-Integer(1))             # optional - mathematica
>>> mobj.Factor()                         # optional - mathematica
(-1 + x)*(1 + x)

In the above example the factorization is done using Mathematica’s Factor[] function.

To see Mathematica’s output you can simply print the Mathematica wrapper object. However if you want to import Mathematica’s output back to Sage, call the Mathematica wrapper object’s sage() method. This method returns a native Sage object:

Sage

sage: # optional - mathematica
sage: mobj = mathematica(x^2-1)
sage: mobj2 = mobj.Factor(); mobj2
(-1 + x)*(1 + x)
sage: mobj2.parent()
Mathematica
sage: sobj = mobj2.sage(); sobj
(x + 1)*(x - 1)
sage: sobj.parent()
Symbolic Ring

Python

>>> from sage.all import *
>>> # optional - mathematica
>>> mobj = mathematica(x**Integer(2)-Integer(1))
>>> mobj2 = mobj.Factor(); mobj2
(-1 + x)*(1 + x)
>>> mobj2.parent()
Mathematica
>>> sobj = mobj2.sage(); sobj
(x + 1)*(x - 1)
>>> sobj.parent()
Symbolic Ring

If you want to run a Mathematica function and don’t already have the input in the form of a Sage object, then it might be simpler to input a string to mathematica(expr). This string will be evaluated as if you had typed it into Mathematica:

Sage

sage: mathematica('Factor[x^2-1]')          # optional - mathematica
(-1 + x)*(1 + x)
sage: mathematica('Range[3]')               # optional - mathematica
{1, 2, 3}

Python

>>> from sage.all import *
>>> mathematica('Factor[x^2-1]')          # optional - mathematica
(-1 + x)*(1 + x)
>>> mathematica('Range[3]')               # optional - mathematica
{1, 2, 3}

If you don’t want Sage to go to the trouble of creating a wrapper for the Mathematica expression, then you can call mathematica.eval(expr), which returns the result as a Mathematica AsciiArtString formatted string. If you want the result to be a string formatted like Mathematica’s InputForm, call repr(mobj) on the wrapper object mobj. If you want a string formatted in Sage style, call mobj._sage_repr():

Sage

sage: mathematica.eval('x^2 - 1')           # optional - mathematica
               2
         -1 + x
sage: repr(mathematica('Range[3]'))         # optional - mathematica
'{1, 2, 3}'
sage: mathematica('Range[3]')._sage_repr()  # optional - mathematica
'[1, 2, 3]'

Python

>>> from sage.all import *
>>> mathematica.eval('x^2 - 1')           # optional - mathematica
               2
         -1 + x
>>> repr(mathematica('Range[3]'))         # optional - mathematica
'{1, 2, 3}'
>>> mathematica('Range[3]')._sage_repr()  # optional - mathematica
'[1, 2, 3]'

Finally, if you just want to use a Mathematica command line from within Sage, the function mathematica_console() dumps you into an interactive command-line Mathematica session. This is an enhanced version of the usual Mathematica command-line, in that it provides readline editing and history (the usual one doesn’t!)

Long Input#

The Mathematica interface reads in even very long input (using files) in a robust manner.

Sage

sage: t = '"%s"'%10^10000   # ten thousand character string.
sage: a = mathematica(t)        # optional - mathematica
sage: a = mathematica.eval(t)   # optional - mathematica

Python

>>> from sage.all import *
>>> t = '"%s"'%Integer(10)**Integer(10000)   # ten thousand character string.
>>> a = mathematica(t)        # optional - mathematica
>>> a = mathematica.eval(t)   # optional - mathematica

Loading and saving#

Mathematica has an excellent InputForm function, which makes saving and loading Mathematica objects possible. The first examples test saving and loading to strings.

Sage

sage: # optional - mathematica
sage: x = mathematica(pi/2)
sage: print(x)
         Pi
         --
         2
sage: loads(dumps(x)) == x
True
sage: n = x.N(50)
sage: print(n)
              1.5707963267948966192313216916397514420985846996876
sage: loads(dumps(n)) == n
True

Python

>>> from sage.all import *
>>> # optional - mathematica
>>> x = mathematica(pi/Integer(2))
>>> print(x)
         Pi
         --
         2
>>> loads(dumps(x)) == x
True
>>> n = x.N(Integer(50))
>>> print(n)
              1.5707963267948966192313216916397514420985846996876
>>> loads(dumps(n)) == n
True

Complicated translations#

The mobj.sage() method tries to convert a Mathematica object to a Sage object. In many cases, it will just work. In particular, it should be able to convert expressions entirely consisting of:

numbers, i.e. integers, floats, complex numbers;
functions and named constants also present in Sage, where:
- Sage knows how to translate the function or constant’s name from Mathematica’s, or
- the Sage name for the function or constant is trivially related to Mathematica’s;
symbolic variables whose names don’t pathologically overlap with objects already defined in Sage.

This method will not work when Mathematica’s output includes:

strings;
functions unknown to Sage;
Mathematica functions with different parameters/parameter order to the Sage equivalent.

If you want to convert more complicated Mathematica expressions, you can instead call mobj._sage_() and supply a translation dictionary:

Sage

sage: m = mathematica('NewFn[x]')       # optional - mathematica
sage: m._sage_(locals={('NewFn', 1): sin})   # optional - mathematica
sin(x)

Python

>>> from sage.all import *
>>> m = mathematica('NewFn[x]')       # optional - mathematica
>>> m._sage_(locals={('NewFn', Integer(1)): sin})   # optional - mathematica
sin(x)

For more details, see the documentation for ._sage_().

OTHER Examples:

Sage

sage: def math_bessel_K(nu,x):
....:     return mathematica(nu).BesselK(x).N(20)
sage: math_bessel_K(2,I)                      # optional - mathematica
-2.59288617549119697817 + 0.18048997206696202663*I

Python

>>> from sage.all import *
>>> def math_bessel_K(nu,x):
...     return mathematica(nu).BesselK(x).N(Integer(20))
>>> math_bessel_K(Integer(2),I)                      # optional - mathematica
-2.59288617549119697817 + 0.18048997206696202663*I

Sage

sage: slist = [[1, 2], 3., 4 + I]
sage: mlist = mathematica(slist); mlist     # optional - mathematica
{{1, 2}, 3., 4 + I}
sage: slist2 = list(mlist); slist2          # optional - mathematica
[{1, 2}, 3., 4 + I]
sage: slist2[0]                             # optional - mathematica
{1, 2}
sage: slist2[0].parent()                    # optional - mathematica
Mathematica
sage: slist3 = mlist.sage(); slist3         # optional - mathematica
[[1, 2], 3.00000000000000, I + 4]

Python

>>> from sage.all import *
>>> slist = [[Integer(1), Integer(2)], RealNumber('3.'), Integer(4) + I]
>>> mlist = mathematica(slist); mlist     # optional - mathematica
{{1, 2}, 3., 4 + I}
>>> slist2 = list(mlist); slist2          # optional - mathematica
[{1, 2}, 3., 4 + I]
>>> slist2[Integer(0)]                             # optional - mathematica
{1, 2}
>>> slist2[Integer(0)].parent()                    # optional - mathematica
Mathematica
>>> slist3 = mlist.sage(); slist3         # optional - mathematica
[[1, 2], 3.00000000000000, I + 4]

Sage

sage: mathematica('10.^80')     # optional - mathematica
1.*^80
sage: mathematica('10.^80').sage()  # optional - mathematica
1.00000000000000e80

Python

>>> from sage.all import *
>>> mathematica('10.^80')     # optional - mathematica
1.*^80
>>> mathematica('10.^80').sage()  # optional - mathematica
1.00000000000000e80

AUTHORS:

William Stein (2005): first version
Doug Cutrell (2006-03-01): Instructions for use under Cygwin/Windows.
Felix Lawrence (2009-08-21): Added support for importing Mathematica lists and floats with exponents.

class sage.interfaces.mathematica.Mathematica(maxread=None, script_subdirectory=None, logfile=None, server=None, server_tmpdir=None, command=None, verbose_start=False)[source]#

Bases: ExtraTabCompletion, Expect

Interface to the Mathematica interpreter.

chdir(dir)[source]#

Change Mathematica’s current working directory.

EXAMPLES:

Sage

sage: mathematica.chdir('/')          # optional - mathematica
sage: mathematica('Directory[]')      # optional - mathematica
"/"

Python

>>> from sage.all import *
>>> mathematica.chdir('/')          # optional - mathematica
>>> mathematica('Directory[]')      # optional - mathematica
"/"

console(readline=True)[source]#

eval(code, strip=True, **kwds)[source]#

get(var, ascii_art=False)[source]#

Get the value of the variable var.

AUTHORS:

William Stein
Kiran Kedlaya (2006-02-04): suggested using InputForm

help(cmd)[source]#

set(var, value)[source]#: Set the variable var to the given value.

class sage.interfaces.mathematica.MathematicaElement(parent, value, is_name=False, name=None)[source]#

Bases: ExpectElement

n(*args, **kwargs)[source]#

Numerical approximation by converting to Sage object first

Convert the object into a Sage object and return its numerical approximation. See documentation of the function sage.misc.functional.n() for details.

EXAMPLES:

Sage

sage: mathematica('Pi').n(10)    # optional -- mathematica
3.1
sage: mathematica('Pi').n()      # optional -- mathematica
3.14159265358979
sage: mathematica('Pi').n(digits=10)   # optional -- mathematica
3.141592654

Python

>>> from sage.all import *
>>> mathematica('Pi').n(Integer(10))    # optional -- mathematica
3.1
>>> mathematica('Pi').n()      # optional -- mathematica
3.14159265358979
>>> mathematica('Pi').n(digits=Integer(10))   # optional -- mathematica
3.141592654

save_image(filename, ImageSize=600)[source]#

Save a mathematica graphics

INPUT:

filename – string. The filename to save as. The extension determines the image file format.
ImageSize – integer. The size of the resulting image.

EXAMPLES:

Sage

sage: P = mathematica('Plot[Sin[x],{x,-2Pi,4Pi}]')   # optional - mathematica
sage: filename = tmp_filename()                      # optional - mathematica
sage: P.save_image(filename, ImageSize=800)                # optional - mathematica

Python

>>> from sage.all import *
>>> P = mathematica('Plot[Sin[x],{x,-2Pi,4Pi}]')   # optional - mathematica
>>> filename = tmp_filename()                      # optional - mathematica
>>> P.save_image(filename, ImageSize=Integer(800))                # optional - mathematica

show(ImageSize=600)[source]#

Show a mathematica expression immediately.

This method attempts to display the graphics immediately, without waiting for the currently running code (if any) to return to the command line. Be careful, calling it from within a loop will potentially launch a large number of external viewer programs.

INPUT:

ImageSize – integer. The size of the resulting image.

OUTPUT:

This method does not return anything. Use save() if you want to save the figure as an image.

EXAMPLES:

Sage

sage: Q = mathematica('Sin[x Cos[y]]/Sqrt[1-x^2]')   # optional - mathematica
sage: show(Q)                                        # optional - mathematica
Sin[x*Cos[y]]/Sqrt[1 - x^2]

Python

>>> from sage.all import *
>>> Q = mathematica('Sin[x Cos[y]]/Sqrt[1-x^2]')   # optional - mathematica
>>> show(Q)                                        # optional - mathematica
Sin[x*Cos[y]]/Sqrt[1 - x^2]

The following example starts a Mathematica frontend to do the rendering (Issue #28819):

Sage

sage: P = mathematica('Plot[Sin[x],{x,-2Pi,4Pi}]')   # optional - mathematica
sage: show(P)                                        # optional - mathematica mathematicafrontend
sage: P.show(ImageSize=800)                          # optional - mathematica mathematicafrontend

Python

>>> from sage.all import *
>>> P = mathematica('Plot[Sin[x],{x,-2Pi,4Pi}]')   # optional - mathematica
>>> show(P)                                        # optional - mathematica mathematicafrontend
>>> P.show(ImageSize=Integer(800))                          # optional - mathematica mathematicafrontend

str()[source]#

class sage.interfaces.mathematica.MathematicaFunction(parent, name)[source]#: Bases: ExpectFunction

class sage.interfaces.mathematica.MathematicaFunctionElement(obj, name)[source]#: Bases: FunctionElement

sage.interfaces.mathematica.clean_output(s)[source]#

sage.interfaces.mathematica.mathematica_console(readline=True)[source]#

sage.interfaces.mathematica.parse_moutput_from_json(page_data, verbose=False)[source]#

Return the list of outputs found in the json (with key 'moutput')

INPUT:

page_data – json obtained from Wolfram Alpha
verbose – bool (default: False)

OUTPUT:

list of unicode strings

EXAMPLES:

Sage

sage: from sage.interfaces.mathematica import request_wolfram_alpha
sage: from sage.interfaces.mathematica import parse_moutput_from_json
sage: page_data = request_wolfram_alpha('integrate Sin[x]') # optional internet
sage: parse_moutput_from_json(page_data)                    # optional internet
['-Cos[x]']

Python

>>> from sage.all import *
>>> from sage.interfaces.mathematica import request_wolfram_alpha
>>> from sage.interfaces.mathematica import parse_moutput_from_json
>>> page_data = request_wolfram_alpha('integrate Sin[x]') # optional internet
>>> parse_moutput_from_json(page_data)                    # optional internet
['-Cos[x]']

Sage

sage: page_data = request_wolfram_alpha('Sin[x]')           # optional internet
sage: L = parse_moutput_from_json(page_data)                # optional internet
sage: sorted(L)                                             # optional internet
['-Cos[x]', '{x == 0}', '{x == Pi C[1], Element[C[1], Integers]}']

Python

>>> from sage.all import *
>>> page_data = request_wolfram_alpha('Sin[x]')           # optional internet
>>> L = parse_moutput_from_json(page_data)                # optional internet
>>> sorted(L)                                             # optional internet
['-Cos[x]', '{x == 0}', '{x == Pi C[1], Element[C[1], Integers]}']

sage.interfaces.mathematica.reduce_load(X)[source]#

sage.interfaces.mathematica.request_wolfram_alpha(input, verbose=False)[source]#

Request Wolfram Alpha website.

INPUT:

input – string
verbose – bool (default: False)

OUTPUT:

json

EXAMPLES:

Sage

sage: from sage.interfaces.mathematica import request_wolfram_alpha
sage: page_data = request_wolfram_alpha('integrate Sin[x]')      # optional internet
sage: [str(a) for a in sorted(page_data.keys())]                 # optional internet
['queryresult']
sage: [str(a) for a in sorted(page_data['queryresult'].keys())]  # optional internet
['datatypes',
 'encryptedEvaluatedExpression',
 'encryptedParsedExpression',
 'error',
 'host',
 'id',
 'inputstring',
 'numpods',
 'parsetimedout',
 'parsetiming',
 'pods',
 'recalculate',
 'related',
 'server',
 'sponsorCategories',
 'success',
 'timedout',
 'timedoutpods',
 'timing',
 'version']

Python

>>> from sage.all import *
>>> from sage.interfaces.mathematica import request_wolfram_alpha
>>> page_data = request_wolfram_alpha('integrate Sin[x]')      # optional internet
>>> [str(a) for a in sorted(page_data.keys())]                 # optional internet
['queryresult']
>>> [str(a) for a in sorted(page_data['queryresult'].keys())]  # optional internet
['datatypes',
 'encryptedEvaluatedExpression',
 'encryptedParsedExpression',
 'error',
 'host',
 'id',
 'inputstring',
 'numpods',
 'parsetimedout',
 'parsetiming',
 'pods',
 'recalculate',
 'related',
 'server',
 'sponsorCategories',
 'success',
 'timedout',
 'timedoutpods',
 'timing',
 'version']

sage.interfaces.mathematica.symbolic_expression_from_mathematica_string(mexpr)[source]#

Translate a mathematica string into a symbolic expression

INPUT:

mexpr – string

OUTPUT:

symbolic expression

EXAMPLES:

Sage

sage: from sage.interfaces.mathematica import symbolic_expression_from_mathematica_string
sage: symbolic_expression_from_mathematica_string('-Cos[x]')
-cos(x)

Python

>>> from sage.all import *
>>> from sage.interfaces.mathematica import symbolic_expression_from_mathematica_string
>>> symbolic_expression_from_mathematica_string('-Cos[x]')
-cos(x)

Interface to Mathematica#

Tutorial#

Syntax#

Some typical input#

Assignments and definitions#

Function definitions#

Numerical Calculations#

Polynomial and Integer Factorization#

Long Input#

Loading and saving#

Complicated translations#