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: 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: mobj = mathematica(x^2-1)             # optional - mathematica
sage: 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: # 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

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: mathematica('Factor[x^2-1]')          # optional - mathematica
(-1 + x)*(1 + x)
sage: 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: 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]'

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!)

Tutorial#

We follow some of the tutorial from http://library.wolfram.com/conferences/devconf99/withoff/Basic1.html/.

For any of this to work you must buy and install the Mathematica program, and it must be available as the command math in your PATH.

Syntax#

Now make 1 and add it to itself. The result is a Mathematica object.

sage: m = mathematica
sage: a = m(1) + m(1); a                # optional - mathematica
2
sage: a.parent()                        # optional - mathematica
Mathematica
sage: m('1+1')                          # optional - mathematica
2
sage: m(3)**m(50)                       # optional - mathematica
717897987691852588770249

The following is equivalent to Plus[2, 3] in Mathematica:

sage: m = mathematica
sage: m(2).Plus(m(3))                   # optional - mathematica
5

We can also compute \(7(2+3)\).

sage: m(7).Times(m(2).Plus(m(3)))       # optional - mathematica
35
sage: m('7(2+3)')                       # optional - mathematica
35

Some typical input#

We solve an equation and a system of two equations:

sage: # optional - mathematica
sage: eqn = mathematica('3x + 5 == 14')
sage: eqn
5 + 3*x == 14
sage: eqn.Solve('x')
{{x -> 3}}
sage: sys = mathematica('{x^2 - 3y == 3, 2x - y == 1}')
sage: print(sys)
           2
         {x  - 3 y == 3, 2 x - y == 1}
sage: sys.Solve('{x, y}')
{{x -> 0, y -> -1}, {x -> 6, y -> 11}}

Assignments and definitions#

If you assign the mathematica \(5\) to a variable \(c\) in Sage, this does not affect the \(c\) in Mathematica.

sage: c = m(5)                          # optional - mathematica
sage: print(m('b + c x'))               # optional - mathematica
             b + c x
sage: print(m('b') + c*m('x'))          # optional - mathematica
         b + 5 x

The Sage interfaces changes Sage lists into Mathematica lists:

sage: m = mathematica
sage: eq1 = m('x^2 - 3y == 3')          # optional - mathematica
sage: eq2 = m('2x - y == 1')            # optional - mathematica
sage: v = m([eq1, eq2]); v              # optional - mathematica
{x^2 - 3*y == 3, 2*x - y == 1}
sage: v.Solve(['x', 'y'])               # optional - mathematica
{{x -> 0, y -> -1}, {x -> 6, y -> 11}}

Function definitions#

Define mathematica functions by simply sending the definition to the interpreter.

sage: m = mathematica
sage: _ = mathematica('f[p_] = p^2');   # optional - mathematica
sage: m('f[9]')                         # optional - mathematica
81

Numerical Calculations#

We find the \(x\) such that \(e^x - 3x = 0\).

sage: eqn = mathematica('Exp[x] - 3x == 0') # optional - mathematica
sage: eqn.FindRoot(['x', 2])                # optional - mathematica
{x -> 1.512134551657842}

Note that this agrees with what the PARI interpreter gp produces:

sage: gp('solve(x=1,2,exp(x)-3*x)')
1.512134551657842473896739678              # 32-bit
1.5121345516578424738967396780720387046    # 64-bit

Next we find the minimum of a polynomial using the two different ways of accessing Mathematica:

sage: mathematica('FindMinimum[x^3 - 6x^2 + 11x - 5, {x,3}]')  # optional - mathematica
{0.6150998205402516, {x -> 2.5773502699629733}}
sage: f = mathematica('x^3 - 6x^2 + 11x - 5')  # optional - mathematica
sage: f.FindMinimum(['x', 3])                  # optional - mathematica
{0.6150998205402516, {x -> 2.5773502699629733}}

Polynomial and Integer Factorization#

We factor a polynomial of degree 200 over the integers.

sage: R.<x> = PolynomialRing(ZZ)
sage: f = (x**100+17*x+5)*(x**100-5*x+20)
sage: f
x^200 + 12*x^101 + 25*x^100 - 85*x^2 + 315*x + 100
sage: g = mathematica(str(f))            # optional - mathematica
sage: print(g)                           # optional - mathematica
                           2       100       101    200
         100 + 315 x - 85 x  + 25 x    + 12 x    + x
sage: g                                  # optional - mathematica
100 + 315*x - 85*x^2 + 25*x^100 + 12*x^101 + x^200
sage: print(g.Factor())                  # optional - mathematica
                      100               100
         (20 - 5 x + x   ) (5 + 17 x + x   )

We can also factor a multivariate polynomial:

sage: f = mathematica('x^6 + (-y - 2)*x^5 + (y^3 + 2*y)*x^4 - y^4*x^3')  # optional - mathematica
sage: print(f.Factor())                  # optional - mathematica
          3                  2    3
         x  (x - y) (-2 x + x  + y )

We factor an integer:

sage: # optional - mathematica
sage: n = mathematica(2434500)
sage: n.FactorInteger()
{{2, 2}, {3, 2}, {5, 3}, {541, 1}}
sage: n = mathematica(2434500)
sage: F = n.FactorInteger(); F
{{2, 2}, {3, 2}, {5, 3}, {541, 1}}
sage: F[1]
{2, 2}
sage: F[4]
{541, 1}

Mathematica’s ECM package is no longer available.

Long Input#

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

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

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

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

OTHER Examples:

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

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]

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

Bases: ExtraTabCompletion, Expect

Interface to the Mathematica interpreter.

chdir(dir)#

Change Mathematica’s current working directory.

EXAMPLES:

sage: mathematica.chdir('/')          # optional - mathematica
sage: mathematica('Directory[]')      # optional - mathematica
"/"
console(readline=True)#
eval(code, strip=True, **kwds)#
get(var, ascii_art=False)#

Get the value of the variable var.

AUTHORS:

  • William Stein

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

help(cmd)#
set(var, value)#

Set the variable var to the given value.

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

Bases: ExpectElement

n(*args, **kwargs)#

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: 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
save_image(filename, ImageSize=600)#

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: 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
show(ImageSize=600)#

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: 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]

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

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
str()#
class sage.interfaces.mathematica.MathematicaFunction(parent, name)#

Bases: ExpectFunction

class sage.interfaces.mathematica.MathematicaFunctionElement(obj, name)#

Bases: FunctionElement

sage.interfaces.mathematica.clean_output(s)#
sage.interfaces.mathematica.mathematica_console(readline=True)#
sage.interfaces.mathematica.parse_moutput_from_json(page_data, verbose=False)#

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: 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]']

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]}']
sage.interfaces.mathematica.reduce_load(X)#
sage.interfaces.mathematica.request_wolfram_alpha(input, verbose=False)#

Request Wolfram Alpha website.

INPUT:

  • input – string

  • verbose – bool (default: False)

OUTPUT:

json

EXAMPLES:

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']
sage.interfaces.mathematica.symbolic_expression_from_mathematica_string(mexpr)#

Translate a mathematica string into a symbolic expression

INPUT:

  • mexpr – string

OUTPUT:

symbolic expression

EXAMPLES:

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