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.

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: mobj = mathematica(x^2-1)             # optional - mathematica
sage: mobj2 = mobj.Factor(); mobj2          # optional - mathematica
(-1 + x)*(1 + x)
sage: mobj2.parent()                        # optional - mathematica
Mathematica
sage: sobj = mobj2.sage(); sobj             # optional - mathematica
(x - 1)*(x + 1)
sage: sobj.parent()                         # optional - mathematica
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: eqn = mathematica('3x + 5 == 14') # optional - mathematica
sage: eqn                               # optional - mathematica
5 + 3*x == 14
sage: eqn.Solve('x')                    # optional - mathematica
{{x -> 3}}
sage: sys = mathematica('{x^2 - 3y == 3, 2x - y == 1}')  # optional - mathematica
sage: print sys                         # optional - mathematica
           2
         {x  - 3 y == 3, 2 x - y == 1}
sage: sys.Solve('{x, y}')               # optional - mathematica
{{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: e = mathematica('Exp[x] - 3x == 0') # optional - mathematica
sage: e.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: n = mathematica(2434500)           # optional - mathematica
sage: n.FactorInteger()                  # optional - mathematica
{{2, 2}, {3, 2}, {5, 3}, {541, 1}}
sage: n = mathematica(2434500)           # optional - mathematica
sage: F = n.FactorInteger(); F           # optional - mathematica
{{2, 2}, {3, 2}, {5, 3}, {541, 1}}
sage: F[1]                               # optional - mathematica
{2, 2}
sage: F[4]                               # optional - mathematica
{541, 1}

We can also load the ECM package and factoring using it:

sage: _ = mathematica.eval("<<NumberTheory`FactorIntegerECM`");  # optional - mathematica
sage: mathematica.FactorIntegerECM('932901*939321')              # optional - mathematica
8396109

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: x = mathematica(pi/2)     # optional - mathematica
sage: print x                   # optional - mathematica
         Pi
         --
         2
sage: loads(dumps(x)) == x      # optional - mathematica
True
sage: n = x.N(50)               # optional - mathematica
sage: print n                   # optional - mathematica
              1.5707963267948966192313216916397514420985846996876
sage: loads(dumps(n)) == n      # optional - mathematica
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': 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
0.180489972066962*I - 2.592886175491197             # 32-bit
-2.59288617549119697817 + 0.18048997206696202663*I  # 64-bit
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], 3.0, I + 4]
sage: mathematica('10.^80')     # optional - mathematica
1.*^80
sage: mathematica('10.^80').sage()  # optional - mathematica
1e+80

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=100, script_subdirectory=None, logfile=None, server=None, server_tmpdir=None)

Bases: sage.interfaces.expect.Expect

Interface to the Mathematica interpreter.

chdir(dir)

Change Mathematica’s current working directory.

EXAMPLES:

sage: mathematica.chdir('/')          # optional
sage: mathematica('Directory[]')      # optional
"/"
console(readline=True)

x.__init__(...) initializes x; see help(type(x)) for signature

eval(code, strip=True, **kwds)

x.__init__(...) initializes x; see help(type(x)) for signature

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)

x.__init__(...) initializes x; see help(type(x)) for signature

set(var, value)

Set the variable var to the given value.

trait_names()

x.__init__(...) initializes x; see help(type(x)) for signature

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

Bases: sage.interfaces.expect.ExpectElement

N(*args)

EXAMPLES:

sage: mathematica('Pi').N(10)    # optional -- mathematica
3.1415926536
sage: mathematica('Pi').N(50)    # optional -- mathematica
3.14159265358979323846264338327950288419716939937511
show(filename=None, ImageSize=600)

Show a mathematica expression or plot in the Sage notebook.

EXAMPLES:

sage: P = mathematica('Plot[Sin[x],{x,-2Pi,4Pi}]')   # optional - mathematica
sage: show(P)                                        # optional - mathematica
sage: P.show(ImageSize=800)                          # optional - mathematica
sage: Q = mathematica('Sin[x Cos[y]]/Sqrt[1-x^2]')   # optional - mathematica
sage: show(Q)                                        # optional - mathematica
<html><div class="math">\frac{\sin (x \cos (y))}{\sqrt{1-x^2}}</div></html>
str()

x.__init__(...) initializes x; see help(type(x)) for signature

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

Bases: sage.interfaces.expect.ExpectFunction

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

Bases: sage.interfaces.expect.FunctionElement

sage.interfaces.mathematica.clean_output(s)

x.__init__(...) initializes x; see help(type(x)) for signature

sage.interfaces.mathematica.mathematica_console(readline=True)

x.__init__(...) initializes x; see help(type(x)) for signature

sage.interfaces.mathematica.reduce_load(X)

x.__init__(...) initializes x; see help(type(x)) for signature