Interface to Mathics#

Mathics is an open source interpreter for the Wolfram Language. From the introduction of its reference manual:

Note

Mathics — to be pronounced like “Mathematics” without the “emat” — is a general-purpose computer algebra system (CAS). It is meant to be a free, light-weight alternative to Mathematica®. It is free both as in “free beer” and as in “freedom”. There are various online mirrors running Mathics but it is also possible to run Mathics locally. A list of mirrors can be found at the Mathics homepage, http://mathics.github.io.

The programming language of Mathics is meant to resemble Wolfram’s famous Mathematica® as much as possible. However, Mathics is in no way affiliated or supported by Wolfram. Mathics will probably never have the power to compete with Mathematica® in industrial applications; yet, it might be an interesting alternative for educational purposes.

The Mathics interface will only work if the optional Sage package Mathics is installed. The interface lets you send certain Sage objects to Mathics, run Mathics functions, import certain Mathics expressions to Sage, or any combination of the above.

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

Sage

sage: from sage.interfaces.mathics import mathics
sage: mobj = mathics(x^2-1); mobj       # optional - mathics
-1 + x ^ 2
sage: mobj.Factor()                     # optional - mathics
(-1 + x) (1 + x)

Python

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

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

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

Sage

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

Python

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

If you want to run a Mathics 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 mathics(expr). This string will be evaluated as if you had typed it into Mathics:

Sage

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

Python

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

If you want work with the internal Mathics expression, then you can call mathics.eval(expr), which returns an instance of mathics.core.expression.Expression. If you want the result to be a string formatted like Mathics’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: mathics.eval('x^2 - 1')           # optional - mathics
'-1 + x ^ 2'
sage: repr(mathics('Range[3]'))         # optional - mathics
'{1, 2, 3}'
sage: mathics('Range[3]')._sage_repr()  # optional - mathics
'[1, 2, 3]'

Python

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

Finally, if you just want to use a Mathics command line from within Sage, the function mathics_console() dumps you into an interactive command-line Mathics session.

Long Input#

The Mathics 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 = mathics(t)        # optional - mathics
sage: a = mathics.eval(t)   # optional - mathics

Python

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

Loading and saving#

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

Sage

sage: # optional - mathics
sage: x = mathics(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 - mathics
>>> x = mathics(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 Mathics 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 Mathics’s, or
- the Sage name for the function or constant is trivially related to Mathics’s;
symbolic variables whose names don’t pathologically overlap with objects already defined in Sage.

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

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

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

Sage

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

Python

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

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

OTHER Examples:

Sage

sage: def math_bessel_K(nu,x):
....:     return mathics(nu).BesselK(x).N(20)
sage: math_bessel_K(2,I)                      # optional - mathics
-2.5928861754911969782 + 0.18048997206696202663 I

Python

>>> from sage.all import *
>>> def math_bessel_K(nu,x):
...     return mathics(nu).BesselK(x).N(Integer(20))
>>> math_bessel_K(Integer(2),I)                      # optional - mathics
-2.5928861754911969782 + 0.18048997206696202663 I

Sage

sage: slist = [[1, 2], 3., 4 + I]
sage: mlist = mathics(slist); mlist         # optional - mathics
{{1, 2}, 3., 4 + I}
sage: slist2 = list(mlist); slist2          # optional - mathics
[{1, 2}, 3., 4 + I]
sage: slist2[0]                             # optional - mathics
{1, 2}
sage: slist2[0].parent()                    # optional - mathics
Mathics
sage: slist3 = mlist.sage(); slist3         # optional - mathics
[[1, 2], 3.00000000000000, 4.00000000000000 + 1.00000000000000*I]

Python

>>> from sage.all import *
>>> slist = [[Integer(1), Integer(2)], RealNumber('3.'), Integer(4) + I]
>>> mlist = mathics(slist); mlist         # optional - mathics
{{1, 2}, 3., 4 + I}
>>> slist2 = list(mlist); slist2          # optional - mathics
[{1, 2}, 3., 4 + I]
>>> slist2[Integer(0)]                             # optional - mathics
{1, 2}
>>> slist2[Integer(0)].parent()                    # optional - mathics
Mathics
>>> slist3 = mlist.sage(); slist3         # optional - mathics
[[1, 2], 3.00000000000000, 4.00000000000000 + 1.00000000000000*I]

Sage

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

Python

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

AUTHORS:

Sebastian Oehms (2021): first version from a copy of the Mathematica interface (see Issue #31778).

Thanks to Rocky Bernstein and Juan Mauricio Matera for their support. For further acknowledgments see this list.

class sage.interfaces.mathics.Mathics(maxread=None, logfile=None, init_list_length=1024, seed=None)[source]#

Bases: Interface

Interface to the Mathics interpreter.

Implemented according to the Mathematica interface but avoiding Pexpect functionality.

EXAMPLES:

Sage

sage: # optional - mathics
sage: t = mathics('Tan[I + 0.5]')
sage: t.parent()
Mathics
sage: ts = t.sage()
sage: ts.parent()
Complex Field with 53 bits of precision
sage: t == mathics(ts)
True
sage: mtan = mathics.Tan
sage: mt = mtan(I+1/2)
sage: mt == t
True
sage: u = mathics(I+1/2)
sage: u.Tan() == mt
True

Python

>>> from sage.all import *
>>> # optional - mathics
>>> t = mathics('Tan[I + 0.5]')
>>> t.parent()
Mathics
>>> ts = t.sage()
>>> ts.parent()
Complex Field with 53 bits of precision
>>> t == mathics(ts)
True
>>> mtan = mathics.Tan
>>> mt = mtan(I+Integer(1)/Integer(2))
>>> mt == t
True
>>> u = mathics(I+Integer(1)/Integer(2))
>>> u.Tan() == mt
True

More examples can be found in the module header.

chdir(dir)[source]#

Change Mathics’s current working directory.

EXAMPLES:

Sage

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

Python

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

console()[source]#

Spawn a new Mathics command-line session.

EXAMPLES:

Sage

sage: mathics.console()  # not tested

Mathics 2.1.1.dev0
on CPython 3.9.2 (default, Mar 19 2021, 22:23:28)
using SymPy 1.7, mpmath 1.2.1, numpy 1.19.5, cython 0.29.21

Copyright (C) 2011-2021 The Mathics Team.
This program comes with ABSOLUTELY NO WARRANTY.
This is free software, and you are welcome to redistribute it
under certain conditions.
See the documentation for the full license.

Quit by evaluating Quit[] or by pressing CONTROL-D.

In[1]:= Sin[0.5]
Out[1]= 0.479426

Goodbye!

sage:

Python

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

Mathics 2.1.1.dev0
on CPython 3.9.2 (default, Mar 19 2021, 22:23:28)
using SymPy 1.7, mpmath 1.2.1, numpy 1.19.5, cython 0.29.21

Copyright (C) 2011-2021 The Mathics Team.
This program comes with ABSOLUTELY NO WARRANTY.
This is free software, and you are welcome to redistribute it
under certain conditions.
See the documentation for the full license.

Quit by evaluating Quit[] or by pressing CONTROL-D.

In[1]:= Sin[0.5]
Out[1]= 0.479426

Goodbye!

sage:

eval(code, *args, **kwds)[source]#

Evaluates a command inside the Mathics interpreter and returns the output in printable form.

EXAMPLES:

Sage

sage: mathics.eval('1+1')  # optional - mathics
'2'

Python

>>> from sage.all import *
>>> mathics.eval('1+1')  # optional - mathics
'2'

get(var)[source]#

Get the value of the variable var.

EXAMPLES:

Sage

sage: mathics.set('u', '2*x +E')        # optional - mathics
sage: mathics.get('u')                  # optional - mathics
'E + 2 x'

Python

>>> from sage.all import *
>>> mathics.set('u', '2*x +E')        # optional - mathics
>>> mathics.get('u')                  # optional - mathics
'E + 2 x'

help(cmd, long=False)[source]#

Return the Mathics documentation of the given command.

EXAMPLES:

Sage

sage: mathics.help('Sin')                   # optional - mathics
"\n  'Sin[z]'\n    returns the sine of z.\n"

sage: print(_)                              # optional - mathics

'Sin[z]'
  returns the sine of z.

sage: print(mathics.help('Sin', long=True)) # optional - mathics

  'Sin[z]'
    returns the sine of z.

Attributes[Sin] = {Listable, NumericFunction, Protected}

sage: print(mathics.Factorial.__doc__)  # optional - mathics

'Factorial[n]'
  'n!'
  computes the factorial of n.

sage: u = mathics('Pi')                 # optional - mathics
sage: print(u.Cos.__doc__)              # optional - mathics

'Cos[z]'
  returns the cosine of z.

Python

>>> from sage.all import *
>>> mathics.help('Sin')                   # optional - mathics
"\n  'Sin[z]'\n    returns the sine of z.\n"

>>> print(_)                              # optional - mathics
<BLANKLINE>
'Sin[z]'
  returns the sine of z.
<BLANKLINE>

>>> print(mathics.help('Sin', long=True)) # optional - mathics
<BLANKLINE>
  'Sin[z]'
    returns the sine of z.
<BLANKLINE>
Attributes[Sin] = {Listable, NumericFunction, Protected}
<BLANKLINE>

>>> print(mathics.Factorial.__doc__)  # optional - mathics
<BLANKLINE>
'Factorial[n]'
  'n!'
  computes the factorial of n.
<BLANKLINE>

>>> u = mathics('Pi')                 # optional - mathics
>>> print(u.Cos.__doc__)              # optional - mathics
<BLANKLINE>
'Cos[z]'
  returns the cosine of z.
<BLANKLINE>

set(var, value)[source]#

Set the variable var to the given value.

EXAMPLES:

Sage

sage: mathics.set('u', '2*x +E')         # optional - mathics
sage: bool(mathics('u').sage() == 2*x+e) # optional - mathics
True

Python

>>> from sage.all import *
>>> mathics.set('u', '2*x +E')         # optional - mathics
>>> bool(mathics('u').sage() == Integer(2)*x+e) # optional - mathics
True

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

Bases: ExtraTabCompletion, InterfaceElement

Element class of the Mathics interface.

Its instances are usually constructed via the instance call of its parent. It wrapes the Mathics library for this object. In a session Mathics methods can be obtained using tab completion.

EXAMPLES:

Sage

sage: # optional - mathics
sage: me=mathics(e); me
E
sage: type(me)
<class 'sage.interfaces.mathics.MathicsElement'>
sage: P = me.parent(); P
Mathics
sage: type(P)
<class 'sage.interfaces.mathics.Mathics'>

Python

>>> from sage.all import *
>>> # optional - mathics
>>> me=mathics(e); me
E
>>> type(me)
<class 'sage.interfaces.mathics.MathicsElement'>
>>> P = me.parent(); P
Mathics
>>> type(P)
<class 'sage.interfaces.mathics.Mathics'>

Access to the Mathics expression objects:

Sage

sage: # optional - mathics
sage: res = me._mathics_result
sage: type(res)
<class 'mathics.core.evaluation.Result'>
sage: expr = res.last_eval; expr
<Symbol: System`E>
sage: type(expr)
<class 'mathics.core.expression.Symbol'>

Python

>>> from sage.all import *
>>> # optional - mathics
>>> res = me._mathics_result
>>> type(res)
<class 'mathics.core.evaluation.Result'>
>>> expr = res.last_eval; expr
<Symbol: System`E>
>>> type(expr)
<class 'mathics.core.expression.Symbol'>

Applying Mathics methods:

Sage

sage: # optional - mathics
sage: me.to_sympy()
E
sage: me.get_name()
'System`E'
sage: me.is_inexact()
False
sage: me.is_symbol()
True

Python

>>> from sage.all import *
>>> # optional - mathics
>>> me.to_sympy()
E
>>> me.get_name()
'System`E'
>>> me.is_inexact()
False
>>> me.is_symbol()
True

Conversion to Sage:

Sage

sage: bool(me.sage() == e)             # optional - mathics
True

Python

>>> from sage.all import *
>>> bool(me.sage() == e)             # optional - mathics
True

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

Python

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

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

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

Python

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

show(ImageSize=600)[source]#

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

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

Python

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

>>> P = mathics('Plot[Sin[x],{x,-2Pi,4Pi}]')   # optional - mathics
>>> show(P)                                    # optional - mathics
>>> P.show(ImageSize=Integer(800))                      # optional - mathics

sage.interfaces.mathics.mathics_console()[source]#

Spawn a new Mathics command-line session.

EXAMPLES:

Sage

sage: mathics_console()  # not tested

Mathics 2.1.1.dev0
on CPython 3.9.2 (default, Mar 19 2021, 22:23:28)
using SymPy 1.7, mpmath 1.2.1, numpy 1.19.5, cython 0.29.21

Copyright (C) 2011-2021 The Mathics Team.
This program comes with ABSOLUTELY NO WARRANTY.
This is free software, and you are welcome to redistribute it
under certain conditions.
See the documentation for the full license.

Quit by evaluating Quit[] or by pressing CONTROL-D.

In[1]:= Sin[0.5]
Out[1]= 0.479426

Goodbye!

Python

>>> from sage.all import *
>>> mathics_console()  # not tested

Mathics 2.1.1.dev0
on CPython 3.9.2 (default, Mar 19 2021, 22:23:28)
using SymPy 1.7, mpmath 1.2.1, numpy 1.19.5, cython 0.29.21

Copyright (C) 2011-2021 The Mathics Team.
This program comes with ABSOLUTELY NO WARRANTY.
This is free software, and you are welcome to redistribute it
under certain conditions.
See the documentation for the full license.

Quit by evaluating Quit[] or by pressing CONTROL-D.

In[1]:= Sin[0.5]
Out[1]= 0.479426

Goodbye!

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

Used in unpickling a Mathics element.

This function is just the __call__ method of the interface instance.

EXAMPLES:

Sage

sage: sage.interfaces.mathics.reduce_load('Denominator[a / b]')  # optional -- mathics
b

Python

>>> from sage.all import *
>>> sage.interfaces.mathics.reduce_load('Denominator[a / b]')  # optional -- mathics
b

Interface to Mathics#

Tutorial#

Syntax#

Some typical input#

Assignments and definitions#

Function definitions#

Numerical Calculations#

Polynomial and Integer Factorization#

Long Input#

Loading and saving#

Complicated translations#