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: 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)
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: # 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
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: mathics('Factor[x^2-1]') # optional - mathics
(-1 + x) (1 + x)
sage: 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: 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]'
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.
Tutorial#
We follow some of the tutorial from http://library.wolfram.com/conferences/devconf99/withoff/Basic1.html/.
Syntax#
Now make 1 and add it to itself. The result is a Mathics object.
sage: m = mathics
sage: a = m(1) + m(1); a # optional - mathics
2
sage: a.parent() # optional - mathics
Mathics
sage: m('1+1') # optional - mathics
2
sage: m(3)**m(50) # optional - mathics
717897987691852588770249
The following is equivalent to Plus[2, 3] in
Mathics:
sage: m = mathics
sage: m(2).Plus(m(3)) # optional - mathics
5
We can also compute \(7(2+3)\).
sage: m(7).Times(m(2).Plus(m(3))) # optional - mathics
35
sage: m('7(2+3)') # optional - mathics
35
Some typical input#
We solve an equation and a system of two equations:
sage: # optional - mathics
sage: eqn = mathics('3x + 5 == 14')
sage: eqn
5 + 3 x == 14
sage: eqn.Solve('x')
{{x -> 3}}
sage: sys = mathics('{x^2 - 3y == 3, 2x - y == 1}')
sage: print(sys)
{x ^ 2 - 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 mathics \(5\) to a variable \(c\) in Sage, this does not affect the \(c\) in Mathics.
sage: c = m(5) # optional - mathics
sage: print(m('b + c x')) # optional - mathics
b + c x
sage: print(m('b') + c*m('x')) # optional - mathics
b + 5 x
The Sage interfaces changes Sage lists into Mathics lists:
sage: m = mathics
sage: eq1 = m('x^2 - 3y == 3') # optional - mathics
sage: eq2 = m('2x - y == 1') # optional - mathics
sage: v = m([eq1, eq2]); v # optional - mathics
{x ^ 2 - 3 y == 3, 2 x - y == 1}
sage: v.Solve(['x', 'y']) # optional - mathics
{{x -> 0, y -> -1}, {x -> 6, y -> 11}}
Function definitions#
Define mathics functions by simply sending the definition to the interpreter.
sage: m = mathics
sage: _ = mathics('f[p_] = p^2'); # optional - mathics
sage: m('f[9]') # optional - mathics
81
Numerical Calculations#
We find the \(x\) such that \(e^x - 3x = 0\).
sage: eqn = mathics('Exp[x] - 3x == 0') # optional - mathics
sage: eqn.FindRoot(['x', 2]) # optional - mathics
{x -> 1.51213}
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 Mathics:
sage: mathics('FindMinimum[x^3 - 6x^2 + 11x - 5, {x,3}]') # not tested (since not supported, so far)
{0.6150998205402516, {x -> 2.5773502699629733}}
sage: f = mathics('x^3 - 6x^2 + 11x - 5') # optional - mathics
sage: f.FindMinimum(['x', 3]) # not tested (since not supported, so far)
{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 = mathics(str(f)) # optional - mathics
sage: print(g) # optional - mathics
100 + 315 x - 85 x ^ 2 + 25 x ^ 100 + 12 x ^ 101 + x ^ 200
sage: g # optional - mathics
100 + 315 x - 85 x ^ 2 + 25 x ^ 100 + 12 x ^ 101 + x ^ 200
sage: print(g.Factor()) # optional - mathics
(5 + 17 x + x ^ 100) (20 - 5 x + x ^ 100)
We can also factor a multivariate polynomial:
sage: f = mathics('x^6 + (-y - 2)*x^5 + (y^3 + 2*y)*x^4 - y^4*x^3') # optional - mathics
sage: print(f.Factor()) # optional - mathics
x ^ 3 (x - y) (-2 x + x ^ 2 + y ^ 3)
We factor an integer:
sage: # optional - mathics
sage: n = mathics(2434500)
sage: n.FactorInteger()
{{2, 2}, {3, 2}, {5, 3}, {541, 1}}
sage: n = mathics(2434500)
sage: F = n.FactorInteger(); F
{{2, 2}, {3, 2}, {5, 3}, {541, 1}}
sage: F[1]
{2, 2}
sage: F[4]
{541, 1}
Long Input#
The Mathics interface reads in even very long input (using files) in a robust manner.
sage: t = '"%s"'%10^10000 # ten thousand character string.
sage: a = mathics(t) # optional - mathics
sage: 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: # 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
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: x = var('x')
sage: m = mathics('NewFn[x]') # optional - mathics
sage: m._sage_(locals={'NewFn': sin, 'x':x}) # optional - mathics
sin(x)
For more details, see the documentation for ._sage_().
OTHER Examples:
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
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]
sage: mathics('10.^80') # optional - mathics
1.*^80
sage: mathics('10.^80').sage() # optional - mathics
1.00000000000000e80
AUTHORS:
Sebastian Oehms (2021): first version from a copy of the Mathematica interface (see github 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)#
Bases:
InterfaceInterface to the Mathics interpreter.
Implemented according to the Mathematica interface but avoiding Pexpect functionality.
EXAMPLES:
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
More examples can be found in the module header.
- chdir(dir)#
Change Mathics’s current working directory.
EXAMPLES:
sage: mathics.chdir('/') # optional - mathics sage: mathics('Directory[]') # optional - mathics /
- console()#
Spawn a new Mathics command-line session.
EXAMPLES:
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:
- eval(code, *args, **kwds)#
Evaluates a command inside the Mathics interpreter and returns the output in printable form.
EXAMPLES:
sage: mathics.eval('1+1') # optional - mathics '2'
- get(var)#
Get the value of the variable var.
EXAMPLES:
sage: mathics.set('u', '2*x +E') # optional - mathics sage: mathics.get('u') # optional - mathics 'E + 2 x'
- help(cmd, long=False)#
Return the Mathics documentation of the given command.
EXAMPLES:
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.
- set(var, value)#
Set the variable var to the given value.
EXAMPLES:
sage: mathics.set('u', '2*x +E') # optional - mathics sage: bool(mathics('u').sage() == 2*x+e) # optional - mathics True
- class sage.interfaces.mathics.MathicsElement(parent, value, is_name=False, name=None)#
Bases:
ExtraTabCompletion,InterfaceElementElement 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: # 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'>
Access to the Mathics expression objects:
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'>
Applying Mathics methods:
sage: # optional - mathics sage: me.to_sympy() E sage: me.get_name() 'System`E' sage: me.is_inexact() False sage: me.is_symbol() True
Conversion to Sage:
sage: bool(me.sage() == e) # optional - mathics True
- 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: 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
- save_image(filename, ImageSize=600)#
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: 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
- show(ImageSize=600)#
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: 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
- sage.interfaces.mathics.mathics_console()#
Spawn a new Mathics command-line session.
EXAMPLES:
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.interfaces.mathics.reduce_load(X)#
Used in unpickling a Mathics element.
This function is just the
__call__method of the interface instance.EXAMPLES:
sage: sage.interfaces.mathics.reduce_load('Denominator[a / b]') # optional -- mathics b