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mathematica.py
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mathematica.py
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r"""
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.
TESTS:
Check that numerical approximations via Mathematica's `N[]` function work
correctly (:issue:`18888`, :issue:`28907`)::
sage: # optional - mathematica
sage: mathematica('Pi/2').N(10)
1.5707963268
sage: mathematica('Pi').N(10)
3.1415926536
sage: mathematica('Pi').N(50)
3.14159265358979323846264338327950288419716939937511
sage: str(mathematica('Pi*x^2-1/2').N())
2
-0.5 + 3.14159 x
Check that Mathematica's `E` exponential symbol is correctly backtranslated
as Sage's `e` (:issue:`29833`)::
sage: x = var('x')
sage: (e^x)._mathematica_().sage() # optional -- mathematica
e^x
sage: exp(x)._mathematica_().sage() # optional -- mathematica
e^x
Check that all trig/hyperbolic functions and their reciprocals are correctly
translated to Mathematica (:issue:`34087`)::
sage: # optional - mathematica
sage: x=var('x')
sage: FL=[sin, cos, tan, csc, sec, cot,
....: sinh, cosh, tanh, csch, sech, coth]
sage: IFL=[arcsin, arccos, arctan, arccsc,
....: arcsec, arccot, arcsinh, arccosh,
....: arctanh, arccsch, arcsech, arccoth]
sage: [mathematica.TrigToExp(u(x)).sage()
....: for u in FL]
[-1/2*I*e^(I*x) + 1/2*I*e^(-I*x),
1/2*e^(I*x) + 1/2*e^(-I*x),
(-I*e^(I*x) + I*e^(-I*x))/(e^(I*x) + e^(-I*x)),
2*I/(e^(I*x) - e^(-I*x)),
2/(e^(I*x) + e^(-I*x)),
-(-I*e^(I*x) - I*e^(-I*x))/(e^(I*x) - e^(-I*x)),
-1/2*e^(-x) + 1/2*e^x,
1/2*e^(-x) + 1/2*e^x,
-e^(-x)/(e^(-x) + e^x) + e^x/(e^(-x) + e^x),
-2/(e^(-x) - e^x),
2/(e^(-x) + e^x),
-(e^(-x) + e^x)/(e^(-x) - e^x)]
sage: [mathematica.TrigToExp(u(x)).sage()
....: for u in IFL]
[-I*log(I*x + sqrt(-x^2 + 1)),
1/2*pi + I*log(I*x + sqrt(-x^2 + 1)),
-1/2*I*log(I*x + 1) + 1/2*I*log(-I*x + 1),
-I*log(sqrt(-1/x^2 + 1) + I/x),
1/2*pi + I*log(sqrt(-1/x^2 + 1) + I/x),
-1/2*I*log(I/x + 1) + 1/2*I*log(-I/x + 1),
log(x + sqrt(x^2 + 1)),
log(sqrt(x + 1)*sqrt(x - 1) + x),
1/2*log(x + 1) - 1/2*log(-x + 1),
log(sqrt(1/x^2 + 1) + 1/x),
log(sqrt(1/x + 1)*sqrt(1/x - 1) + 1/x),
1/2*log(1/x + 1) - 1/2*log(-1/x + 1)]
"""
# ****************************************************************************
# Copyright (C) 2005 William Stein <wstein@gmail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# This code is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# The full text of the GPL is available at:
#
# https://www.gnu.org/licenses/
# ****************************************************************************
import os
import re
from sage.misc.cachefunc import cached_method
from sage.interfaces.expect import (Expect, ExpectElement, ExpectFunction,
FunctionElement)
from sage.interfaces.interface import AsciiArtString
from sage.interfaces.tab_completion import ExtraTabCompletion
from sage.misc.instancedoc import instancedoc
from sage.structure.richcmp import rich_to_bool
def clean_output(s):
if s is None:
return ''
i = s.find('Out[')
j = i + s[i:].find('=')
s = s[:i] + ' ' * (j + 1 - i) + s[j + 1:]
s = s.replace('\\\n', '')
return s.strip('\n')
def _un_camel(name):
"""
Convert `CamelCase` to `camel_case`.
EXAMPLES::
sage: sage.interfaces.mathematica._un_camel('CamelCase')
'camel_case'
sage: sage.interfaces.mathematica._un_camel('EllipticE')
'elliptic_e'
sage: sage.interfaces.mathematica._un_camel('FindRoot')
'find_root'
sage: sage.interfaces.mathematica._un_camel('GCD')
'gcd'
"""
s1 = re.sub('(.)([A-Z][a-z]+)', r'\1_\2', name)
return re.sub('([a-z0-9])([A-Z])', r'\1_\2', s1).lower()
class Mathematica(ExtraTabCompletion, Expect):
"""
Interface to the Mathematica interpreter.
"""
def __init__(self, maxread=None, script_subdirectory=None, logfile=None, server=None,
server_tmpdir=None, command=None, verbose_start=False):
r"""
TESTS:
Test that :issue:`28075` is fixed::
sage: repr(mathematica.eval("Print[1]; Print[2]; Print[3]")) # optional - mathematica
'1\n2\n3'
"""
# We use -rawterm to get a raw text interface in Mathematica 9 or later.
# This works around the following issues of Mathematica 9 or later
# (tested with Mathematica 11.0.1 for Mac OS X x86 (64-bit))
#
# 1) If TERM is unset and input is a pseudoterminal, Mathematica shows no
# prompts, so pexpect will not work.
#
# 2) If TERM is set (to dumb, lpr, vt100, or xterm), there will be
# prompts; but there is bizarre echoing behavior by Mathematica (not
# the terminal driver). For example, with TERM=dumb, many spaces and
# \r's are echoed. With TERM=vt100 or better, in addition, many escape
# sequences are printed.
#
if command is None:
command = os.getenv('SAGE_MATHEMATICA_COMMAND') or 'math -rawterm'
eval_using_file_cutoff = 1024
# Removing terminal echo using "stty -echo" is not essential but it slightly
# improves performance (system time) and eliminates races of the terminal echo
# as a possible source of error.
if server:
command = 'stty -echo; {}'.format(command)
else:
command = 'sh -c "stty -echo; {}"'.format(command)
Expect.__init__(self,
name='mathematica',
terminal_echo=False,
command=command,
prompt=r'In\[[0-9]+\]:= ',
server=server,
server_tmpdir=server_tmpdir,
script_subdirectory=script_subdirectory,
verbose_start=verbose_start,
logfile=logfile,
eval_using_file_cutoff=eval_using_file_cutoff)
def _read_in_file_command(self, filename):
return '<<"%s"' % filename
def _keyboard_interrupt(self):
print("Interrupting %s..." % self)
e = self._expect
e.sendline(chr(3)) # send ctrl-c
e.expect('Interrupt> ')
e.sendline("a") # a -- abort
e.expect(self._prompt)
return e.before
def _install_hints(self):
"""
Hints for installing mathematica on your computer.
AUTHORS:
- William Stein and Justin Walker (2006-02-12)
"""
return """
In order to use the Mathematica interface you need to have Mathematica
installed and have a script in your PATH called "math" that runs the
command-line version of Mathematica. Alternatively, you could use a
remote connection to a server running Mathematica -- for hints, type
print(mathematica._install_hints_ssh())
(1) You might have to buy Mathematica (https://www.wolfram.com/), or
install a currently (Feb 2022) free for personal use Wolfram Engine
(https://www.wolfram.com/engine/).
(2) * LINUX: The math script usually comes standard with your Mathematica install.
However, on some systems it may be called wolfram,
or, in case of Wolfram Engine, wolframengine, while math is absent.
In this case, assuming wolfram, respectively, wolframengine,
is in your PATH,
(a) create a file called math (in your PATH):
#!/bin/sh
/usr/bin/env wolfram $@
respectively,
(a') create a file called math (in your PATH):
#!/bin/sh
/usr/bin/env wolframengine $@
(b) Make the file executable.
chmod +x math
* Apple macOS: for Mathematica,
(a) create a file called math (in your PATH):
#!/bin/sh
/Applications/Mathematica.app/Contents/MacOS/MathKernel $@
(a') for Wolfram Engine, follow the Linux step (a') above.
The path in the above script must be modified if you installed
Mathematica elsewhere or installed an old version of
Mathematica that has the version in the .app name.
(b) Make the file executable.
chmod +x math
* WINDOWS:
Install Mathematica for Linux into the VMware virtual machine, or in
a WSL/WSL2 Linux installation with Sage installed there (sorry,
that's the only ways at present).
"""
def eval(self, code, strip=True, **kwds):
s = Expect.eval(self, code, **kwds)
if strip:
return AsciiArtString(clean_output(s))
else:
return AsciiArtString(s)
def set(self, var, value):
"""
Set the variable var to the given value.
"""
cmd = '%s=%s;' % (var, value)
out = self._eval_line(cmd, allow_use_file=True)
if len(out) > 8:
raise TypeError("Error executing code in Mathematica\nCODE:\n\t%s\nMathematica ERROR:\n\t%s" % (cmd, out))
def get(self, var, ascii_art=False):
"""
Get the value of the variable var.
AUTHORS:
- William Stein
- Kiran Kedlaya (2006-02-04): suggested using InputForm
"""
if ascii_art:
return self.eval(var, strip=True)
return self.eval('InputForm[%s, NumberMarks->False]' % var, strip=True)
def _eval_line(self, line, allow_use_file=True, wait_for_prompt=True, restart_if_needed=False):
s = Expect._eval_line(self, line,
allow_use_file=allow_use_file, wait_for_prompt=wait_for_prompt)
return str(s).strip('\n')
def _function_call_string(self, function, args, kwds):
"""
Returns the string used to make function calls.
EXAMPLES::
sage: mathematica._function_call_string('Sin', ['x'], [])
'Sin[x]'
"""
return "%s[%s]" % (function, ",".join(args))
def _left_list_delim(self):
return "{"
def _right_list_delim(self):
return "}"
def _left_func_delim(self):
return "["
def _right_func_delim(self):
return "]"
###########################################
# System -- change directory, etc
###########################################
def chdir(self, dir):
"""
Change Mathematica's current working directory.
EXAMPLES::
sage: mathematica.chdir('/') # optional - mathematica
sage: mathematica('Directory[]') # optional - mathematica
"/"
"""
self.eval('SetDirectory["%s"]' % dir)
def _true_symbol(self):
return 'True'
def _false_symbol(self):
return 'False'
def _equality_symbol(self):
return '=='
def _assign_symbol(self):
return ":="
def _exponent_symbol(self):
"""
Returns the symbol used to denote the exponent of a number in
Mathematica.
EXAMPLES::
sage: mathematica._exponent_symbol() # optional - mathematica
'*^'
::
sage: bignum = mathematica('10.^80') # optional - mathematica
sage: repr(bignum) # optional - mathematica
'1.*^80'
sage: repr(bignum).replace(mathematica._exponent_symbol(), 'e').strip() # optional - mathematica
'1.e80'
"""
return "*^"
def _object_class(self):
return MathematicaElement
def console(self, readline=True):
mathematica_console(readline=readline)
def _tab_completion(self):
a = self.eval('Names["*"]')
return a.replace('$', '').replace('\n \n>', '').replace(',', '').replace('}', '').replace('{', '').split()
def help(self, cmd):
return self.eval('? %s' % cmd)
def __getattr__(self, attrname):
if attrname[:1] == "_":
raise AttributeError
return MathematicaFunction(self, attrname)
@instancedoc
class MathematicaElement(ExpectElement):
def __getitem__(self, n):
return self.parent().new('%s[[%s]]' % (self._name, n))
def __getattr__(self, attrname):
self._check_valid()
if attrname[:1] == "_":
raise AttributeError
return MathematicaFunctionElement(self, attrname)
def __float__(self, precision=16):
P = self.parent()
return float(P.eval('N[%s,%s]' % (self.name(), precision)))
def _reduce(self):
return self.parent().eval('InputForm[%s]' % self.name()).strip()
def __reduce__(self):
return reduce_load, (self._reduce(), )
def _latex_(self):
z = self.parent().eval('TeXForm[%s]' % self.name())
i = z.find('=')
return z[i + 1:].strip()
def _repr_(self):
P = self.parent()
return P.get(self._name, ascii_art=False).strip()
def _sage_(self, locals={}):
r"""
Attempt to return a Sage version of this object.
This method works successfully when Mathematica returns a result
or list of results that consist only 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 naming scheme, or
- you provide a translation dictionary `locals`, or
- the Sage name for the function or constant is simply the
Mathematica name in lower case;
- symbolic variables whose names do not pathologically overlap with
objects already defined in Sage.
This method will not work when Mathematica's output includes:
- strings;
- functions unknown to Sage that are not specified in `locals`;
- Mathematica functions with different parameters/parameter order to
the Sage equivalent. In this case, define a function to do the
parameter conversion, and pass it in via the locals dictionary.
EXAMPLES:
Mathematica lists of numbers/constants become Sage lists of
numbers/constants::
sage: # optional - mathematica
sage: m = mathematica('{{1., 4}, Pi, 3.2e100, I}')
sage: s = m.sage(); s
[[1.00000000000000, 4], pi, 3.20000000000000*e100, I]
sage: s[1].n()
3.14159265358979
sage: s[3]^2
-1
::
sage: m = mathematica('x^2 + 5*y') # optional - mathematica
sage: m.sage() # optional - mathematica
x^2 + 5*y
::
sage: m = mathematica('Sin[Sqrt[1-x^2]] * (1 - Cos[1/x])^2') # optional - mathematica
sage: m.sage() # optional - mathematica
(cos(1/x) - 1)^2*sin(sqrt(-x^2 + 1))
::
sage: m = mathematica('NewFn[x]') # optional - mathematica
sage: m._sage_(locals={('NewFn', 1): sin}) # optional - mathematica
sin(x)
::
sage: var('bla') # optional - mathematica
bla
sage: m = mathematica('bla^2') # optional - mathematica
sage: bla^2 - m.sage() # optional - mathematica
0
::
sage: # optional - mathematica
sage: m = mathematica('bla^2')
sage: mb = m.sage()
sage: var('bla')
bla
sage: bla^2 - mb
0
AUTHORS:
- Felix Lawrence (2010-11-03): Major rewrite to use ._sage_repr() and
sage.calculus.calculus.symbolic_expression_from_string() for greater
compatibility, while still supporting conversion of symbolic
expressions.
TESTS:
Check that :issue:`28814` is fixed::
sage: mathematica('Exp[1000.0]').sage() # optional - mathematica
1.97007111401700e434
sage: mathematica('1/Exp[1000.0]').sage() # optional - mathematica
5.07595889754950e-435
sage: mathematica(RealField(100)(1/3)).sage() # optional - mathematica
0.3333333333333333333333333333335
"""
from sage.symbolic.expression import symbol_table
from sage.symbolic.constants import constants_name_table as constants
from sage.calculus.calculus import symbolic_expression_from_string
from sage.calculus.calculus import _find_func as find_func
# Get Mathematica's output and perform preliminary formatting
res = self._sage_repr()
if '"' in res:
raise NotImplementedError("String conversion from Mathematica \
does not work. Mathematica's output was: %s" % res)
# Find all the mathematica functions, constants and symbolic variables
# present in `res`. Convert MMA functions and constants to their
# Sage equivalents (if possible), using `locals` and
# `sage.symbolic.pynac.symbol_table['mathematica']` as translation
# dictionaries. If a MMA function or constant is not in either
# dictionary, then we use a variety of tactics listed in `autotrans`.
# If a MMA variable is not in any dictionary, then create an
# identically named Sage equivalent.
# Merge the user-specified locals dictionary and the symbol_table
# (locals takes priority)
lsymbols = symbol_table['mathematica'].copy()
lsymbols.update(locals)
# Strategies for translating unknown functions/constants:
autotrans = [str.lower, # Try it in lower case
_un_camel, # Convert `CamelCase` to `camel_case`
lambda x: x] # Try the original name
# Find the MMA funcs/vars/constants - they start with a letter.
# Exclude exponents (e.g. 'e8' from 4.e8)
p = re.compile(r'(?<!\.)[a-zA-Z]\w*')
for m in p.finditer(res):
# If the function, variable or constant is already in the
# translation dictionary, then just move on.
if m.group() in lsymbols:
pass
# Now try to translate all other functions -- try each strategy
# in `autotrans` and check if the function exists in Sage
elif m.end() < len(res) and res[m.end()] == '(':
for t in autotrans:
f = find_func(t(m.group()), create_when_missing=False)
if f is not None:
lsymbols[m.group()] = f
break
else:
raise NotImplementedError("Don't know a Sage equivalent \
for Mathematica function '%s'. Please specify one \
manually using the 'locals' dictionary" % m.group())
# Check if Sage has an equivalent constant
else:
for t in autotrans:
if t(m.group()) in constants:
lsymbols[m.group()] = constants[t(m.group())]
break
# If Sage has never heard of the variable, then
# symbolic_expression_from_string will automatically create it
try:
return symbolic_expression_from_string(res, lsymbols,
accept_sequence=True)
except Exception:
raise NotImplementedError("Unable to parse Mathematica \
output: %s" % res)
def __str__(self):
P = self._check_valid()
return P.get(self._name, ascii_art=True)
def __len__(self):
"""
Return the object's length, evaluated by mathematica.
EXAMPLES::
sage: len(mathematica([1,1.,2])) # optional - mathematica
3
AUTHORS:
- Felix Lawrence (2009-08-21)
"""
return int(self.Length())
@cached_method
def _is_graphics(self):
"""
Test whether the mathematica expression is graphics
OUTPUT:
Boolean.
EXAMPLES::
sage: P = mathematica('Plot[Sin[x],{x,-2Pi,4Pi}]') # optional - mathematica
sage: P._is_graphics() # optional - mathematica
True
"""
P = self._check_valid()
return P.eval('InputForm[%s]' % self.name()).strip().startswith('Graphics[')
def save_image(self, filename, ImageSize=600):
r"""
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
"""
P = self._check_valid()
if not self._is_graphics():
raise ValueError('mathematica expression is not graphics')
filename = os.path.abspath(filename)
s = 'Export["%s", %s, ImageSize->%s]' % (filename, self.name(),
ImageSize)
P.eval(s)
def _rich_repr_(self, display_manager, **kwds):
"""
Rich Output Magic Method
See :mod:`sage.repl.rich_output` for details.
EXAMPLES::
sage: from sage.repl.rich_output import get_display_manager
sage: dm = get_display_manager()
sage: P = mathematica('Plot[Sin[x],{x,-2Pi,4Pi}]') # optional - mathematica
The following test requires a working X display on Linux so that the
Mathematica frontend can do the rendering (:issue:`23112`)::
sage: P._rich_repr_(dm) # optional - mathematica mathematicafrontend
OutputImagePng container
"""
if self._is_graphics():
OutputImagePng = display_manager.types.OutputImagePng
if display_manager.preferences.graphics == 'disable':
return
if OutputImagePng in display_manager.supported_output():
return display_manager.graphics_from_save(
self.save_image, kwds, '.png', OutputImagePng)
else:
OutputLatex = display_manager.types.OutputLatex
dmp = display_manager.preferences.text
if dmp is None or dmp == 'plain':
return
if dmp == 'latex' and OutputLatex in display_manager.supported_output():
return OutputLatex(self._latex_())