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 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 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: m = mathematica('{{1., 4}, Pi, 3.2e100, I}') # optional - mathematica
sage: s = m.sage(); s # optional - mathematica
[[1.0, 4], pi, 3.2*e100, I]
sage: s[1].n() # optional - mathematica
3.14159265358979
sage: s[3]^2 # optional - mathematica
-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': 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: m = mathematica('bla^2') # optional - mathematica
sage: mb = m.sage() # optional - mathematica
sage: var('bla') # optional - mathematica
bla
sage: bla^2 - mb # optional - mathematica
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.
"""
from sage.symbolic.pynac 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 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('(?<!\.)[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 != 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 StandardError:
raise NotImplementedError, "Unable to parse Mathematica \
output: %s" % res
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 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 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: m = mathematica('{{1., 4}, Pi, 3.2e100, I}') # optional - mathematica
sage: s = m.sage(); s # optional - mathematica
[[1.0, 4], pi, 3.2*e100, I]
sage: s[1].n() # optional - mathematica
3.14159265358979
sage: s[3]^2 # optional - mathematica
-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': 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: m = mathematica('bla^2') # optional - mathematica
sage: mb = m.sage() # optional - mathematica
sage: var('bla') # optional - mathematica
bla
sage: bla^2 - mb # optional - mathematica
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.
"""
from sage.symbolic.pynac 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 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('(?<!\.)[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 symbolic_expression_from_mathematica_string(mexpr):
r"""
Translate a mathematica string into a symbolic expression
INPUT:
- ``mexpr`` -- string
OUTPUT:
symbolic expression
EXAMPLES::
sage: from sage.symbolic.integration.external import symbolic_expression_from_mathematica_string
sage: symbolic_expression_from_mathematica_string(u'-Cos[x]')
-cos(x)
"""
import re
from sage.libs.pynac.pynac import symbol_table
from sage.interfaces.mathematica import _un_camel as un_camel
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
expr = mexpr.replace('\n', ' ').replace('\r', '')
expr = expr.replace('[', '(').replace(']', ')')
expr = expr.replace('{', '[').replace('}', ']')
lsymbols = symbol_table['mathematica'].copy()
autotrans = [
lambda x: x.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(expr):
# 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(expr) and expr[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'."
% 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
return symbolic_expression_from_string(expr,
lsymbols,
accept_sequence=True)
def mma_free_integrator(expression, v, a=None, b=None):
"""
Integration using Mathematica's online integrator
EXAMPLES::
sage: from sage.symbolic.integration.external import mma_free_integrator
sage: mma_free_integrator(sin(x), x) # optional - internet
-cos(x)
TESTS:
Check that :trac:`18212` is resolved::
sage: var('y') # optional - internet
y
sage: integral(sin(y)^2, y, algorithm='mathematica_free') # optional - internet
-1/2*cos(y)*sin(y) + 1/2*y
sage: mma_free_integrator(exp(-x^2)*log(x), x) # optional - internet
1/2*sqrt(pi)*erf(x)*log(x) - x*hypergeometric((1/2, 1/2), (3/2, 3/2), -x^2)
"""
import re
# import compatible with py2 and py3
from six.moves.urllib.request import urlopen
from six.moves.urllib.parse import urlencode
# We need to integrate against x
vars = [str(x) for x in expression.variables()]
if any(len(x)>1 for x in vars):
raise NotImplementedError("Mathematica online integrator can only handle single letter variables.")
x = SR.var('x')
if repr(v) != 'x':
for i in range(ord('a'), ord('z')+1):
if chr(i) not in vars:
shadow_x = SR.var(chr(i))
break
expression = expression.subs({x:shadow_x}).subs({v: x})
params = urlencode({'expr': expression._mathematica_init_(), 'random': 'false'})
page = urlopen("http://integrals.wolfram.com/home.jsp", params).read()
page = page[page.index('"inputForm"'):page.index('"outputForm"')]
page = re.sub("\s", "", page)
mexpr = re.match(r".*Integrate.*==</em><br/>(.*)</p>", page).groups()[0]
try:
from sage.libs.pynac.pynac import symbol_table
from sage.interfaces.mathematica import _un_camel as un_camel
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
expr = mexpr.replace('\n',' ').replace('\r', '')
expr = expr.replace('[', '(').replace(']', ')')
expr = expr.replace('{', '[').replace('}', ']')
lsymbols = symbol_table['mathematica'].copy()
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('(?<!\.)[a-zA-Z]\w*')
for m in p.finditer(expr):
# 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(expr) and expr[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'." % 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
ans = symbolic_expression_from_string(expr, lsymbols, accept_sequence=True)
if repr(v) != 'x':
ans = ans.subs({x:v}).subs({shadow_x:x})
return ans
except TypeError:
raise ValueError("Unable to parse: %s" % mexpr)
