Source code for sympy.utilities.lambdify
"""
This module provides convenient functions to transform sympy expressions to
lambda functions which can be used to calculate numerical values very fast.
"""
from sympy.external import import_module
from sympy.core.compatibility import is_sequence
import inspect
# These are the namespaces the lambda functions will use.
MATH = {}
MPMATH = {}
NUMPY = {}
SYMPY = {}
# Default namespaces, letting us define translations that can't be defined
# by simple variable maps, like I => 1j
# These are separate from the names above because the above names are modified
# throughout this file, whereas these should remain unmodified.
MATH_DEFAULT = {}
MPMATH_DEFAULT = {}
NUMPY_DEFAULT = {"I": 1j}
SYMPY_DEFAULT = {}
# Mappings between sympy and other modules function names.
MATH_TRANSLATIONS = {
"Abs":"fabs",
"ceiling":"ceil",
"E":"e",
"ln":"log",
}
MPMATH_TRANSLATIONS = {
"ceiling":"ceil",
"chebyshevt":"chebyt",
"chebyshevu":"chebyu",
"E":"e",
"I":"j",
"ln":"log",
#"lowergamma":"lower_gamma",
"oo":"inf",
#"uppergamma":"upper_gamma",
"LambertW":"lambertw",
"Matrix":"matrix",
"conjugate":"conj",
"dirichlet_eta":"altzeta",
"Ei":"ei",
"Shi":"shi",
"Chi":"chi",
"Si":"si",
"Ci":"ci"
}
NUMPY_TRANSLATIONS = {
"Abs":"abs",
"acos":"arccos",
"acosh":"arccosh",
"arg":"angle",
"asin":"arcsin",
"asinh":"arcsinh",
"atan":"arctan",
"atan2":"arctan2",
"atanh":"arctanh",
"ceiling":"ceil",
"E":"e",
"im":"imag",
"ln":"log",
"Matrix":"matrix",
"Max":"amax",
"Min":"amin",
"oo":"inf",
"re":"real",
}
# Available modules:
MODULES = {
"math" : (MATH, MATH_DEFAULT, MATH_TRANSLATIONS, ("from math import *",)),
"mpmath" : (MPMATH, MPMATH_DEFAULT, MPMATH_TRANSLATIONS, ("from sympy.mpmath import *",)),
"numpy" : (NUMPY, NUMPY_DEFAULT, NUMPY_TRANSLATIONS, ("import_module('numpy')",)),
"sympy" : (SYMPY, SYMPY_DEFAULT, {}, ("from sympy.functions import *",
"from sympy.matrices import Matrix",
"from sympy import Integral, pi, oo, nan, zoo, E, I",)),
}
def _import(module, reload="False"):
"""
Creates a global translation dictionary for module.
The argument module has to be one of the following strings: "math",
"mpmath", "numpy", "sympy".
These dictionaries map names of python functions to their equivalent in
other modules.
"""
try:
namespace, namespace_default, translations, import_commands = MODULES[module]
except KeyError:
raise NameError("'%s' module can't be used for lambdification" % module)
# Clear namespace or exit
if namespace != namespace_default:
# The namespace was already generated, don't do it again if not forced.
if reload:
namespace.clear()
namespace.update(namespace_default)
else:
return
for import_command in import_commands:
if import_command.startswith('import_module'):
module = eval(import_command)
if module is not None:
namespace.update(module.__dict__)
continue
else:
try:
exec(import_command, {}, namespace)
continue
except ImportError:
pass
raise ImportError("can't import '%s' with '%s' command" % (module, import_command))
# Add translated names to namespace
for sympyname, translation in translations.items():
namespace[sympyname] = namespace[translation]
[docs]def lambdify(args, expr, modules=None, printer=None, use_imps=True):
"""
Returns a lambda function for fast calculation of numerical values.
Usage:
>>> from sympy import sqrt, sin
>>> from sympy.utilities.lambdify import lambdify
>>> from sympy.abc import x, y, z
>>> f = lambdify(x, x**2)
>>> f(2)
4
>>> f = lambdify((x,y,z), [z,y,x])
>>> f(1,2,3)
[3, 2, 1]
>>> f = lambdify(x, sqrt(x))
>>> f(4)
2.0
>>> f = lambdify((x,y), sin(x*y)**2)
>>> f(0, 5)
0.0
If not specified differently by the user, SymPy functions are replaced as
far as possible by either python-math, numpy (if available) or mpmath
functions - exactly in this order.
To change this behavior, the "modules" argument can be used.
It accepts:
- the strings "math", "mpmath", "numpy", "sympy"
- any modules (e.g. math)
- dictionaries that map names of sympy functions to arbitrary functions
- lists that contain a mix of the arguments above. (Entries that are first
in the list have higher priority)
Examples
========
(1) Use one of the provided modules:
>> f = lambdify(x, sin(x), "math")
Attention: Functions that are not in the math module will throw a name
error when the lambda function is evaluated! So this would
be better:
>> f = lambdify(x, sin(x)*gamma(x), ("math", "mpmath", "sympy"))
(2) Use some other module:
>> import numpy
>> f = lambdify((x,y), tan(x*y), numpy)
Attention: There are naming differences between numpy and sympy. So if
you simply take the numpy module, e.g. sympy.atan will not be
translated to numpy.arctan. Use the modified module instead
by passing the string "numpy":
>> f = lambdify((x,y), tan(x*y), "numpy")
>> f(1, 2)
-2.18503986326
>> from numpy import array
>> f(array([1, 2, 3]), array([2, 3, 5]))
[-2.18503986 -0.29100619 -0.8559934 ]
(3) Use own dictionaries:
>> def my_cool_function(x): ...
>> dic = {"sin" : my_cool_function}
>> f = lambdify(x, sin(x), dic)
Now f would look like:
>> lambda x: my_cool_function(x)
Functions present in `expr` can also carry their own numerical
implementations, in a callable attached to the ``_imp_``
attribute. Usually you attach this using the
``implemented_function`` factory:
>>> from sympy.abc import x, y, z
>>> from sympy.utilities.lambdify import lambdify, implemented_function
>>> from sympy import Function
>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> func = lambdify(x, f(x))
>>> func(4)
5
``lambdify`` always prefers ``_imp_`` implementations to
implementations in other namespaces, unless the ``use_imps`` input
parameter is False.
"""
from sympy.core.symbol import Symbol
# If the user hasn't specified any modules, use what is available.
if modules is None:
# Use either numpy (if available) or python.math where possible.
# XXX: This leads to different behaviour on different systems and
# might be the reason for irreproducible errors.
modules = ["math", "mpmath", "sympy"]
try:
_import("numpy")
except ImportError:
pass
else:
modules.insert(1, "numpy")
# Get the needed namespaces.
namespaces = []
# First find any function implementations
if use_imps:
namespaces.append(_imp_namespace(expr))
# Check for dict before iterating
if isinstance(modules, (dict, str)) or not hasattr(modules, '__iter__'):
namespaces.append(modules)
else:
namespaces += list(modules)
# fill namespace with first having highest priority
namespace = {}
for m in namespaces[::-1]:
buf = _get_namespace(m)
namespace.update(buf)
if hasattr(expr, "atoms"):
#Try if you can extract symbols from the expression.
#Move on if expr.atoms in not implemented.
syms = expr.atoms(Symbol)
for term in syms:
namespace.update({str(term): term})
# Create lambda function.
lstr = lambdastr(args, expr, printer=printer)
return eval(lstr, namespace)
def _get_namespace(m):
"""
This is used by _lambdify to parse its arguments.
"""
if isinstance(m, str):
_import(m)
return MODULES[m][0]
elif isinstance(m, dict):
return m
elif hasattr(m, "__dict__"):
return m.__dict__
else:
raise TypeError("Argument must be either a string, dict or module but it is: %s" % m)
[docs]def lambdastr(args, expr, printer=None):
"""
Returns a string that can be evaluated to a lambda function.
>>> from sympy.abc import x, y, z
>>> from sympy.utilities.lambdify import lambdastr
>>> lambdastr(x, x**2)
'lambda x: (x**2)'
>>> lambdastr((x,y,z), [z,y,x])
'lambda x,y,z: ([z, y, x])'
"""
if printer is not None:
if inspect.isfunction(printer):
lambdarepr = printer
else:
if inspect.isclass(printer):
lambdarepr = lambda expr: printer().doprint(expr)
else:
lambdarepr = lambda expr: printer.doprint(expr)
else:
#XXX: This has to be done here because of circular imports
from sympy.printing.lambdarepr import lambdarepr
# Transform everything to strings.
expr = lambdarepr(expr)
if isinstance(args, str):
pass
elif hasattr(args, "__iter__"):
args = ",".join(str(a) for a in args)
else:
args = str(args)
return "lambda %s: (%s)" % (args, expr)
def _imp_namespace(expr, namespace=None):
""" Return namespace dict with function implementations
We need to search for functions in anything that can be thrown at
us - that is - anything that could be passed as `expr`. Examples
include sympy expressions, as well as tuples, lists and dicts that may
contain sympy expressions.
Parameters
----------
expr : object
Something passed to lambdify, that will generate valid code from
``str(expr)``.
namespace : None or mapping
Namespace to fill. None results in new empty dict
Returns
-------
namespace : dict
dict with keys of implemented function names within `expr` and
corresponding values being the numerical implementation of
function
Examples
--------
>>> from sympy.abc import x, y, z
>>> from sympy.utilities.lambdify import implemented_function, _imp_namespace
>>> from sympy import Function
>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> g = implemented_function(Function('g'), lambda x: x*10)
>>> namespace = _imp_namespace(f(g(x)))
>>> sorted(namespace.keys())
['f', 'g']
"""
# Delayed import to avoid circular imports
from sympy.core.function import FunctionClass
if namespace is None:
namespace = {}
# tuples, lists, dicts are valid expressions
if is_sequence(expr):
for arg in expr:
_imp_namespace(arg, namespace)
return namespace
elif isinstance(expr, dict):
for key, val in list(expr.items()):
# functions can be in dictionary keys
_imp_namespace(key, namespace)
_imp_namespace(val, namespace)
return namespace
# sympy expressions may be Functions themselves
func = getattr(expr, 'func', None)
if isinstance(func, FunctionClass):
imp = getattr(func, '_imp_', None)
if imp is not None:
name = expr.func.__name__
if name in namespace and namespace[name] != imp:
raise ValueError('We found more than one '
'implementation with name '
'"%s"' % name)
namespace[name] = imp
# and / or they may take Functions as arguments
if hasattr(expr, 'args'):
for arg in expr.args:
_imp_namespace(arg, namespace)
return namespace
[docs]def implemented_function(symfunc, implementation):
""" Add numerical ``implementation`` to function ``symfunc``.
``symfunc`` can be an ``UndefinedFunction`` instance, or a name string.
In the latter case we create an ``UndefinedFunction`` instance with that
name.
Be aware that this is a quick workaround, not a general method to create
special symbolic functions. If you want to create a symbolic function to be
used by all the machinery of sympy you should subclass the ``Function``
class.
Parameters
----------
symfunc : ``str`` or ``UndefinedFunction`` instance
If ``str``, then create new ``UndefinedFunction`` with this as
name. If `symfunc` is a sympy function, attach implementation to it.
implementation : callable
numerical implementation to be called by ``evalf()`` or ``lambdify``
Returns
-------
afunc : sympy.FunctionClass instance
function with attached implementation
Examples
--------
>>> from sympy.abc import x, y, z
>>> from sympy.utilities.lambdify import lambdify, implemented_function
>>> from sympy import Function
>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> lam_f = lambdify(x, f(x))
>>> lam_f(4)
5
"""
# Delayed import to avoid circular imports
from sympy.core.function import UndefinedFunction
# if name, create function to hold implementation
if isinstance(symfunc, str):
symfunc = UndefinedFunction(symfunc)
elif not isinstance(symfunc, UndefinedFunction):
raise ValueError('symfunc should be either a string or'
' an UndefinedFunction instance.')
# We need to attach as a method because symfunc will be a class
symfunc._imp_ = staticmethod(implementation)
return symfunc
