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 __future__ import print_function, division

from sympy.external import import_module
from sympy.core.compatibility import exec_, is_sequence, iterable, string_types

import inspect

# These are the namespaces the lambda functions will use.
MATH = {}
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.

# Mappings between sympy and other modules function names.
    "Abs": "fabs",
    "ceiling": "ceil",
    "E": "e",
    "ln": "log",

    "elliptic_k": "ellipk",
    "elliptic_f": "ellipf",
    "elliptic_e": "ellipe",
    "elliptic_pi": "ellippi",
    "ceiling": "ceil",
    "chebyshevt": "chebyt",
    "chebyshevu": "chebyu",
    "E": "e",
    "I": "j",
    "ln": "log",
    "oo": "inf",
    "LambertW": "lambertw",
    "Matrix": "matrix",
    "MutableDenseMatrix": "matrix",
    "ImmutableMatrix": "matrix",
    "conjugate": "conj",
    "dirichlet_eta": "altzeta",
    "Ei": "ei",
    "Shi": "shi",
    "Chi": "chi",
    "Si": "si",
    "Ci": "ci"

    "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",
    "MutableDenseMatrix": "matrix",
    "ImmutableMatrix": "matrix",
    "Max": "amax",
    "Min": "amin",
    "oo": "inf",
    "re": "real",

# Available 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 *",
        "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.
        namespace, namespace_default, translations, import_commands = MODULES[
    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:

    for import_command in import_commands:
        if import_command.startswith('import_module'):
            module = eval(import_command)

            if module is not None:
                exec_(import_command, {}, namespace)
            except ImportError:

        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. 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, with higher priority given to entries appearing first. The default behavior is to substitute all arguments in the provided expression with dummy symbols. This allows for applied functions (e.g. f(t)) to be supplied as arguments. Call the function with dummify=False if dummy substitution is unwanted. If you want to view the lambdified function or provide "sympy" as the module, you should probably set dummify=False. Usage ===== (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) Examples ======== >>> from sympy.utilities.lambdify import implemented_function, lambdify >>> from sympy import sqrt, sin, Matrix >>> from sympy import Function >>> 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 >>> f = lambdify((x, y), Matrix((x, x + y)).T, modules='sympy') >>> f(1, 2) Matrix([[1, 3]]) 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: >>> 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. module_provided = True if modules is None: module_provided = False # 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, dummify=True) 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, dummify=False): """ 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])' """ # Transforming everything to strings. from sympy.matrices import DeferredVector from sympy import Dummy, sympify, Symbol, Function 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 def sub_args(args, dummies_dict): if isinstance(args, str): return args elif isinstance(args, DeferredVector): return str(args) elif iterable(args): flatten = lambda *n: (e for a in n for e in (flatten(*a) if iterable(a) else (a,))) dummies = flatten([sub_args(a, dummies_dict) for a in args]) return ",".join(str(a) for a in dummies) else: if isinstance(args, Function): dummies = Dummy() dummies_dict.update({args : dummies}) return str(dummies) else: return str(args) def sub_expr(expr, dummies_dict): try: expr = sympify(expr).xreplace(dummies_dict) except: if isinstance(expr, DeferredVector): pass elif isinstance(expr, dict): k = [sub_expr(sympify(a), dummies_dict) for a in expr.keys()] v = [sub_expr(sympify(a), dummies_dict) for a in expr.values()] expr = dict(zip(k, v)) elif isinstance(expr, tuple): expr = tuple(sub_expr(sympify(a), dummies_dict) for a in expr) elif isinstance(expr, list): expr = [sub_expr(sympify(a), dummies_dict) for a in expr] return expr # Transform args dummies_dict = {} if dummify: args = sub_args(args, dummies_dict) else: if isinstance(args, str): pass elif iterable(args, exclude=DeferredVector): args = ",".join(str(a) for a in args) # Transform expr if dummify: if isinstance(expr, str): pass else: expr = sub_expr(expr, dummies_dict) expr = lambdarepr(expr) 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 >>> 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 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 >>> 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, string_types): 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