Lambdify

This module provides convenient functions to transform diofant expressions to lambda functions which can be used to calculate numerical values very fast.

diofant.utilities.lambdify.implemented_function(symfunc, implementation)[source]

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 Diofant 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 diofant function, attach implementation to it.
  • implementation (callable) – numerical implementation to be called by evalf() or lambdify
Returns:

afunc (diofant.FunctionClass instance) – function with attached implementation

Examples

>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> lam_f = lambdify(x, f(x))
>>> lam_f(4)
5
diofant.utilities.lambdify.lambdastr(args, expr, printer=None, dummify=False)[source]

Returns a string that can be evaluated to a lambda function.

Examples

>>> lambdastr(x, x**2)
'lambda x: (x**2)'
>>> lambdastr((x, y, z), [z, y, x])
'lambda x,y,z: ([z, y, x])'

Although tuples may not appear as arguments to lambda in Python 3, lambdastr will create a lambda function that will unpack the original arguments so that nested arguments can be handled:

>>> lambdastr((x, (y, z)), x + y)
'lambda _0,_1: (lambda x,y,z: (x + y))(*list(__flatten_args__([_0,_1])))'
diofant.utilities.lambdify.lambdify(args, expr, modules=None, printer=None, use_imps=True, dummify=True)[source]

Returns a lambda function for fast calculation of numerical values.

If not specified differently by the user, modules defaults to ["numpy"] if NumPy is installed, and ["math", "mpmath", "sympy"] if it isn’t, that is, Diofant functions are replaced as far as possible by either numpy functions if available, and Python’s standard library math, or mpmath functions otherwise. To change this behavior, the “modules” argument can be used. It accepts:

  • the strings “math”, “mpmath”, “numpy”, “numexpr”, “diofant”
  • any modules (e.g. math)
  • dictionaries that map names of diofant 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 (and \(args\) is not a string). If you want to view the lambdified function or provide “diofant” as the module, you should probably set dummify=False.

For functions involving large array calculations, numexpr can provide a significant speedup over numpy. Please note that the available functions for numexpr are more limited than numpy but can be expanded with implemented_function and user defined subclasses of Function. If specified, numexpr may be the only option in modules. The official list of numexpr functions can be found at: https://numexpr.readthedocs.io/en/latest/user_guide.html#supported-functions

In previous releases lambdify replaced Matrix with numpy.matrix by default. As of release 0.7.7 numpy.array is the default. To get the old default behavior you must pass in [{'ImmutableMatrix': numpy.matrix}, 'numpy'] to the modules kwarg.

  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", "diofant"))
    
  2. Use some other module:

    >>> import numpy
    >>> f = lambdify((x, y), tan(x*y), numpy)
    
    Attention: There are naming differences between numpy and diofant. So if

    you simply take the numpy module, e.g. diofant.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 a dictionary defining custom functions:

    >>> def my_cool_function(x): return 'sin(%s) is cool' % x
    >>> myfuncs = {"sin" : my_cool_function}
    >>> f = lambdify(x, sin(x), myfuncs); f(1)
    'sin(1) is cool'
    

Examples

>>> from diofant.abc import w
>>> 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
>>> row = lambdify((x, y), Matrix((x, x + y)).T, modules='diofant')
>>> row(1, 2)
Matrix([[1, 3]])

Tuple arguments are handled and the lambdified function should be called with the same type of arguments as were used to create the function.:

>>> f = lambdify((x, (y, z)), x + y)
>>> f(1, (2, 4))
3

A more robust way of handling this is to always work with flattened arguments:

>>> args = w, (x, (y, z))
>>> vals = 1, (2, (3, 4))
>>> f = lambdify(flatten(args), w + x + y + z)
>>> f(*flatten(vals))
10

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.