Lambdify¶
This module provides convenient functions to transform SymPy expressions to lambda functions which can be used to calculate numerical values very fast.
- sympy.utilities.lambdify.implemented_function(symfunc, implementation)[source]¶
Add numerical
implementation
to functionsymfunc
.symfunc
can be anUndefinedFunction
instance, or a name string. In the latter case we create anUndefinedFunction
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
orUndefinedFunction
instanceIf
str
, then create newUndefinedFunction
with this as name. Ifsymfunc
is an Undefined function, create a new function with the same name and the implemented function attached.implementation : callable
numerical implementation to be called by
evalf()
orlambdify
- Returns:
afunc : sympy.FunctionClass instance
function with attached implementation
Examples
>>> from sympy.abc import x >>> from sympy.utilities.lambdify import implemented_function >>> from sympy import lambdify >>> f = implemented_function('f', lambda x: x+1) >>> lam_f = lambdify(x, f(x)) >>> lam_f(4) 5
- sympy.utilities.lambdify.lambdastr(args, expr, printer=None, dummify=None)[source]¶
Returns a string that can be evaluated to a lambda function.
Examples
>>> 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])'
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))(_0,_1[0],_1[1])'
- sympy.utilities.lambdify.lambdify(args, expr, modules=None, printer=None, use_imps=True, dummify=False, cse=False, docstring_limit=1000)[source]¶
Convert a SymPy expression into a function that allows for fast numeric evaluation.
Warning
This function uses
exec
, and thus should not be used on unsanitized input.Deprecated since version 1.7: Passing a set for the args parameter is deprecated as sets are unordered. Use an ordered iterable such as a list or tuple.
- Parameters:
args : List[Symbol]
A variable or a list of variables whose nesting represents the nesting of the arguments that will be passed to the function.
Variables can be symbols, undefined functions, or matrix symbols.
>>> from sympy import Eq >>> from sympy.abc import x, y, z
The list of variables should match the structure of how the arguments will be passed to the function. Simply enclose the parameters as they will be passed in a list.
To call a function like
f(x)
then[x]
should be the first argument tolambdify
; for this case a singlex
can also be used:>>> f = lambdify(x, x + 1) >>> f(1) 2 >>> f = lambdify([x], x + 1) >>> f(1) 2
To call a function like
f(x, y)
then[x, y]
will be the first argument of thelambdify
:>>> f = lambdify([x, y], x + y) >>> f(1, 1) 2
To call a function with a single 3-element tuple like
f((x, y, z))
then[(x, y, z)]
will be the first argument of thelambdify
:>>> f = lambdify([(x, y, z)], Eq(z**2, x**2 + y**2)) >>> f((3, 4, 5)) True
If two args will be passed and the first is a scalar but the second is a tuple with two arguments then the items in the list should match that structure:
>>> f = lambdify([x, (y, z)], x + y + z) >>> f(1, (2, 3)) 6
expr : Expr
An expression, list of expressions, or matrix to be evaluated.
Lists may be nested. If the expression is a list, the output will also be a list.
>>> f = lambdify(x, [x, [x + 1, x + 2]]) >>> f(1) [1, [2, 3]]
If it is a matrix, an array will be returned (for the NumPy module).
>>> from sympy import Matrix >>> f = lambdify(x, Matrix([x, x + 1])) >>> f(1) [[1] [2]]
Note that the argument order here (variables then expression) is used to emulate the Python
lambda
keyword.lambdify(x, expr)
works (roughly) likelambda x: expr
(see How It Works below).modules : str, optional
Specifies the numeric library to use.
If not specified, modules defaults to:
["scipy", "numpy"]
if SciPy is installed["numpy"]
if only NumPy is installed["math", "mpmath", "sympy"]
if neither is installed.
That is, SymPy functions are replaced as far as possible by either
scipy
ornumpy
functions if available, and Python’s standard librarymath
, ormpmath
functions otherwise.modules can be one of the following types:
The strings
"math"
,"mpmath"
,"numpy"
,"numexpr"
,"scipy"
,"sympy"
, or"tensorflow"
or"jax"
. This uses the corresponding printer and namespace mapping for that module.A module (e.g.,
math
). This uses the global namespace of the module. If the module is one of the above known modules, it will also use the corresponding printer and namespace mapping (i.e.,modules=numpy
is equivalent tomodules="numpy"
).A dictionary that maps names of SymPy functions to arbitrary functions (e.g.,
{'sin': custom_sin}
).A list that contains a mix of the arguments above, with higher priority given to entries appearing first (e.g., to use the NumPy module but override the
sin
function with a custom version, you can use[{'sin': custom_sin}, 'numpy']
).
dummify : bool, optional
Whether or not the variables in the provided expression that are not valid Python identifiers are substituted with dummy symbols.
This allows for undefined functions like
Function('f')(t)
to be supplied as arguments. By default, the variables are only dummified if they are not valid Python identifiers.Set
dummify=True
to replace all arguments with dummy symbols (ifargs
is not a string) - for example, to ensure that the arguments do not redefine any built-in names.cse : bool, or callable, optional
Large expressions can be computed more efficiently when common subexpressions are identified and precomputed before being used multiple time. Finding the subexpressions will make creation of the ‘lambdify’ function slower, however.
When
True
,sympy.simplify.cse
is used, otherwise (the default) the user may pass a function matching thecse
signature.docstring_limit : int or None
When lambdifying large expressions, a significant proportion of the time spent inside
lambdify
is spent producing a string representation of the expression for use in the automatically generated docstring of the returned function. For expressions containing hundreds or more nodes the resulting docstring often becomes so long and dense that it is difficult to read. To reduce the runtime of lambdify, the rendering of the full expression inside the docstring can be disabled.When
None
, the full expression is rendered in the docstring. When0
or a negativeint
, an ellipsis is rendering in the docstring instead of the expression. When a strictly positiveint
, if the number of nodes in the expression exceedsdocstring_limit
an ellipsis is rendered in the docstring, otherwise a string representation of the expression is rendered as normal. The default is1000
.
Explanation
For example, to convert the SymPy expression
sin(x) + cos(x)
to an equivalent NumPy function that numerically evaluates it:>>> from sympy import sin, cos, symbols, lambdify >>> import numpy as np >>> x = symbols('x') >>> expr = sin(x) + cos(x) >>> expr sin(x) + cos(x) >>> f = lambdify(x, expr, 'numpy') >>> a = np.array([1, 2]) >>> f(a) [1.38177329 0.49315059]
The primary purpose of this function is to provide a bridge from SymPy expressions to numerical libraries such as NumPy, SciPy, NumExpr, mpmath, and tensorflow. In general, SymPy functions do not work with objects from other libraries, such as NumPy arrays, and functions from numeric libraries like NumPy or mpmath do not work on SymPy expressions.
lambdify
bridges the two by converting a SymPy expression to an equivalent numeric function.The basic workflow with
lambdify
is to first create a SymPy expression representing whatever mathematical function you wish to evaluate. This should be done using only SymPy functions and expressions. Then, uselambdify
to convert this to an equivalent function for numerical evaluation. For instance, above we createdexpr
using the SymPy symbolx
and SymPy functionssin
andcos
, then converted it to an equivalent NumPy functionf
, and called it on a NumPy arraya
.Examples
>>> from sympy.utilities.lambdify import implemented_function >>> from sympy import sqrt, sin, Matrix >>> from sympy import Function >>> from sympy.abc import w, 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 >>> row = lambdify((x, y), Matrix((x, x + y)).T, modules='sympy') >>> row(1, 2) Matrix([[1, 3]])
lambdify
can be used to translate SymPy expressions into mpmath functions. This may be preferable to usingevalf
(which uses mpmath on the backend) in some cases.>>> f = lambdify(x, sin(x), 'mpmath') >>> f(1) 0.8414709848078965
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
The
flatten
function can be used to always work with flattened arguments:>>> from sympy.utilities.iterables import flatten >>> 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. This can be used with undefined functions using theimplemented_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 theuse_imps
input parameter is False.Usage with Tensorflow:
>>> import tensorflow as tf >>> from sympy import Max, sin, lambdify >>> from sympy.abc import x
>>> f = Max(x, sin(x)) >>> func = lambdify(x, f, 'tensorflow')
After tensorflow v2, eager execution is enabled by default. If you want to get the compatible result across tensorflow v1 and v2 as same as this tutorial, run this line.
>>> tf.compat.v1.enable_eager_execution()
If you have eager execution enabled, you can get the result out immediately as you can use numpy.
If you pass tensorflow objects, you may get an
EagerTensor
object instead of value.>>> result = func(tf.constant(1.0)) >>> print(result) tf.Tensor(1.0, shape=(), dtype=float32) >>> print(result.__class__) <class 'tensorflow.python.framework.ops.EagerTensor'>
You can use
.numpy()
to get the numpy value of the tensor.>>> result.numpy() 1.0
>>> var = tf.Variable(2.0) >>> result = func(var) # also works for tf.Variable and tf.Placeholder >>> result.numpy() 2.0
And it works with any shape array.
>>> tensor = tf.constant([[1.0, 2.0], [3.0, 4.0]]) >>> result = func(tensor) >>> result.numpy() [[1. 2.] [3. 4.]]
Notes
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-functionsIn the above examples, the generated functions can accept scalar values or numpy arrays as arguments. However, in some cases the generated function relies on the input being a numpy array:
>>> import numpy >>> from sympy import Piecewise >>> from sympy.testing.pytest import ignore_warnings >>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "numpy")
>>> with ignore_warnings(RuntimeWarning): ... f(numpy.array([-1, 0, 1, 2])) [-1. 0. 1. 0.5]
>>> f(0) Traceback (most recent call last): ... ZeroDivisionError: division by zero
In such cases, the input should be wrapped in a numpy array:
>>> with ignore_warnings(RuntimeWarning): ... float(f(numpy.array([0]))) 0.0
Or if numpy functionality is not required another module can be used:
>>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "math") >>> f(0) 0
How It Works
When using this function, it helps a great deal to have an idea of what it is doing. At its core, lambdify is nothing more than a namespace translation, on top of a special printer that makes some corner cases work properly.
To understand lambdify, first we must properly understand how Python namespaces work. Say we had two files. One called
sin_cos_sympy.py
, with# sin_cos_sympy.py from sympy.functions.elementary.trigonometric import (cos, sin) def sin_cos(x): return sin(x) + cos(x)
and one called
sin_cos_numpy.py
with# sin_cos_numpy.py from numpy import sin, cos def sin_cos(x): return sin(x) + cos(x)
The two files define an identical function
sin_cos
. However, in the first file,sin
andcos
are defined as the SymPysin
andcos
. In the second, they are defined as the NumPy versions.If we were to import the first file and use the
sin_cos
function, we would get something like>>> from sin_cos_sympy import sin_cos >>> sin_cos(1) cos(1) + sin(1)
On the other hand, if we imported
sin_cos
from the second file, we would get>>> from sin_cos_numpy import sin_cos >>> sin_cos(1) 1.38177329068
In the first case we got a symbolic output, because it used the symbolic
sin
andcos
functions from SymPy. In the second, we got a numeric result, becausesin_cos
used the numericsin
andcos
functions from NumPy. But notice that the versions ofsin
andcos
that were used was not inherent to thesin_cos
function definition. Bothsin_cos
definitions are exactly the same. Rather, it was based on the names defined at the module where thesin_cos
function was defined.The key point here is that when function in Python references a name that is not defined in the function, that name is looked up in the “global” namespace of the module where that function is defined.
Now, in Python, we can emulate this behavior without actually writing a file to disk using the
exec
function.exec
takes a string containing a block of Python code, and a dictionary that should contain the global variables of the module. It then executes the code “in” that dictionary, as if it were the module globals. The following is equivalent to thesin_cos
defined insin_cos_sympy.py
:>>> import sympy >>> module_dictionary = {'sin': sympy.sin, 'cos': sympy.cos} >>> exec(''' ... def sin_cos(x): ... return sin(x) + cos(x) ... ''', module_dictionary) >>> sin_cos = module_dictionary['sin_cos'] >>> sin_cos(1) cos(1) + sin(1)
and similarly with
sin_cos_numpy
:>>> import numpy >>> module_dictionary = {'sin': numpy.sin, 'cos': numpy.cos} >>> exec(''' ... def sin_cos(x): ... return sin(x) + cos(x) ... ''', module_dictionary) >>> sin_cos = module_dictionary['sin_cos'] >>> sin_cos(1) 1.38177329068
So now we can get an idea of how
lambdify
works. The name “lambdify” comes from the fact that we can think of something likelambdify(x, sin(x) + cos(x), 'numpy')
aslambda x: sin(x) + cos(x)
, wheresin
andcos
come from thenumpy
namespace. This is also why the symbols argument is first inlambdify
, as opposed to most SymPy functions where it comes after the expression: to better mimic thelambda
keyword.lambdify
takes the input expression (likesin(x) + cos(x)
) andConverts it to a string
Creates a module globals dictionary based on the modules that are passed in (by default, it uses the NumPy module)
Creates the string
"def func({vars}): return {expr}"
, where{vars}
is the list of variables separated by commas, and{expr}
is the string created in step 1., thenexec``s that string with the module globals namespace and returns ``func
.
In fact, functions returned by
lambdify
support inspection. So you can see exactly how they are defined by usinginspect.getsource
, or??
if you are using IPython or the Jupyter notebook.>>> f = lambdify(x, sin(x) + cos(x)) >>> import inspect >>> print(inspect.getsource(f)) def _lambdifygenerated(x): return sin(x) + cos(x)
This shows us the source code of the function, but not the namespace it was defined in. We can inspect that by looking at the
__globals__
attribute off
:>>> f.__globals__['sin'] <ufunc 'sin'> >>> f.__globals__['cos'] <ufunc 'cos'> >>> f.__globals__['sin'] is numpy.sin True
This shows us that
sin
andcos
in the namespace off
will benumpy.sin
andnumpy.cos
.Note that there are some convenience layers in each of these steps, but at the core, this is how
lambdify
works. Step 1 is done using theLambdaPrinter
printers defined in the printing module (seesympy.printing.lambdarepr
). This allows different SymPy expressions to define how they should be converted to a string for different modules. You can change which printerlambdify
uses by passing a custom printer in to theprinter
argument.Step 2 is augmented by certain translations. There are default translations for each module, but you can provide your own by passing a list to the
modules
argument. For instance,>>> def mysin(x): ... print('taking the sin of', x) ... return numpy.sin(x) ... >>> f = lambdify(x, sin(x), [{'sin': mysin}, 'numpy']) >>> f(1) taking the sin of 1 0.8414709848078965
The globals dictionary is generated from the list by merging the dictionary
{'sin': mysin}
and the module dictionary for NumPy. The merging is done so that earlier items take precedence, which is whymysin
is used above instead ofnumpy.sin
.If you want to modify the way
lambdify
works for a given function, it is usually easiest to do so by modifying the globals dictionary as such. In more complicated cases, it may be necessary to create and pass in a custom printer.Finally, step 3 is augmented with certain convenience operations, such as the addition of a docstring.
Understanding how
lambdify
works can make it easier to avoid certain gotchas when using it. For instance, a common mistake is to create a lambdified function for one module (say, NumPy), and pass it objects from another (say, a SymPy expression).For instance, say we create
>>> from sympy.abc import x >>> f = lambdify(x, x + 1, 'numpy')
Now if we pass in a NumPy array, we get that array plus 1
>>> import numpy >>> a = numpy.array([1, 2]) >>> f(a) [2 3]
But what happens if you make the mistake of passing in a SymPy expression instead of a NumPy array:
>>> f(x + 1) x + 2
This worked, but it was only by accident. Now take a different lambdified function:
>>> from sympy import sin >>> g = lambdify(x, x + sin(x), 'numpy')
This works as expected on NumPy arrays:
>>> g(a) [1.84147098 2.90929743]
But if we try to pass in a SymPy expression, it fails
>>> g(x + 1) Traceback (most recent call last): ... TypeError: loop of ufunc does not support argument 0 of type Add which has no callable sin method
Now, let’s look at what happened. The reason this fails is that
g
callsnumpy.sin
on the input expression, andnumpy.sin
does not know how to operate on a SymPy object. As a general rule, NumPy functions do not know how to operate on SymPy expressions, and SymPy functions do not know how to operate on NumPy arrays. This is why lambdify exists: to provide a bridge between SymPy and NumPy.However, why is it that
f
did work? That’s becausef
does not call any functions, it only adds 1. So the resulting function that is created,def _lambdifygenerated(x): return x + 1
does not depend on the globals namespace it is defined in. Thus it works, but only by accident. A future version oflambdify
may remove this behavior.Be aware that certain implementation details described here may change in future versions of SymPy. The API of passing in custom modules and printers will not change, but the details of how a lambda function is created may change. However, the basic idea will remain the same, and understanding it will be helpful to understanding the behavior of lambdify.
In general: you should create lambdified functions for one module (say, NumPy), and only pass it input types that are compatible with that module (say, NumPy arrays). Remember that by default, if the
module
argument is not provided,lambdify
creates functions using the NumPy and SciPy namespaces.