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bundles / scipy 1.17.1 / scipy / special / _lambertw / lambertw

function

scipy.special._lambertw:lambertw

source: /scipy/special/_lambertw.py :6

Signature

def   lambertw ( z ,  k = 0 ,  tol = 1e-08 )

Summary

Lambert W function.

Extended Summary

The Lambert W function W(z) is defined as the inverse function of w * exp(w). In other words, the value of W(z) is such that z = W(z) * exp(W(z)) for any complex number z.

The Lambert W function is a multivalued function with infinitely many branches. Each branch gives a separate solution of the equation z = w exp(w). Here, the branches are indexed by the integer k.

Parameters

z : array_like

Input argument.

k : int, optional

Branch index.

tol : float, optional

Evaluation tolerance.

Returns

w : array

w will have the same shape as z.

Notes

All branches are supported by lambertw:

  • lambertw(z) gives the principal solution (branch 0)

  • lambertw(z, k) gives the solution on branch k

The Lambert W function has two partially real branches: the principal branch (k = 0) is real for real z > -1/e, and the k = -1 branch is real for -1/e < z < 0. All branches except k = 0 have a logarithmic singularity at z = 0.

Possible issues

The evaluation can become inaccurate very close to the branch point at -1/e. In some corner cases, lambertw might currently fail to converge, or can end up on the wrong branch.

Algorithm

Halley's iteration is used to invert w * exp(w), using a first-order asymptotic approximation (O(log(w)) or O(w)) as the initial estimate.

The definition, implementation and choice of branches is based on [2].

Examples

The Lambert W function is the inverse of ``w exp(w)``: 
import numpy as np
from scipy.special import lambertw
w = lambertw(1)
✓
w
w * np.exp(w)
✗
Any branch gives a valid inverse: 
w = lambertw(1, k=3)
✓
w
w*np.exp(w)
✗
**Applications to equation-solving** The Lambert W function may be used to solve various kinds of equations. We give two examples here. First, the function can be used to solve implicit equations of the form :math:`x = a + b e^{c x}` for :math:`x`. We assume :math:`c` is not zero. After a little algebra, the equation may be written :math:`z e^z = -b c e^{a c}` where :math:`z = c (a - x)`. :math:`z` may then be expressed using the Lambert W function :math:`z = W(-b c e^{a c})` giving :math:`x = a - W(-b c e^{a c})/c` For example, 
a = 3
b = 2
c = -0.5
✓
The solution to :math:`x = a + b e^{c x}` is: 
x = a - lambertw(-b*c*np.exp(a*c))/c
✓
x
✗
Verify that it solves the equation: 
a + b*np.exp(c*x)
✗
The Lambert W function may also be used find the value of the infinite power tower :math:`z^{z^{z^{\ldots}}}`: 
def tower(z, n):
    if n == 0:
        return z
    return z ** tower(z, n-1)
tower(0.5, 100)
✓
-lambertw(-np.log(0.5)) / np.log(0.5)
✗

See also

wrightomega

the Wright Omega function

Aliases

  • scipy.special.lambertw

Referenced by

This package