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scipy.special.lambertw#

scipy.special.lambertw(z, k=0, tol=1e-8)[source]#

Lambert W function.

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:
zarray_like

Input argument.

kint, optional

Branch index.

tolfloat, optional

Evaluation tolerance.

Returns:
warray

w will have the same shape as z.

See also

wrightomega

the Wright Omega function

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].

References

[2]

Corless et al, “On the Lambert W function”, Adv. Comp. Math. 5 (1996) 329-359. https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf

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
(0.56714329040978384+0j)
>>> w * np.exp(w)
(1.0+0j)

Any branch gives a valid inverse:

>>> w = lambertw(1, k=3)
>>> w
(-2.8535817554090377+17.113535539412148j)
>>> w*np.exp(w)
(1.0000000000000002+1.609823385706477e-15j)

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

\(x = a + b e^{c x}\)

for \(x\). We assume \(c\) is not zero. After a little algebra, the equation may be written

\(z e^z = -b c e^{a c}\)

where \(z = c (a - x)\). \(z\) may then be expressed using the Lambert W function

\(z = W(-b c e^{a c})\)

giving

\(x = a - W(-b c e^{a c})/c\)

For example,

>>> a = 3
>>> b = 2
>>> c = -0.5

The solution to \(x = a + b e^{c x}\) is:

>>> x = a - lambertw(-b*c*np.exp(a*c))/c
>>> x
(3.3707498368978794+0j)

Verify that it solves the equation:

>>> a + b*np.exp(c*x)
(3.37074983689788+0j)

The Lambert W function may also be used find the value of the infinite power tower \(z^{z^{z^{\ldots}}}\):

>>> def tower(z, n):
...     if n == 0:
...         return z
...     return z ** tower(z, n-1)
...
>>> tower(0.5, 100)
0.641185744504986
>>> -lambertw(-np.log(0.5)) / np.log(0.5)
(0.64118574450498589+0j)
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