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 ofW(z)
is such thatz = W(z) * exp(W(z))
for any complex numberz
.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 thek = -1
branch is real for-1/e < z < 0
. All branches exceptk = 0
have a logarithmic singularity atz = 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)