lambertw(z, k=0, tol=1e-08)¶
r lambertw(z, k=0, tol=1e-8)
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
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.
- z : array_like
- Input argument.
- k : int, optional
- Branch index.
- tol : float, optional
- Evaluation tolerance.
- w : array
- w will have the same shape as z.
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 = -1branch is real for
-1/e < z < 0. All branches except
k = 0have a logarithmic singularity at
z = 0.
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.
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 .
wrightomega : the Wright Omega function
 http://en.wikipedia.org/wiki/Lambert_W_function  Corless et al, “On the Lambert W function”, Adv. Comp. Math. 5 (1996) 329-359. http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf
The Lambert W function is the inverse of
>>> 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, such as finding the value of the infinite power tower :
>>> 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)