Skip to main content
`Ctrl`+`K`
scipy.special.

# lambertw

scipy.special.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 `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.

  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

  [1]

  https://en.wikipedia.org/wiki/Lambert_W_function

  [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)