Vous êtes un professionnel et vous avez besoin d'une formation ? RAG (Retrieval-Augmented Generation)
et Fine Tuning d'un LLM Voir le programme détaillé

Module « scipy.special »

Fonction lambertw - module scipy.special

Signature de la fonction lambertw

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

Description

help(scipy.special.lambertw)

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

    :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
(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 :math:`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)

Vous êtes un professionnel et vous avez besoin d'une formation ? Deep Learning avec Python
et Keras et Tensorflow Voir le programme détaillé