(28/48/50) Lambert W Function

[John Keith]

(03-21-2023 05:15 PM)Albert Chan Wrote:  I am unfamilar with RPL, can you post actual formula for W(a) guess, and the iteration formula?


Python mpmath lambertw(z, k=0) support all branches.

https://mpmath.org/doc/0.19/functions/po...w-function
https://github.com/mpmath/mpmath/blob/ma...ns.py#L464

Many thanks for the mpmath github reference, that solved my branch problem! Updated program will follow soon.

Here are the formulas that you requested:

Formula for global approximation of W(z) for branch 0.
From "New approximations to the principal real-valued
branch of the Lambert W-function", by Iacono and Boyd.

-1 + a*ln((1 + b*y)/(1 + c*ln(1+y))

where

y = sqrt(1+e*x)

c = (e^(1/a) - 1 - sqrt(2)/a)/(1 - e^(1/a)*ln(2))

b = sqrt(2)/a + c


With 'a' determined empirically to be 2.036, final formula is:

-1 + 2.036*ln((1 + 1.14956131*y)/(1 + 0.4549574*ln(1 + y))


Recurrence from same paper, also from "Guaranteed- and high-precision
evaluation of the Lambert W function" by Lajos Lóczi.

B'(x) = (B(x)/(1 + B(x))*(1 + ln(x/B(x))

where

B(x) is current estimate
B'(x) is new estimate