(28/48/50) Lambert W Function

[Albert Chan]

(04-02-2023 11:12 PM)Albert Chan Wrote:  lua> a, k = 3+4*I, 5
lua> x = 2*k*pi*I -- guess for W_k(a)

With Newton's rapid convergence, it might be overkill for better W guess ...

x = r cis(θ)
e^x = e^(r*cos(θ)) cis(r*sin(θ))
x*e^x = r*e^(r*cos(θ)) cis(θ+r*sin(θ)) = a

r*e^(r*cos(θ)) = |a|
θ+r*sin(θ) = arg(a) + 2*k*pi = T           // for W branch k

cos(θ) = ln(|a|/r)/r
sin(θ) = (T-θ)/r

We assume k positive --> both r and x imag part positive too.

lua> a, k = 3+4*I, 5
lua> A, T = I.abs(a), 2*k*pi + I.arg(a)
lua> A, T
5      32.34322175389954

lua> c = 0 -- = cos(θ)
lua> r = (T-acos(c)) / sqrt(1-c*c) -- = T - pi/2
lua> c = log(A/r)/r
lua> x = r * I.new(c, sqrt(1-c*c))
lua> x -- W_k(a) guess
(-1.81718109820366+30.71872424960138*I)

Rinse and repeat ...

lua> r = (T-acos(c)) / sqrt(1-c*c)
lua> c = log(A/r)/r
lua> x2 = r * I.new(c, sqrt(1-c*c))
lua> x2
(-1.8170057678766207+30.7133303234809*I)

lua> r = (T-acos(c)) / sqrt(1-c*c)
lua> c = log(A/r)/r
lua> x3 = r * I.new(c, sqrt(1-c*c))
lua> x3
(-1.8170058919343504+30.713334139703402*I)

lua> aitken(x, x2, x3)
(-1.8170058918466334+30.71333413700532*I)

We don't even need machine to support complex numbers.
We could just iterate for (r, c), improve with Aitken's delta-squared process
Assuming K is positve, after (r, c) converged, W_k(a) = complex(r*c, r*sqrt(1-c*c))