(28/48/50) Lambert W Function
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04-02-2023, 11:12 PM
Post: #23
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RE: (28/48/50) Lambert W Function
(03-26-2023 06:43 PM)Albert Chan Wrote: Because solving for y = e^W(a) is easier, W code is not recommended. W code is perfect for other branches! (expW code for W branch 0, -1) Numbering the branches of the Lambert W function Except for 2*k*pi*I term, it is basically same formula for W code! lua> f = fn'x: x + I.log(x) - I.log(a) - 2*k*pi*I' lua> df = fn'x: 1 + 1/x' Let X = |x|, A = |a| log(x) - log(a) = log(X/A) + I*(arg(x) - arg(a)) log(X/A) = 1 + log(X*(r+err)/A) ≈ 1 + log1p((X*err - (A+r) + r*(1+X)) / A) This means we can evaluate f accurately around branch point! f=0 guaranteed we get the correct branch, which is nice. There is no need to use complicated W guesses. lua> a, k = 3+4*I, 5 lua> x = 2*k*pi*I -- guess for Wk(a) lua> repeat h=f(x)/df(x); x=x-h; print(x, I.abs(h)) until x == x+1e-6*h (-1.815554248884793+30.714634540472343*I) 1.9462907525577031 (-1.817005891456528+30.71333413897562*I) 0.0019489253984594729 (-1.8170058918466274+30.713334137004896*I) 2.0089622485451688e-009 (-1.8170058918466274+30.713334137004896*I) 0 >>> from mpmath import * >>> lambertw(3+4j, k=5) mpc(real='-1.8170058918466274', imag='30.713334137004896') |
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