lambertw, all branches

[Albert Chan]

(01-28-2024 11:18 PM)Albert Chan Wrote:  But let's try a *bad* guess, x0=B+j, crossed discontinuity!
(this guess for for W(a,-1) would get stuck in endless loops.)

This is a safe way to fix stuck in loops problem for bad guess.
We know ahead of time final sign of imag part is s or 0 (if x is real)
If any iterations does not match the sign, LN will nudge it back.

Code:

def LN(x, bad=-s):
    x = ln(x)
    return conj(x) if sign(im(x))==bad else x

For debugging purpose, lets add a way to input user-supplied guess.
Put this right before Newton while 1 loop

Code:

if verbal and verbal!=True: x = verbal

Repeat the same test, with bad guess

p2> a = mpc(-0.1)
p2> B = ln(-a)
p2> W(a, -1, B+j)

p2> W.W(a, -1, B+j)
(-2.30258509299405 + 1.0j)
(-3.14484755109194 - 1.43195611385943j)
(-3.45450302775194 - 0.017653789718918j)
(-3.57799843751793 + 0.000256987554174169j)
(-3.57715209366168 - 0.000199346248602327j)
(-3.57715206180199 - 8.48877473305296e-13j)
(-3.5771520639573 + 5.31405873378874e-16j)
(-3.5771520639573 + 0.0j)

It is hard to nudge back to real line, but it work!

I think guess is good enough without this trick, so it is not updated to version 3.
Perhaps I would put it to next version.