lambertw, all branches

[Albert Chan]

(04-07-2023 07:54 PM)Albert Chan Wrote:  It is just here, "equivalent" may mean a difference of +0*I vs -0*I

Just to show how sensitive this ±0*I business is ...

lua> I.W(-1e-9,0,true)
(-1.0000000005000001e-009+0*I)
(-1.000000001e-009+0*I)
(-1.0000000010000002e-009+0*I)

lua> I.W(I.conj(-1e-9),0,true)
(-1.0000000005000001e-009+0*I)
(-1.000000001e-009+6.283185316604365e-009*I)
(3.1450880362038825e-008-4.729032266472273e-010*I)
(-7.848729544901978e-008-9.717506161329017e-008*I)
...

2nd should be complex conjugate of first. W(-1E-9 ± 0*I) ≈ -1.000000001E-9 ± 0*I
Imaginary part sign of (a, W₀(a)) should match.

At first, I thought it is also a case of bad W guess.
To keep x imag part intact, x+1 should be written as x-(-1)
I added +0 to correction too, just to be sure. Technically, it is not needed.

Signed zero: (±0) - 0 = (±0)      ,       (±0) + 0 = 0

Code:

< function complex.log1p(x) local y = x+1; return I.log(y) - (y-1-x)/y end
> function complex.log1p(x) local y = x-(-1); return I.log(y) - ((y-1-x)/y+0) end

This get W₀ guess phase correct, however iterations still diverges.
It is interesting Newton's 1st iteration essentially undo log1p patches.

lua> I.W(I.conj(-1e-9),0,true)
(-1.0000000005000001e-009-0*I)
(-1.000000001e-009+0*I)
(-1.0000000010000002e-009+6.283185319745956e-009*I)
(3.145088037733907e-008-4.729032324603788e-010*I)
...

The problem is Newton correction: 1/slope = x/(x+1). We fix it the same way:

Code:

< h = x/(x+1) * s(x)
> h = x/(x-(-1)) * s(x)

lua> I.W(I.conj(-1e-9),0,true)
(-1.0000000005000001e-009-0*I)
(-1.000000001e-009-0*I)
(-1.0000000010000002e-009-0*I)

I had applied above 2 patches to previous post.