lambertw, all branches

[Albert Chan]

(04-09-2023 03:59 AM)Albert Chan Wrote:  [*] if |Im(x)| < pi, we are on W branch 0, or branch ±1 with small imag part.

Another way to look at this, if above is true, we can simplify f:

< f = x - (ln(a) - ln(x)) - 2*k*pi*i
> f = x - ln(a/x)

f general form has issues with catastrophic cancellation, simplifed f is much better.

The problem is we don't know |Im(x)| < pi, until we solve it.
Also, f simplifed form have multiple roots. A good guess is needed to get correct branch.
For f general form, with 1 root, guess good enough for convergence is all that is needed.

We use f general form for *good* guess, correct with using simplified f (if applicable)

lua> a = 1e-100-1e-10*I
lua> I.W(a, 0, true)
(1e-100-1e-010*I)
(1.000000082740371e-020-1e-010*I)
(1.000000082740371e-020-1e-010*I)        -- general f=0 solution
(1.0000000000000002e-020-1e-010*I)      true

Here, initial guess log1p(a) ≈ a is not good (I.log code need work ...)
But, it doesn't matter. Guess is good enough for general f to converge.
f simplified form, with 1 Newton step, we get W(a) ≈ a - a*a