lambertw, all branches

[Albert Chan]

(01-23-2024 04:57 PM)Gil Wrote:  We accept...
You decided to accept that the real part in question is indeed zero...

That's not what I mean.

I specifically say real part will *not* be zero, even with more precision.
This is because of numerical limits, a = 3/2*pi is only an approximation.

I accepted limited precision limits, and confirm symbolic result (if pi is true pi, ε is true 0 ...)

see https://en.wikipedia.org/wiki/Experimental_mathematics

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Test for W_k(-pi/2), k=0 or -1, is to make sure we don't get stuck in loops.

That's the reason for f = x + ln(x) - T, where T = ln_k(a) = (ln(a) + 2*k*pi*I)

(04-09-2023 03:59 AM)Albert Chan Wrote:  * f formula rearranged, to cause catastrophic cancellation, on purpose!

With cancellation, final x might not be as accurate, but loops will terminate.
This is then used as an extremely good guess, with another f.

(04-19-2023 01:30 AM)Albert Chan Wrote:  We can use Newton's method, f = x*e^x - a, for finishing touches