Lambert W Function (hp-42s)

[lyuka]

I'm sorry the word fail was wrong. It meant go bad from a convergence point of view, even if it might or might not fail.

In my opinion, it is not appropriate to use ln(x + 1) as the initial guess of e^W(x) if x is very close to -1 / e because it would require so many itterations very close to -1 / e (e.g. -1 / e + 1E-11).

Using Free42 with the example value of x = -1 / e + 1E-11 ~= -0.3678794411614423215955237701614609,
it requires 22 itterations if you start with the guess of y0 = ln(x + 1) ~= -0.4586751453712621239530755133385323.

On the other hand, starting with the guess of y0 = 1 / e + sqrt ((2 / e) * (x + 1 / e)) + 0.3 * (x + 1 / e) ~= 0.3678821536620134320783140520913751, only two itterations are needed.
In this case the guess is already correct upto 12 digits.