LambertW function

[Albert Chan]

(11-07-2020 08:41 AM)Stevetuc Wrote:  Edit: add initial guess of 0 to avoid terminal screen in cas.
This forces going direct to iterative solver rather than first trying and failing with bisection solver.

You might want to mention guess of 0 will iterate for W0(x), i.e. principle branch. (*)

Also, guess 0 is same as guess x, but wasted 1 Newton iteration.
We might as well use guess = x

Newton: w - (w*exp(w) - x) / (w*exp(w) + exp(w))

With guess 0, first iteration of w = 0 - (0 - x) / (0 + 1) = x


(*) Assumed W0 is not complex (x ≥ -1/e), see comment below.

Comment: For fsolve, some randomization of guess is going on.
With complex ON, if we fsolve again and again, we got different solutions.

XCas> fsolve(w*e^w = -1.78, w=0)

0.0892180498562+1.62562367443*i
-1.4781113814-7.66344321151*i
-3.68225172433+70.6337502365*i
-2.07259091944+13.9900896316*i
-2.9207293675-32.8981741284*i
...

Without randomization, fsolve should not even converge. (w will not flip to complex)
Maybe this is the reason guess randomization kick in ...