lambertw, all branches

[Albert Chan]

(04-10-2023 04:44 PM)Albert Chan Wrote:  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.

As expected, better complex.log make little difference.
I considered Kahan's clog(z), but Python's cmath.log code is simpler.

Code:

function complex.log(z)
    local x, y = I.real(z), I.imag(z)
    local h = hypot(x, y)
    if 0.71 <= h and h <= 1.73 then
        if abs(x) < abs(y) then x,y = y,x end
        h = log1p((x+1)*(x-1) + y*y)/2
    else
        h = log(h)
    end
    return I.new(h, I.arg(z))
end

lua> a = 1e-100-1e-10*I
lua> I.W(a, 0, true)
(5.0000000000000005e-021-1e-010*I)      -- log1p(a) ≈ a - a^2/2
(1.0000000413701856e-020-1e-010*I)
(1.0000000413701856e-020-1e-010*I)
(1.0000000000000001e-020-1e-010*I)      true