Lambert W Function (hp-42s)

[Albert Chan]

(10-01-2020 09:37 AM)Werner Wrote:  nitpicking: it won't recognize (-1/e,0).

I don't like testing special cases, if I can avoid it.
Better approach is to not returning guess y0 = 1/e, even if x ≈ -1/e

One way is to take advantage of calculator internal extra precision of PI (lets call it π)

Free42: [PI] [SIN]       → -1.158028306006248941790250554076922e-34

SIN(PI) = SIN(π - PI) ≈ π - PI

Instead of adding an ε to 1/e, we get more bang for the buck if ε added to (x+1/e)

Thus, another approach is to remove testing x = -1/e altogether, replacing Line 1 to 11 to this:

Code:

01▸LBL "eW"
02 -1
03 E↑X
04 ENTER
05 RCL+ ST Z    # x+r   r   x
06 PI
07 SIN
08 -            # x+r+ε r   x   x

To have guess returning r = 1/e, we need:

y0 = r + √(2r*(x+r+ε)) + 0.3 (x+r+ε) = r

→ √(x+r+ε) [ √(2r) + 0.3 √(x+r+ε) ] = 0
→ √(x+r+ε) = 0 or (-√(2r)/0.3 ≈ -2.859213)

When (x+r) approaching -ε, (x+r) will suffer massive cancellations, thus x+r+ε ≠ 0

√(x+r+ε) > 0 or gone complex.