HP49-50G: MLSV OK for -1E-495, not OK for -1E-496, ROOT OK for both

[Albert Chan]

(01-28-2024 12:59 AM)Gil Wrote:  Though, the ROOT solver seems to find all solutions for negative values close to a — up to -1E-495 —, the values it finds for a= -1E-496,-1E-497, -1E-498 and -1E-496 are all the same (-1148.98996141)... and incorrect!

x*e^x = a

min(e^x) = 1e-499      // if e^x = 0 then x*e^x = -inf * 0 = nan
min(x) = ln(1e-499) = -1148.9899614

(01-28-2024 02:01 AM)Gil Wrote:  But your method on my HP50G gives

{ { for -1.E-497: -1151.2925465 } { for -1.E-498: -1151.2925465 } {for -1.E-499: -1151.2925465 } }.

Three same results, and incorrect.

x + ln(x/a) = 0

max(x/a) = 9.99999999999E499
min(x) = -ln(9.99999999999E499) = -1151.2925465

Quote:We should try and alternative solution.

Just changing your formula Ln (x/a) by Ln (x) - Ln(a):
'(1-(LN(x)-LN(a)))/(1+1/x)'

Since x and a are negative real, try (ln(-x) - ln(-a))
Or, with (1+1/x) ≈ 1, just iterate x = ln(-a) - ln(-x)

Cas> c := ln(10) * -100000.; /* = ln(-a) */
−230258.509299

Cas> c - ln(-Ans)
−230270.856257
−230270.856311
−230270.856311 /* = W_-1(-1e-100000) */