At large x the fixed recurrence start makes BESSELI badly wrong

FN-BESSELI-003: At large x the fixed recurrence start makes BESSELI badly wrong

Excel differs from the mathematical answer · measured · recorded 2026-11-04

Excel's recurrence for orders above 1 starts at a fixed j = 2*(n + int(sqrt(40*n))), which is far too low at large x. At x = 709.7 BESSELI of order 2 is 0.20% high, and order 3 is 53% low, where I3(709.7) is 2.4631E306. Orders 0 and 1, which use the polynomial and not the recurrence, are within about 1.6E-7 at the same x.

Correct value: mpmath besseli at 800 bits, order the integer part of n, correctly rounded.

Correct value: mpmath besseli at 800 bits, order the integer part of n, correctly rounded.

Correct value: mpmath besseli at 800 bits, order the integer part of n, correctly rounded.

Basis: Readings at x = 709.7 for orders 0, 1, 2 and 3, compared with mpmath.

Inferred: That the fixed start is the cause is inferred: a binary64 transcription of the published body gives the same bits for orders 0, 2 and 3 at this x.

Open: How far the error extends at other x is not measured.