ELF and GNOME: Two tiny codes to evaluate the real zeros of the Bessel functions of the first kind for real orders

✍ Scribed by J. Segura; A. Gil

• Book ID: 104109892
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 666 KB
• Volume: 117
• Category: Article
• ISSN: 0010-4655
• DOI: 10.1016/s0010-4655(98)00193-3

✦ Synopsis

Two codes to evaluate the real zeros (j~..~.) of the Bessel functions of the first kind J~ (x) for real orders v are presented. The codes are based on a Newton-Raphson iteration over the monotonic function f~ (x) = x2~-JJ~ (x)/J,~_ t(x).

The code ELF is a remarkably short program for finding, given any starting value x0 > 0 and any real order, the zero of J,~(x) in the neighborhood of x0 (x0 and the zero in the same branch of f,(x)). GNOME is a modification of ELF for finding the zeros of .1, (x) inside a given interval [Xmi,, Xm,~]; for simplicity, we restrict the code GNOME to work for ~, > -1, which is the region of greatest practical use, where all the zeros of J~(x) are real.

The method is especially efficient for moderate values of v and for small zeros, where asymptotic expansions tend to fail and, besides, contrary to existing algorithms, enables the search of the re~l zeros for real orders, including negative orders. @