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

Two codes to evaluate the real zeros (jv.s) of the Bessel functions of the first kind Jv(x) for real orders v are presented. The codes are based on a Newton-Raphson iteration over the monotonic function ƒv(x) = x2v−1Jv(x)/Jv−1(x). The code ELF is a remarkably short program for finding, given any starting value x0 > 0 and any real order, the zero of Jv(x) in the neighborhood of x0 (x0 and the zero in the same branch of ƒv(x)). GNOME is a modification of ELF for finding the zeros of Jv(x) inside a given interval [xmin, xmax; for simplicity, we restrict the code GNOME to work for v > −1, which is the region of greatest practical use, where all the zeros of Jv(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 real zeros for real orders, including negative orders.