Amparo Gil

Algorithm 822: GIZ, HIZ: two Fortran 77 routines for the computation of complex Scorer functions

Two Fortran 77 routines for the evaluation of Airy functions of complex arguments Ai(z), Bi(z) and their first derivatives are presented. The routines are based on the use of Gaussian quadrature, Maclaurin series and asymptotic expansions. Comparison with a previous code by D. E. Amos [1986] is provided.

Computation of the modified Bessel function of the third kind of imaginary orders: uniform Airy-type asymptotic expansion

The use of a uniform Airy-type asymptotic expansion for the computation of the modified Bessel functions of the third kind of imaginary orders (Kia(x)) near the transition point x = a, is discussed. In A. Gil et al., Evaluation of the modified Bessel functions of the third kind of imaginary orders, J. Comput. Phys. 17 (2002) 398-411, an algorithm for the evaluation of Kia(x) was presented, which made use of series, a continued fraction method and nonoscillating integral representations. The range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point. We show how uniform Airy-type asymptotic expansions fill the gap left by the steepest descent method.

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

Abstract Two codes to evaluate the real zeros ( j v.s ) of the Bessel functions of the first kind J v ( x ) for real orders v are presented. The codes are based on a Newton-Raphson iteration over the monotonic function ƒ v ( x ) = x 2 v −1 J v ( x )/ J v −1 ( x ). The code ELF is a remarkably short program for finding, given any starting value x 0 > 0 and any real order, the zero of J v ( x ) in the neighborhood of x 0 ( x 0 and the zero in the same branch of ƒ v ( x )). GNOME is a modification of ELF for finding the zeros of J v ( x ) inside a given interval [ x min , x max ; 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 J v ( 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.

Efficient computation of Laguerre polynomials

An efficient algorithm and a Fortran 90 module (LaguerrePol) for computing Laguerre polynomials Ln(α)(z) are presented. The standard three-term recurrence relation satisfied by the polynomials and different types of asymptotic expansions valid for n large and α small, are used depending on the parameter region. Based on tests of contiguous relations in the parameter α and the degree n satisfied by the polynomials, we claim that a relative accuracy close to or better than 10−12 can be obtained using the module LaguerrePol for computing the functions Ln(α)(z) in the parameter range z≥0, −1