An improved algorithm and a Fortran 90 module for computing the conical function fm-1/2+iτ(x)

In this paper we describe an algorithm and a Fortran 90 module (Conical) for the computation of the conical function P −1 2+iτ (x) for x > −1, m ≥ 0, τ > 0. These functions appear in the solution of Dirichlet problems for domains bounded by cones; because of this, they are involved in a large number of applications in Engineering and Physics. In the Fortran 90 module, the admissible parameter ranges for computing the conical functions in standard IEEE double precision arithmetic are restricted to (x,m, τ) ∈ (−1, 1) × [0, 40] × [0, 100] and (x,m, τ) ∈ (1, 100) × [0, 100] × [0, 100]. Based on tests of the three-term recurrence relation satisfied by these functions and direct comparison with Maple, we claim a relative accuracy close to 10−12 in the full parameter range, although a mild loss of accuracy can be found at some points of the oscillatory region of the conical functions. The relative accuracy increases to 10−13 − 10−14 in the region of the monotonic regime of the functions where integral representations are computed (−1 < x < 0). PROGRAM SUMMARY Manuscript Title: An improved algorithm and a module for computing the conical function Pm −1/2+iτ (x) Authors: Amparo Gil, Javier Segura, Nico M. Temme Program Title: Preprint submitted to Computer Physics Communications November 21, 2011 Module Conical Journal Reference: Catalogue identifier: Licensing provisions: Programming language: Fortran 90 Computer: Any supporting a FORTRAN compiler. Operating system: Any supporting a FORTRAN compiler. RAM: a few MB Number of processors used: