Algorithm xxx: Modified Bessel functions of imaginary order and positive argument

TL;DR

This paper presents Fortran 77 programs for accurately computing modified Bessel functions of imaginary order and positive argument, covering a wide range of parameters with high precision.

Contribution

It introduces computational routines for these special functions, including scaled versions, with extensive accuracy and range analysis.

Findings

Relative accuracy better than 10^{-13} in key ranges

Computational range extends to (0,1500]×[-1500,1500] for scaled functions

Codes effectively compute derivatives of the functions

Abstract

Fortran 77 programs for the computation of modified Bessel functions of purely imaginary order are presented. The codes compute the functions $K_{ia}(x)$, $L_{ia}(x)$ and their derivatives for real $a$ and positive $x$; these functions are independent solutions of the differential equation $x^2 w'' +x w' +(a^2 -x^2)w=0$. The code also computes exponentially scaled functions. The range of computation is $(x,a)\in (0,1500]\times [-1500,1500]$ when scaled functions are considered and it is larger than $(0,500]\times [-400,400]$ for standard IEEE double precision arithmetic. The relative accuracy is better than $10^{-13}$ in the range $(0,200]\times [-200,200]$ and close to $10^{-12}$ in $(0,1500]\times [-1500,1500]$.