Software for Computing the Spheroidal Wave Functions Using Arbitrary Precision Arithmetic

TL;DR

This paper introduces 'spheroidal', a software tool that accurately computes spheroidal wave functions using arbitrary precision arithmetic, addressing a long-standing computational challenge in physics and engineering.

Contribution

The paper presents novel software that employs arbitrary precision arithmetic and adaptive methods to compute spheroidal wave functions with improved accuracy.

Findings

Software successfully computes spheroidal functions with high precision.

Uses Wronskian to select optimal computation methods.

Freely available online for researchers.

Abstract

The spheroidal wave functions, which are the solutions to the Helmholtz equation in spheroidal coordinates, are notoriously difficult to compute. Because of this, practically no programming language comes equipped with the means to compute them. This makes problems that require their use hard to tackle. We have developed computational software for calculating these special functions. Our software is called spheroidal and includes several novel features, such as: using arbitrary precision arithmetic; adaptively choosing the number of expansion coefficients to compute and use; and using the Wronskian to choose from several different methods for computing the spheroidal radial functions to improve their accuracy. There are two types of spheroidal wave functions: the prolate kind when prolate spheroidal coordinates are used; and the oblate kind when oblate spheroidal coordinate are used. In this paper, we describe both, methods for computing them, and our software. We have made our software freely available on our webpage.