On the numerical evaluation of the prolate spheroidal wave functions of order zero

TL;DR

This paper presents a new numerical method for evaluating prolate spheroidal wave functions of order zero, leveraging the Riccati equation for improved efficiency over traditional algorithms.

Contribution

It introduces a Riccati equation-based algorithm that enhances the efficiency of computing prolate spheroidal wave functions compared to existing methods.

Findings

The new algorithm has slower growth in running time with increasing bandlimit.

Numerical experiments demonstrate the efficiency and accuracy of the method.

The approach outperforms standard algorithms in computational speed.

Abstract

We describe a method for the numerical evaluation of the angular prolate spheroidal wave functions of the first kind of order zero. It is based on the observation that underlies the WKB method, namely that many second order differential equations admit solutions whose logarithms can be represented much more efficiently than the solutions themselves. However, rather than exploiting this fact to construct asymptotic expansions of the prolate spheroidal wave functions, our algorithm operates by numerically solving the Riccati equation satisfied by their logarithms. Its running time grows much more slowly with bandlimit and characteristic exponent than standard algorithms. We illustrate this and other properties of our algorithm with numerical experiments.