On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations

TL;DR

This paper introduces a novel numerical method for accurately computing roots of special functions governed by second order linear ODEs, leveraging nonoscillatory phase functions to achieve high precision and efficiency.

Contribution

The authors present a general algorithm that computes roots of special functions with near machine precision accuracy, independent of oscillation frequency, and competitive with specialized methods.

Findings

Achieves near machine precision accuracy.

Time to compute roots is independent of oscillation frequency.

Performs well in constructing Gaussian quadrature rules.

Abstract

We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented via nonoscillatory phase functions, even in the high-frequency regime. Our algorithm achieves near machine precision accuracy and the time required to compute one root of a solution is independent of the frequency of oscillations of that solution. Moreover, despite its great generality, our approach is competitive with specialized, state-of-the-art methods for the construction of Gaussian quadrature rules of large orders when it used in such a capacity. The performance of the scheme is illustrated with several numerical experiments and a Fortran implementation of our algorithm is available at the author's website.