High precision series solution of differential equations: Ordinary and regular singular point of second order ODEs

TL;DR

This paper introduces a subroutine for high-precision numerical solutions of second-order ordinary differential equations, emphasizing efficient memory and computational scaling for large precision requirements.

Contribution

The paper presents a novel high-precision numerical method with linear memory and computational complexity for solving second-order ODEs with regular singular points.

Findings

Memory requirement scales linearly with precision P

Algebraic operations scale roughly linearly with P

Extensive testing demonstrates efficiency and accuracy

Abstract

A subroutine for very-high-precision numerical solution of a class of ordinary differential equations is provided. For given evaluation point and equation parameters the memory requirement scales linearly with precision $P$, and the number of algebraic operations scales roughly linearly with $P$ when $P$ becomes sufficiently large. We discuss results from extensive tests of the code, and how one for a given evaluation point and equation parameters may estimate precision loss and computing time in advance.