Rigorous numerical computation of the Stokes multipliers for linear differential equations with single level one

TL;DR

This paper presents a practical, high-precision algorithm for computing Stokes multipliers of linear differential equations with polynomial coefficients at irregular singular points of single level one, including an open-source implementation.

Contribution

It introduces a novel algorithm based on Borel summation and numerical ODE solving that is efficient, applicable to arbitrary order equations, and provides rigorous error bounds with minimal prior knowledge required.

Findings

Algorithm successfully computes Stokes multipliers with high precision.

Open-source SageMath implementation supports arbitrary-precision and error bounds.

Method applicable to a wide class of differential equations without genericity assumptions.

Abstract

We describe a practical algorithm for computing the Stokes multipliers of a linear differential equation with polynomial coefficients at an irregular singular point of single level one. The algorithm follows a classical approach based on Borel summation and numerical ODE solving, but avoids a large amount of redundant work compared to a direct implementation. It applies to differential equations of arbitrary order, with no genericity assumption, and is suited to high-precision computations. In addition, we present an open-source implementation of this algorithm in the SageMath computer algebra system and illustrate its use with several examples. Our implementation supports arbitrary-precision computations and automatically provides rigorous error bounds. The article assumes minimal prior knowledge of the asymptotic theory of meromorphic differential equations and provides an elementary introduction to the linear Stokes phenomenon that may be of independent interest.