Symbolic-Numeric Factorization of Differential Operators

TL;DR

This paper introduces a hybrid symbolic-numeric algorithm for factoring Fuchsian differential operators with rational coefficients, effectively handling various complexity levels by combining existing methods.

Contribution

It develops a new hybrid algorithm that adapts between local-to-global and analytic approaches for efficient factorization of differential operators.

Findings

Successfully combines two methods for different difficulty cases

Improves efficiency over existing algorithms in intermediate cases

Handles a broad class of Fuchsian differential operators

Abstract

We present a symbolic-numeric Las Vegas algorithm for factoring Fuchsian ordinary differential operators with rational function coefficients. The new algorithm combines ideas of van Hoeij's "local-to-global" method and of the ''analytic'' approach proposed by van der Hoeven. It essentially reduces to the former in ''easy'' cases where the local-to-global method succeeds, and to an optimized variant of the latter in the "hardest" cases, while handling intermediate cases more efficiently than both.