Symbolic-Numeric Factorization of 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-toglobal" 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. CCS CONCEPTS• Computing methodologies → Hybrid symbolic-numeric methods.

On the Complexity of Symbolic Computation

“…What about relying more heavily on homotopy methods for polynomial system solving? What about numerical differential Galois theory [23]?…”