Fast algorithm for factoring difference operators

Abstract

Beke (1894) gave an algorithm that factors any differential operator and that algorithm can be used for difference operators as well. Bronstein improved Beke’s algorithm (see [2]). To find an order-m right-hand factor of an order-n operator using Beke-Bronstein’s algorithm, we need to create a difference system of order N = ( n m ) and solve that system. Experiments show that this is practical for n ≤ 8, or n = 9 if m ≤ 3, but beyond that N becomes too large. Our goal is a new method to find factors without increasing the order.