Algorithms for Linear Ordinary Differential Operators
arXiv:2607.27003
The paper presents algorithms and a Scratchpad II implementation for representing and manipulating linear ordinary differential operators over differential rings, providing constructive methods for division, gcd, lcm, factorization, and related operations.
Abstract
We describe an implementation in Scratchpad II of linear ordinary differential operators over a differential ring, acting on a module equipped with a compatible derivation. The abstract data type facilities of the system allow such operators to be represented and manipulated as first-class objects while retaining the usual notation for operator application. For coefficients in a field, we give constructive algorithms for left and right division, greatest common divisors, least common multiples, and an extended Euclidean algorithm; the right-hand constructions may be obtained from the corresponding left-hand constructions in the opposite ring. We also discuss pseudo-division over polynomial coefficient rings and an Ore localization yielding a right field of fractions. Finally, we apply this operator arithmetic to factorization of ordinary differential equations, using the associated Riccati equation and Newton polygons to analyze possible singular parts of factors. Examples include operators with constant, elementary-function, rational-function, and matrix coefficients.
This article is posted for historical archival. It was accepted for presentation and publication in the "Conference on Computers and Mathematics'', held at Stanford, California, 30 July--1 August 1986. The proceedings, however, did not appear. This is a translation of the contemporaneous Script/VM to LaTeX