Complexity Estimates for Two Uncoupling Algorithms

TL;DR

This paper analyzes two algorithms for transforming linear differential systems into scalar equations, providing tight complexity bounds, structural insights, and demonstrating the efficiency of a new fast variant of the cyclic-vector method.

Contribution

It offers tight size bounds for CVM, introduces a quasi-optimal fast variant, and establishes a structural link between CVM and DBZ with complexity comparisons.

Findings

CVM produces scalar equations with tight size bounds.

A fast variant of CVM has quasi-optimal complexity.

CVM is nearly 100 times faster than DBZ in practice.

Abstract

Uncoupling algorithms transform a linear differential system of first order into one or several scalar differential equations. We examine two approaches to uncoupling: the cyclic-vector method (CVM) and the Danilevski-Barkatou-Z\"urcher algorithm (DBZ). We give tight size bounds on the scalar equations produced by CVM, and design a fast variant of CVM whose complexity is quasi-optimal with respect to the output size. We exhibit a strong structural link between CVM and DBZ enabling to show that, in the generic case, DBZ has polynomial complexity and that it produces a single equation, strongly related to the output of CVM. We prove that algorithm CVM is faster than DBZ by almost two orders of magnitude, and provide experimental results that validate the theoretical complexity analyses.