Analysis of Normal-Form Algorithms for Solving Systems of Polynomial Equations

TL;DR

This paper evaluates various normal-form algorithms for solving polynomial systems, analyzing their complexity, speed, and accuracy, especially in challenging cases with closely clustered roots, highlighting their limitations and stability issues.

Contribution

It provides a comparative analysis of multiple polynomial rootfinding methods, including their performance and stability in difficult scenarios, which was previously not thoroughly examined.

Findings

Algorithms struggle with systems having closely spaced roots.

Performance varies significantly across different variants.

Resultant-based methods show instability in challenging cases.

Abstract

We examine several of the normal-form multivariate polynomial rootfinding methods of Telen, Mourrain, and Van Barel and some variants of those methods. We analyze the performance of these variants in terms of their asymptotic temporal complexity as well as speed and accuracy on a wide range of numerical experiments. All variants of the algorithm are problematic for systems in which many roots are very close together. We analyze performance on one such system in detail, namely the 'devastating example' that Noferini and Townsend used to demonstrate instability of resultant-based methods.