Analysis of normal-form algorithms for solving systems of polynomial equations

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 the performance on one such system in detail, namely the “devastating example” that Noferini and Townsend used to demonstrate instability of resultant-based methods. We demonstrate that the problems with the devastating example arise from having a large number of roots very close to each other. We also show that a small number of clustered roots does not cause numerical problems for these methods. We conjecture that the clustering of many roots is the primary source of problematic examples.