Fast and backward stable computation of roots of polynomials, Part II: backward error analysis; companion matrix and companion pencil

TL;DR

This paper presents an improved backward error analysis for fast algorithms computing polynomial roots via companion matrices and pencils, demonstrating enhanced stability and accuracy over traditional methods.

Contribution

It introduces a companion QZ algorithm with superior backward error properties and refined analysis leveraging the problem's structure, advancing polynomial root-finding stability.

Findings

Companion QR algorithm has linear backward error growth with polynomial coefficient norm.

Companion QZ algorithm achieves similar backward error performance when coefficients are scaled.

Improved turnover operation enhances the overall accuracy of the algorithms.

Abstract

This work is a continuation of "Fast and backward stable computation of roots of polynomials" by J.L. Aurentz, T. Mach, R. Vandebril, and D.S. Watkins, SIAM Journal on Matrix Analysis and Applications, 36(3): 942--973, 2015. In that paper we introduced a companion QR algorithm that finds the roots of a polynomial by computing the eigenvalues of the companion matrix in $O(n^{2})$ time using $O(n)$ memory. We proved that the method is backward stable. Here we introduce, as an alternative, a companion QZ algorithm that solves a generalized eigenvalue problem for a companion pencil. More importantly, we provide an improved backward error analysis that takes advantage of the special structure of the problem. The improvement is also due, in part, to an improvement in the accuracy (in both theory and practice) of the turnover operation, which is the key component of our algorithms. We prove that for the companion QR algorithm, the backward error on the polynomial coefficients varies linearly with the norm of the polynomial's vector of coefficients. Thus the companion QR algorithm has a smaller backward error than the unstructured QR algorithm (used by MATLAB's \texttt{roots} command, for example), for which the backward error on the polynomial coefficients grows quadratically with the norm of the coefficient vector. The companion QZ algorithm has the same favorable backward error as companion QR, provided that the polynomial coefficients are properly scaled.