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

Jared L. Aurentz, Thomas Mach, Leonardo Robol, Raf Vandebril, David S. Watkins

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)

Abstract

This work is a continuation of work by [J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, J. Matrix Anal. Appl., 36 (2015), pp. 942–973]. 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(n2) 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 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.

Original languageEnglish
Pages (from-to)1245-1269
Number of pages25
JournalSIAM Journal on Matrix Analysis and Applications
Volume39
Issue number3
DOIs
Publication statusPublished - Jan 1 2018

Keywords

  • Backward stability
  • Companion matrix
  • Companion pencil
  • Core transformation
  • Eigenvalue
  • Francis algorithm
  • Polynomial
  • QR algorithm
  • QZ algorithm
  • Root

ASJC Scopus subject areas

  • Analysis

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