Paper Info
Title
ON THE STABILITY OF COMPUTING POLYNOMIAL ROOTS VIA CONFEDERATE LINEARIZATIONS
Abstract
A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems to be known for other polynomial bases. This paper studies the stability of algorithms that compute the roots via linearization in nonmonomial bases, and has three goals. First we prove normwise stability when the polynomial is properly scaled and the QZ algorithm (as opposed to the more commonly used QR algorithm) is applied to a comrade pencil associated with a Jacobi orthogonal polynomial. Second, we extend a result by Arnold that leads to a first-order expansion of the backward error when the eigenvalues are computed via QR, which shows that the method can be unstable. Based on the analysis we suggest how to choose between QR and QZ. Finally, we focus on the special case of the Chebyshev basis and finding real roots of a general function on an interval, and discuss how to compute accurate roots. The main message is that to guarantee backward stability QZ applied to a properly scaled pencil is necessary.
Year
DOI
Venue
2016
10.1090/mcom3049
MATHEMATICS OF COMPUTATION
Field
DocType
Volume
Characteristic polynomial,Mathematical optimization,Stable polynomial,Polynomial matrix,Algebra,Mathematical analysis,Kharitonov's theorem,Monomial basis,Monic polynomial,Monomial,Matrix polynomial,Mathematics
Journal
85
Issue
ISSN
Citations 
301
0025-5718
4
PageRank 
References 
Authors
0.55
14
2
Name
Order
Citations
PageRank
Yuji Nakatsukasa19717.74
Vanni Noferini2266.83