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On the stability of computing polynomial roots via confederate linearizations

Authors: Yuji Nakatsukasa and Vanni Noferini
Journal: Math. Comp. 85 (2016), 2391-2425
MSC (2010): Primary 65H04; Secondary 65F15
Published electronically: November 10, 2015
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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.

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Additional Information

Yuji Nakatsukasa
Affiliation: Department of Mathematical Informatics, University of Tokyo, Tokyo 113-8656, Japan

Vanni Noferini
Affiliation: School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom
Address at time of publication: Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom

Received by editor(s): September 25, 2014
Received by editor(s) in revised form: February 12, 2015
Published electronically: November 10, 2015
Additional Notes: The first author was supported by JSPS Scientific Research Grant No. 26870149.
The second author was supported by ERC Advanced Grant MATFUN (267526)
Article copyright: © Copyright 2015 American Mathematical Society