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Chebyshev rootfinding via computing eigenvalues of colleague matrices: when is it stable?

Authors: Vanni Noferini and Javier Pérez
Journal: Math. Comp. 86 (2017), 1741-1767
MSC (2010): Primary 65H04, 65H17, 65F15, 65G50
Published electronically: December 7, 2016
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Abstract: Computing the roots of a scalar polynomial, or the eigenvalues of a matrix polynomial, expressed in the Chebyshev basis $ \{ T_k(x)\}$ is a fundamental problem that arises in many applications. In this work, we analyze the backward stability of the polynomial rootfinding problem solved with colleague matrices. In other words, given a scalar polynomial $ p(x)$ or a matrix polynomial $ P(x)$ expressed in the Chebyshev basis, the question is to determine whether or not the whole set of computed eigenvalues of the colleague matrix, obtained with a backward stable algorithm, like the QR algorithm, are the set of roots of a nearby polynomial. In order to do so, we derive a first order backward error analysis of the polynomial rootfinding algorithm using colleague matrices adapting the geometric arguments in [A. Edelman and H. Murakami, Polynomial roots for companion matrix eigenvalues, Math. Comp. 210, 763-776, 1995] to the Chebyshev basis. We show that, if the absolute value of the coefficients of $ p(x)$ (respectively, the norm of the coefficients of $ P(x)$) are bounded by a moderate number, computing the roots of $ p(x)$ (respectively, the eigenvalues of $ P(x)$) via the eigenvalues of its colleague matrix using a backward stable eigenvalue algorithm is backward stable. This backward error analysis also expands on the very recent work [Y. Nakatsukasa and V. Noferini, On the stability of computing polynomial roots via confederate linearizations, Math. Comp. 85 (2016), no. 301, 2391-2425] that already showed that this algorithm is not backward normwise stable if the coefficients of the polynomial $ p(x)$ do not have moderate norms.

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Additional Information

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

Javier Pérez
Affiliation: School of Mathematics, The University of Manchester, Manchester, England, M13 9PL
Address at time of publication: Department of Rehabilitation Sciences, Biomedical Sciences Group, KU Leuven – University, Tervuursevest 101 – box 1500, 3001 Leuven, Belgium

Keywords: Polynomial, roots, Chebyshev basis, matrix polynomial, colleague matrix, backward stability, polynomial eigenvalue problem, Arnold transversality theorem
Received by editor(s): April 5, 2015
Received by editor(s) in revised form: January 13, 2016, and February 9, 2016
Published electronically: December 7, 2016
Additional Notes: The work of the first-named author was supported by European Research Council Advanced Grant MATFUN (267526)
The work of the second-named author was supported by Engineering and Physical Sciences Research Council grant EP/I005293.
Article copyright: © Copyright 2016 American Mathematical Society