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• W1533206967 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." @default.
• W1533206967 created "2016-06-24" @default.
• W1533206967 creator A5022647438 @default.
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• W1533206967 date "2016-12-07" @default.
• W1533206967 modified "2023-10-10" @default.
• W1533206967 title "Chebyshev rootfinding via computing eigenvalues of colleague matrices: when is it stable?" @default.
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• W1533206967 doi "https://doi.org/10.1090/mcom/3149" @default.
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