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## 2015.24: Chebyshev rootfinding via computing eigenvalues of colleague matrices: when is it stable?

2015.24: Vanni Noferini and Javier Perez Alvaro (2015) Chebyshev rootfinding via computing eigenvalues of colleague matrices: when is it stable?

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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 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 or not. 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, \emph{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, \emph{On the stability of computing polynomial roots via confederate linearizations}, To appear in Math. Comp.] that already showed that this algorithm is not backward normwise stable if the coefficients of the polynomial p(x) do not have moderate norms.

Item Type: MIMS Preprint polynomial, roots, Chebyshev basis, matrix polynomial, colleague matrix, backward stability, polynomial eigenvalue problem, Arnold trasnversality theorem MSC 2000 > 65 Numerical analysis 2015.24 Javier Perez Alvaro 13 April 2015