Technical report on backward stability of polynomial root-finding using Fiedler companion matrices

De Terán, Fernando and Dopico, Froilán M. and Pérez, Javier (2014) Technical report on backward stability of polynomial root-finding using Fiedler companion matrices. [MIMS Preprint]

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Abstract

{Computing roots of scalar polynomials as the eigenvalues of Frobenius companion matrices using backward stable eigenvalue algorithms is a classical approach. The introduction of new families of companion matrices allows for the use of other matrices in the root-finding problem. In this paper, we analyze the backward stability of polynomial root-finding algorithms via Fiedler companion matrices. In other words, given a polynomial $p(z)$, the question is to determine whether the whole set of computed eigenvalues of the companion matrix, obtained with a backward stable algorithm for the standard eigenvalue problem, are the set of roots of a nearby polynomial or not. We show that, if the coefficients of $p(z)$ are bounded in absolute value by a moderate number, then algorithms for polynomial root-finding using Fiedler matrices are backward stable, and Fiedler matrices are as good as the Frobenius companion matrices. This allows us to use Fiedler companion matrices with favorable structures in the polynomial root-finding problem. However, when some of the coefficients of the polynomial is large, companion Fiedler matrices may produce larger backward errors than Frobenius companion matrices, although in this case neither Frobenius nor Fiedler matrices lead to backward stable computations. To prove this we obtain explicit expressions for the change, to first order, of the characteristic polynomial coefficients of Fielder matrices under small perturbations. We show that, for all Fiedler matrices except the Frobenius ones, this change involves quadratic terms in the coefficients of the characteristic polynomial of the original matrix, while for the Frobenius matrices it only involves linear terms. We present extensive numerical experiments that support these theoretical results. The effect of balancing these matrices is also investigated.

Item Type:	MIMS Preprint
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Fernando De Terán
Date Deposited:	28 Jul 2014
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2162

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