RT Conference Proceedings
T1 Backward stability of polynomial root-finding using Fiedler companion matrices
A1 Terán Vergara, Fernando de
A1 Martínez Dopico, Froilán César
A1 Pérez Álvaro, Javier
AB 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 are large, Fiedler companion 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.
PB Oxford University Press
SN 0272-4979 (Print)
SN 1464-3642 (Online)
YR 2014
FD 2014-09-08
LK https://hdl.handle.net/10016/22060
UL https://hdl.handle.net/10016/22060
LA eng
NO The proceeding at: 6th Conference on Structured Numerical Linear and Multilinear Algebra: Analysis, Algorithms and Applications (SLA 2014), took place at 2014, Septembe 8-12, in Kalamata (Grece),
NO This work has been supported by the Ministerio de Economía y Competitividad of Spain throughgrant MTM2012-32542
DS e-Archivo
RD 6 ago. 2024