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Fast and backward stable computation of eigenvalues and eigenvectors of matrix polynomials


Authors: Jared Aurentz, Thomas Mach, Leonardo Robol, Raf Vandebril and David S. Watkins
Journal: Math. Comp.
MSC (2010): Primary 65F15, 65L07
DOI: https://doi.org/10.1090/mcom/3338
Published electronically: April 12, 2018
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Abstract:

In the last decade matrix polynomials have been investigated with the primary focus on adequate linearizations and good scaling techniques for computing their eigenvalues and eigenvectors. In this article we propose a new method for computing a factored Schur form of the associated companion pencil. The algorithm has a quadratic cost in the degree of the polynomial and a cubic one in the size of the coefficient matrices. Also the eigenvectors can be computed at the same cost.

The algorithm is a variant of Francis's implicitly shifted QR algorithm applied on the companion pencil. A preprocessing unitary equivalence is executed on the matrix polynomial to simultaneously bring the leading matrix coefficient and the constant matrix term to triangular form before forming the companion pencil. The resulting structure allows us to stably factor each matrix of the pencil as a product of $ k$ matrices of unitary-plus-rank-one form, admitting cheap and numerically reliable storage. The problem is then solved as a product core chasing eigenvalue problem. A backward error analysis is included, implying normwise backward stability after a proper scaling. Computing the eigenvectors via reordering the Schur form is discussed as well.

Numerical experiments illustrate stability and efficiency of the proposed methods.


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Additional Information

Jared Aurentz
Affiliation: Instituto de Ciencias Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain
Email: jared.aurentz@icmat.es

Thomas Mach
Affiliation: Department of Mathematics, Nazarbayev University, Astana 010000, Kazakhstan
Email: thomas.mach@nu.edu.kz

Leonardo Robol
Affiliation: ISTI, Area della ricerca CNR, Pisa, Italy
Email: leonardo.robol@isti.cnr.it

Raf Vandebril
Affiliation: Department of Computer Science, KU Leuven, 3001 Leuven, Belgium
Email: raf.vandebril@cs.kuleuven.be

David S. Watkins
Affiliation: Department of Mathematics, Washington State University, Pullman, Washington 99164-3113
Email: watkins@math.wsu.edu

DOI: https://doi.org/10.1090/mcom/3338
Keywords: Matrix polynomial, product eigenvalue problem, core chasing algorithm, eigenvalues, eigenvectors
Received by editor(s): November 16, 2016
Received by editor(s) in revised form: June 14, 2017, and October 31, 2017
Published electronically: April 12, 2018
Additional Notes: This research was partially supported by the Research Council KU Leuven, project C14/16/056 (Inverse-free Rational Krylov Methods: Theory and Applications), and by the GNCS/INdAM project “Metodi numerici avanzati per equazioni e funzioni di matrici con struttura”.
Article copyright: © Copyright 2018 American Mathematical Society

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