Articolo in rivista, 2019, ENG, 10.1090/mcom/3338
Aurentz J.; Mach T.; Robol L.; Vandebril R.; Watkins D. S.
Instituto de Ciencias Mathematics, Universidad Autonoma de Madrid, Madrid, Spain; Department of Mathematics, Nazarbayev University, Kazakhstan; CNR-ISTI, Pisa, Italy; Department of Computer Science, KU Leuven, Belgium; Department of Mathematics, Washington State University, USA
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.
Mathematics of computation (Online) 88 , pp. 313–347
Matrix polynomial, Product eigenvalue problem, Core chasing algorithm, Eigenvalues, Eigenvectors
ISTI – Istituto di scienza e tecnologie dell'informazione "Alessandro Faedo"
ID: 385202
Year: 2019
Type: Articolo in rivista
Creation: 2018-03-16 13:47:25.000
Last update: 2021-03-26 12:34:48.000
CNR authors
External links
OAI-PMH: Dublin Core
OAI-PMH: Mods
OAI-PMH: RDF
DOI: 10.1090/mcom/3338
URL: https://www.ams.org/journals/mcom/2019-88-315/S0025-5718-2018-03338-5/
External IDs
CNR OAI-PMH: oai:it.cnr:prodotti:385202
DOI: 10.1090/mcom/3338
ISI Web of Science (WOS): 000444769600012
Scopus: 2-s2.0-85053325863