An algorithm for computing the eigenvalues of block companion matrices

Authors:
Katrijn Frederix;Steven Delvaux;Marc Barel
Affiliations:
Department of Computer Science, KU Leuven, Leuven (Heverlee), Belgium 3001;Department of Mathematics, KU Leuven, Leuven (Heverlee), Belgium 3001;Department of Computer Science, KU Leuven, Leuven (Heverlee), Belgium 3001
Venue:
Numerical Algorithms
Year:
2013

Citing 10
Cited 0

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Abstract

In this paper we propose a method for computing the roots of a monic matrix polynomial. To this end we compute the eigenvalues of the corresponding block companion matrix C. This is done by implementing the QR algorithm in such a way that it exploits the rank structure of the matrix. Because of this structure, we can represent the matrix in Givens-weight representation. A similar method as in Chandrasekaran et al. (Oper Theory Adv Appl 179:111---143, 2007), the bulge chasing, is used during the QR iteration. For practical usage, matrix C has to be brought in Hessenberg form before the QR iteration starts. During the QR iteration and the transformation to Hessenberg form, the property of the matrix being unitary plus low rank numerically deteriorates. A method to restore this property is used.