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An algorithm for computing the eigenvalues of block companion matrices

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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.

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Correspondence to Marc Van Barel.

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The second author is a Postdoctoral Fellow of the Fund for Scientific Research - Flanders (Belgium).

The research was partially supported by the Research Council KU Leuven, project OT/05/40 (Large rank structured matrix computations), OT/10/038 (Multi-parameter model order reduction and its applications), CoE EF/05/006 Optimization in Engineering (OPTEC), and by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization). The scientific responsibility rests with its authors.

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Frederix, K., Delvaux, S. & Van Barel, M. An algorithm for computing the eigenvalues of block companion matrices. Numer Algor 62, 261–287 (2013).

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