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Fast computation of eigenvalues of companion, comrade, and related matrices

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Abstract

The class of eigenvalue problems for upper Hessenberg matrices of banded-plus-spike form includes companion and comrade matrices as special cases. For this class of matrices a factored form is developed in which the matrix is represented as a product of essentially 2×2 matrices and a banded upper-triangular matrix. A non-unitary analogue of Francis’s implicitly-shifted QR algorithm that preserves the factored form and consequently computes the eigenvalues in O(n 2) time and O(n) space is developed. Inexpensive a posteriori tests for stability and accuracy are performed as part of the algorithm. The results of numerical experiments are mixed but promising in certain areas. The single-shift version of the code applied to companion matrices is much faster than the nearest competitor.

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  1. http://www.math.wsu.edu/students/jaurentz/publications/code.html.

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Authors and Affiliations

  1. Department of Mathematics, Washington State University, Pullman, WA, 99164-3113, USA

    Jared L. Aurentz & David S. Watkins

  2. Department of Computer Science, KU Leuven, 3001, Leuven, (Heverlee), Belgium

    Raf Vandebril

Authors
  1. Jared L. Aurentz
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  2. Raf Vandebril
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  3. David S. Watkins
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Correspondence to David S. Watkins.

Additional information

Communicated by Peter Benner.

The research was partially supported by the Research Council KU Leuven, projects OT/11/055 (Spectral Properties of Perturbed Normal Matrices and their Applications), CoE EF/05/006 Optimization in Engineering (OPTEC), by the Fund for Scientific Research–Flanders (Belgium) project G034212N (Reestablishing smoothness for matrix manifold optimization via resolution of singularities) and by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization).

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Aurentz, J.L., Vandebril, R. & Watkins, D.S. Fast computation of eigenvalues of companion, comrade, and related matrices. Bit Numer Math 54, 7–30 (2014). https://doi.org/10.1007/s10543-013-0449-x

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  • DOI: https://doi.org/10.1007/s10543-013-0449-x

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