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Singular quadratic eigenvalue problems: linearization and weak condition numbers

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Abstract

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of \(\delta \)-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases \(\delta \)-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.

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Acknowledgements

The authors thank Bor Plestenjak for helpful discussions on the numerical solution of singular eigenvalue problems, and Petra Lazić for discussions regarding the theory of \(\delta \)–weak condition numbers. The authors also thank the reviewers for their careful reading and useful comments.

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Authors and Affiliations

  1. Institute of Mathematics, EPFL, 1015, Lausanne, Switzerland

    Daniel Kressner

  2. Faculty of Science, Department of Mathematics, University of Zagreb, HR-10000, Zagreb, Croatia

    Ivana Šain Glibić

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  1. Daniel Kressner
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  2. Ivana Šain Glibić
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Correspondence to Ivana Šain Glibić.

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Communicated by Elias Jarlebring.

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The work of the first author was supported by the SNSF research project Probabilistic methods for joint and singular eigenvalue problems, grant number: 200021 L_192049. The work of the second author was supported by the CSF research project Randomized low rank algorithms and applications to parameter dependent problems, grant number: IP-2019-04-6268. Part of this work was done while the second author was a PostDoctoral researcher at EPFL.

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Kressner, D., Šain Glibić, I. Singular quadratic eigenvalue problems: linearization and weak condition numbers. Bit Numer Math 63, 18 (2023). https://doi.org/10.1007/s10543-023-00960-4

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