Abstract
Pole-swapping algorithms are generalizations of bulge-chasing algorithms for the generalized eigenvalue problem. Structure-preserving pole-swapping algorithms for the palindromic and alternating eigenvalue problems, which arise in control theory, are derived. A refinement step that guarantees backward stability of the algorithms is included. This refinement can also be applied to bulge-chasing algorithms that had been introduced previously, thereby guaranteeing their backward stability in all cases.
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Notes
Here and in what follows, the notation A − ρB is shorthand for βA − αB, where α and β are any scalars satisfying ρ = α/β. This allows us to include the case \(\rho = \infty \) by taking β = 0.
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This research was partially supported by the Research Council KU Leuven, project C14/16/056 (Inverse-free Rational Krylov Methods: Theory and Applications).
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Dedicated to Professor Volker Mehrmann on his 65th birthday.
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Mach, T., Steel, T., Vandebril, R. et al. Pole-Swapping Algorithms for Alternating and Palindromic Eigenvalue Problems. Vietnam J. Math. 48, 679–701 (2020). https://doi.org/10.1007/s10013-020-00408-0
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DOI: https://doi.org/10.1007/s10013-020-00408-0