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Implicit QR algorithms for palindromic and even eigenvalue problems

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Abstract

In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques.

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Authors and Affiliations

  1. Seminar für angewandte Mathematik, ETH Zürich, Zürich, Switzerland

    Daniel Kressner

  2. Institut für Mathematik, MA 4-5, Technische Universität Berlin, Berlin, Germany

    Christian Schröder

  3. Department of Mathematics, Washington State University, Pullman, WA, 99164, USA

    David S. Watkins

Authors
  1. Daniel Kressner
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  2. Christian Schröder
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  3. David S. Watkins
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Correspondence to Christian Schröder.

Additional information

C. Schröder supported by Deutsche Forschungsgemeinschaft through Matheon, the DFG Research Center Mathematics for key technologies in Berlin.

D. S. Watkins partly supported by Deutsche Forschungsgemeinschaft through Matheon, the DFG Research Center Mathematics for key technologies in Berlin.

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Kressner, D., Schröder, C. & Watkins, D.S. Implicit QR algorithms for palindromic and even eigenvalue problems. Numer Algor 51, 209–238 (2009). https://doi.org/10.1007/s11075-008-9226-3

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  • DOI: https://doi.org/10.1007/s11075-008-9226-3

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