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Rational Krylov Algorithms for Nonsymmetric Eigenvalue Problems

  • Conference paper
Recent Advances in Iterative Methods

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 60))

Abstract

The Rational Krylov algorithm for the nonsymmetric matrix pencil eigenvalue problem is described. It is a generalization of the shifted and inverted Lanczos (or Araoldi) algorithm, in which several shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil, gives Ritz approximations to the solution of the original pencil.

Rational Krylov is the natural alternative when factorization of the matrix is not much more expensive than solution, as e. g. when a multigrid algorithm is used to solve systems.

In one variant several iterations with different shifts are started on the same starting vector. Such iterations can be performed in parallel, yielding a p degree Krylov vector in one iteration on p processors. An analogy to a method of experimentally verifying stability of aircraft structures is shown.

The algorithm is demonstrated on two applied test problems.

The work was partly supported by the Swedish National Board for Technical Development, STU grant 91-1210.

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Abbreviations

AMS(MOS):

subject classifications. 65F15

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Author information

Authors and Affiliations

  1. Department of Computer Sciences, Chalmers University of Technology, S-41296, Göteborg, Sweden

    Axel Ruhe

  2. Department of Computer Sciences, University of Goteborg, S-41296, Göteborg, Sweden

    Axel Ruhe

Authors
  1. Axel Ruhe
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Stanford University, 94035, Stanford, CA, USA

    Gene Golub

  2. School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, 55455, Minneapolis, MN, USA

    Mitchell Luskin

  3. Courant Institute of Mathematical Sciences, 251 Mercer Street, 10012, New York, NY, USA

    Anne Greenbaum

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© 1994 Springer-Verlag New York, Inc

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Ruhe, A. (1994). Rational Krylov Algorithms for Nonsymmetric Eigenvalue Problems. In: Golub, G., Luskin, M., Greenbaum, A. (eds) Recent Advances in Iterative Methods. The IMA Volumes in Mathematics and its Applications, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9353-5_10

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  • DOI: https://doi.org/10.1007/978-1-4613-9353-5_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9355-9

  • Online ISBN: 978-1-4613-9353-5

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