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Rational Krylov algorithms for eigenvalue computation and model reduction

  • Axel Ruhe
  • Daniel Skoogh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1541)

Abstract

Rational Krylov is an extension of the Lanczos or Arnoldi eigenvalue algorithm where several shifts (matrix factorizations) are performed in one run. A variant has been developed, where these factorizations are performed in parallel.

It is shown how Rational Krylov can be used to find a reduced order model of a large linear dynamical system. In Electrical Engineering, it is important that the reduced model is accurate over a wide range of frequencies, and then Rational Krylov with several shifts comes to advantage.

Results for numerical examples coming from Electrical Engineering applications are demonstrated.

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References

  1. 1.
    I. M. Elfadel and D. D. Ling, A block rational Arnoldi algorithm for multipoint passive model-order reduction of multiport RLC networks, in Technical Digest of the 1997 IEEE/ACM International Conference on Computer-Aided Design, IEEE Computer Society Press, 1997, pp. 66–71.Google Scholar
  2. 2.
    I. M. Elfadel and D. D. Ling, Zeros and passivity of Arnoldi-reduced-order models for interconnect networks, in Proc 34nd IEEE/ACM Design Automation Conference, ACM, New York, 1997, pp. 28–33.Google Scholar
  3. 3.
    T. Ericsson and A. Ruhe, The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems, Mathematics of Computation, 35 (1980), pp. 1251–1268.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    R. W. Freund, Circuit simulation techniques based on Lanczos-type algorithms, in Systems and Control in the Twenty-First Century, C. I. Byrnes, B. N. Datta, D. S. Gilliam, and C. F. Martin, eds., Birkäuser, 1997, pp. 171–184.Google Scholar
  5. 5.
    R. W. Freund, Reduced-order modeling techniques based on Krylov subspaces and their use in circuit simulation, Applied and Computational Control, Signals, and Circuits, (1998, to appear).Google Scholar
  6. 6.
    K. Gallivan, E. Grimme, and P. Van Dooren, A rational Lanczos algorithm for model reduction, Numerical Algorithms, 12 (1996), pp. 33–64.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    H. O. Karlsson, Atomic and Molecular Density-of-States by Direct Lanczos Methods, PhD thesis, Uppsala University, Department of Quantum Chemistry, 1994.Google Scholar
  8. 8.
    A. Ruhe, Eigenvalue algorithms with several factorizations—a unified theory yet? to appear, 1998.Google Scholar
  9. 9.
    -, Rational Krylov, a practical algorithm for large sparse nonsymmetric matrix pencils, SIAM J. Sci. Comp., 19 (1998), pp. 1535–1551.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Skoogh, Model order reduction by the rational Krylov method. work in progress, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Axel Ruhe
    • 1
  • Daniel Skoogh
    • 1
  1. 1.Department of MathematicsChalmers Institute of Technology and the University of GöteborgGöteborgSweden

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