Rational Krylov for Large Nonlinear Eigenproblems

  • Axel Ruhe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3732)

Abstract

Rational Krylov is an extension of the Lanczos or Arnoldi eigenvalue algorithm where several shifts (matrix factorizations) are used in one run. It corresponds to multipoint moment matching in model reduction. A variant applicable to nonlinear eigenproblems is described.

This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., Mckenney, A., Ostrouchov, S., Sorensen, D.: LAPACK Users’ Guide, 2nd edn. SIAM, Philadelphia (1995)Google Scholar
  2. 2.
    Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., Van Der Vorst, H. (eds.): Templates for the solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia (2000)MATHGoogle Scholar
  3. 3.
    Betcke, T., Voss, H.: A Jacobi–Davidson–type projection method for nonlinear eigenvalue problems. Future Generation Computer Systems 20, 363–372 (2004)CrossRefGoogle Scholar
  4. 4.
    Demmel, J., Eisenstat, S.C., Gilbert, J.R., Li, X.S., Liu, J.W.H.: A supernodal approach to sparse partial pivoting. SIAM J. Matr. Anal. Appl. 20, 720–755 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hager, P.: Eigenfrequency Analysis - FE-Adaptivity and a Nonlinear Eigenproblem Algorithm, PhD thesis, Chalmers University of Technology, Göteborg, Sweden (2001)Google Scholar
  6. 6.
    Jarlebring, E.: Krylov methods for nonlinear eigenvalue problems,Master’s thesis, NADA KTH, Stockholm Sweden, TRITA-NA-E03042 (2003)Google Scholar
  7. 7.
    Kublanovskaja, V.N.: On an applicaton ofNewton’smethod to the determination of eigenvalues of λ-matrices, Dokl. Akad. Nauk SSSR, 188, pp. 1240–1241. Page numbers may refer to AMS translation series (1969)Google Scholar
  8. 8.
    Olsson, K.H.A.: Model Order Reduction in FEMLAB by Dual Rational Arnoldi. Licentiate thesis, Chalmers University of Technology, Göteborg, Sweden (2002)Google Scholar
  9. 9.
    Ruhe, A.: Algorithms for the nonlinear algebraic eigenvalue problem. SIAMJ. Num. Anal 10, 674–689 (1973)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ruhe, A.: Rational Krylov, a practical algorithm for large sparse nonsymmetric matrix pencils. SIAM J. Sci. Comp. 19, 1535–1551 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ruhe, A.: A Rational Krylov algorithm for nonlinear matrix eigenvalue problems. Zapiski Nauchnyh Seminarov POMI, vol. 268, pp. 176–180. S:t Petersburg (2000)Google Scholar
  12. 12.
    Ruhe, A., Skoogh, D.: Rational Krylov algorithms for eigenvalue computation and model reduction. In: Kågström, B., Elmroth, E., Waśniewski, J., Dongarra, J. (eds.) PARA 1998. LNCS, vol. 1541, pp. 491–502. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. 13.
    Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matr. Anal. Appl. 13, 357–385 (1992)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Voss, H.: An Arnoldi method for nonlinear eigenvalue problems. BIT Numerical Mathematics 44, 387–401 (2004)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Voss, H.: A Jacobi–Davidsonmethod for nonlinear eigenproblems. In: Bubak, M., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2004. LNCS, vol. 3037, pp. 34–41. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Axel Ruhe
    • 1
  1. 1.Department of Numerical Analysis and Computer ScienceRoyal Institute of TechnologyStockholmSweden

Personalised recommendations