BIT Numerical Mathematics

, Volume 34, Issue 1, pp 165–176

The rational Krylov algorithm for nonsymmetric eigenvalue problems. III: Complex shifts for real matrices

  • Axel Ruhe

DOI: 10.1007/BF01935024

Cite this article as:
Ruhe, A. BIT (1994) 34: 165. doi:10.1007/BF01935024


A new algorithm for the computation of eigenvalues of a nonsymmetric matrix pencil is described. It is a generalization of the shifted and inverted Lanczos (or Arnoldi) 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. It is shown how complex shifts can be used to compute a real block Hessenberg pencil to a real matrix pair.

Two applicationx, one coming from an aircraft stability problem and the other from a hydrodynamic bifurcation, have been tested and results are reported.

AMS subject classifications


Key words

eigenvalue matrix pencil nonsymmetric Krylov double shift inverse iteration Lanczos method Arnoldi method 

Copyright information

© the BIT Foundation 1994

Authors and Affiliations

  • Axel Ruhe
    • 1
  1. 1.Department of Computing ScienceChalmers Institute of Technology and the University of GöteborgGöteborgSweden

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