Rational Krylov, A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils

Axel Ruhe

EECS Department, University of California, Berkeley

Technical Report No. UCB/CSD-95-871

1995

This publication is archived. It is kept only for reference purposes, so it is no longer being updated and may not meet accessibility standards. If you need this content in a different format, please email webteam@eecs.berkeley.edu.

http://www2.eecs.berkeley.edu/Pubs/TechRpts/1995/Archive/CSD-95-871.pdf

The Rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil. It is a generalization of the shifted and inverted Arnoldi algorithm, where several factorizations with different shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil approximates the solution of the original pencil. Different types of Ritz values and harmonic Ritz values are described and compared. Periodical purging of uninteresting directions reduces the size of the basis, and makes it possible to get many linearly independent eigenvectors and principal vectors to pencils with multiple eigenvalues. Relations to iterative methods are established.

<

p>Results are reported for two large test examples. One is a symmetric pencil coming from a finite element approximation of a membrane, the other a nonsymmetric matrix modeling an idealized aircraft stability problem.

BibTeX citation:

@techreport{Ruhe:CSD-95-871,
    Author= {Ruhe, Axel},
    Title= {Rational Krylov, A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils},
    Year= {1995},
    Month= {Apr},
    Url= {http://www2.eecs.berkeley.edu/Pubs/TechRpts/1995/5203.html},
    Number= {UCB/CSD-95-871},
    Abstract= {The Rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil. It is a generalization of the shifted and inverted Arnoldi algorithm, where several factorizations with different shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil approximates the solution of the original pencil. Different types of Ritz values and harmonic Ritz values are described and compared. Periodical purging of uninteresting directions reduces the size of the basis, and makes it possible to get many linearly independent eigenvectors and principal vectors to pencils with multiple eigenvalues. Relations to iterative methods are established. <p>Results are reported for two large test examples. One is a symmetric pencil coming from a finite element approximation of a membrane, the other a nonsymmetric matrix modeling an idealized aircraft stability problem.},
}

EndNote citation:

%0 Report
%A Ruhe, Axel 
%T Rational Krylov, A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils
%I EECS Department, University of California, Berkeley
%D 1995
%@ UCB/CSD-95-871
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/1995/5203.html
%F Ruhe:CSD-95-871