Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

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
April 1995

http://www.eecs.berkeley.edu/Pubs/TechRpts/1995/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.

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},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {1995},
    Month = {Apr},
    URL = {http://www.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://www.eecs.berkeley.edu/Pubs/TechRpts/1995/5203.html
%F Ruhe:CSD-95-871