Abstract
A rational Krylov algorithm for eigenvalue computation and model order reduction is described. It is shown how to implement it as a modified shift-and-invert spectral transformation Arnoldi decomposition. It is shown how to do deflation, locking converged eigenvalues and purging irrelevant approximations. Computing reduced order models of linear dynamical systems by moment matching of the transfer function is considered.
Results are reported from one illustrative toy example and one practical example, a linear descriptor system from a computational fluid dynamics application.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, 2000.
D. Bindel, J. Demmel, W. Kahan, and O. Marques, On computing Givens rotations reliably and efficiently, ACM Trans. Math. Softw., 28 (2002), pp. 206–238.
K. Gallivan, E. Grimme, and P. V. Dooren, A rational Lanczos algorithm for model reduction, Numer. Algorithms, 12 (1995), pp. 33–63.
G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, Maryland, 3rd edn., 1996.
Oberwolfach model reduction benchmark collection, http://www.imtek.uni-freiburg.de/simulation/benchmark/.
K. H. A. Olsson and A. Ruhe, Rational Krylov for model order reduction and eigenvalue computation, technical report TRITA-NA-0520, NADA, Royal Institute of Technology, Stockholm, 2005.
A. Ruhe, An algorithm for numerical determination of the structure of a general matrix, BIT, 10 (1970), pp. 196–216.
A. Ruhe, Eigenvalue algorithms with several factorizations – a unified theory yet?, technical report 1998:11, Department of Mathematics, Chalmers University of Technology, Göteborg, Sweden, 1998.
A. Ruhe, Rational Krylov, a practical algorithm for large sparse nonsymmetric matrix pencils, SIAM J. Sci. Comput., 19 (1998), pp. 1535–1551.
D. C. Sorensen, Implicit application of polynomial filters in a k -step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357–385.
G. W. Stewart, A Krylov–Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 601–614.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Björn Engquist on the occasion of his 60th birthday.
AMS subject classification (2000)
65F15, 65F50, 65P99, 93A30
Rights and permissions
About this article
Cite this article
Olsson, K., Ruhe, A. Rational Krylov for eigenvalue computation and model order reduction . Bit Numer Math 46 (Suppl 1), 99–111 (2006). https://doi.org/10.1007/s10543-006-0085-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-006-0085-9