Computation of smallest eigenvalues using spectral Schur complements

SIAM Journal on Scientific Computing

Volumn 27, Issue 2, 2006, Pages 458-481

  Bekas, Constantine ^a   Saad, Yousef ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

AMLS; Eigenvalues; Krylov subspaces; Spectral Schur complements; Substructuring

Indexed keywords

APPROXIMATION THEORY; EIGENVALUES AND EIGENFUNCTIONS; MATRIX ALGEBRA; PROBLEM SOLVING;

AMLS; EIGENVALUES; KRYLOV SUBSPACES; SPECTRAL SCHUR COMPLEMENTS; SUBSTRUCTURING;

COMPUTATION THEORY;

EID: 33644936546     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/040603528     Document Type: Article

References (25)

1. A. ABRAMOV, On the separation of the principal part of some algebraic problems, Zh. Vych. Mat. Fiz., 2 (1962), pp. 141-145.
2. A. ABRAMOV, Remarks on finding the eigenvalues and eigenvectors of matrices which arise in the application of Ritz's method or in the difference method, Zh. Vych. Mat. Fiz., 7 (1962), pp. 644-647.
3. P. ARBENZ AND R. LEHOUCQ, A Comparison of Algorithms for Modal Analysis in the Absence of a Sparse Direct Method, Technical report 394, Institute of Scientific Computing, ETH Zurich, March 2003.
4. Also available as Sandia National Laboratories, Technical report SAND2003-1028J.
5. C. BEKAS AND Y. SAAD, Accurate computation of eigenvalues via multiple-shift AMLS, in preparation.
6. J. K. BENNIGHOF AND R. B. LEHOUCQ, An automated multilevel substructuring method for eigenspace computation in linear elastodynamics, SIAM. J. Sci. Comput., 25 (2004), pp. 2084-2106.
7. R. F. BOISVERT, R. POZO, K. REMINGTON, R. BARRETT, AND J. DONGARRA, The matrix market: A web repository for test matrix data, in The Quality of Numerical Software, Assessment and Enhancement, R. Boisvert, ed., Chapman & Hall, London, 1997, pp. 125-137.
8. V. S. CHICHOV, A method for partitioning a high order matrix into blocks in order to compute its eigenvalues, Zh. Vych. Mat., 1 (1961), pp. 169-173.
9. K. ELSSEL AND H. VOSS, An a priori bound for automated multi-level substructuring, submitted, 2004.
10. J. A. GEORGE AND J. W-H. LIU, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981.
11. R. R. CRAIG, JR. AND M. C. C. BAMPTON, Coupling of substructures for dynamic analysis, AIAA Journal, 6 (1968), pp. 1313-1319.
12. R. GRIMES, J. G. LEWIS, AND H. SIMON, A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 228-272.
13. W. C. HURTY, Vibrations of structural systems by component-mode synthesis, Journal of the Engineering Mechanics Division, ASCE, 86 (1960), pp. 51-69.
14. M. KAPLAN, Implementation of Automated Multilevel Substructuring for Frequency Response Analysis of Structures, Ph.D. thesis, The University of Texas at Austin, Austin, TX, 2001.
15. T. KATO, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin, 1980.
16. L. KOMZSIK AND T. ROSE, Substructuring in MSC.NASTRAN for large scale parallel applications, Computing Systems in Engineering, 2 (1991), pp. 167-173.
17. A. KROPP AND D. HEISERER, Efficient Broadband Vibro-acoustic Analysis of Passenger Car Bodies using FE-based Component Mode Synthesis Approach, in Fifth World Congress on Computational Mechanics (WCCM V), H. A. Mang, G. H. Rammerstorfer, and J. Eber-hardsteiner, eds., Vienna University of Technology, Vienna, Austria, 2002.
18. R. LEHOUCQ, D. C. SORENSEN, AND C. YANG, Arpack User's Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia, 1998; also available online from http://www.caam.rice.edu/ software/ARPACK.
19. B. N. PARLETT, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.
20. Y. SAAD, Shifts of origin for the LR and QR algorithms, C.R.A.S. Proceedings of the French Academy of Sciences, Ser. A278, 1973, pp. 93-96. In French.
21. Y. SAAD, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia, 2003.
22. B. SMITH, P. BJØRSTAD, AND W. GROPP, Domain decomposition: Parallel multilevel methods for elliptic partial differential equations, Cambridge University Press, Cambridge, 1996.
23. F. TISSEUR AND K. MEERBERGEN, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235-286.
24. C. YANG, Private communication, October 2004.
25. C. YANG, W. G. GAO, Z. BAI, X. LI, L. LEE, P. HUSBANDS AND E. G. No, An Algebraic Sub-structuring Algorithm for Large-scale Eigenvalue Calculation, LBNL Technical report, No. 55050, Lawrence Berkeley National Laboratory, Berkeley, CA, 2004.