Arnoldi-riccati method for large eigenvalue problems

BIT Numerical Mathematics

Volumn 36, Issue 3, 1996, Pages 579-594

  Simoncini, V ^a   Sadkane, M ^b  

^a CNR   (Italy)

^b CAMPUS DE BEAULIEU   (France)

Author keywords

Arnoldi; Eigenvalue approximation; Iterative methods; Krylov subspace; Perturbation theory; Riccati equation

Indexed keywords

EID: 1542603280     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF01731935     Document Type: Article

References (30)

1. W. E. Arnoldi, The principle of minimized iteration in the solution of the matrix eigenvalue problem, Quart. Appl. Math., 9 (1951), pp. 17-29.
2. G. Angelidis and A. Semlyen, Efficient calculation of critical eigenvalue clusters in the small signal stability analysis of large power systems, IEEE Trans. Power Systems, 10:1 (1995), pp. 427-432.
3. R. H. Bartels and G. W. Stewart, Algorithm 432: Solution of the matrix equation AX+XB=C, Comm. ACM, 15:9 (1972), pp. 820-826.
4. M. Bennani and T. Braconnier, Stopping criteria for eigensolvers, manuscript, February 1995.
5. D. L. Boley, Krylov space methods on state-space control models, Circ. Syst. Signal Proc., 13:6 (1994), pp. 733-758.
6. A. Chapman and Y. Saad, Deflated and augmented Krylov subspace techniques, manuscript, September 1995.
7. F. Chatelin, Simultaneous Newton's iteration for the eigenproblem, Computing. Suppl., 5 (1984), pp. 67-74.
8. F. Chaitin-Chatelin and V. Frayssè, Lectures on Finite Precision Computations, SIAM, Philadelphia, 1996.
9. J. Demmel, Three methods for refining estimates of invariant subspaces, Computing, 38 (1987), pp. 43-57.
10. I. S. Duff, R. G. Grimes, and J. G. Lewis, User's guide for the Harwell-Boeing sparse matrix collection (release I), Tech. Report TR/PA/92/86, CERFACS, Toulouse, France, Oct. 1992.
11. I. S. Duff and J. A. Scott, Computing selected eigenvalues of sparse unsymmetric matrices using subspace iteration, ACM Trans. Math. Software, 19 (1993), pp. 137-159.
12. G. H. Golub, S. Nash, and C. Van Loan, A Hessenberg-Schur method for the Problem AX+XB=C, IEEE Trans. Autom. Control, AC-24, 6 (1979), pp. 909-913.
13. D. Ho, Tchebychev acceleration technique for large scale nonsymmetric matrices, Numer. Math., 86 (1990), pp. 721-734.
14. D. Ho, F. Chatelin, and M. Bennani, Arnoldi-Tchebychev procedure for large scale nonsymmetric matrices. Math. Modelling Numer. Anal., (1990), pp. 53-65.
15. M. Hochbruck and G. Starke. Preconditioned Krylov subspace methods for Lyapunov matrix equations, SIAM J. Matrix Anal. Appl., 16:1 (1995), pp. 156-171.
16. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
17. D. Y. Hu and L. Reichel, Krylov subspace methods for the Sylvester equation, Linear Algebra Appl., 174 (1992), pp. 283-314.
18. I. M. Jaimoukha and E. M. Kasenally, Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal., 31 (1994), pp. 227-251.
19. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Standards, 45 (1950), pp. 255-282.
20. A. J. Laub, Invariant subspace methods for the numerical solution of algebraic Riccati equations, in The Riccati Equation, S. Bittanti, A. J. Laub, and J. Willems, eds., Springer-Verlag, 1991, pp. 163-196.
21. H.. Rutishauser, Computation aspects of F. L. Bauer's simultaneous iteration method, Numer. Math., 13 (1969), pp. 4-13.
22. Y. Saad, Projection method for solving large sparse eigenvalue problems, in Matrix Pencils, B. Kagstrom and A. Ruhe, eds., Springer-Verlag, 1982, pp. 121-144.
23. Y. Saad, Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems, Math. Comp., 42 (1984), pp. 567-588.
24. Y. Saad, Numerical solution of large Lyapunov equations, in Signal Processing, Scattering, Operator Theory, and Numerical Methods. Proceedings of the International Symposium MTNS-89, vol III, M. A. Kaashoek, J. H. van Schuppen, and A. C. Ran eds., Birkhaüser, Boston (1990), pp. 503-511.
25. M. Sadkane, A block Arnoldi-Chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices, Numer. Math., 64 (1993), pp. 181-193.
26. M. Sadkane, Block Arnoldi and Davidson methods for unsymmetric large eigenvalue problems, Numer. Math., 64 (1993), pp. 195-211.
27. D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13:1 (1992), pp. 357-385.
28. G. W. Stewart, Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Review, 15 (1973), pp. 727-764.
29. G. W. Stewart, Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices, Numer. Math. 25, (1976) pp. 123-136.
30. G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Academic Press, 1990.