Ritz and Harmonic Ritz Values and the Convergence of FOM and GMRES

Numerical Linear Algebra with Applications

Volumn 6, Issue 4, 1999, Pages 281-293

  Goossens, Serge ^a   Roose, Dirk ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

FOM; GMRES; Harmonic Ritz values; Krylov subspaces; Ritz values

Indexed keywords

EID: 0033443621     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1099-1506(199906)6:4<281::AID-NLA158>3.0.CO;2-B     Document Type: Article

References (26)

1. W. E. Arnoldi. The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math., 9(1), 17-29, 1951.
2. R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. Van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, PA, 1994.
3. P. N. Brown. A theoretical comparison of the Arnoldi and GMRES algorithms. SIAM J. Sci. Stat. Comput., 12(1), 58-78, 1991.
4. A. M. Bruaset. A Survey of Preconditioned Iterative Methods. Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Essex, UK, 1995.
5. J. Cullum and A. Greenbaum. Relations between Galerkin and norm-minimising iterative methods for solving linear systems. SIAM J. Matrix Anal. Appl., 17(2), 223-246, 1996.
6. H. C. Elman, D. J. Silvester and A. J. Wathen. Iterative methods for problems in computational fluid dynamics. Technical Report 96/19, Oxford University Computing Laboratory, 1996.
7. B. Fischer. Polynomial Based Iteration Methods for Symmetric Linear Systems. John Wiley & Sons, Ltd., Chicester, UK, 1996.
8. R. W. Freund. Quasi-kernel polynomials and their use in non-Hermitian matrix iterations. J. Comput. Appl. Math., 43; 135-158, 1992.
9. R. W. Freund, G. H. Golub and N. M. Nachtigal. Iterative Solution of Linear Systems, pages 57-100. Acta Numerica 1992. Cambridge University Press, 1992.
10. S. Goossens and D. Roose. Ritz and Harmonic Ritz values and the convergence of FOM and GMRES. Technical Report TW257, Department of Computer Science, K. U. Leuven, 1997.
11. A. Greenbaum. Iterative Methods for Solving Linear Systems. SIAM, Philadelphia, PA, 1997.
12. A. Greenbaum, V. Pták and Z. Strakos. Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl., 17(3), 465-469, 1996.
13. T. A. Manteuffel and J. S. Otto. On the roots of the orthogonal polynomials and residual polynomials associated with a conjugate gradient method. Num. Lin. Alg. Appl., 1(5), 449-475, 1994.
14. T. A. Manteuffel and G. Starke. On hybrid iterative methods for nonsymmetric systems of linear equations. Numer. Math., 73, 489-506, 1996.
15. R. B. Morgan and M. Zeng. Harmonic projection methods for large nonsymmetric eigenvalue problems. Num. Lin. Alg. Appl., 5(1), 33-55, 1998.
16. N.M. Nachtigal, L. Reichel and L. N. Trefethen. A hybrid GMRES algorithm for nonsymmetric linear systems. SIAM J. Matrix Anal. Appl., 13(3), 796-825, 1992.
17. C. C. Paige, B. N. Parlett and H. A. Van der Vorst. Approximate solutions and eigenvalue bounds from Krylov subspaces. Num. Lin. Alg. Appl., 2(2):115-134, 1995.
18. B. N. Parlett. The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, NJ, 1980.
19. Y. Saad. Projection methods for solving large sparse eigenvalue problems. In B. Kågström and A. Ruhe, editors, Matrix Pencils, Lecture Notes in Mathematics 973, pages 121-144. Springer-Verlag, 1983.
20. Y. Saad. Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester, UK, 1992.
21. Y. Saad. Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston, MA, 1996.
22. Y. Saad and M. H. Schultz. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7(3), 856-869, 1986.
23. P. E. Saylor and D. C. Smolarski. Implementation of an adaptive algorithm for Richardson's method. Lin. Alg. Appl., 154-156, 615-646, 1991.
24. G. L. G. Sleijpen and H. A. Van der Vorst. A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl., 17(2), 401-425, 1996.
25. K. C. Toh and L. N. Trefethen. Calculation of pseudospectra by the Arnoldi iteration. SIAM J. Sci. Comput., 17(1), 1-15, 1996.
26. H. F. Walker and L. Zhou. A Simpler GMRES. Num. Lin. Alg. Appl., 1(6), 571-581, 1994.