Expressions and bounds for the GMRES residual

BIT Numerical Mathematics

Volumn 40, Issue 3, 2000, Pages 524-535

  Ipsen, I C F ^a  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

Departure from normality; Eigenvalues; GMRES; Krylov space methods; Linear system; MINRES; Vandermonde matrix

Indexed keywords

EID: 0013133887     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022371814205     Document Type: Article

References (32)

1. M. Arioli, V. Pták, and Z. Strakoš, Krylov sequences of maximal length and convergence of GMRES, BIT, 38 (1998), pp. 636-643.
2. P. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM J. Sci. Stat. Comput., 12 (1991), pp. 58-78.
3. P. Brown and H. Walker, GMRES on (nearly) singular systems, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 37-51.
4. S. Campbell, I. Ipsen, C. Kelley, and C. Meyer, GMRES and the minimal polynomial BIT, 36 (1996), pp. 664-675.
5. Z. Cao, A note on the convergence behaviour of GMRES, Appl. Numer. Math., 25 (1997), pp. 13-20.
6. S. Chandrasekaran and I. Ipsen, Perturbation theory for the solution of systems of linear equations, Research Report 866, Department of Computer Science, Yale University, New Haven, CT, 1991.
7. S. Chandrasekaran and I. Ipsen, On the sensitivity of solution components in linear systems of equations, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 93-112.
8. T. Driscoll, K. Toh, and L. Trefethen, From potential theory to matrix iterations in six steps, SIAM Rev., 40 (1998), pp. 547-578.
9. V. Faber, W. Joubert, E. Knill, and T. Manteuffel, Minimal residual method stronger than polynomial preconditioning, SIAM J. Matrix Anal. Appl, 17 (1996), pp. 707-729.
10. G. Golub and C. van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
11. W. Govaerts and J. Pryce, A singular value inequality for block matrices, Linear Algebra Appl., 125 (1989), pp. 141-148.
12. A. Greenbaum, Comparison of splittings used with the conjugate gradient algorithm, Numer. Math., 33 (1979), pp. 181-194.
13. A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, PA, 1997.
14. A. Greenbaum and L. Gurvits, Max-min propres of matrix factor norms, SIAM J. Sci. Comput., 15 (1994), pp. 348-358.
15. A. Greenbaum V. Pták, and Z. Strakoš, Any nonincreasing convergence curve is possible for GMRES, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 465-469.
16. A. Greenbaum and L. Trefethen, GMRES/CR and Arnoldi/Lanczos as matrix approximation problems, SIAM J. Sci. Comput., 15 (1994), pp. 359-368.
17. M. Gu and S. Eisenstat, Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM J. Sci. Comput., 17 (1996), pp. 848-69.
18. P. Henrici, Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices, Numer. Math., 4 (1962), pp. 24-40.
19. N. Higham, Accuracy and Stability of Numeral Algorithms, SIAM, Philadelphia, 1996.
20. M. Hochbruck and C. Lubich, Error analysis of Krylov methods in a nutshell, SIAM J Sci. Comput., 19 (1998), pp. 695-701.
21. I. Ipsen, A different approach to bounding the minimal residual norm in Krylov methods, Tech. Rep. CRSC-TR98-19, Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, 1998. http-//www4.ncsu.edu/∼ipsen/linsys.html.
22. I. Ipsen and C. Meyer, The idea behind Krylov methods, Amer. Math. Monthly, 105 (1998), pp. 889-899.
23. I. Moret, A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal., 34 (1997), pp. 513-516.
24. N. Nachtigal, S. Reddy, and L. Trefethen, How fast are nonsymmetric matrix iterations?, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 778-795.
25. N. Nachtigal, L. Reichel, and L. Trefethen, A hybrid GMRES algorithm for nonsymmetric linear systems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 796-825.
26. C. Paige, B. Parlett, and H. Van der Vorst, Approximate solutions and eigenvalue bounds from Krylov subspaces, Numer. Linear Algebra, 2 (1995), pp. 115-133.
27. C. Paige and M. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629.
28. Y. Saad and M. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Sci. Stat. Comput., 7 (1986), pp. 856-869.
29. G. Stewart, Collinearity and least squares regression, Statist. Sci., 2 (1987), pp. 68-100.
30. K. Toh, GMRES vs. ideal GMRES, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 30-36.
31. H. Van der Vorst and C. Vuik, The superlinear convergence behaviour of GMRES, J. Comput. Appl. Math., 48 (1993), pp. 327-341.
32. H. Walker and L. Zhou, A simpler GMRES, Numer. Linear Algebra Appl., 1 (1994), pp. 571-581.