×

zbMATH — the first resource for mathematics

GMRES-type methods for inconsistent systems. (English) Zbl 0963.65042
The authors study the behaviour of GMRES methods when the coefficient matrix is singular, particulary when the linear system is inconsistent. Some conditions are given under which these methods converge to the least squares solution of the system. An error bound for the computed iterates is proved.

MSC:
65F22 Ill-posedness and regularization problems in numerical linear algebra
65F10 Iterative numerical methods for linear systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994 · Zbl 0815.15016
[2] Å. Björck, Numerical Methods for Least-squares Problems, SIAM, Philadelphia, PA, 1996
[3] Brown, P.N.; Walker, H.F., GMRES on nearly singular systems, SIAM J. matrix anal. appl., 18, 37-51, (1997) · Zbl 0876.65019
[4] Ipsen, I.C.F.; Meyer, C.D., The idea behind Krylov methods, Amer. math. monthly, 105, 889-899, (1998) · Zbl 0982.65034
[5] Hochbruck, M.; Lubich, C., Error analysis of Krylov methods in a nutshell, SIAM J. sci. comput., 19, 695-701, (1998) · Zbl 0914.65024
[6] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual method for solving nonsymmetric linear systems, SIAM J. sci. statist. comput., 7, 856-869, (1986) · Zbl 0599.65018
[7] Y. Saad, Iterative Methods for Sparse Linear Systems, PWS, Boston, MA, 1996 · Zbl 1031.65047
[8] Walker, H.F.; Zhou, L., A simpler GMRES, Numer. linear algebra appl., 1, 571-581, (1994) · Zbl 0838.65030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.