Abstract
In the paper we deal with some computational aspects of the Generalized minimal residual method (GMRES) for solving systems of linear algebraic equations. The key question of the paper is the importance of the orthogonality of computed vectors and its influence on the rate of convergence, numerical stability and accuracy of different implementations of the method. Practical impact on the efficiency in the parallel computer environment is considered.
Part of this work was performed while the second author visited Department of Mathematics and Computer Science, Emory University, Atlanta, USA
This work was supported by the GA AS CR under grant 230401 and by the NSF grant Int 921824
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
R. Barret, 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. SIAM, Philadelphia, 1994.
M. Benzi, M. Tůma: A sparse approximate inverse preconditioner for nonsymmetric linear systems. to appear in SIAM J. Sci. Comput.
T. Davis: Sparse matrix collection. NA Digest, Volume 94, Issue 42, October 1994.
J. Drkošová, A. Greenbaum, M. Rozložník, Z. Strakoš, Numerical Stability of the GMRES Method, BIT 3, pp. 309–330, 1995
R.W. Freund, G.H. Golub, N.M. Nachtigal, Iterative Solution of Linear Systems, Acta Numerica 1, pp. 1–44, 1992
A. Greenbaum. M. Rozložník, Z. Strakoš, Numerical Behaviour of the Modified Gram Schmidt GMRES Method, (submitted to BIT)
Y. Saad, M.H. Schultz, GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM J. Sci. Stat. Comput. 7, pp. 856–869, 1986
H. F. Walker, Implementation of the GMRES method using Householder transformations, SIAM J. Sci. Stat. Comput. 9, 1 (1988), pp. 152–163.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Rozložník, M., Strakoš, Z., Tůma, M. (1996). On the role of orthogonality in the GMRES method. In: Jeffery, K.G., Král, J., Bartošek, M. (eds) SOFSEM'96: Theory and Practice of Informatics. SOFSEM 1996. Lecture Notes in Computer Science, vol 1175. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0037424
Download citation
DOI: https://doi.org/10.1007/BFb0037424
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61994-9
Online ISBN: 978-3-540-49588-8
eBook Packages: Springer Book Archive