×

Two new modified Gauss-Seidel methods for linear system with M-matrices. (English) Zbl 1181.65049

The authors propose two preconditioners involving elements from the upper triangular part of the system matrix. They establish convergence and comparison theorems for the modified Gauss-Seidel iteration with these two preconditioners.

MSC:

65F10 Iterative numerical methods for linear systems
65F08 Preconditioners for iterative methods
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gunawardena, A. D.; Jain, S. K.; Snyder, L., Modified iterative methods for consistent linear systems, Linear Algebra Appl., 154-156, 123-143 (1991) · Zbl 0731.65016
[2] Kohno, T.; Kotakemori, H.; Niki, H., Improving the modified Gauss-Seidel method for \(Z\)-matrices, Linear Algebra Appl., 267, 113-123 (1997) · Zbl 0886.65030
[3] Niki, H.; Harada, K.; Morimoto, M.; Sakakihara, M., The survey of preconditioners used for accelerating the rate of convergence in the Gauss-Seidel method, J. Comput. Appl. Math., 164-165, 587-600 (2004) · Zbl 1057.65022
[4] Li, W., A note on the preconditioned Gauss-Seidel (GS) method for linear systems, J. Comput. Appl. Math., 182, 81-90 (2005) · Zbl 1072.65042
[5] Kotakemori, H.; Niki, H.; Okamoto, N., Accerated iterative method for \(Z\)-matrices, J. Comput. Appl. Math., 75, 87-97 (1996) · Zbl 0872.65027
[6] Milaszewicz, J. P., Impriving Jacobi and Guass-Seidel iterations, Linear Algebra Appl., 93, 161-170 (1987) · Zbl 0628.65022
[7] Kotakemori, H.; Harada, K.; Morimoto, M.; Niki, H., A comparison theorem for the iterative method with the preconditioner \((I + S_{\max})\), J. Comput. Appl. Math., 145, 373-378 (2002) · Zbl 1003.65029
[8] Niki, H.; Kohno, T.; Morimoto, M., The preconditioned Gauss-Seidel method faster than the SOR method, J. Comput. Appl. Math., 219, 59-71 (2008) · Zbl 1158.65023
[9] Young, D. M., Iterative solution of large linear systems (1971), Academic Press: Academic Press New York · Zbl 0204.48102
[10] Varga, R. S., Matrix Iterative Analysis, Prentice-Hall (1981), Englewood Cliffs: Englewood Cliffs NJ · Zbl 0133.08602
[11] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), Academic Press: Academic Press New York · Zbl 0484.15016
[12] Song, Y. Z., Comparisons of nonnegative splttings of matrices, Linear Algebra Appl., 154-156, 433-455 (1991) · Zbl 0732.65024
[13] Schneider, H., Theorems on \(M\)-splittings of a singular \(M\)-matrix which depend on graph structure, Linear Algebra Appl., 58, 407-424 (1984) · Zbl 0561.65020
[14] Li, W.; Sun, W. W., Modified Gauss-Seidel type methods and Jacobi type methods for \(Z\)-matrices, Linear Algebra Appl., 317, 227-240 (2000) · Zbl 0966.65032
[15] Zhang, F., Matrix theory (1999), Springer
[16] Woźniki, Z. I., Nonnegative splitting theory, Japan J. Industrial Appl. Math., 11, 289-342 (1994) · Zbl 0805.65033
[17] Bai, Z.-Z.; Golub, G.; Ng, M., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24, 603-626 (2003) · Zbl 1036.65032
[18] Bai, Z.-Z.; Golub, G.; Li, C.-K., Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput., 28, 583-603 (2006) · Zbl 1116.65039
[19] Bai, Z.-Z.; Golub, G.; Li, C.-K., Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comp., 76, 287-298 (2007) · Zbl 1114.65034
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.