Browse > Article
http://dx.doi.org/10.4134/CKMS.2013.28.3.603

ON CONVERGENCE OF THE MODIFIED GAUSS-SEIDEL ITERATIVE METHOD FOR H-MATRIX LINEAR SYSTEM  

Miao, Shu-Xin (School of Mathematics and Statistics Lanzhou University, Department of Mathematics Northwest Normal University)
Zheng, Bing (School of Mathematics and Statistics Lanzhou University)
Publication Information
Communications of the Korean Mathematical Society / v.28, no.3, 2013 , pp. 603-613 More about this Journal
Abstract
In 2009, Zheng and Miao [B. Zheng and S.-X. Miao, Two new modified Gauss-Seidel methods for linear system with M-matrices, J. Comput. Appl. Math. 233 (2009), 922-930] considered the modified Gauss-Seidel method for solving M-matrix linear system with the preconditioner $P_{max}$. In this paper, we consider the modified Gauss-Seidel method for solving the linear system with the generalized preconditioner $P_{max}({\alpha})$, and study its convergent properties when the coefficient matrix is an H-matrix. Numerical experiments are performed with different examples, and the numerical results verify our theoretical analysis.
Keywords
H-matrix; preconditioner; modified Gauss-Seidel method; convergence;
Citations & Related Records
연도 인용수 순위
  • Reference
1 B. Zheng and S.-X. Miao, Two new modified Gauss-Seidel methods for linear system with M-matrices, J. Comput. Appl. Math. 233 (2009), no. 4, 922-930.   DOI   ScienceOn
2 A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
3 K. Chen, Matrix Preconditioning Techniques and Applications, Cambridge University Press, Cambridge, 2005.
4 K. Fan, Topological proofs for cretain theorems on matrices with non-negative elements, Monatsh. Math. 62 (1958), 219-237.   DOI
5 A. Frommer and D. B. Szyld, H-splitting and two-stage iterative methods, Numer. Math. 63 (1992), no. 3, 345-356.   DOI   ScienceOn
6 R. M. Gray, Toeplitz and Circulant Matrices: A Review, Foundations and Trends in Communications and Information Theory 2 (2006), 155-239.
7 A. D. Gunawardena, S. K. Jain, and L. Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl. 154/156 (1991), 123-143.   DOI   ScienceOn
8 T. Kohno, H. Kotakemori, and H. Niki, Improving the modified Gauss-Seidel method for Z-matrices, Linear Algebra Appl. 267 (1997), 113-123.   DOI   ScienceOn
9 T. Kohno and H. Niki, A note on the preconditioner (I + $S_{max}$), J. Comput. Appl. Math. 225 (2009), no. 1, 316-319.   DOI   ScienceOn
10 H. Kotakemori, K. Harada, M. Morimoto, and H. Niki, A comparison theorem for the iterative method with the preconditioner (I+$S_{max}$), J. Comput. Appl. Math. 145 (2002), no. 2, 373-378.   DOI   ScienceOn
11 H. Kotakemori, H. Niki, and N. Okamoto, Accerated iterative method for Z-matrices, J. Comput. Appl. Math. 75 (1996), 87-97.   DOI   ScienceOn
12 W. Li, A note on the preconditioned Gauss-Seidel method for linear systems, J. Comput. Appl. Math. 182, (2005) 81-90.   DOI   ScienceOn
13 W. Li and W. W. Sun, Modified Gauss-Seidel type methods and Jacobi type methods for Z-matrices, Linear Algebra Appl. 317 (2000), no. 1-3, 227-240.   DOI   ScienceOn
14 J. P. Milaszewicz, Impriving Jacobi and Guass-Seidel iterations, Linear Algebra Appl. 93 (1987), 161-170.   DOI   ScienceOn
15 M. Morimoto, Study on the preconditioner (I + $S_{max}$), J. Comput. Appl. Math. 234 (2010), no. 1, 209-214.   DOI   ScienceOn
16 H. Niki, K. Harada, M. Morimoto, and M. Sakakihara, The survey of preconditioners used for accelerating the rate of convergence in the Gauss-Seidel method, J. Comput. Appl. Math. 164/165 (2004), 587-600.   DOI   ScienceOn
17 H. Niki, T. Kohno, and M. Morimoto, The preconditioned Gauss-Seidel method faster than the SOR method, J. Comput. Appl. Math. 219 (2008), no. 1, 59-71.   DOI   ScienceOn
18 R. S. Varga, Matrix Iterative Analysis, 2nd edition, Springer, 2000.
19 X. Z. Wang, T. Z. Huang, and Y. D. Fu, Preconditioned diagonally dominant property for linear systems with H-matrices, Appl. Math. E-Notes 6 (2006), 235-243.