# PRECONDITIONED GAUSS-SEIDEL ITERATIVE METHOD FOR Z-MATRICES LINEAR SYSTEMS

Shen, Hailong;Shao, Xinhui;Huang, Zhenxing;Li, Chunji

• Published : 2011.03.31
• 23 4

#### Abstract

For Ax = b, it has recently been reported that the convergence of the preconditioned Gauss-Seidel iterative method which uses a matrix of the type P = I + S (${\alpha}$) to perform certain elementary row operations on is faster than the basic Gauss-Seidel method. In this paper, we discuss the adaptive Gauss-Seidel iterative method which uses P = I + S (${\alpha}$) + $\bar{K}({\beta})$ as a preconditioner. We present some comparison theorems, which show the rate of convergence of the new method is faster than the basic method and the method in [7] theoretically. Numerical examples show the effectiveness of our algorithm.

#### Keywords

Gauss-Seidel iterative method;preconditioned method;Z-matrix;diagonal dominant matrix

#### References

1. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994.
2. A. D. Gunawardena, S. K. Jain, and L. Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl. 154/156 (1991), 123-143. https://doi.org/10.1016/0024-3795(91)90376-8
3. A. Hadjidimos, D. Noutsos, and M. Tzoumas, More on modifications and improvements of classical iterative schemes for M-matrices, Linear Algebra Appl. 364 (2003), 253-279. https://doi.org/10.1016/S0024-3795(02)00570-0
4. J. Hu, The iterative methods of the linear equations, Beijing: Science Press (1997), 63-64.
5. T. Kohno, H. Kotakemori, H. Niki, and M. Usui, Improving the modified Gauss-Seidel method for Z-matrices, Linear Algebra Appl. 267 (1997), 113-123. https://doi.org/10.1016/S0024-3795(97)80045-6
6. H. Kotakemori, H. Niki, and N. Okamoto, Accelerated iterative method for Z-matrices, J. Comput. Appl. Math. 75 (1996), no. 1, 87-97. https://doi.org/10.1016/S0377-0427(96)00061-1
7. J. Li and T. Huang, Preconditioning methods for Z-matrices, Acta Math. Sci. Ser. A Chin. Ed. 25 (2005), no. 1, 5-10.
8. J. P. Milaszewicz, Improving Jacobi and Gauss-Seidel iterations, Linear Algebra Appl. 93 (1987), 161-170. https://doi.org/10.1016/S0024-3795(87)90321-1
9. X. Shao, Z. Li, and C. Li, Modified SOR-like method for the augmented system, Int. J. Comput. Math. 84 (2007), no. 11, 1653-1662. https://doi.org/10.1080/00207160601117313

#### Acknowledgement

Supported by : National Natural Science Foundation of China, Central Universities