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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)
  • Received : 2012.06.09
  • Published : 2013.07.31

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.

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
  2. K. Chen, Matrix Preconditioning Techniques and Applications, Cambridge University Press, Cambridge, 2005.
  3. K. Fan, Topological proofs for cretain theorems on matrices with non-negative elements, Monatsh. Math. 62 (1958), 219-237. https://doi.org/10.1007/BF01303967
  4. A. Frommer and D. B. Szyld, H-splitting and two-stage iterative methods, Numer. Math. 63 (1992), no. 3, 345-356. https://doi.org/10.1007/BF01385865
  5. R. M. Gray, Toeplitz and Circulant Matrices: A Review, Foundations and Trends in Communications and Information Theory 2 (2006), 155-239.
  6. 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
  7. T. Kohno, H. Kotakemori, and H. Niki, 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
  8. T. Kohno and H. Niki, A note on the preconditioner (I + $S_{max}$), J. Comput. Appl. Math. 225 (2009), no. 1, 316-319. https://doi.org/10.1016/j.cam.2008.07.042
  9. 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. https://doi.org/10.1016/S0377-0427(01)00588-X
  10. H. Kotakemori, H. Niki, and N. Okamoto, Accerated iterative method for Z-matrices, J. Comput. Appl. Math. 75 (1996), 87-97. https://doi.org/10.1016/S0377-0427(96)00061-1
  11. W. Li, A note on the preconditioned Gauss-Seidel method for linear systems, J. Comput. Appl. Math. 182, (2005) 81-90. https://doi.org/10.1016/j.cam.2004.11.041
  12. 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. https://doi.org/10.1016/S0024-3795(00)00140-3
  13. J. P. Milaszewicz, Impriving Jacobi and Guass-Seidel iterations, Linear Algebra Appl. 93 (1987), 161-170. https://doi.org/10.1016/S0024-3795(87)90321-1
  14. M. Morimoto, Study on the preconditioner (I + $S_{max}$), J. Comput. Appl. Math. 234 (2010), no. 1, 209-214. https://doi.org/10.1016/j.cam.2009.12.016
  15. 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. https://doi.org/10.1016/j.cam.2003.11.012
  16. 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. https://doi.org/10.1016/j.cam.2007.07.002
  17. R. S. Varga, Matrix Iterative Analysis, 2nd edition, Springer, 2000.
  18. 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.
  19. 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. https://doi.org/10.1016/j.cam.2009.08.056