Research and Innovation Articles
An extension of the I + Smax preconditioner for the Gauss-Seidel method
Published 2013-07-29
Keywords
- Preconditioning,
- Gauss-Seidel method,
- regular splitting,
- point and block preconditioners
How to Cite
Arenas, I., Castillo, P., & Yong, X. (2013). An extension of the I + Smax preconditioner for the Gauss-Seidel method. Revista Integración, Temas De matemáticas, 31(1), 1–14. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/3361
Abstract
A preconditioning technique based on the application of a fixedbut arbitrary number of I + Smax steps is proposed. A reduction of the spectral radius of the Gauss-Seidel iteration matrix is theoretically analyzed fordiagonally dominant Z-matrices. In particular, it is shown that after a finitenumber of steps this matrix reduces to null matrix. To illustrate the performance of the proposed technique numerical experiments on a wide variety ofmatrices are presented. Point and block versions of the preconditioner are numerically studied.
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References
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[16] Shao J.L., Huang T.Z. and Zhang G.F., “Linear system based approach for solving some related problems of M-matrices”, Linear Algebra Appl. 432 (2010), 327–337.
[17] Shen H., Shao X., Huang Z. and Li C., “Preconditioned Gauss-Seidel iterative method for Z matrices linear systems”, Bull. Korean. Math. Soc. 48 (2011), 303–314.
[18] Varga R.S., Matrix Iterative Analysis, Springer Series in Computational Mathematics, Springer, Berlin, 2000.
[19] Zheng B. and Miao S.X., “Two new modified Gauss-Seidel methods for linear systems with M-matrices”, J. Comput. Appl. Math. 233 (2009), 922–930.
[2] Berman A. and Plemmons R.J., Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9, SIAM, 1994.
[3] Castillo P. and Sequeira F.A., “Computational aspects of the Local Discontinuous Galerkin method on unstructured grids in three dimensions”, Mathematical and Computer Modelling 57 (2013), 2279–2288.
[4] Du J., Zheng B. and Wang L., “New iterative methods for solving linear systems”, J. Appl. Anal. Comput. 1 (2011), no. 3, 351–360.
[5] Durlofsky L.J., “Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities”, Water Resources Research 30 (1994), no.4, 965–973.
[6] Gunawardena A.D., Jain S.K. and Snyder L., “Modified iterative methods for consistent linear systems”, Linear Algebra Appl. 154-156 (1991), 123–143.
[7] Hadjidimos A., Noutsos D. and Tzoumas M., “More on modifications and improvements of classical iterative schemes for M-matrices”, Linear Algebra Appl. 364 (2003), 253–279.
[8] Kohno T., Kotakemori H., Niki H. and Usui M., “Improving the modified Gauss-Seidel method for Z-matrices”, Linear Algebra Appl. 267 (1997), 113–123.
[9] Kotakemori H., Harada K., Morimoto M. and Niki H., “A comparison theorem for the iterative method with preconditioner (I + Smax)”, J. Comput. Appl. Math. 145 (2002), no. 2, 373–378.
[10] Kotakemori H., Niki H. and Okamoto N., “Accelerated iterative method for Z-matrices”, J. Comput. Appl. Math. 75 (1996), no. 1, 87–97.
[11] Liu Q., Chen G. and Cai J., “Convergence analysis of the preconditioned Gauss-Seidel method for H-matrices”, Comput. Math. Appl. 56 (2008), no. 8, 2048–2053.
[12] Milaszewicz J.P., “Improving Jacobi and Gauss-Seidel iterations”, Linear Algebra Appl. 93 (1987), 161–170.
[13] Mokari-Bolhassan M.E. and Trick T.N., “A new iterative algorithm for the solutions of large scale systems”, 28th Midwest Symposium on Circuits and Systems Louisville, Kentucky, 1985.
[14] Morimoto M., Harada K., Sakakihara M. and Sawami H., “The Gauss-Seidel iterative method with the preconditioning matrix (I + S + SM)”, Japan J. Appl. Math. 21 (2004), 25–34.
[15] Niki H., Harada K., Morimoto M. and Sakakihara M., “The survey of preconditioners used for accelerating the rate of convergence in the Gauss-Seidel method”, J. Comput. Applied. Math. 164-165 (2004), 587–600.
[16] Shao J.L., Huang T.Z. and Zhang G.F., “Linear system based approach for solving some related problems of M-matrices”, Linear Algebra Appl. 432 (2010), 327–337.
[17] Shen H., Shao X., Huang Z. and Li C., “Preconditioned Gauss-Seidel iterative method for Z matrices linear systems”, Bull. Korean. Math. Soc. 48 (2011), 303–314.
[18] Varga R.S., Matrix Iterative Analysis, Springer Series in Computational Mathematics, Springer, Berlin, 2000.
[19] Zheng B. and Miao S.X., “Two new modified Gauss-Seidel methods for linear systems with M-matrices”, J. Comput. Appl. Math. 233 (2009), 922–930.