IMPROVING PRECONDITIONED GAUSS-SEIDEL ITERATIVE METHOD FOR L-MATRICES
Keywords:
Gauss-Seidel method, L--matrix, iteration matrix, convergence, spectral radius
Abstract
The Gauss-Seidel is a well-known iterative method for solving the linear system Ax = b. Convergence of this method is guaranteed for linear systems whose coefficient matrix is strictly or irreducibly diagonally dominant, Hermitian positive definite and invertible H -matrix. In this current work, a preconditioned version of the Gauss-Seidel method is used to accelerate the convergence of this iterative method towards the solution of linear system under mild conditions imposed on A . Convergence theorems on preconditioned Gauss-Seidel iterative method are advanced and proved. The superiority of Preconditioned Gauss-Seidel method is demonstrated by solving some numerical examples.
References
Dehghan, M. and Hajarian, M. (2009). Improving preconditioned SOR-type iterative methods for L-matrices. International Journal for Numerical Methods in Biomedical Engineering, 27:774-784.
Gunawardena, A. D., Jain, S. K. and Snyder, L. (1991). Modified iterative methods for consistent linear systems. Linear Algebra and its Applications, 154-156: 123-143.
Kohno, T., Kotakemori, H., Niki, H. and Usui, M. (1997). Improving modified Gauss–Seidel method for Z-matrices. Linear Algebra Appl., 267: 113–123.
Kotakemori, H., Harada, K., Morimoto, M. and Niki, H. (2002). A comparison theorem for the iterative method with the preconditioner (I+S_max). Journal of Computational and Applied Mathematics, 145: 373-378.
Kotakemori, H., Niki, H. and Okamoto, N. (1996). Accelerated iteration method for Z-matrices. Journal of Computational and Applied Mathematics, 75: 87–97.
Li, W. and Sun, W. (2000). Modified Gauss–Seidel type methods and Jacobi type methods for Z - matrices. Linear Algebra and its Applications, 317: 227-240.
Milaszewicz, J. P. (1987). Improving Jacobi and Gauss-Seidel iterations. Linear Algebra and its Applications, 93: 161-170.
Morimoto, M., Harada, K., Sakakihara, M. and Sawami, H. (2004). The Gauss–Seidel iterative method with the preconditioning matrix (I+S_max+S_m). Japan J. Indust. Appl. Math., 21: 25-34.
Ndanusa, A. and Adeboye, K. R. (2012). Preconditioned SOR iterative methods for L-matrices. American Journal of Computational and Applied Mathematics, 2(6): 300-305.
Niki, H., Harada, K., Morimoto, M. and Sakakihara, M. (2004). The survey of preconditioners used for accelerating the rate of convergence in the Gauss-Seidel method. Journal of Computational and Applied Mathematics, 164-165: 587-600.
Varga, R. S. (1981). Matrix iterative analysis (2nd ed.). Englewood Cliffs, New Jersey: Prentice-Hall. pp 358.
Gunawardena, A. D., Jain, S. K. and Snyder, L. (1991). Modified iterative methods for consistent linear systems. Linear Algebra and its Applications, 154-156: 123-143.
Kohno, T., Kotakemori, H., Niki, H. and Usui, M. (1997). Improving modified Gauss–Seidel method for Z-matrices. Linear Algebra Appl., 267: 113–123.
Kotakemori, H., Harada, K., Morimoto, M. and Niki, H. (2002). A comparison theorem for the iterative method with the preconditioner (I+S_max). Journal of Computational and Applied Mathematics, 145: 373-378.
Kotakemori, H., Niki, H. and Okamoto, N. (1996). Accelerated iteration method for Z-matrices. Journal of Computational and Applied Mathematics, 75: 87–97.
Li, W. and Sun, W. (2000). Modified Gauss–Seidel type methods and Jacobi type methods for Z - matrices. Linear Algebra and its Applications, 317: 227-240.
Milaszewicz, J. P. (1987). Improving Jacobi and Gauss-Seidel iterations. Linear Algebra and its Applications, 93: 161-170.
Morimoto, M., Harada, K., Sakakihara, M. and Sawami, H. (2004). The Gauss–Seidel iterative method with the preconditioning matrix (I+S_max+S_m). Japan J. Indust. Appl. Math., 21: 25-34.
Ndanusa, A. and Adeboye, K. R. (2012). Preconditioned SOR iterative methods for L-matrices. American Journal of Computational and Applied Mathematics, 2(6): 300-305.
Niki, H., Harada, K., Morimoto, M. and Sakakihara, M. (2004). The survey of preconditioners used for accelerating the rate of convergence in the Gauss-Seidel method. Journal of Computational and Applied Mathematics, 164-165: 587-600.
Varga, R. S. (1981). Matrix iterative analysis (2nd ed.). Englewood Cliffs, New Jersey: Prentice-Hall. pp 358.
Published
2020-04-14
How to Cite
NdanusaA., DavidB. E., AyantolaB. B., & AbubakarA. W. (2020). IMPROVING PRECONDITIONED GAUSS-SEIDEL ITERATIVE METHOD FOR L-MATRICES. FUDMA JOURNAL OF SCIENCES, 4(1), 453 - 459. Retrieved from https://fjs.fudutsinma.edu.ng/index.php/fjs/article/view/67
Section
Research Articles
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