IMPROVING PRECONDITIONED GAUSS-SEIDEL ITERATIVE METHOD FOR L-MATRICES
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.
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 Gaussâ€“Seidel method for Z-matrices. Linear Algebra Appl., 267: 113â€“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: 87â€“97.
Li, W. and Sun, W. (2000). Modified Gaussâ€“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 Gaussâ€“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.
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