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2011•Bulletin of the Korean Mathematical SocietyOpen access

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

Hai-Long Shen, Xin-Hui Shao, Zhenxing Huang, Chunji Li

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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.

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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.

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Available 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.

Key concepts: Gauss–Seidel method, Preconditioner, Mathematics, Iterative method, Convergence (economics), Rate of convergence, Applied mathematics, Matrix (chemical analysis)

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