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YuPeng Liang,XinHui Shao 한국전산응용수학회 2020 Journal of applied mathematics & informatics Vol.38 No.5
The spatial fractional diffusion equation can be discretized by employing the implicit finite difference scheme using the shifted Gr"unwald formula. The discretized linear system is obtained, whose the coefficient matrix has a diagonal-plus-Toeplitz structure. In order to solve the diagonal-plus-Toeplitz linear system, on the basis of circulant and skew-circulant splitting (CSCS splitting), we construct a new and efficient iterative method, called DSCS iterative methods, which have two parameters. Than we prove the convergence of DSCS methods. As a focus, we derive the simple and effective values of two optimal parameters under some restrictions. Some numerical experiments are carried out to illustrate the validity and accuracy of the new methods.
PRECONDITIONED GAUSS-SEIDEL ITERATIVE METHOD FOR Z-MATRICES LINEAR SYSTEMS
Shen, Hailong,Shao, Xinhui,Huang, Zhenxing,Li, Chunji Korean Mathematical Society 2011 대한수학회보 Vol.48 No.2
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.
Preconditioned Gauss-Seidel iterative method for Z-matrices linear systems
Hailong Shen,Xinhui Shao,Zhenxing Huang,Chunji Li 대한수학회 2011 대한수학회보 Vol.48 No.2
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(α) 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(α) +[기호](β) 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.
PRECONDITIONED AOR ITERATIVE METHODS FOR SOLVING MULTI-LINEAR SYSTEMS WITH 𝓜-TENSOR
QI, MENG,SHAO, XINHUI The Korean Society for Computational and Applied M 2021 Journal of applied mathematics & informatics Vol.39 No.3
Some problems in engineering and science can be equivalently transformed into solving multi-linear systems. In this paper, we propose two preconditioned AOR iteration methods to solve multi-linear systems with -tensor. Based on these methods, the general conditions of preconditioners are given. We give the convergence theorem and comparison theorem of the two methods. The results of numerical examples show that methods we propose are more effective.