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HSS METHOD WITH A complex PARAMETER FOR THE SOLUTION OF complex linear system

Journal of Computational Mathematics 2011年第4期29卷 441-457页

作者： Guiding Gu Department of Applied Mathematics Shanghai University of Finance and Economics Shanghai 200433 China

In this paper, a complex parameter is employed in the Hermitian and skew-Hermitian splitting （HSS） method （Bai, Golub and Ng： SIAM J. Matrix Anal. Appl., 24（2003）, 603-626） for solving the complex linear system Ax = f. The convergence of the resulting method is proved when the spectrum of the matrix A lie in the right upper （or lower） part of the complex plane. We also derive an upper bound of the spectral radius of the HSS iteration matrix, and a estimated optimal parameter a （denoted by a^st） of this upper bound is presented. Numerical experiments on two modified model problems show that the HSS method with a est has a smaller spectral radius than that with the real parameter which minimizes the corresponding upper hound. In particular, for the ＇dominant＇ imaginary part of the matrix A, this improvement is considerable. We also test the GMRES method preconditioned by the HSS preconditioning matrix with our parameter a est.

关键词： Hermitian matrix Skew-Hermitian matrix Splitting iteration method complex linear system complex parameter.

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A Variant Modified Skew-Normal Splitting Iterative Method for Non-Hermitian Positive Definite linear systems

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Numerical Mathematics(Theory,Methods and Applications) 2022年第2期15卷 510-529页

作者： Rui Li Jun-Feng Yin Zhi-Lin Li College of Data Science Jiaxing UniversityZhejiang 314001P.R.China School of Mathematical Sciences Tongji UniversityShanghai 200092P.R.China Center for Research in Scientific Computation and Department of Mathematics North Carolina State UniversityRaleighNC 27695USA

We propose a variant modified skew-normal splitting iterative method to solve a class of large sparse non-Hermitian positive definite linear systems.Applying the preconditioning technique we also construct the preconditioned version of the proposed method.Theoretical analysis shows that the proposed method is unconditionally convergent even when the real part and the imaginary part of the coefficient matrix are non-symmetric.Meanwhile,when the real part and the imaginary part of the coefficient matrix are symmetric positive definite,we prove that the preconditioned variant modified skew-normal splitting iterative method will also unconditionally converge.Numerical experiments are presented to illustrate the efficiency of the proposed method and show better performance of it when compared with some other methods.

关键词： Non-Hermitian matrix skew-normal splitting precondition complex linear system

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