# American Institute of Mathematical Sciences

2012, 2(4): 811-821. doi: 10.3934/naco.2012.2.811

## A generalization of the positive-definite and skew-Hermitian splitting iteration

 1 School of Transportation, Nantong University, Nantong, 226019, China 2 School of Mathematical Sciences, Soochow University, Suzhou, 215006, China, China

Received  December 2011 Revised  September 2012 Published  November 2012

In this paper, a generalization of the positive-definite and skew-Hermitian splitting (GPSS) iteration is considered for solving non-Hermitian and positive definite systems of linear equations. Theoretical analysis shows that the GPSS method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the positive-definite splitting matrices. In some situations, this new scheme can outperform the Hermitian and skew-Hermitian splitting (HSS) method, the positive-definite and skew-Hermitian splitting (PSS) method, and the generalized HSS method (GHSS) and can be used as an efficient preconditioner. Numerical experiments using discretization of convection-diffusion-reaction equations demonstrate the effectiveness of the new method.
Citation: Yang Cao, Wei- Wei Tan, Mei-Qun Jiang. A generalization of the positive-definite and skew-Hermitian splitting iteration. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 811-821. doi: 10.3934/naco.2012.2.811
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##### References:
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