Volume 28, Issue 6
On Contraction and Semi-Contraction Factors of GSOR Method for Augmented Linear Systems

Fang Chen, Yao-Lin Jiang & Bing Zheng

J. Comp. Math., 28 (2010), pp. 901-912.

Published online: 2010-12

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  • Abstract

The generalized successive overrelaxation (GSOR) method was presented and studied by Bai, Parlett and Wang [Numer. Math. 102(2005), pp.1-38] for solving the augmented system of linear equations, and the optimal iteration parameters and the corresponding optimal convergence factor were exactly obtained. In this paper, we further estimate the contraction and the semi-contraction factors of the GSOR method. The motivation of the study is that the convergence speed of an iteration method is actually decided by the contraction factor but not by the spectral radius in finite-step iteration computations. For the nonsingular augmented linear system, under some restrictions we obtain the contraction domain of the parameters involved, which guarantees that the contraction factor of the GSOR method is less than one. For the singular but consistent augmented linear system, we also obtain the semi-contraction domain of the parameters in a similar fashion. Finally, we use two numerical examples to verify the theoretical results and the effectiveness of the GSOR method.

  • Keywords

Contraction and semi-contraction factors Augmented linear system GSOR method Convergence

  • AMS Subject Headings

65F10 65F50 CR: G1.3.

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COPYRIGHT: © Global Science Press

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@Article{JCM-28-901, author = {Fang Chen, Yao-Lin Jiang and Bing Zheng}, title = {On Contraction and Semi-Contraction Factors of GSOR Method for Augmented Linear Systems}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {6}, pages = {901--912}, abstract = {

The generalized successive overrelaxation (GSOR) method was presented and studied by Bai, Parlett and Wang [Numer. Math. 102(2005), pp.1-38] for solving the augmented system of linear equations, and the optimal iteration parameters and the corresponding optimal convergence factor were exactly obtained. In this paper, we further estimate the contraction and the semi-contraction factors of the GSOR method. The motivation of the study is that the convergence speed of an iteration method is actually decided by the contraction factor but not by the spectral radius in finite-step iteration computations. For the nonsingular augmented linear system, under some restrictions we obtain the contraction domain of the parameters involved, which guarantees that the contraction factor of the GSOR method is less than one. For the singular but consistent augmented linear system, we also obtain the semi-contraction domain of the parameters in a similar fashion. Finally, we use two numerical examples to verify the theoretical results and the effectiveness of the GSOR method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1004-m3180}, url = {http://global-sci.org/intro/article_detail/jcm/8557.html} }
TY - JOUR T1 - On Contraction and Semi-Contraction Factors of GSOR Method for Augmented Linear Systems AU - Fang Chen, Yao-Lin Jiang & Bing Zheng JO - Journal of Computational Mathematics VL - 6 SP - 901 EP - 912 PY - 2010 DA - 2010/12 SN - 28 DO - http://doi.org/10.4208/jcm.1004-m3180 UR - https://global-sci.org/intro/article_detail/jcm/8557.html KW - Contraction and semi-contraction factors KW - Augmented linear system KW - GSOR method KW - Convergence AB -

The generalized successive overrelaxation (GSOR) method was presented and studied by Bai, Parlett and Wang [Numer. Math. 102(2005), pp.1-38] for solving the augmented system of linear equations, and the optimal iteration parameters and the corresponding optimal convergence factor were exactly obtained. In this paper, we further estimate the contraction and the semi-contraction factors of the GSOR method. The motivation of the study is that the convergence speed of an iteration method is actually decided by the contraction factor but not by the spectral radius in finite-step iteration computations. For the nonsingular augmented linear system, under some restrictions we obtain the contraction domain of the parameters involved, which guarantees that the contraction factor of the GSOR method is less than one. For the singular but consistent augmented linear system, we also obtain the semi-contraction domain of the parameters in a similar fashion. Finally, we use two numerical examples to verify the theoretical results and the effectiveness of the GSOR method.

Fang Chen, Yao-Lin Jiang & Bing Zheng. (1970). On Contraction and Semi-Contraction Factors of GSOR Method for Augmented Linear Systems. Journal of Computational Mathematics. 28 (6). 901-912. doi:10.4208/jcm.1004-m3180
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