ON CONTRACTION AND SEMI-CONTRACTION FACTORS OF GSOR METHOD FOR AUGMENTED LINEAR SYSTEMS *
ON CONTRACTION AND SEMI-CONTRACTION FACTORS OF GSOR METHOD FOR AUGMENTED LINEAR SYSTEMS *
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DOI：
10.4208/jcm.1004-m3180
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发表时间：
2010
影响因子：
0.9
通讯作者：
Fang;Chen;Yao-lin;Jiang;Zheng
中科院分区：
文献类型：
--
作者：
Fang;Chen;Yao-lin;Jiang;Zheng
Bai， Parlett和Wang [number]提出并研究了广义逐次超松弛（GSOR）方法。求解线性方程组的增广系统，得到了最优迭代参数和相应的最优收敛因子。数学学报，102(2005),pp.1-38。本文进一步估计了GSOR方法的收缩因子和半收缩因子。研究的动机是，在快步迭代计算中，迭代方法的收敛速度实际上是由收缩因子而不是由谱半径决定的。对于非奇异增广线性系统，在一定的约束条件下得到了所涉及参数的收缩域，从而保证了GSOR方法的收缩因子小于1。对于奇异但一致的增广线性系统，我们也用类似的方法得到了参数的半收缩域。最后，通过两个数值算例验证了理论结果和GSOR方法的有效性。
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 flnite-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 efiectiveness of the GSOR method.