关于广义鞍点问题的HSS迭代方法的收缩因子(英文)

On contraction factors of Hermitian and skew-Hermitian splitting iteration method for generalized saddle point problems

白中治等提出了解非埃尔米特正定线性方程组的埃尔米特和反埃尔米特分裂(HSS)迭代方法(Bai Z Z,Golub G H,Ng M K.Hermitian and skew-Hermitian splitting methodsfor non-Hermitian positive definite linear systems.SIAM J.Matrix Anal.Appl.,2003,24:603-626).本文精确地估计了用HSS迭代方法求解广义鞍点问题时在加权2-范数和2-范数下的收缩因子.在实际的计算中,正是这些收缩因子而不是迭代矩阵的谱半径,本质上控制着HSS迭代方法的实际收敛速度.根据文中的分析,求解广义鞍点问题的HSS迭代方法的收缩因子在加权2-范数下等于1,在2-范数下它会大于等于1,而在某种适当选取的范数之下,它则会小于1.最后,用数值算例说明了理论结果的正确性.

The Hermitian and skew-Hermitian splitting （HSS） iteration method was presented and studied by Bai, et al. for solving non-Hermitian positive defi- nite linear systems （Bai Z Z, Golub G H, Ng M K. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Ma- trix Anal. Appl., 2003, 24：603 626）. In this paper, contraction factors of the HSS iteration method in terms of the weighted 2-norm and the 2-norm are given, respec- tively, for the generalized saddle point problems. These contraction factors rather than the spectral radius of the iteration matrix essentially control the actual conver- gent speed of the HSS iteration method in practical computations. According to the analyses, the contraction factor of the HSS iteration method for the generalized sad- dle point problem is one in the weighted 2-norm. However, it may be greater than or equal to one in the 2-norm and less than one in other suitable norms. Finally, numerical examples are used to examine the correctness of the theoretical results.