求解鞍点问题的一般加速超松弛方法

GENERALIZED AOR METHOD FOR AUGMENTED SYSTEMS

针对大型稀疏鞍点问题给出了一种含有待定参数的新迭代解法,将其称之为一般加速松弛方法,简记为GAOR方法．当参数α=时,新迭代方法是变成由Golub等人给出的SOR-Like方法．该迭代法的构成是基于对系数矩阵进行的一种分裂．迭代法需要选择一个预处理矩阵和待定参数,通过适当选取预处理矩阵和待定参数,新迭代法是收敛的,并且以定理的形式给出了新迭代方法的迭代矩阵的特征值和参数之间的基本等式,从而也导出了迭代法收敛的充分和必要条件．理论结果表明新方法更具有广泛性,并且适当的选择参数可以使新方法较SOR-Like方法具有更快的收敛速度．在文中的最后给出了迭代法的数值试验结果．

In this article, the new method with the uncertain parameter is considered for solving the augmented system. The new method is called the Generalized AOR method （GAOR）. The Generalized AOR method becomes SOR-like method given by Golub et al. when a = 0. The new method is based on the splitting form of the coefficient matrix. The iterative method need choose a precondition matrix and the uncertain parameter . The functional equation between the parameters and the eigenvalues of the iteration matrix of the Generalized AOR method is given. Therefore, the necessary and sufficient condition for the convergence of the Generalized AOR method is derived by giving the restrictions imposed on the parameters. Finally, numerical computation based on a particular linear system is given, which clearly show the Generalized AOR method outperforms the SOR-like method.