解线性方程组的预条件SOR型迭代法

针对大型线性方程组问题构造了一种含有待定参数和预条件因子的新迭代解法,将其称为预条件SOR型迭代法.当待定参数ω=1时,预条件SOR迭代法就变成程光辉等人给出的预条件Gauss-Seidel型方法.讨论了当系数矩阵是不可约Z-矩阵时,SOR法和预条件SOR法的迭代矩阵所具有的性质,并通过定理将这两种迭代矩阵的谱半径进行了比较,同时给出了收敛最快时参数的取值范围.另外也将预条件SOR型迭代法和预条件Gauss-Seidel型方法进行了比较,显示了新方法的优越性.最后通过数值例子说明,选取合适的预条件因子可以使求解线性方程组的预条件SOR方法变得更有效.

Preconditioned SOR-Type Iterative Methods for Solving Linear Systems

A new iterative method is suggested for large linear equation sets including the undetermined parameter and preconditioner, named the preconditioned SOR-type iterative method. The method becomes the preconditioned Gauss-Seidel type iterative method given by Cheng Guang-hui if ω = 1. When the coefficient matrix is irreducible Z-matrix, the iterative matrices of SOR method and preconditioned SOR-type iterative method are both Z-matrices as well, and their spectral radii are compared with each other via relevant theorem with the range of fetching parameter values given for the quickest convergence. Moreover, the preconditioned SOR iterative method is compared with the preconditioned Gauss-Seidel iterative method, and the result indicates that the former is superior to the latter. Numerical examples show that the proper choice of preconditioner can make the preconditioned SOR-type iterative methods more efficient in solving linear systems.