H-矩阵方程组的预条件迭代法

针对系数矩阵A为H-矩阵的线性方程组Ax=b,引入了预条件矩阵I＋S_α~β,通过对系数矩阵施行初等行变换,提出了求解线性方程组Ax=b的一种新的预条件Gauss-Seidel方法.论文中首先证明了若A为H-矩阵,则（I＋S_α~β）A仍然是H-矩阵;其次,以定理的形式给出了新的预条件Gauss-Seidel方法收敛的充分条件,即给出了为保证新的预条件Gauss-Seidel方法收敛时参数所需满足的条件;然后从理论上证明了新的预条件Gauss-Seidel迭代方法较经典的Gauss-Seidel迭代方法收敛速度快,论文中提出的新的预条件Gauss-Seidel迭代方法推广了文[1-2]中提出的预条件方法;最后又通过数值算例说明了新的预条件Gauss-Seidel迭代方法的有效性.

A PRECONDITIONED ITERATIVE METHOD FOR H-MATRICES SYSTEMS

For solving a H- matrix linear system Ax = b, the new preconditioning Gauss-Seidel iterative method is presented. The preconditioning matrix I + S_α~β is introduced. Certain elementary row operations axe performed on A before applying the Ganss-Seidel iterative method. If the matrix A is an H- matrix, then (I + S_α~β) A is also an H- matrix. The suf-ficient conditions for guaranteeing the convergence of the new preconditioning Gauss-Seidel iterative method is obtained. In other words, The conditions are given, which is satisfied by the parameters for guaranteeing the convergence of the new preconditioning Gauss-Seidel iterative method . The convergence analysis of the Gauss-Seidel iterative method with a preconditioning matrix (I + S_α~β) is given. It is shown that convergence rate of the new preconditioning Gauss-Seidel iterative method is superior to that of the basic Gauss-Seidel iterative method, and the new preconditioning Gauss-Seidel iterative method contain the methods in [1-2]. At last, numerical examples show the effectiveness of the new precondi-tioning Gauss-Seidel iterative method.