H-矩阵方程组的预条件迭代法  

A PRECONDITIONED ITERATIVE METHOD FOR H-MATRICES SYSTEMS

作　　者：沈海龙[1,2] 邵新慧[1] 张铁[1] 李长军[1] 

机构地区：[1]东北大学理学院,沈阳110819 [2]东北大学信息科学与工程学院,沈阳110819

出　　处：《数值计算与计算机应用》2009年第4期266-276,共11页Journal on Numerical Methods and Computer Applications

基　　金：国家自然科学基金资助项目(No.10771031)

摘　　要：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.For solving a H- matrix linear system Ax = b,the new preconditioning Gauss-Seidel iterative method is presented.The preconditioning matrix I＋S_α~βis introduced.Certain elementary row operations are performed on A before applying the Gauss-Seidel iterative method.If the matrix A is an H-matrix,then（I＋S_α~β） A is also an H-matrix.The sufficient 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 （1 ＋ Sα^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-Seidet iterative method contain the methods in [1-2]. At last, numerical examples show the effectiveness of the new precondi- tionin~ Causs-Seidel iterative method.

关 键 词：H-矩阵 Gauss-Seidel迭代方法 预条件矩阵 收敛性 

分 类 号：O241.6[理学—计算数学] TP301.6[理学—数学]