The Convergence of Semi-iterative Methods for Large Linear Systems

迭代法是求解大规模稀疏线性方程组的常用方法之一.迭代方法的健壮性和收敛速度是影响迭代法有效使用的两大因素,因此在使用中对迭代法加速是非常必要的.半迭代法对加快迭代法的的收敛速度,增加迭代法的健壮性等方面是有效和实用的.本文在迭代矩阵是亏损阵的情况下,讨论影响半迭代法的加速效果的几个因素.结论表明,如果迭代矩阵的特征值分布不理想,或迭代矩阵的特征值的指标大,或迭代矩阵的Jordan基矩阵病态时,都会对半迭代的加速效果产生较大的影响.Iterative methods are one of the common used methods for solving large linear systems.Two main keys effecting the use of the iteartive methods are the robustness and the convergence speed.It is necessary to accelerate the iterative methods.The semi-iterative methods are one important accelerative methods.This paper discusses the convergence of the semi-iterative methods in case the iterative matrixs are defective.The results show the semi-iterative methods are likely to converge slowly whenever one of three cases occurs: the iteative matrixs is defective,the distribution of its spectrum is not favorable,or the Jordan basis of the iterative matrix is ill-conditioned