线性方程组的迭代解法

Iterative Methods for Solving the Linear Systems

线性方程组的数值求解常见于许多科学与工程计算领域,介绍了求解大型线性方程组的主要迭代算法。首先,对一些经典迭代法(Jacobi方法、Gauss-Seidel方法、SOR方法、SSOR方法和CG方法等)进行了详细的讨论,并从理论上对收敛性进行分析。其次,讨论了最新的Hermitian/Skew-Hermitian splitting(HSS)迭代理论,给出了迭代公式和收敛性定理。最后,通过数值实验对所有迭代法的有效性进行了验证。

Numerical methods for linear systems are very important in many areas. Several iterative methods for solving the large linear systems are presented. Firstly, some classical iterative methods such as Jacobi, Gauss- Seidel, SOR, SSOR, and CG iterative method are discussed from the iterative formulas and convergence. Secondly, the Hermitian/Skew-Hermitian splitting （HSS） iteration is given, which is a new iterative methods for linear systems. Its convergence theorems are obtained. Lastly, the effectiveness of all the iterative methods is proved by numerical examples.