解线性方程组的松弛方法及其应用

Relaxation Methods for Systems of Linear Equations and Applications

中文关键词:  迭代法  松弛法  线性方程组  鞍点问题  PageRank问题

英文关键词:iterative methods  relaxation methods  linear systems  saddle point problem  PageRank problem

中文摘要:

      线性方程组在科学和工程领域中有着重要的应用,松弛方法是求解线性方程组的有效算法之一.本文在著名的Gauss-Seidel迭代法的基础上,研究了一种有效的松弛方法.理论分析表明,该方法能收敛到线性方程组的唯一解.此外,我们还将该方法应用在鞍点问题和PageRank问题的求解上,并得出了相应的数值结果.结果表明该方法比现有的松弛方法更有效.

英文摘要:

      The relaxation methods have served as very efficient tools for solving linear system and have many important applications in the field of science and engineering. In this paper, we study an efficient relaxation method based on the well-known Gauss-Seidel iteration method. Theoretical analysis shows our method can converge to the unique solution of the linear system. In addition, our method is applied to solve the saddle point problem and PageRank problem, and the numerical results show our method is more powerful than the existent relaxation methods.