主元加权迭代法求解病态线性方程组

摘要

When the iterative method is devoted to solving for the ill-conditioned linear equations, convergence of iterative method and accuracy of the numerical solution are quited bad because of the large condition number of ill -conditioned linear equations. Aiming at this problem, an algorithm of pivot element weighting iterative algo-rithm ( PEWI) is proposed. The pivot elements of coefficient matrix are added to a weight and whereby their condi-tion number will be decreased. Finally, the PEWI proposes, Gauss-Seidel iterative and Jacobi iterative are tested on linear equations which their coefficient matrix is constituted by Hilbert matrix. Contrast experimental results show that the PEWI is able to increase the accuracy of the numerical solution.%由于病态线性方程组的系数矩阵条件数很大,使用迭代法求解病态线性方程组时,收敛速度慢且数值解的精度很低.针对此问题,设计了一种主元加权迭代算法.该算法在系数矩阵主元上叠加一个权值,以此来降低系数矩阵的条件数.最后以希尔伯特矩阵构成的病态线性方程组为例,对提出的主元加权迭代算法和高斯-赛德尔迭代法以及雅克比迭代法进行了测试.对比试验结果表明:主元加权迭代算法能有效地提高数值解的精度.