自适应谱修正LU分解法解算高病态法方程

SELF-ADAPTIVE SPECTRUM CORRECTION LU DECOMPOSITION ALGORITHM FOR SOLVING A NORMAL EQUATION WITH SEVERELY ILL-CONDITIONED MATRIX

谱修正迭代算法及其改正算法均没有很好地解决收敛速度与降低矩阵病态性之间的矛盾,且改进算法的阻尼因子缺乏有效的确定方法。针对该问题,提出自适应谱修正LU分解算法。该算法基于谱修正方程,采用LU分解法解算法方程,回避了矩阵求逆问题;在迭代过程中,阻尼因子不固定,而是根据残差下降速度自适应地调整其大小。该算法能极大降低高病态法方程的解算迭代次数,提高收敛速度和计算精度。

The spectrum correction iterative algorithm and its correction algorithm are not a good solution to se- rious ill-posed problem, for the contradiction between convergent speed and ill-condition reducing is difficult to deal with. At currently, the damping factor used by the improved algorithm still lacks an effective method to determinate. Aiming at this problem, the self-adaptive spectrum correction LU decomposition algorithm was proposed in the pa- per. The algorithm, which based on the spectrum correction equation, can avoid the matrix inverse problem by using the LU decomposition method for solving a normal equation. In the process of iteration, the damping factor is not fixed, and its value is adjusted adaptively according to the rate of residual decline. Examples show that, the algorithm can greatly reduce the iteration times to solve a normal equation with severely ill-conditioned matrix, and it also en- hances the convergent speed and calculation accuracy greatly.