求解超定病态线性方程组的一种正则化迭代算法

A Regularized Iterative Algorithm for Solving Over-determined Ill-conditioned Linear Equations

首先将超定病态线性方程组转化为正则方程组,在对其病态系数矩阵改良的同时,构造出与正则方程组同解的方程组和一个一般迭代公式。为了保证迭代算法的有效性,对迭代公式的收敛性进行了证明。同时对正则参数选取进行了判定。最后,以核磁共振弛豫反演模型对应的大型超定病态线性方程组为测试实例,对正则化迭代算法进行了验证。实验结果表明,该算法对大型超定病态线性方程组求解是有效的。

First,an over-determined ill-conditioned linear equation group is converted into regularized equations. An equation group with the same solution as the regular equations as well as a general iterative formula are constructed when the ill-conditioned coefficient matrix of the regularized equations are improved. Then the convergence of the algorithm is proved to ensure the effectiveness of the iterative algorithm. At the same time,The value of regularized parameter is determined. Finally,taking large-scale over-determined ill-conditioned linear equations from the NMR relaxation inversion model for test case,effectiveness of the regularized iterative algorithm is verified. Experimental results show that the proposed algorithm is effective for addressing large-scale over-determined ill-conditioned linear equations.