病态问题中奇异值修正法初探

Discussion on Modified SVD in the Ill-posed Problems

在法矩阵的条件数相对较小时,基于奇异值分解理论、截断奇异值分解方法和一种全部修正奇异值分解方法,提出了一种部分修正的奇异值分解方案和一种截断奇异值修正方案,以改善设计阵的病态性,提高参数估值的稳定性和准确性。经过算例验证,当法矩阵的条件数小于106,截断奇异值修正方案比较有效,与其他方法相比,得到的结果精度更高。

On the basis of the theory of SVD,truncated SVD and an entirely modified SVD method,a partly modified SVD method and a modified truncated SVD have been proposed when the conditioning number of the normal matrix is comparatively small.The results show that the proposed methods can resist the ill-conditioned effect and improve the numerical stability and accuracy of parameter estimation.Compared with other methods,modified truncated SVD is proved to be very effective and the precision of its results is higher on the premise that the conditioning number of the normal matrix is less than 106.