最小二乘配置的SVD分解解法

An algorithm for least-square collocation by singular value decomposition

最小二乘配置最初是在组合各种资料来研究地球形状与重力场的一种数学方法,目前最小二乘配置已经在测绘数据处理中得到广泛应用。本文首先分析了目前采用的最小二乘配置法解算方法,在讨论了矩阵的奇异值分解(Singular Value Decomposition,SVD)方法的基础上,推导得出了矩阵SVD分解与广义逆矩阵的关系,得出了可以直接利用SVD分解求解矩阵的Moore-Penrose广义逆,并推导了应用SVD分解求解最小二乘配置的估值计算公式和精度估算公式,最后通过重力异常实例进行了计算,得出矩阵的SVD分解用于最小二乘配置解算的正确性和可行性,为最小二乘配置的求解提供了一种新方法。

Least Square Collocation is the mathematics method that is initially used to study the earth shape and gravity field together with the various gravity data. At present, Least Square Collocation has been widely applied in surveying data processing. Based on the analysis of the algorithm of least square collocation and discussion of the matrix singular value decomposition （ SVD）, the relationshi Pbetween SVD decomposition and Moore - Penrose generalized inverse is deduced. Then the calculation formulaes of Least Square Collocation and Moore -Penrose generalized inverse based on SVD decomposition are provided. Finally practical computations of the gravity anomaly have shown that the SVD method is correct and validity in Least - Square collocation calculation. From this new collocation algorithm in this paper, the special function of matrix decomposition in surveying data processing, as well as the relationshi Pbetween matrix decomposition and adjustment, are found out.