非等间距GM(1,1)模型的总体最小二乘算法及其病态问题

在非等间距GM(1,1)模型中，系数矩阵中有无误差的常数项和有误差的随机项，并且系数矩阵与观测向量误差同源，即系数矩阵与观测向量中有相同的元素存在，这些相同元素应该有相同的改正数，为此本文推导了一种适合非等间距GM(1,1)模型求解的总体最小二乘算法。同时，考虑到非等间距GM(1,1)模型中存在病态问题时影响总体最小二乘计算结果的稳定性，提出对系数矩阵常数列乘以某一常数的方法，以改善病态问题。

A Total Least Squares Algorithm for Non-Equidistant GM(1,1) Model and Its Ill-Posed Problem

In non-equidistant GM(1,1) model there are constant terms without error and random terms with errors in the coefficient matrix. The errors of the coefficient matrix and observation vector are from the same source; the same elements are in the coefficient matrix and observation vector. These same elements ought to have the same corrected value. Therefore, a total least squares algorithm that is suitable to solve non-equidistant GM(1,1) model is deduced in this paper. The ill-posed problem in the non-equidistant GM(1,1) model is taken into consideration, which has an influence on the stability of the calculated results of total least squares. The method, which is to multiply the constant column in coefficient matrix by a constant, is proposed to alleviate the ill-posed problem.