In this paper first, the method used for solving the weighted total least squares is discussed in two cases; (1) The parameter corresponding to the erroneous column in the design matrix is a scalar, model \(({\mathbf{H}} + {\mathbf{G}})^{T} {\mathbf{r}} + \delta \, = {\mathbf{q}} + {\mathbf{e}}\), (2) The parameter corresponding to the erroneous column in the design matrix is a vector, model \(({\mathbf{H}} + {\mathbf{G}})^{T} {\mathbf{r}} + {\varvec{\updelta}}\, = {\mathbf{q}} + {\mathbf{e}}\). Available techniques for solving TLS are based on the SVD and have a high computational burden. Besides, for the other presented methods that do not use SVD, there is need for large matrices, and it is needed to put zero in the covariance matrix of the design matrix, corresponding to errorless columns. This in turn increases the matrix size and results in increased volume of the calculations. However, in the proposed method, problem-solving is done without the need for SVD, and without introducing Lagrange multipliers, thus avoiding the error-free introducing of some columns of the design matrix by entering zero in the covariance matrix of the design matrix. It needs only easy equations based on the principles of summation, which will result in very low computing effort and high speed. Another advantage of this method is that, due to the similarity between this solving method and the ordinary least squares method, one can determine the covariance matrix of the estimated parameters by the error propagation law and use of other advantages of the ordinary least squares method.

中文翻译：

---

加权总最小二乘的改进迭代算法

在本文中，首先讨论了在两种情况下用于求解加权总最小二乘法的方法：（1）设计矩阵中与错误列对应的参数是标量模型\（（{\ mathbf {H}} + {\ mathbf {G}}）^ {T} {\ mathbf {r}} + \ delta \，= {\ mathbf {q}} + {\ mathbf {e}} \），）（2）设计矩阵中与错误列对应的参数是向量\（（{{mathbf {H }} + {\ mathbf {G}}）^ {T} {\ mathbf {r}} + {\ varvec {\ updelta}} \，= {\ mathbf {q}} + {\ mathbf {e}} \\ ）。解决TLS的可用技术基于SVD，并且具有很高的计算负担。此外，对于其他未使用SVD的方法，需要使用大型矩阵，并且需要在设计矩阵的协方差矩阵中放入零（对应于无误列）。反过来，这会增加矩阵大小，并导致计算量增加。但是，在提出的方法中，不需要SVD即可解决问题，并且无需引入Lagrange乘法器，从而通过在设计矩阵的协方差矩阵中输入零来避免无错误地引入设计矩阵的某些列。它仅需要基于求和原理的简单方程式，这将导致非常少的计算工作量和很高的速度。这种方法的另一个优点是，