A modified iterative algorithm for the weighted total least squares

Abstract

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.