A universal and fast method to solve linear systems with correlated coefficients using weighted total least squares

Especially in metrology and geodesy, but also in many other disciplines, the solution of overdetermined linear systems of the form ${\boldsymbol{Ax}} \approx {\boldsymbol{b}}$ with individual uncertainties not only in ${\boldsymbol{b}}$ but also in ${\boldsymbol{A}}$ is an important task. The problem is known in literature as weighted total least squares. In the most general case, correlations between the elements of $\left[ {{\boldsymbol{A}},{\boldsymbol{b}}} \right]$ exist as well. The problem becomes more complicated and can—except for special cases—only be solved numerically. While the formulation of this problem and even its solution is straightforward, its implementation—when the focus is on reliability and computational costs—is not. In this paper, a robust, fast, and universal method for computing the solution of such linear systems as well as their covariance matrix is presented. The results were confirmed by applying the method to several special cases for which an analytical or numerical solution is available. If individual coefficients can be considered to be free of errors, this can be taken into account in a simple way. An implementation of the code in MATLAB is provided.