The most reliable method for calculating linear equations of the least squares principle, which can be used to solve incorrect geodetic problems, is based on matrix factorization, which is called a singular expansion. There are other methods that require less machine time and memory. But they are less effective in taking into account the errors of the source information, rounding errors and linear dependence. The methodology of such research is that for any matrix A and any two orthogonal matrices U and V there is a matrix Σ, which is determined from the ratio. The idea of a singular decomposition is that by choosing the right matrices U and V, you can convert most elements of the matrix to zero and make it diagonal with non-negative elements. The novelty and relevance of scientific solutions lies in the feasibility of using a singular decomposition of the matrix to obtain linear equations of the least squares method, which can be used to solve incorrect geodetic problems. The purpose of scientific research is to obtain a stable solution of parametric equations of corrections to the results of measurements in incorrect geodetic problems. Based on the performed research on the application of the singular decomposition method in solving incorrect geodetic problems, we can summarize the following. A singular expansion of a real matrix is any factorization of a matrix with orthogonal columns , an orthogonal matrix and a diagonal matrix , the elements of which are called singular numbers of the matrix , and the columns of matrices and left and right singular vectors. If the matrix has a full rank, then its solution will be unique and stable, which can be obtained by different methods. But the method of singular decomposition, in contrast to other methods, makes it possible to solve problems with incomplete rank. Research shows that the method of solving normal equations by sequential exclusion of unknowns (Gaussian method), which is quite common in geodesy, does not provide stable solutions for poorly conditioned or incorrect geodetic problems. Therefore, in the case of unstable systems of equations, it is proposed to use the method of singular matrix decomposition, which in computational mathematics is called SVD. The SVD singular decomposition method makes it possible to obtain stable solutions of both stable and unstable problems by nature. This possibility to solve incorrect geodetic problems is associated with the application of some limit τ, the choice of which can be made by the relative errors of the matrix of coefficients of parametric equations of corrections and the vector of results of geodetic measurements . Moreover, the solution of the system of normal equations obtained by the SVD method will have the shortest length. Thus, applying the apparatus of the singular decomposition of the matrix of coefficients of parametric equations of corrections to the results of geodetic measurements, we obtained new formulas for estimating the accuracy of the least squares method in solving incorrect geodetic problems. The derived formulas have a compact form and make it possible to easily calculate the elements and estimates of accuracy, almost ignoring the complex procedure of rotation of the matrix of coefficients of normal equations.
On the Evaluation of Determinants, the Formation of their Adjugates, and the Practical Solution of Simultaneous Linear Equations
