APPLICATION OF THE SINGULAR DECOMPOSITION OF THE MATRIX IN SOLVING INCORRECT GEODESIC PROBLEMS
Authors:   Institute of Geodesy Lviv Polytechnic National University, Sohor A. , Sidorov I. ,   Institute of Geodesy Lviv Polytechnic National University, Smirnova O. ,   Institute of Geodesy Lviv Polytechnic National University
Published: August 2024

The most reliable method for solving 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. Some other methods require less machine time and memory. However, 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 , which is determined as T U AV  . 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 lie 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 mea- surements in incorrect geodetic problems. Based on the performed research on the application of the singular decomposition method for solving incorrect geodetic problems, we can summarize the following results. A singular expansion of a real matrix A is any factorization T A U W  of a matrix with orthogonal columns U, an orthogonal matrix W , and a diagonal matrix , the elements of which are called singular numbers of the matrix A, and the columns of matrices U and W — left and right singular vectors. If the matrix A has a full rank, then its solution will be unique and stable, which can be obtained by different methods. However, 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 geod- esy, 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 to both stable and by nature unstable problems. This possibility to solve incorrect geodetic problems is associated with the application of some limit , the choice of which can be made by the relative errors of the matrix of coefficients of parametric equations of corrections A and the vector of results of geodetic measurements L . 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 correc- tions 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 allow the easy calculation of elements  and XQ estimates of accuracy, almost ignoring the complex procedure of rotation of the matrix of coefficients of normal equations.