Pseudoinversion of matrices through the search method of nonlinear programming in the equalization of free geodesic networks

Shevshenko, G.G.; Bryn, M.J.; Naumova, N.A.

doi:10.22389/0016-7126-2023-991-1-20-28

G&C

2023

DOI: 10.22389/0016-7126-2023-991-1-20-28

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Pseudoinversion of matrices through the search method of nonlinear programming in the equalization of free geodesic networks

G.G. Shevshenko

M.J. Bryn

N.A. Naumova

Abstract: The authors consider the possibility of using the search method to perform pseudo-rotation of matrices at equalizing free geodetic networks. Two methods of pseudo-circulation of matrices are proposed

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2024

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Cited by 3 publications

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Development of a methodology for determining the parameters of the transformation of coordinate systems based on the first-order nonlinear programming method when accompanying the construction of buildings and structures

Schenyavskaya,

Shevchenko,

Kovalenko

2024

Proceedings of Petersburg Transport University

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Objective: to analyze the advantages and disadvantages of existing nonlinear programming methods for solving problems of coordinate system transformation (SC). To consider the need to convert the coordinates of points from local SC to urban. To determine the possibility of using the generalized reduced gradient method to transform the coordinates of points located both at a distance between points of about 150 m and at a distance of about 1.5 km. Methods: the theoretical foundations of this method are revealed. Information is provided on the procedure for performing transformations of coordinate systems. The algorithm of the generalized reduced gradient (OPG) method is given. A study has been conducted on the transformation of coordinate systems of starting points from SK1 to SK2 using the OPG method. The requirement for the value of the mean square error (SQR) of determining the position of the starting points when converting them from one coordinate system to another is justified. Results: a check was performed on the possibility of applying the obtained transformation parameters to recalculate the coordinates of points from the local SC to the city SC. Based on the verification of the obtained transformation parameters, the possibility of using them to recalculate the coordinates of points located at a distance of about 1,5 km was revealed. The need for additional research on the application of the generalized reduced gradient method to solve coordinate system transformation problems has been identified. Practical significance: the necessity of transformation of coordinate systems in the field of construction of buildings and structures is shown. The possibility of applying the generalized reduced gradient method to transform coordinate systems is presented. Using this method will allow you to convert the coordinates of the starting points from the local coordinate system to the coordinate system of the city with geodetic support for the construction of buildings and structures of various categories.

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Development of a methodology for determining the parameters of the transformation of coordinate systems based on the first-order nonlinear programming method when accompanying the construction of buildings and structures

Schenyavskaya,

Shevchenko,

Kovalenko

2024

Proceedings of Petersburg Transport University

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Autonomous free geodetic 3D-construction as a coordinate basis of the space under study

Padve,

Barliani,

Salnikov

et al. 2024

G&C

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Looking for the global minimum of the objective function to determine the pseudoinverse matrix through the search method at equalizing free geodesic networks

Shevshenko,

Naumova,

Bryn

2024

G&C

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When evaluating the accuracy of equalized free geodetic networks based on the search method of nonlinear programming, a degenerate matrix of unknowns’ normal equations coefficients is formed. It must be pseudoinverse in order to calculate the inverse weight one of the equalized parameters. Performing a mathematical procedure for calculating a pseudoinverse matrix based on the mentioned technology of nonlinear programming at equalizing free geodetic constructions, a situation may arise when the minimization step is incorrectly selected or the approximate values in the desired array of numbers are set roughly. It can lead to finding a local, not a global minimum, as well as to cycling the algorithm. The authors provide the conclusion of the formula and substantiate the conditions on which the elements of the required matrix change, so that the value of the objective function either comes as close as possible to the global minimum, or indicates that it is impossible to achieve it. The correctness of the proposed conditions and the derived formula are confirmed by a test example

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Pseudoinversion of matrices through the search method of nonlinear programming in the equalization of free geodesic networks

Abstract: The authors consider the possibility of using the search method to perform pseudo-rotation of matrices at equalizing free geodetic networks. Two methods of pseudo-circulation of matrices are proposed

Cited by 3 publications

References 8 publications

Development of a methodology for determining the parameters of the transformation of coordinate systems based on the first-order nonlinear programming method when accompanying the construction of buildings and structures

Development of a methodology for determining the parameters of the transformation of coordinate systems based on the first-order nonlinear programming method when accompanying the construction of buildings and structures

Autonomous free geodetic 3D-construction as a coordinate basis of the space under study

Looking for the global minimum of the objective function to determine the pseudoinverse matrix through the search method at equalizing free geodesic networks

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