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<p>The research presented in this paper concerns the determination of the attraction basins of Newton’s iterative method, which was used to solve the non-linear systems of observational equations associated with the geodetic measurements. The simple observation systems corresponding to the intersections or linear and angular resections used in practice were considered. The main goal was to investigate the properties of the sets of convergent initial points of the applied iterative method. Therefore, the answers to the questions regarding the geometric structure of the basins, their limitations, connectedness, or self-similarity were sought. The research also concerned the iterative structures of the basin: maps of the number of iterations which are necessary to achieve the convergence of the Newton’s method. The determined basins were compared with the areas of convergence that result from theorems on the convergence of the Newton’s method: the conditions imposed on the eigenvalues and norms of the matrices of the studied iterative systems. One of the significant results is the indication that the obtained basins of attraction contain areas resulting from the theoretical premises. Their diameters can be comparable with the sizes of the analyzed geodetic structures. Consequently, in the analyzed cases, it is possible to construct methods that enable quick selection of the initial starting points or automation of such selection. The paper also characterizes the global convergence mechanism of the Newton’s method for disconnected basins and, as a consequence, the non-local initial points located far from the solution points.</p>
</abstract>

The research presented in this paper concerns the determination of the attraction basins of Newton’s iterative method, which was used to solve the non-linear systems of observational equations associated with the geodetic measurements. The simple observation systems corresponding to the intersections or linear and angular resections used in practice were considered. The main goal was to investigate the properties of the sets of convergent initial points of the applied iterative method. Therefore, the answers to the questions regarding the geometric structure of the basins, their limitations, connectedness, or self-similarity were sought. The research also concerned the iterative structures of the basin: maps of the number of iterations which are necessary to achieve the convergence of the Newton’s method. The determined basins were compared with the areas of convergence that result from theorems on the convergence of the Newton’s method: the conditions imposed on the eigenvalues and norms of the matrices of the studied iterative systems. One of the significant results is the indication that the obtained basins of attraction contain areas resulting from the theoretical premises. Their diameters can be comparable with the sizes of the analyzed geodetic structures. Consequently, in the analyzed cases, it is possible to construct methods that enable quick selection of the initial starting points or automation of such selection. The paper also characterizes the global convergence mechanism of the Newton’s method for disconnected basins and, as a consequence, the non-local initial points located far from the solution points.

Some properties of the basins of attraction of the Newton’s method for simple nonlinear geodetic systems

Department of Civil Engineering and Geodesy

Reports on Geodesy and Geoinformatics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The research presented in this paper concerns the determination of the attraction basins of Newton’s iterative method, which was used to solve the non-linear...

Some properties of the basins of attraction of the Newton’s method for simple nonlinear geodetic systems

Data publikacji: 17 lip 2024

Zakres stron: 22 - 40

Otrzymano: 11 lut 2024

Przyjęty: 10 cze 2024

DOI: https://doi.org/10.2478/rgg-2024-0012

Słowa kluczowe
basin of attraction, convergence of the Newton’s iterative method, planar intersection/resection

© 2024 Ireneusz Winnicki et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Some properties of the basins of attraction of the Newton’s method for simple nonlinear geodetic systems

Data publikacji: 17 lip 2024

Zakres stron: 22 - 40

Otrzymano: 11 lut 2024

Przyjęty: 10 cze 2024

DOI: https://doi.org/10.2478/rgg-2024-0012

Słowa kluczowebasin of attraction, convergence of the Newton’s iterative method, planar intersection/resection

© 2024 Ireneusz Winnicki et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Słowa kluczowe
basin of attraction, convergence of the Newton’s iterative method, planar intersection/resection