Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Some Properties of the Basins of Attraction of the Newton’s Method for Simple Nonlinear Geodetic Systems

Version 1 : Received: 7 December 2021 / Approved: 9 December 2021 / Online: 9 December 2021 (10:55:53 CET)

How to cite: Kroszczynski, K.; Kiliszek, D.; Winnicki, I. Some Properties of the Basins of Attraction of the Newton’s Method for Simple Nonlinear Geodetic Systems. Preprints 2021, 2021120151 (doi: 10.20944/preprints202112.0151.v1). Kroszczynski, K.; Kiliszek, D.; Winnicki, I. Some Properties of the Basins of Attraction of the Newton’s Method for Simple Nonlinear Geodetic Systems. Preprints 2021, 2021120151 (doi: 10.20944/preprints202112.0151.v1).

Abstract

The research presented in this paper concerns the determination of the attraction basins of the Newton’s iterative method which was used to solve the non-linear systems of observational equations associated with the geodetic measurements. The authors considered simple observation systems corresponding to the intersections, or linear and angular resections, used in practice. The main goal was to investigate the properties of the sets of convergent in the initial points of the applied iterative method. An important issue regarding the possibility of automatic and quick selection of such points was also considered. 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 basins, i.e. 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, i.e. the conditions imposed on the eigenvalues and norms of the matrices of the studied iterative systems. One of the essential results of the research is the indication that the obtained basins of attraction contain areas resulting from the theoretical premises and 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, i.e. located far from the solution points.

Keywords

basin of attraction; convergence of the Newton’s iterative method; planar intersection/resection.

Subject

MATHEMATICS & COMPUTER SCIENCE, Computational Mathematics

Comments (0)

We encourage comments and feedback from a broad range of readers. See criteria for comments and our diversity statement.

Leave a public comment
Send a private comment to the author(s)
Views 0
Downloads 0
Comments 0
Metrics 0


×
Alerts
Notify me about updates to this article or when a peer-reviewed version is published.
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here.