On the efficiency of Newton and Broyden numerical methods in the investigation of the regular polygon problem of (N + 1) bodies

Keywords: Celestial Mechanics (N + 1)-Ring-body problem Non-linear algebraic equations Newton and quasi-Newton methods Comparison of numerical methods a b s t r a c t Numerical methods of finding the roots of a system of non-linear algebraic equations are treated in this paper. This paper attempts to give an answer to the selection of the most efficient method in a complex problem of Celestial Dynamics, the so-called ring problem of (N + 1) bodies. We apply Newton and Broyden's method to these problems and we investigate , by means of their use, the planar equilibrium points, the five equilibrium zones, which are symbolized by A 1 , A 2 , B, C 2 , and C 1 (by order of appearance from the center O to the periphery of the imaginary circle on which the primaries lie) [T.J. Kalvouridis, A pla-nar case of the N + 1 body problem: the ring problem. Astrophys. Space Sci. 260 (3) (1999) 309–325], and the attracting regions of the system. The efficiency of these methods is studied through a comparative process. The obtained results are demonstrated in figures and are discussed. In this paper, we apply two numerical methods for finding the roots of a system of non-linear algebraic equations. Although the developed methodology has traced a rather long historical path, the number of the methods that have been proposed in the course of these years is rather small. Some of them are mentioned for purely historical reasons, as for example the simple iteration method and the steepest descent one; others are difficult to apply in real problems or to be used by people who are not familiar with the sophisticated numerical analysis (see for example [2–4], etc.). Here, we must emphasize the fact, that the wide use of computational software like Matlab, Mathematica, Mapple, etc., has alienated the ordinary users from the fundamental concepts and procedures of Numerical Analysis and the relevant programming environment, and converted them to simple operators of these tools, where their personal interference is trivial. Nevertheless, this software is extremely useful since it gives solutions to many difficult problems, and a lot of complicated scientific research would not have been realized without them. Through this paper we intend to bring forward the importance of the comparative study of homologous numerical methods applicable to realistic problems. Here we note that comparison processes have been widely used in the past …