A Newton-like method for nonlinear system of equations (Q1039287)

The paper is concerned with iteratively solving a system of nonlinear equations by a method called centered Newton method. This method was proposed first by \textit{K. Tanake} [Centered Newton method. In: Yamamoto, Proceedings of the winter institute , C/ 1-10, World of Computational Mathematics (1988)]. The centered Newton method is a variation of the classical Newton method and it consists in modifying the Newton direction towards a variety called ''central variety''. The central variety can be understood as an extension of the central path trajectory used in the study of the interior point method for linear programming and also, for nonlinear programming. The authors make an analysis of some properties of the centered Newton method, including local convergence of the method and the rate of convergence. Also, they propose a new variant of the method and prove the global convergence of this method. It is shown that the centered Newton method is an inexact Newton method. Many practical choices of the involved parameters are proposed. The numerical performance of the centered Newton method is presented in a variety of problems. The tests show that, for some problems, in particular those highly nonlinear, the centered Newton method improves the global behaviour of the Newton method and reduces the number of iterations.