A new approach based on the Newton's method to solve systems of nonlinear equations

An alternative strategy for solving systems of nonlinear equations when the classical Newton's method fails is presented. The proposed strategy is an extension for systems based on the idea presented in the article (Ramos and Vigo-Aguiar, 2015). It relies in obtaining the approximate solutions of a given system of equations through solving an associated system obtained through the theory underlying the Newton's method. In this way, the solutions of the associated system that are not 2-cycles of the Newton iteration function provide solutions of the original system. As it is usual, in most cases, the associated system cannot be solved exactly, and some iterative procedure must be Used. For particular starting values, solving the associated system with the Newton's method results to be more efficient than the application of the Newton's method to the original system. Some examples are given to illustrate the performance of the proposed strategy. Performance profiles are evaluated in terms of number of iterations, error and CPU time.The second author has been supported by the ALGORITMI R&D Center and project PEst-UID/CEC/00319/2013. The authors are grateful to two referees for several constructive comments and remarks.