Copyright © 2013 Gustavo Fernández-Torres and Juan Vásquez-Aquino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.