On a third order family of methods for solving nonlinear equations

Abstract

In this article we present a third-order family of methods for solving nonlinear equations. Some well-known methods belong to our family, for example Halley's method, method (24) from [M. Basto, V. Semiao, and F.L. Calheiros, A new iterative method to compute nonlinear equations, Appl. Math. Comput. 173 (2006), pp. 468-483] and the super-Halley method from [J.M. Gutierrez and M.A. Hernandez, An acceleration of Newton's method: Super-Halley method, Appl. Math. Comput. 117 (2001), pp. 223-239]. The convergence analysis shows the third order of our family. We also give sufficient conditions for the stopping inequality |xn+1-α|≤|xn+1-xn| for this family. Comparison of the family members shows that there are no significant differences between them. Several examples are presented and compared. © 2010 Taylor & Francis.