Means based modifications of Newton’s method for solving nonlinear equations

In this paper we consider a family of six sets of means based modifications of Newton’s method for solving nonlinear equations. Each set is a parametric class of methods. Some well-known methods belong to our family, for example, the arithmetic mean Newton’s method [S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third order convergence, Appl. Math. Lett. 13 (2000) 87–93], the harmonic mean Newton’s method, [A.Y. Özban, Some new variants of Newton’s method, Appl. Math. Lett. 17 (2004) 677–682], the geometric mean Newton’s method, [T. Lukić, N.M. Ralević, Geometric mean Newton’s method for simple and multiple roots, Appl. Math. Lett. 21 (2008) 30–36], the power mean Newton’s method [X. Zhou, A class of Newton’s methods with third-order convergence, Appl. Math. Lett. 20 (2007) 1026–1030]. The convergence analysis shows third order of our family. Comparison of the family members shows that there are no big differences between them. Twelve numerical examples were tested, and two characteristic ones are presented.