Sixth-order modifications of Newton's method based on Stolarsky and Gini means

In this article we present sixth order methods developed by extending third order methods of Herceg and Herceg (2013) for solving nonlinear equations. The methods require only four function evaluations per iteration. In this regard the efficiency index of our methods is 61/4≈1.56508. Considered methods are based on Stolarsky and Gini means and depend on two parameters. Sixth order convergence of considered methods is proved, and corresponding asymptotic error constants are expressed in terms of two parameters. Numerical examples, obtained using Mathematica with high precision arithmetic, are included to demonstrate convergence and efficacy of our methods. For some combinations of parameter values, the new sixth order methods produce very good results on tested examples, compared to the results produced by some of the sixth order methods existing in the related literature.