A stable family with high order of convergence for solving nonlinear equations

Recently, Li et al. (2014) have published a new family of iterative methods, without memory, with order of convergence five or six, which are not optimal in the sense of Kung and Traub’s conjecture. Therefore, we attempt to modify this suggested family in such a way that it becomes optimal. To this end, we consider the same two first steps of the mentioned family, and furthermore, we introduce a better approximation for f′(z)f′(z) in the third step based on interpolation idea as opposed to the Taylor’s series used in the work of Li et al. Theoretical, dynamical and numerical aspects of the new family are described and investigated in details.