A family of higher order iterations free from second derivative for nonlinear equations in R

The aim of this article is to develop a family of higher order iterations free from second derivative for solving nonlinear equations in R. Their theoretical, computational and dynamical aspects are fully investigated and theorems are established to provide their order of convergence and asymptotic error constant. It is observed that the family includes sixth order methods and for a particular case its eighth order can be achieved. In this family, methods use three functions and one first derivative evaluations. The family of methods can be shown to be optimal by using Kung-Traub conjecture. A number of numerical examples are worked out to demonstrate the applicability of these methods. The improved results are obtained in comparison to some of the existing robust methods on the considered test examples. Local convergence analysis and dynamical study of the proposed family of methods are also carried out.