Robust cubically and quartically iterative techniques free from derivative

Abstract

Constructing of a technique which is both accurate and derivativefree is one of the most important tasks in the field of iterative processes. Hence in this study, convergent iterative techniques are suggested for solving single variable nonlinear equations. Their error equations are given theoretically to show that they have cubic and quartical convergence. Per iteration the novel schemes include three evaluations of the function while they are free from derivative as well. In viewpoint of optimality, the developed quartically class reaches the optimal efficiency index 41/3 H" 1.587 based on the Kung-TraubHypothesis regarding the optimality of multi-point iterations without memory. In the end, the theoretical results are supported by numerical examples to elucidate the accuracy of the developed schemes.