On improved three‐step schemes with high efficiency index and their dynamics

This paper presents an improvement of the sixth‐order method of Chun and Neta as a class of three‐step iterations with optimal efficiency index, in the sense of Kung‐Traub conjecture. Each member of the presented class reaches the highest possible order using four functional evaluations. Error analysis will be studied and numerical examples are also made to support the theoretical results. We then present results which describe the dynamics of the presented optimal methods for complex polynomials. The basins of attraction of the existing optimal methods and our methods are presented and compared to illustrate their performances.