Optimal fourth order family of iterative methods

Abstract

In this work, we construct a family of optimal fourth orderiterative methods requiring three evaluations. During each iterativestep, methods need evaluation of two derivatives and one function.According to the Kung and Traub conjecture an optimal iterativemethod without memory based on $3$ evaluations could achieve anoptimal convergence order of $4$. The proposed iterative family ofmethods are especially appropriate for finding zeros of functionswhose derivative is easy to evaluate. For example, polynomialfunctions and functions defined via integrals.