A new highly efficient and optimal family of eighth-order methods for solving nonlinear equations

The principle aim of this manuscript is to present a new highly efficient and optimal eighth-order family of iterative methods to solve nonlinear equations in the case of simple roots. The derivation of this scheme is based on weight function and rational approximation approaches. The proposed family requires only four functional evaluations (viz. f(xn) f′(xn) f(yn) and f(zn)) per iteration. Therefore, the proposed family is optimal in the sense of Kung–Traub hypotheses. In addition, we given a theorem which describing the order of convergence of the proposed family. Moreover, we present a local convergence analysis using hypotheses only on the first-order derivative, since in our preceding theorem we used hypotheses on higher-order derivatives that do not appear in these methods. In this way, we expand the applicability of these methods even further. Furthermore, a variety of nonlinear equations is considered for the numerical experiments. It is observed from the numerical experiments that our proposed methods perform better than the existing optimal methods of same order, when the accuracy is checked in multi precision digits.