A new class of three-point methods with optimal convergence order eight and its dynamics

Abstract

We establish a new class of three-point methods for the computation of simple zeros of a scalar function. Based on the two-point optimal method by Ostrowski (1966), we construct a family of order eight methods which use three evaluations of f and one of f? and therefore have an efficiency index equal to 8 4 ? 1 . 682 $\sqrt [4]{8}\approx 1.682$ and are optimal in the sense of the Kung and Traub conjecture (Kung and Traub J. Assoc. Comput. Math. 21, 634---651, 1974). Moreover, the dynamics of the proposed methods are shown with some comparisons to other existing methods. Numerical comparison with existing optimal schemes suggests that the new class provides a valuable alternative for solving nonlinear equations.