In this paper, we present a family of optimal, in the sense of Kung-Traub's conjecture, iterative methods for solving nonlinear equations with eighth-order convergence. Our methods are based on Chun's fourth-order method. We use the Ostrowski's efficiency index and several numerical tests in order to compare the new methods with other known eighth-order ones. We also extend this comparison to the dynamical study of the different methods. [ABSTRACT FROM AUTHOR]
In this paper, we present some new third-order iterative methods for finding a simple root α of nonlinear scalar equation f( x)=0 in R. A geometric approach based on the circle of curvature is used to construct the new methods. Analysis of convergence shows that the new methods have third-order convergence, that is, the sequence { x n} generated by each of the presented methods converges to α with the order of convergence three. The efficiency of the methods are tested on several numerical examples. It is observed that our methods can compete with Newton’s method and the classical third-order methods. [ABSTRACT FROM AUTHOR]