Efficient methods of optimal eighth and sixteenth order convergence for solving nonlinear equations

Abstract

We present simple yet efficient three- and four-point iterative methods for solving nonlinear equations. The methodology is based on fourth order Kung–Traub method and further developed by using rational Hermite interpolation. Three-point method requires four function evaluations and has the order of convergence eight, whereas the four-point method requires the evaluation of five functions and has the order of convergence sixteen, that means, the methods are optimal in the sense of Kung–Traub hypothesis (Kung and Traub, J ACM 21:643–651, 1974). The methods are tested through numerical experimentation. Their performance is compared with already established methods in literature. It is observed that new algorithms are well-behaved and very effective in high precision computations. Moreover, the presented basins of attraction also confirm stable nature of the algorithms.