Efficient Ostrowski-like methods of optimal eighthand sixteenth order convergence and their dynamics

Abstract

We present simple yet efficient three- and four-point iterative methods for solving nonlinear equations. The methodology is based on fourth order Ostrowski’s method and further developed by using inverse rational function approximation. 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 (J ACM 21:643–651, 1974) hypothesis. The methods are tested through numerical experimentation. It is observed that new algorithms in general are more accurate than existing counterparts and very effective in high precision computations. Moreover, the presented basins of attraction also confirm stable nature of the algorithms as compared to existing ones.