Optimal fourth order methods with its multi-step version for nonlinear equation and their Basins of attraction

Abstract

Construction of iterative processes without memory, which is optimal according to the Kung–Traub hypothesis is considered in this paper. A new class of method by improving Ezquerro et al. method having three function evaluations per iteration, which reaches the optimal order four is discussed. The efficiency index of this method is found to be 1.587 which is better than the efficiency index of Newton’s method (1.414) and equals the efficiency of Jarratt and Chun’s methods. This procedure also provides an n-step family of methods using n + 1 function evaluations per iteration cycle to possess the order (2r + 4). The theoretical proof of the main contributions are given and numerical examples are included to confirm the convergence order of the presented methods. The extraneous fixed points of the presented fourth order method and other existing fourth order methods are discussed. Basins of attraction study is also done for the presented methods and few other methods. We apply the new methods to find the optimal launch angle in a projectile motion problem as an application.