New efficient bi-parametric families of iterative methods with engineering applications and their basins of attraction

Abstract

We present in this paper two new families of bi-parametric multipoint higher order iterative methods of optimal order for determining simple roots of the nonlinear equation Ω(s)=0. The proposed families of methods are derivative-free with the optimal three-point fourth order methods requiring only three function evaluations per iteration and the optimal four-point eighth order methods consuming four function evaluations per iteration. Taylor’s series expansion and divided difference techniques are employed for the formulation of the methods. Their theoretical convergence properties are thoroughly analysed through the main theorems. Numerical experiments on nonlinear functions with some engineering applications are presented and are compared with some existing methods to demonstrate the effectiveness, applicability and validity of the proposed families of methods. Finally, graphical comparison is made through the basins of attraction which further provide useful information about their dynamical behaviour in the complex plane.