An optimal eighth order derivative free multiple root finding scheme and its dynamics

The problem of solving a nonlinear equation is considered to be one of the significant domain. Motivated by the requirement to achieve more optimal derivative-free schemes, we present an eighth-order optimal derivative-free method to find multiple zeros of the nonlinear equation by weight function approach in this paper. This family of methods requires four functional evaluations. The technique is based on a three-step method including the first step as a Traub-Steffensen iteration and the next two as Traub-Steffensen-like iterations. Our proposed scheme is optimal in the sense of Kung-Traub conjecture. The applicability of the proposed schemes is shown by using different nonlinear functions that verify the robust convergence behavior. Convergence of the presented family of methods is demonstrated through the graphical regions by drawing basins of attraction.