On a new efficient Steffensen-like iterative class by applying a suitable self-accelerator parameter.

It is attempted to present an efficient and free derivative class of Steffensen-like methods for solving nonlinear equations. To this end, firstly, we construct an optimal eighth-order three-step uniparameter without memory of iterative methods. Then the self-accelerator parameter is estimated using Newton's interpolation in such a way that it improves its convergence order from 8 to 12 without any extra function evaluation. Therefore, its efficiency index is increased from 8(1/4) to 12(1/4) which is the main feature of this class. To show applicability of the proposed methods, some numerical illustrations are presented.

On generalization based on bi et Al. Iterative methods with eighth-order convergence for solving nonlinear equations.

The primary goal of this work is to provide a general optimal three-step class of iterative methods based on the schemes designed by Bi et al. (2009). Accordingly, it requires four functional evaluations per iteration with eighth-order convergence. Consequently, it satisfies Kung and Traub's conjecture relevant to construction optimal methods without memory. Moreover, some concrete methods of this class are shown and implemented numerically, showing their applicability and efficiency.