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In this paper, we consider two-step fourth-order and three-step sixth-order derivative free iterative methods and study their convergence in Banach spaces to approximate a locally-unique solution of nonlinear equations. Study of convergence analysis provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions using higher order derivatives. Furthermore, in quest of fast algorithms, a generalized  q -step scheme with increasing convergence order 2 q  + 2 is proposed and analyzed. Novelty of the  q -step algorithm is that, in each step, order of convergence is increased by an amount of two at the cost of only one additional function evaluation. To maximize the computational efficiency, the optimal number of steps is calculated. Theoretical results regarding convergence and computational efficiency are verified through numerical experimentation.

Higher order Traub–Steffensen type methods and their convergence analysis in Banach spaces