On an efficient family of derivative free three-point methods for solving nonlinear equations

New three-step derivative free families of three-point methods for solving nonlinear equations are presented. First, a new family without memory of optimal order eight, consuming four function evaluations per iteration, is proposed by using two weight functions. The improvement of the convergence rate of this basic family, even up to 50%, is obtained without any additional function evaluation using a self-accelerating parameter. This varying parameter is calculated in each iterative step employing only information from the current and the previous iteration, defining in this way a family with memory. The self-accelerating parameter is calculated applying Newton’s interpolating polynomials of degree scaling from 1 to 4. The corresponding R  -orders of convergence are increased from 8 to 10, 11, 6+42≈11.66 and 12, providing very high computational efficiency of the proposed methods with memory. Another convenient fact is that these methods do not use derivatives. Numerical examples and comparison with the existing three-point methods are included to confirm theoretical results.