A New Two Step Class of Methods with Memory for Solving Nonlinear Equations with High Efficiency Index

It is attempted to extend a two-step without memory method to it''s with memory. Then, a new two-step derivative free class of without memory methods, requiring three function evaluations per step, is suggested by using a convenient weight function for solving nonlinear equations. Eventually, we obtain a new class of methods by employing a self-accelerating parameter calculated in each iterative step applying only information from the current and the previous iteration, defining a with memory class.Although these improvements are achieved without any additional function evaluations, the $ R $-order of convergence are boosted from 4 to 5.24 and 6, respectively, and it is demonstrated that the proposed with memory classes provide a very high computational efficiency.Numerical examples are put forward and the performances are compared with the basic two-step without memory methods.