Structure of an Adaptive with Memory Method with Efficiency Index 2

The current research develops derivative-free family with memory methods with 100% improvement in the order of convergence. They have three parameters. The parameters are approximated and increase the convergence order from 4 to 6, 7, 7.5 and 8, respectively. Additionally, the new self-accelerating parameters are constructed by a new way. They have the properties of simple structures and they are calculated easily. The parameters do not increase the computational cost of the iterative methods. Numerical examples of the new schemes are given to support the theoretical results.