without memory

In this paper, there has been developed an adaptive method with memory based on the on the first two steps of the general optimal method without memory by Kung and Traub. Included is convergence analysis along with numerical implementations. The proposed adaptive method with memory can increase the efficiency index of the optimal method without memory as high as possible while reusing all the available information of the functional evaluations in the previous iteration for this purpose. Also, we discuss the numerical stability from the point of basins of attraction.