adaptive method with memory

The adaptive technique enables us to achieve the highest efficiency index theoretically and practically. The idea of introducing an adaptive self-accelerator (via all the old information for Steffensen-type methods) is new and efficient to obtain the highest efficiency index. In this work, we have used four self-accelerating parameters and have increased the order of convergence from 8 to 16, i. e. any new function evaluations improve the convergence order up to 100%. The numerical results are compared without and with memory methods. It confirms that the proposed methods have more efficiency index.

An efficient family of the recursive methods of adaptive is proposed for solving nonlinear equations, is developed such that all previous information are applied. These methods have reached the maximum degree of convergence improvement of 100%, and also have an efficiency index of 2. Three families have been examined from Steffensen-Like single, two, and three-step methods that have used 2, 3 and 4 parameters respectively. Numerical comparisons are made with other existing methods one-, two-, three-, and four-point to show the performance of the convergence speed of the proposed method and confirm theoretical results.

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.