A new family of adaptive methods with memory for solving nonlinear equations.

In this work, an adaptive method with memory is developed such that all previous information are applied. The importance of the proposed method can be seen because of the optimization in important effecting factors, i.e., least number of iterations steps, least number of functional evaluations, least value of absolute error, and maximum efficiency index in final as well as in individual step as compared with the other methods. Indeed, it is proved that this adaptive method with memory has efficiency index 2 and competes all the existing methods without and with memory in the literature. The order of convergence is obtained by using two self-accelerating parameters, which is increased from 2 to 4 without any new function evaluation. It means that, the order of convergence can be improved until 100%. Numerical examples and the comparison with existing methods are included to demonstrate exceptional convergence speed of the proposed method and confirm theoretical results. [ABSTRACT FROM AUTHOR]