High-order Newton-type iterative methods with memory for solving nonlinear equations

In this paper, we present a new family of two-step Newton-type iterative methods with memory for solving nonlinear equations. In order to obtain a Newton-type method with memory, we first present an optimal two-parameter fourth-order Newton-type method without memory. Then, based on the two-parameter method without memory, we present a new two-parameter Newton-type method with memory. Using two self-correcting parameters calculated by Hermite interpolatory polynomials, the R-order of convergence of a new Newton-type method with memory is increased from 4 to 5.7016 without any additional calculations. Numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the presented methods.