Two- and three-point with memory methods for solving nonlinear equations

The main objective and inspiration in the construction of two- and three-point with memory methods is to attain the utmost computational efficiency without any additional function evaluations. At this juncture, we have modified the existing fourth and eighth order without memory methods with optimal order of convergence by means of different approximations of self-accelerating parameters. The parameters are calculated by a Hermite interpolating polynomial, which accelerates the order of convergence of the without memory methods. In particular, the $R$-order convergence of the proposed two- and three-step with memory methods is increased from four to five and eight to ten. One more advantage of these methods is that the condition $f'(x)\ne0$ in the neighborhood of the required root, imposed on Newton's method, can be removed. Numerical comparison is also stated to confirm the theoretical results.