On novel classes of iterative methods for solving nonlinear equations

In this paper, we establish two new classes of derivative-involved methods for solving single valued nonlinear equations of the form $f(x)=0$. The first contributed two-step class includes two evaluations of the function and one of its first derivative where its error analysis shows a fourth-order convergence. Next, we construct a three-step high-order class of methods including four evaluations per full cycle to achieve the seventh order of convergence. Numerical examples are included to re-verify the theoretical results and moreover put on show the efficiency of the new methods from our classes.