A family of optimal iterative methods with fifth and tenth order convergence for solving nonlinear equations

In this paper, we propose a two-step method with fifth-order convergence and a family of optimal three-step method with tenth-order convergence for finding the simple roots of nonlinear equations. The optimal efficiency indices are all found to be $5^{frac{1}{3}}approx1.71$ and $10^{frac{1}{4}}approx1.78.$ Some numerical examples illustrate that the algorithms are more efficient and performs better than other methods.