New high-order convergence iteration methods without employing derivatives for solving nonlinear equations

A family of new iteration methods without employing derivatives is proposed in this paper. We have proved that these new methods are quadratic convergence. Their efficiency is demonstrated by numerical experiments. The numerical experiments show that our algorithms are comparable to well-known methods of Newton and Steffensen. Furthermore, combining the new method with bisection method we construct another new high-order iteration method with nice asymptotic convergence properties of the diameters (bn − an).