Convergence ball and error analysis of a family of iterative methods with cubic convergence

Abstract

A family of iterative methods with cubic convergence to find approximate solutions of nonlinear equations is proposed. The radius of the convergence ball of any iterative method of the family is estimated and quadratic convergence is obtained under the assumption that the first order Fréchet derivative of the operator involved is Lipschitz continuous. If the second order Fréchet derivative of the operator involved is assumed also to be Lipschitz continuous, cubic convergence of any iterative method of the family is obtained. Finally, some examples are provided to show applications of our theorems.