The purpose of this paper is to study local convergence and applications of a new seventh-order iterative method.
The order of convergence for the method is proved by using Taylor expansions. In addition, local convergence is studied under Lipschitz conditions with the first derivative.
By using Taylor expansions, we can show the convergence order of the method is seven. The specific domains of convergence and the solutions of nonlinear equations can be obtained by applying the method to practical physics problems and nonlinear systems. In this way, the uniqueness of the solution and error estimates also are analyzed.
In the proof of convergence order, Taylor expansions require third or higher derivatives. The applicability of the method is restricted. In order to extend the applicability of the method, local convergence is studied under Lipschitz conditions with the first derivative. Finally, in order to prove the applicability of the method, the method is applied to some physical problems and nonlinear systems.
