Increasing the convergence order of an iterative method for nonlinear systems

In this work we introduce a technique for solving nonlinear systems that improves the order of convergence of any given iterative method which uses the Newton iteration as a predictor. The main idea is to compose a given iterative method of order p with a modification of the Newton method that introduces just one evaluation of the function, obtaining a new method of order p+2. By applying this procedure to known methods of order three and four, we obtain new methods of order five and six, respectively. The efficiency index and the computational effort of the new methods are checked. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with the ones from which have been derived and with the classical Newton method.