Achieving higher order of convergence for solving systems of nonlinear equations

In this paper, we develop a class of third order methods which is a generalization of the existing ones and a method of fourth order method, then introduce a technique that improves the order of convergence of any given iterative method for solving systems of nonlinear equations. Based on a given iterative method of order p ≥ 2 which uses the extended Newton iteration as a predictor, a new method of order p+2 is proposed with only one additional evaluation of the function. Moreover, if the given iterative method of order p ≥ 3 uses the Newton iteration as a predictor, then a new method of order p+3 can be developed. Applying this procedure, we obtain some new methods with higher order of convergence. Moreover, computational efficiency is analyzed and comparisons are made between these new methods and the ones from which have been derived. Finally, several numerical tests are performed to show the asymptotic behaviors which confirm the theoretical results.