Accelerating the convergence speed of iterative methods for solving nonlinear systems

In this paper, for solving systems of nonlinear equations, we develop a family of two-step third order methods and introduce a technique by which the order of convergence of many iterative methods can be improved. Given an iterative method of order p ≥ 2 which uses the extended Newton iteration as a predictor, a new method of order p+2 is constructed by introducing only one additional evaluation of the function. In addition, for an iterative method of order p ≥ 3 using the Newton iteration as a predictor, a new method of order p+3 can be extended. Applying this procedure, we develop some new efficient methods with higher order of convergence. For comparing these new methods with the ones from which they have been derived, we discuss the computational efficiency in detail. Several numerical examples are given to justify the theoretical results by the asymptotic behaviors of the considered methods.