Improved Newton-like methods for solving systems of nonlinear equations

We present the iterative methods of fifth and eighth order of convergence for solving systems of nonlinear equations. Fifth order method is composed of two steps namely, Newton’s and Newton-like steps and requires the evaluations of two functions, two first derivatives and one matrix inversion in each iteration. The eighth order method is composed of three steps, of which the first two steps are that of the proposed fifth order method whereas the third is Newton-like step. This method requires one extra function evaluation in addition to the evaluations of fifth order method. Computational efficiency of proposed techniques is discussed and compared with the existing methods. Some numerical examples are considered to test the performance of the new methods. Moreover, theoretical results concerning order of convergence and computational efficiency are confirmed in numerical examples. Numerical results have confirmed the robust and efficient character of the proposed techniques. © 2016, Sociedad Española de Matemática Aplicada.