On efficient weighted-Newton methods for solving systems of nonlinear equations

In this study, we present iterative methods of convergence order four and six for solving systems of nonlinear equations. The fourth order scheme is composed of two steps, namely; Newton iteration as the first step and weighted-Newton iteration as the second step. The sixth order scheme is composed of three steps; the first two steps are same as that of fourth order scheme whereas the third step is again based on weighted-Newton iteration. Computational efficiency in its general form is discussed. Comparison between the efficiencies of proposed techniques and existing techniques is made. It is proved that for large systems the new methods are more efficient. Numerical tests are performed, which confirm the theoretical results. Moreover, theoretical order of convergence is verified in the examples.