Abstract.

We present a generalized k-step scheme of weighted-Newton methods with increasing convergence order 2k +2 for nonlinear equations. The novelty of the scheme is that in each step the order of convergence is increased by an amount of two at the cost of only one additional function evaluation. The algorithm requires only single evaluation of Fr´echet derivative which points to the name ‘methods with frozen derivative’. Local convergence including radius of convergence, error bounds and estimates on the uniqueness of the solution is presented. To maximize the computational efficiency, the optimal number of steps is computed. Theoretical results regarding convergence and computational efficiency are verified through numerical experimentation.

Keywords:

nonlinear equations, weighted-Newton methods, fast algorithms, convergence, computational efficiency.