Publication

Abstract : We generalize an optimal fourth order Kung–Traub method to Banach spaces and study its local convergence to approximate a locally-unique solution of a system of nonlinear equations. New analysis also provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions of higher order derivatives. Furthermore, based on the Kung–Traub method a -step scheme of Kung–Traub-like methods with increasing convergence order is proposed. The novelty of the scheme is that in each step the order of convergence is increased by an amount two at the cost of only one additional function evaluation. Numerical examples are provided to verify the theoretical results and to show the convergence behavior.