Approximating Solutions of Nonlinear Equations Using an Extended Traub Method

Abstract

The Traub iterates generate a sequence that converges to a solution of a nonlinear equation given certain conditions. The order of convergence has been shown provided that the fifth Fréchet-derivative exists. Notice that this derivative does not appear on the Traub method. Therefore, according to the earlier results, there is no guarantee that the Traub method converges if the operator is not five times Fréchet-differentiable or more. However, the Traub method can converge, since these assumptions are only sufficient. The novelty of our new technique is the fact that only the Fréchet-derivative on the method is assumed to exist to prove convergence. Moreover, the new results does not depend on the Traub method. Consequently, the same technique can be applied on other methods. The dynamics of this method are also studied. Examples further explain the theoretical results.