A fractional Traub method with $(2\alpha +1)$th-order of convergence and its stability

Abstract

Some fractional Newton methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper we introduce a fractional Newton method with order $\alpha +1$ and compare with another fractional Newton method with order $2\alpha$. We also introduce a fractional Traub method with order $2\alpha +1$ and compare with its first step (fractional Newton method with order $\alpha +1$). Some tests and analysis of the dependence on the initial estimations are made for each case.