On the local convergence of a family of Euler-Halley type iterations with a parameter

Abstract

This paper aims to study the local convergence of a family of EulerHalley type methods with a parameter α for solving nonlinear operator equations under the second-order generalized Lipschitz assumption. The radius r α of the optimal convergence ball and the error estimation of the method corresponding to α are estimated for each α ∈ (−∞, +∞). For each α > 0, we get r α ≥ r −α and the upper bound of the error estimation of the method with α > 0 is not larger than the one with α < 0. For each α ≤ 0, we get the precise value of r α , which is closely linked to the dynamical property of the method applied to a real or a complex function, and the optimal error estimation, which decreases when α → 0 − . Results show that the method corresponding to α is better than the one corresponding to −α for each α > 0 and the ChebyshevEuler method is the best among all methods in the family with α ∈ (−∞, 0] from the view of both safe choice of the initial point and error estimation.