A Dynamic and Analytical Study of the Damped Newton's Method

Abstract

This work has two targets. First we study the damped Newton's method: x n+1 = x n - λF′(x n) -1F(x n) 0 < λ≤ 1 , n ≥ 0 where F is an operator defined between two Banach spaces X and Y . We study the semilocal convergence of the method under the Kantorovich-like conditions: 1. x 0 ∈ X is a point where the operator Γ 0 = F′(x 0) -1 is defined. 2. ∥Γ 0F(x 0)|| ≥ a. 3. ∥Γ 0[F′(x)- F′(y)]∥ ≤ b∥x-y∥ ∀ x, y ∈ B(x 0,R) = {x : ∥x-x 0∥ ≤ R}. We obtain the recurrent sequence that majorizes {x n} and we give conditions for its convergence in terms of the parameters h = ab and λ. We use a new skill that allows us to generalize the result given initially by Kantorovich with λ = 1. In addition we make a dynamical study of the damped Newton'smethod in the complex case. Taking into account its relationship with the continuous Newton's method we obtain some conclusions about the fractal structure of the basins of atraction of the roots of a nonlinear equation. © 2010 Civil-Comp Press.