A Steffensen type method of two steps in Banach spaces with applications

Abstract

Please cite this article as: S. Amat, S. Busquier, J.A. Ezquerro, M.A. Hernández-Verón, A Steffensen type method of two steps in Banach spaces with applications, Journal of Computational and Applied Mathematics (2015), http://dx. AbstractThis paper is devoted to the analysis of a Steffensen-type of two steps with order of convergence at least three. The main advantage of this method is that it does not need to evaluate any Fréchet derivative or any bilinear operator. The method includes extra parameters in the divided difference in order to ensure a good approximation to the first derivative in each iteration. We prove, using recurrence relations, a semilocal convergence result in Banach spaces and do a detailed study of the domain of parameters associated to this result. Finally, some numerical results, including differentiable and nondifferentiable operators, are presented. Special attention is paid in the approximation of solutions of boundary problems using the multiple shooting method and in the approximation of a nonlinear model related with image processing.

On semilocal convergence of three-step Kurchatov method under weak condition

“…All of such methods require the differentiability of the involved operator which is a drawback for some certain problems where the operator is nondifferentiable. To overcome these types of problems, many researchers [6,18,22,26] have established and studied the iterative methods using divided differences. One of such method which does not use the differentiability of the involved operator is Kurchatov's method, proposed by Kurchatov [37] and given by:…”

“…The operator [e, f ; B] is a first-order divided difference operator B at the points e and f (e = f ) and satisfy [e, f ; B](e − f ) = B(e) − B( f ). In the finite-dimensional vector space R m , the divided difference of first order [35] To access the solution quickly, some authors [6,19,28,29] also considered two-step derivative-free iterative methods. One such method is two-step Kurchtov method [28,29] defined by the help of divided difference:…”

Mathematical modeling and computational methods

“…S. Amat et al in [25] analyze a Steffensen-type method of two steps with order of convergence at least three. A semilocal result in Banach spaces and a detailed study of the domain of parameters associated is presented.…”