Journal of Applied Mathematics
Volume 2011 (2011), Article ID 786306, 15 pages
http://dx.doi.org/10.1155/2011/786306
Research Article

Semilocal Convergence for a Fifth-Order Newton's Method Using Recurrence Relations in Banach Spaces

Liang Chen,1,2 Chuanqing Gu,1 and Yanfang Ma3

1Department of Mathematics, Shanghai University, Shanghai 200444, China
2School of Mathematical Science, Huaibei Normal University, Anhui, Huaibei 235000, China
3School of Computer Science and Technology, Huaibei Normal University, Anhui, Huaibei 235000, China

Received 13 August 2011; Accepted 3 October 2011

Academic Editor: Ke Chen

Copyright © 2011 Liang Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a modified Newton's method with fifth-order convergence for nonlinear equations in Banach spaces. We make an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived, and then an existence-uniqueness theorem is given to establish the R-order of the method to be five and a priori error bounds. Finally, a numerical application is presented to demonstrate our approach.