Majorizing sequences for Newton's method under centred conditions for the derivative

  1. Argyros, I.K. 1
  2. González, D. 2
  3. Magreñán, Á.A. 2
  1. 1 Cameron University
    info

    Cameron University

    Lawton, Estados Unidos

    ROR https://ror.org/00rgv0036

  2. 2 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Journal:
International Journal of Computer Mathematics

ISSN: 0020-7160

Year of publication: 2014

Volume: 91

Issue: 12

Pages: 2568-2583

Type: Article

DOI: 10.1080/00207160.2014.880782 SCOPUS: 2-s2.0-84914664512 WoS: WOS:000345505700009 GOOGLE SCHOLAR

More publications in: International Journal of Computer Mathematics

Abstract

We present semi-local and local convergence results for Newton's method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our technique is more flexible than in earlier studies such that [J.A. Ezquerro, D. González, and M.A. Hernández, Majorizing sequences for Newton's method from initial value problems, J. Comput. Appl. Math. 236 (2012), pp. 2246-2258; J.A. Ezquerro, D. González, and M.A. Hernández, A general semi-local convergence result for Newton's method under centred conditions for the second derivative, ESAIM: Math. Model. Numer. Anal. 47 (2013), pp. 149-167]. The operator involved is twice Fréchet-differentiable. We also assume certain centred Lipschitz-type conditions for the derivative which are more precise than the Lipschitz conditions used in earlier works. Numerical examples are used to show that our results apply to solve equations but earlier ones do not in the semi-local case. In the local case we obtain a larger convergence ball. These advantages are obtained under the same computational cost as before [17,18]. © 2014 © 2014 Taylor & Francis.