Toward a unified theory for third R-order iterative methods for operators with unbounded second derivative

  1. Hernández, M.A. 1
  2. Romero, N. 1
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Journal:
Applied Mathematics and Computation

ISSN: 0096-3003

Year of publication: 2009

Volume: 215

Issue: 6

Pages: 2248-2261

Type: Article

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DOI: 10.1016/J.AMC.2009.08.017 SCOPUS: 2-s2.0-70349994124 WoS: WOS:000270972400029 GOOGLE SCHOLAR

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Abstract

In this paper, we provide a semilocal convergence analysis for a family of Newton-like methods, which contains the best-known third-order iterative methods for solving a nonlinear equation F (x) = 0 in Banach spaces. It is assumed that the operator F is twice Fréchet differentiable and F″ satisfies a Lipschitz type condition but it is unbounded. By using majorant sequences, we provide sufficient convergence conditions to obtain cubic semilocal convergence. Results on existence and uniqueness of solutions, and error estimates are also given. Finally, a numerical example is provided. © 2009 Elsevier Inc. All rights reserved.