On the local convergence study for an efficient k-step iterative method

  1. Amat, S. 1
  2. Argyros, I.K. 2
  3. Busquier, S. 1
  4. Hernández-Verón, M.A. 3
  5. Martínez, E. 4
  1. 1 Universidad Politécnica de Cartagena
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    Universidad Politécnica de Cartagena

    Cartagena, España

    ROR https://ror.org/02k5kx966

    Geographic location of the organization Universidad Politécnica de Cartagena
  2. 2 Cameron University
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    Cameron University

    Lawton, Estados Unidos

    ROR https://ror.org/00rgv0036

    Geographic location of the organization Cameron University
  3. 3 Universidad de La Rioja
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    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

    Geographic location of the organization Universidad de La Rioja
  4. 4 Universidad Politécnica de Valencia
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    Universidad Politécnica de Valencia

    Valencia, España

    ROR https://ror.org/01460j859

    Geographic location of the organization Universidad Politécnica de Valencia
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Journal:
Journal of Computational and Applied Mathematics

ISSN: 0377-0427

Year of publication: 2018

Type: Article

DOI: 10.1016/J.CAM.2018.02.028 SCOPUS: 2-s2.0-85046091824 GOOGLE SCHOLAR

More publications in: Journal of Computational and Applied Mathematics

Abstract

This paper is devoted to a family of Newton-like methods with frozen derivatives used to approximate a locally unique solution of an equation. The methods have high order of convergence but only using first order derivatives. Moreover only one LU decomposition is required in each iteration. In particular, the methods are real alternatives to the classical Newton method. We present a local convergence analysis based on hypotheses only on the first derivative. These types of local results were usually proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these types of methods (Bermúdez et al., 2012; Petkovic et al., 2013; Traub, 1964). We apply these methods to an equation related to the nonlinear complementarity problem. Finally, we find the most efficient method in the family for this problem and we perform a theoretical and a numerical study for it. © 2018 Elsevier B.V.