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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 17, Issue 2

Issues

On the Efficiency of a Family of Steffensen-Like Methods with Frozen Divided Differences

Sergio Amat
  • Corresponding author
  • Department of Applied Mathematics and Statistics, Technical University of Cartagena, 30203 Cartagena (Murcia), Spain
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/ Sonia Busquier
  • Department of Applied Mathematics and Statistics, Technical University of Cartagena, 30203 Cartagena (Murcia), Spain
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  • Other articles by this author:
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/ Miquel Grau-Sánchez / Miguel A. Hernández-Verón
Published Online: 2016-12-21 | DOI: https://doi.org/10.1515/cmam-2016-0039

Abstract

A generalized k-step iterative method from Steffensen’s method with frozen divided difference operator for solving a system of nonlinear equations is studied and the maximum computational efficiency is computed. Moreover, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the method and the computational efficiency are both well deduced. By using a technique based on recurrence relations, the semilocal convergence of the family is studied. Finally, some numerical experiments related to the approximation of nonlinear elliptic equations are reported. A comparison with other derivative-free families of iterative methods is carried out.

Keywords: Nonlinear Equations; Iterative Methods; Frozen Divided Difference; Order of Convergence, Efficiency

MSC 2010: 65H10; 65Y20

References

  • [1]

    Alarcón V., Amat S., Busquier S. and López D. J., A Steffensen’s type method in Banach spaces with applications on boundary-value problems, J. Comput. Appl. Math. 216 (2008), 243–250. Google Scholar

  • [2]

    Amat S. and Busquier S., Convergence and numerical analysis of a family of two-step Steffensen’s methods, Comput. Math. Appl. 49 (2005), 13–22. Google Scholar

  • [3]

    Amat S. and Busquier S., A two-step Steffensen’s method under modified convergence conditions, J. Math. Anal. Appl. 324 (2006), 1084–1092. Google Scholar

  • [4]

    Amat S. and Busquier S., On a Steffensen’s type method and its behavior for semismooth equations, Appl. Math. Comput. 177 (2006), 819–823. Google Scholar

  • [5]

    Argyros I. K., The secant method and fixed points of nonlinear operators, Monatsh. Math. 106 (1988), 85–94. Google Scholar

  • [6]

    Argyros I. K., On the secant method, Publ. Math. Debrecen 43 (1993), 223–238. Google Scholar

  • [7]

    Argyros I. K., Computational Theory of Computational Methods, Stud. Comput. Math. 15, Elsevier, New York, 2007. Google Scholar

  • [8]

    Argyros I. K. and Ren H., Efficient Steffensen-type algorithms for solving nonlinear equations, Int. J. Comput. Math. 90 (2013), 691–704. Google Scholar

  • [9]

    Ezquerro J. A. and Hernández-Verón M. A., How to improve the domain of starting points for Steffensen’s method, Stud. Appl. Math. 132 (2014), 354–380. Google Scholar

  • [10]

    Grau-Sánchez M., Gutiérrez J. M. and Peris J. M., Accelerated iterative methods for finding solutions of a system of nonlinear equations, Appl. Math. Comput. 190 (2007), 1815–1823. Google Scholar

  • [11]

    Grau-Sánchez M., Noguera M. and Gutiérrez J. M., On some computational orders of convergence, Appl. Math. Lett. 23 (2010), 472–478. Google Scholar

  • [12]

    Grau-Sánchez M., Noguera M. and Gutiérrez J. M., Frozen divided difference scheme for solving systems of nonlinear equations, J. Comput. Appl. Math. 235 (2011), 1739–1743. Google Scholar

  • [13]

    Grau-Sánchez M., Noguera M. and Gutiérrez J. M., Frozen iterative methods sing divided differences “à la Schmidt–Schwetlick”, J. Optim. Theory Appl. 160 (2014), 931–948. Google Scholar

  • [14]

    Herceg D. and Herceg D., Means based modifications of Newton’s method for solving nonlinear equations, Appl. Math. Comput. 219 (2013), 6126–6133. Google Scholar

  • [15]

    Potra F. A. and Pták V., Nondiscrete Induction and Iterarive Processes, Pitman, Boston, 1984. Google Scholar

  • [16]

    Ralston A. and Rabinowitz P., A First Course in Numerical Analysis, McGraw-Hill, Ohio, 1978. Google Scholar

  • [17]

    Schmidt J. W. and Schwetlick H., Ableitungsfreie Verfahren mit höherer Konvergenzgeschwindigkeit, Computing 3 (1968), 215–226. Google Scholar

  • [18]

    Weerakoon S. and Fernando T. G. I., A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000), 87–93. Google Scholar

About the article

Received: 2016-07-20

Revised: 2016-11-09

Accepted: 2016-11-15

Published Online: 2016-12-21

Published in Print: 2017-04-01


This work was supported in part by the project MTM2011-28636-C02-01 of the Spanish Ministry of Science and Innovation. Research supported in part by Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI/14 and MTM2015-64382-P (MINECO/FEDER).


Citation Information: Computational Methods in Applied Mathematics, Volume 17, Issue 2, Pages 187–199, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2016-0039.

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