Journal of Mathematical Chemistry

, Volume 55, Issue 7, pp 1427–1442 | Cite as

Local convergence of a relaxed two-step Newton like method with applications

  • I. K. Argyros
  • Á. A. Magreñán
  • L. Orcos
  • J. A. Sicilia
Original Paper
  • 155 Downloads

Abstract

We present a local convergence analysis for a relaxed two-step Newton-like method. We use this method to approximate a solution of a nonlinear equation in a Banach space setting. Hypotheses on the first Fréchet derivative and on the center divided-difference of order one are used. In earlier studies such as Amat et al. (Numer Linear Algebra Appl 17:639–653, 2010, Appl Math Lett 25(12):2209–2217, 2012, Appl Math Comput 219(24):11341–11347, 2013, Appl Math Comput 219(15):7954–7963, 2013, Reducing Chaos and bifurcations in Newton-type methods. Abstract and applied analysis. Hindawi Publishing Corporation, Cairo, 2013) these methods are analyzed under hypotheses up to the second Fréchet derivative and divided differences of order one. Numerical examples are also provided in this work.

Keywords

Two-step Newton’s method Banach space Fréchet derivative Divided difference of first order Local–semilocal convergence 

Mathematics Subject Classification

65D10 65D99 

Notes

Acknowledgements

This research was supported by Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Plan Propio de Investigación, Desarrollo e Innovación 3 [2015–2017]. Research group: Modelación matemática aplicada a la ingeniería(MOMAIN), by the the grant SENECA 19374/PI/14 and by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-\(\{01\}\)-P.

References

  1. 1.
    S. Amat, C. Bermúdez, S. Busquier, S. Plaza, On a third-order Newton-type method free of bilinear operators. Numer. Linear Algebra Appl. 17, 639–653 (2010)Google Scholar
  2. 2.
    I.K. Argyros, Computational theory of iterative methods, in Series: Studies in Computational Mathematics, vol. 15, ed. by C.K. Chui, L. Wuytack (Elsevier Publ. Co., New York, 2007)Google Scholar
  3. 3.
    I.K. Argyros, S. Hilout, Computational Methods in Nonlinear Analysis: Efficient Algorithms, Fixed Point Theory and Applications (World Scientific, Singapore, 2013)Google Scholar
  4. 4.
    E. Catinas, The inexact, inexact perturbed, and quasi-Newton methods are equivalent models. Math. Comput. 74, 291–301 (2005)Google Scholar
  5. 5.
    J.S. Kou, Y.T. Li, X.H. Wang, A modification of Newton method with third-order convergence. Appl. Math. Comput. 181, 1106–1111 (2006)Google Scholar
  6. 6.
    L.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970)Google Scholar
  7. 7.
    P.K. Parida, D.K. Gupta, Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206, 873–887 (2007)Google Scholar
  8. 8.
    W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Polish Academy of Science. Banach Cent. Publ. bf 3, 129–142 (1977)Google Scholar
  9. 9.
    S. Amat, S. Busquier, Third order methods under Kantorovich conditions. J. Math. Anal. Appl. 336(1), 243–261 (2007)Google Scholar
  10. 10.
    I.K. Argyros, S. Hilout, Weaker conditions for the convergence of Newton’s method. J. Complex. 28, 364–387 (2012)Google Scholar
  11. 11.
    V. Candella, A. Marquina, Recurrence relations for rational cubic methods II: the Chebyshev method. Computing 45, 355–367 (1990)Google Scholar
  12. 12.
    V. Candella, A. Marquina, Recurrence relations for rational cubic methods I: the Halley method. Computing 44, 169–184 (1990)Google Scholar
  13. 13.
    C. Chun, P. Stanica, B. Neta, Third-order family of methods in Banach spaces. Comput. Math. Appl. 61, 1665–1675 (2011)Google Scholar
  14. 14.
    J.A. Ezquerro, M.A. Hernández, Recurrence relations for Chebyshev-type methods. Appl. Math. Optim. 41, 227–236 (2000)Google Scholar
  15. 15.
    J.A. Ezquerro, M.A. Hernández, Third-order iterative methods for operators with bounded second derivative. J. Comput. Math. Appl. 82, 171–183 (1997)Google Scholar
  16. 16.
    M.A. Hernández, M.A. Salanova, Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method. J. Comput. Appl. Math. 126, 131–143 (2000)Google Scholar
  17. 17.
    J.A. Gutiérrez, Á.A. Magreñán, N. Romero, On the semilocal convergence of Newton Kantorovich method under center-Lipschitz conditions. Appl. Math. Comput. 221, 79–88 (2013)Google Scholar
  18. 18.
    F.A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes. Research Notes in Mathematics, vol. 103 (Pitman, Boston, 1984)Google Scholar
  19. 19.
    S. Amat, Á.A. Magreñán, N. Romero, On a two-step relaxed Newton-type method. Appl. Math. Comput. 219(24), 11341–11347 (2013)Google Scholar
  20. 20.
    S. Amat, S. Busquier, A. Grau, M. Grau-Sánchez, Maximum efficiency for a family of Newton-like methods with frozen derivatives and some applications. Appl. Math. Comput. 219(15), 7954–7963 (2013)Google Scholar
  21. 21.
    S. Amat, C. Bermúdez, S. Busquier, S. Plaza, On two families of high order Newton type methods. Appl. Math. Lett. 25(12), 2209–2217 (2012)Google Scholar
  22. 22.
    S. Amat, S. Busquier, Á.A. Magreñán, Reducing Chaos and Bifurcations in Newton-type methods. Abstract and applied analysis, vol. 2013 (Hindawi Publishing Corporation. Cairo (2013). doi: 10.1155/2013/726701
  23. 23.
    T. Yamashita, H. Yabe, T. Tanabe, A globally and superlinearly convergent primal-dual interior point trust region method for large scale constrained optimization. Math. Program. Ser. A 102(1), 111–151 (2005)Google Scholar
  24. 24.
    L.V. Kantorovich, G.P. Akilov, Functional Analysis (Pergamon Press, Oxford, 1982)Google Scholar
  25. 25.
    J.A. Ezquerro, M.A. Hernández, On the R-order of the Halley method. J. Math. Anal. Appl. 303, 591–601 (2005)Google Scholar
  26. 26.
    I.K. Argyros, A semilocal convergence analysis for directional Newton methods. Math. Comput. 80, 327–343 (2011)Google Scholar
  27. 27.
    P.D. Proinov, General local convergence theory for a class of iterative processes and its applications to Newton’s method. J. Complex. 25, 38–62 (2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • I. K. Argyros
    • 1
  • Á. A. Magreñán
    • 2
  • L. Orcos
    • 2
  • J. A. Sicilia
    • 2
  1. 1.Department of Mathematical SciencesCameron UniversityLawtonUSA
  2. 2.Universidad Internacional de La Rioja (UNIR)Logroño, La RiojaSpain

Personalised recommendations