Skip to main content
SpringerLink
Log in

Expanding the Applicability of High-Order Traub-Type Iterative Procedures

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We propose a collection of hybrid methods combining Newton’s method with frozen derivatives and a family of high-order iterative schemes. We present semilocal convergence results for this collection on a Banach space setting. Using a more precise majorizing sequence and under the same or weaker convergence conditions than the ones in earlier studies, we expand the applicability of these iterative procedures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amat, S., Bermúdez, C., Busquier, S., Legaz, M.J., Plaza, S.: On a family of high order iterative methods under Kantorovich conditions and some applications. Abstr. Appl. Anal. 2012, 782170 (2012), 14 pp.

    Article  Google Scholar 

  2. Argyros, I.K., Cho, Y.J., Hilout, S.: Numerical Methods for Equations and Its Applications. CRC Press/Taylor & Francis, New York/London (2012)

    MATH  Google Scholar 

  3. Argyros, I.K., Hilout, S.: Computational Methods in Nonlinear Analysis: Efficient Algorithms, Fixed Point Theory and Applications. World Scientific, Singapore (2013)

    Book  Google Scholar 

  4. Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)

    MATH  Google Scholar 

  5. Ortega, L.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  6. Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall Series in Automatic Computation. Prentice Hall International, Englewood Cliffs (1964)

    MATH  Google Scholar 

  7. Rheinboldt, W.C.: A unified convergence theory for a class of iterative processes. SIAM J. Numer. Anal. 5, 42–63 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ezquerro, J.A., Hernández, M.A.: An improvement of the region of accessibility of Chebyshev’s method from Newton’s method. Math. Comput. 78, 1613–1627 (2009)

    Article  MATH  Google Scholar 

  9. Potra, F.A., Ptǎk, V.: On a class of modified Newton processes. Numer. Funct. Anal. Optim. 2, 107–120 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. Potra, F.A., Ptǎk, V.: A generalization of regula falsi. Numer. Math. 36, 333–346 (1981)

    Article  MATH  Google Scholar 

  11. Potra, F.A., Ptǎk, V.: Nondiscrete Induction and Iterative Processes. Research Notes in Mathematics, vol. 103. Pitman, London (1984)

    MATH  Google Scholar 

  12. Argyros, I.K.: Approximating solutions of equations using Newton’s method with a modified Newton’s method iterate as a starting point. Rev. Anal. Numér. Théor. Approx. 36, 123–138 (2007)

    MATH  MathSciNet  Google Scholar 

  13. Argyros, I.K., Hilout, S.: Weaker conditions for the convergence of Newton’s method. J. Complex. 28, 364–387 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Deuflhard, P.: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics, vol. 35. Springer, Berlin (2004)

    MATH  Google Scholar 

  15. Potra, F.A.: The rate of convergence of a modified Newton’s process. Apl. Mat. 26, 13–17 (1981). With a loose Russian summary

    MATH  MathSciNet  Google Scholar 

  16. Proinov, P.D.: New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems. J. Complex. 26, 3–42 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rall, L.B.: Computational Solution of Nonlinear Operator Equations. Wiley, New York (1969)

    MATH  Google Scholar 

Download references

Acknowledgements

Research supported for the first and third authors in part by MINECO-FEDER MTM2010-17508 and 08662/PI/08.

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain

    Sergio Amat & Sonia Busquier

  2. Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

  3. Laboratoire de Mathématiques et Applications, Poitiers University, Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179, 86962, Futuroscope Chasseneuil Cedex, France

    Saïd Hilout

Authors
  1. Sergio Amat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ioannis K. Argyros
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sonia Busquier
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Saïd Hilout
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ioannis K. Argyros.

Additional information

Communicated by Florian Potra.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Amat, S., Argyros, I.K., Busquier, S. et al. Expanding the Applicability of High-Order Traub-Type Iterative Procedures. J Optim Theory Appl 161, 837–852 (2014). https://doi.org/10.1007/s10957-013-0440-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-013-0440-3

Keywords

Search

Navigation