Skip to main content
SpringerLink
Log in

On the semilocal convergence of a three steps Newton-type iterative process under mild convergence conditions

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper the semilocal convergence for an alternative to the three steps Newton’s method with frozen derivative is presented. We analyze the generalization of convergence conditions given by w-conditioned non-decreasing functions instead of the first derivative Lipschitz or Holder continuous given by other authors. A nonlinear integral equation of mixed Hammerstein type is considered for illustrating the new theoretical results obtained in this paper, where previous results can not be satisfied.

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., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Comput. 25, 2209–2217 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Argyros, I.K.: The Newton-Kantorovich method under mild differentiability conditions and the Pták error estimates. Monatsh. Math. 101, 175–193 (1990)

    Article  MATH  Google Scholar 

  3. Argyros, I.K.: Remarks on the convergence of Newton’s method under Hölder continuity conditions. Tamkang J. Math. 23(4), 269–277 (1992)

    MathSciNet  MATH  Google Scholar 

  4. Bruns, D.D., Bailey, J.E.: Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci. 32, 257–264 (1977)

    Article  Google Scholar 

  5. Ezquerro, J.A., Hernández, M.A.: Generalized differentiability conditions for Newton’s method. IMA J. Numer. Anal. 22(2), 187–205 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ganesh, M., Joshi, M.C.: Numerical solvability of Hammerstein integral equations of mixed type. IMA J. Numer. Anal. 11, 21–31 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hueso, J.L., Martínez, E., Torregrosa, J.R.: Third and fourth order iterative methods free from second derivative for nonlinear systems. Appl. Math. Comput. 211, 190–197 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hueso, J.L., Martínez, E.: Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algor. In Press. doi:10.1007/s11075-013-9795-7

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

    MATH  Google Scholar 

  10. Ostrowski, A.M.: Solutions of equations and system of equations. Academic Press, New York (1960)

    MATH  Google Scholar 

  11. Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  12. Yamamoto, T.: A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions. Numer. Math. 49, 203–220 (1986)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computation, University of La Rioja, 26004, Logroño, Spain

    M. A. Hernández-Verón

  2. Instituto de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera, s/n, 4602, Valencia, Spain

    Eulalia Martínez

Authors
  1. M. A. Hernández-Verón
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eulalia Martínez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eulalia Martínez.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hernández-Verón, M.A., Martínez, E. On the semilocal convergence of a three steps Newton-type iterative process under mild convergence conditions. Numer Algor 70, 377–392 (2015). https://doi.org/10.1007/s11075-014-9952-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-014-9952-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation