Skip to main content
SpringerLink
Log in

Convergence of Newton’s method under Vertgeim conditions: new extensions using restricted convergence domains

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

We present new sufficient convergence conditions for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear equation in a Banach space. We use Hölder and center Hölder conditions, instead of just Hölder conditions, for the first derivative of the operator involved in combination with our new idea of restricted convergence domains. This way, we find a more precise location where the iterates lie, leading to at least as small Hölder constants as in earlier studies. The new convergence conditions are weaker, the error bounds are tighter and the information on the solution at least as precise as before. These advantages are obtained under the same computational cost. Numerical examples show that our results can be used to solve equations where older results cannot.

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

    Google Scholar 

  2. I.K. Argyros, Concerning the “terra incognita” between convergence regions of two Newton methods. Nonlinear Anal. 62, 179–194 (2005)

    Google Scholar 

  3. I.K. Argyros, S. Hilout, On the weakening of the convergence of Newton’s method using recurrent functions. J. Complex. 25(6), 530–543 (2009)

    Google Scholar 

  4. I.K. Argyros, S.K. Khattri, Some development for Newton’s method under mild differentiability conditions. J. Nonlinear Anal. Optim. 5(2), 81–90 (2014)

    Google Scholar 

  5. F. Cianciaruso, E. De Pascale, Newton–Kantorovich aproximations when the derivative is Hölderian: old and new results. Numer. Funct. Anal. Optim. 24, 713–723 (2003)

    Google Scholar 

  6. F. Cianciaruso, A further journey in the “terra incognita” of the Newton–Kantorovich method. Nonlinear Funct. Anal. Appl. 15, 173–183 (2010)

    Google Scholar 

  7. E. De Pascale, P.P. Zabrejko, Convergence of the Newton–Kantorovich method under Vertgeim conditions: a new improvement. Z. Anal. Anwendvugen 17, 271–280 (1998)

    Google Scholar 

  8. M.A. Hernández, The Newton method for operators with Hölder continuous first derivative. J. Optim. Theory Appl. 109, 631–648 (2001)

    Google Scholar 

  9. D. Jain, Families of Newton-like method with fourth-order convergence. Int. J. Comput. Math. 90(5), 1072–1082 (2013)

    Google Scholar 

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

    Google Scholar 

  11. H.B. Keller, Numerical Methods for Two-Point Boundary Value Problems (Dover Publications, New York, 1992)

    Google Scholar 

  12. J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations. Dokl. Akad. Nauk BSSR 38, 20–24 (1994). (in Russian)

    Google Scholar 

  13. J.M. Ortega, The Newton–Kantorovich theorem. Am. Math. Mon. 75, 658–660 (1968)

    Google Scholar 

  14. J. Rokne, Newton’s method under mild differentiability conditions with error analysis. Numer. Math. 18, 401–412 (1972)

    Google Scholar 

  15. B.A. Vertgeim, On conditions for the applicability of Newton’s method. Dokl. Akad. Nauk SSSR 110, 719–722 (1956). (in Russian)

    Google Scholar 

  16. B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces. Uspekhi Mat. Nauk 12, 166–169 (1957) (in Russian); English transl.: Am. Math. Soc. Transl., 16, 378–382 (1960)

  17. P.P. Zabrejko, D.F. Nguen, The majorant method in the theory of Newton–Kantorovich approximations and the Pták error estimates. Numer. Funct. Anal. Optim. 9, 671–684 (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics Sciences, Cameron University, Lawton, OK, 73505, USA

    I. K. Argyros

  2. Department of Mathematics and Computation, University of La Rioja, Calle Luis de Ulloa s/n, 26004, Logroño, Spain

    J. A. Ezquerro & M. A. Hernández-Verón

  3. Escuela Superior de Ingeniería y Tecnología, International University of La Rioja, Avenida de la Paz 137, 26002, Logroño, Spain

    Á. A. Magreñán

Authors
  1. I. K. Argyros
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. Ezquerro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. A. Hernández-Verón
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Á. A. Magreñán
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. Hernández-Verón.

Additional information

This work has been partially supported by Ministerio de Ciencia y Tecnología MTM2014-52016-C02-1-P, UNIR Research (http://research.unir.net), Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Research Support Strategy 3 [2015–2017], Research Group: MOdelación Matemática Aplicada a la INgeniería (MOMAIN) and by the Grant SENECA 19374/PI/14.

This is one of several papers published together in Journal of Mathematical Chemistry on the “Special Issue: CMMSE”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Argyros, I.K., Ezquerro, J.A., Hernández-Verón, M.A. et al. Convergence of Newton’s method under Vertgeim conditions: new extensions using restricted convergence domains. J Math Chem 55, 1392–1406 (2017). https://doi.org/10.1007/s10910-016-0720-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-016-0720-x

Keywords

Mathematics Subject Classification

Search

Navigation