Skip to main content
SpringerLink
Log in

On the convergence of inexact two-step Newton-like algorithms using recurrent functions

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

Using our new concept of recurrent functions, we approximate a locally unique solution of a nonlinear equation by an inexact two-step Newton-like algorithm in a Banach space setting. Our semilocal analysis provides tighter error bounds than before, and in many interesting cases, weaker sufficient convergence conditions. Applications including the solution of a nonlinear Chandrasekhar-type integral equation appearing in radiative transfer, and a two point boundary value problem with a Green kernel are also provided in this study.

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. Argyros, I.K.: On a class of nonlinear integral equations arising in neutron transport. Aequ. Math. 36, 99–111 (1988)

    Article  MATH  Google Scholar 

  2. Argyros, I.K.: A unified approach for constructing fast two-step Newton-like methods. Monatshefte Math. 119, 1–22 (1995)

    Article  MATH  Google Scholar 

  3. Argyros, I.K.: A unified approach for constructing fast two-step methods in Banach space and their applications. Panam. Math. J. 13(3), 59–108 (2003)

    MATH  Google Scholar 

  4. Argyros, I.K.: A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space. J. Math. Anal. Appl. 298, 374–397 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Argyros, I.K.: Convergence and Applications of Newton-Type Iterations. Springer, New York (2008)

    MATH  Google Scholar 

  6. Argyros, I.K.: On the semilocal convergence of inexact methods in Banach spaces. J. Comput. Appl. Math. 228, 434–443 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Argyros, I.K., Hilout, S.: Enclosing roots of polynomial equations and their applications to iterative processes. Surv. Math. Appl. 4, 119–132 (2009)

    MATH  MathSciNet  Google Scholar 

  8. Argyros, I.K., Hilout, S.: Efficient Methods for Solving Equations and Variational Inequalities. Polimetrica, Milano (2009)

    Google Scholar 

  9. Argyros, I.K., Hilout, S.: Inexact Newton methods and recurrent functions. Appl. Math. 37(1), 113–126 (2010)

    MATH  MathSciNet  Google Scholar 

  10. Argyros, I.K., Hilout, S.: On Newton-like methods of “bounded deterioration” using recurrent functions. Aequ. Math. 79(1–2), 61–82 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Argyros, I.K., Hilout, S.: A convergence analysis of Newton-like method for singular equations using recurrent functions. Numer. Funct. Anal. Optim. 31(2), 112–130 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Argyros, I.K., Hilout, S.: Extending the Newton–Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 234, 2993–3006 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Candela, V., Marquina, A.: Recurrence relations for rational cubic methods. II. The Chebyshev method. Computing 45(4), 355–367 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chandrasekhar, S.: Radiative Transfer. Dover, New York (1960)

    Google Scholar 

  15. Chen, X., Nashed, M.Z.: Convergence of Newton-like methods for singular operator equations using outer inverses. Numer. Math. 66, 235–257 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ezquerro, J.A., Hernández, M.A., Salanova, M.A.: Solving a boundary value problem by a Newton-like method. Int. J. Comput. Math. 79(10), 1113–1120 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for the super-Halley method. Comput. Math. Appl. 36(7), 1–8 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hernández, M.A.: Second-derivative-free variant of the Chebyshev method for nonlinear equations. J. Optim. Theory Appl. 104(3), 501–515 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kantorovich, L.V., Akilov, G.P.: Functional Analysis in Normed Spaces. Pergamon Press, Oxford (1982)

    Google Scholar 

  20. Lambert, J.D.: Computational methods in ordinary differential equations. In: Introductory Mathematics for Scientists and Engineers. Wiley, London (1973)

    Google Scholar 

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

    MATH  Google Scholar 

  22. Parida, P.K., Gupta, D.K.: Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces. J. Math. Anal. Appl. 345(1), 350–361 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Proinov, P.D.: General local convergence theory for a class of iterative processes and its applications to Newton’s method. J. Complex. 25, 38–62 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. 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 

  25. Werner, W.: Newton-like method for the computation of fixed points. Comput. Math. Appl. 10(1), 77–86 (1984)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Ioannis K. Argyros

  2. 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. Ioannis K. Argyros
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saïd Hilout
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ioannis K. Argyros.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Argyros, I.K., Hilout, S. On the convergence of inexact two-step Newton-like algorithms using recurrent functions. J. Appl. Math. Comput. 38, 41–61 (2012). https://doi.org/10.1007/s12190-010-0461-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-010-0461-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation