Skip to main content
SpringerLink
Log in

An hybrid method that improves the accessibility of Steffensen’s method

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

Abstract

Steffensen’s method is known for its fast speed of convergence and its difficulty in applying it in Banach spaces. From the analysis of the accessibility of this method, we see that we can improve it by using the simplified secant method for predicting the initial approximation of Steffensen’s method. So, from both methods, we construct an hybrid iterative method which guarantees the convergence of Steffensen’s method from approximations given by the simplified secant method. We also emphasize that the study presented in this work is valid for equations with differentiable operators and non-differentiable operators.

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.: Convergence and numerical analysis of a family of two-step Steffensen’s methods. Comput. Math. Appl. 49(2), 13–22 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Amat, S., Busquier, S.: On a Steffensen’s type method and its behavior for semismooth equations. Appl. Math. Comput. 177(2), 819–823 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Amat, S., Busquier, S.: A two-step Steffensen’s method under modified convergence conditions. J. Math. Anal. Appl. 324(2), 1084–1092 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Andronow, A.A., Chaikin, C.E.: Theory of Oscillations. Princenton University Press, New Jersey (1949)

    Google Scholar 

  5. Argyros, I.K.: On the secant method. Publ. Math. Debrecen 43, 223–238 (1993)

    MATH  MathSciNet  Google Scholar 

  6. Argyros, I.K.: A new convergence theorem for Steffensen’s method on Banach spaces and applications. Southwest J. Pure Appl. Math. 1, 23–29 (1997)

    MATH  Google Scholar 

  7. Ezquerro, J.A., Grau-Sánchez, M., Hernández, M.A.: Solving non-differentiable equations by a new one-point iterative method with memory. J. Complexity 28, 48–58 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ezquerro, J.A., Grau-Sánchez, M., Hernández, M.A., Noguera, M.: Semilocal convergence of secant-like methots for differentiable and non-differentiable operator equations. J. Math. Anal. Appl. 398, 100–112 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ezquerro, J.A., Hernández, M.A., Romero N., Velasco, A.I.: On Steffensen’s method on Banach spaces. J. Comput. Appl. Math. 249, 9–23 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  11. Potra, F.A., Pták, V.: Nondiscrete Induction and Iterative Methods. Pitman Publishing Limited, London (1984)

    Google Scholar 

  12. Stoker, J.J.: Nonlinear Vibrations. Interscience-Wiley, New York (1950)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    J.A. Ezquerro, M.A. Hernández-Verón, M.J. Rubio & A.I. Velasco

Authors
  1. J.A. Ezquerro
    View author publications

    You can also search for this author in PubMed Google Scholar

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

    You can also search for this author in PubMed Google Scholar

  3. M.J. Rubio
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A.I. Velasco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J.A. Ezquerro.

Additional information

This work was supported in part by the project MTM2011-28636-C02-01 of the Spanish Ministry of Science and Innovation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ezquerro, J., Hernández-Verón, M., Rubio, M. et al. An hybrid method that improves the accessibility of Steffensen’s method. Numer Algor 66, 241–267 (2014). https://doi.org/10.1007/s11075-013-9732-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-013-9732-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation