Skip to main content
SpringerLink
Log in

On an improved local convergence analysis for the Secant method

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

Abstract

We provide a local convergence analysis for the Secant method in a Banach space setting under Hölder continuous conditions. Using more precise estimates, and under the same computational cost, we enlarge the radius of convergence obtained in Ren and Wu (J Comput Appl Math 194:284–293, 2006).

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.: The Secant method and fixed points of nonlinear operators. Mh. Math. 106, 85–94 (1988)

    Article  MATH  Google Scholar 

  2. 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 

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

    Google Scholar 

  4. Argyros, I.K.: The Theory and Application of Abstract Polynomial Equations. St. Lucie/CRC/Lewis Publ. Mathematics Series, Boca Raton (1998)

    Google Scholar 

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

    MATH  Google Scholar 

  6. Argyros, I.K.: On the radius of convergence of Newton’s method under average mild differentiability conditions. Nonlinear Funct. Anal. Appl. 13(3), 409–415 (2008)

    MATH  MathSciNet  Google Scholar 

  7. Hernandez, M.A., Rubio, M.J.: The Secant method and divided differences Hölder continuous. Appl. Math. Comput. 15, 139–149 (2001)

    Article  MathSciNet  Google Scholar 

  8. Hernandez, M.A., Rubio, M.J.: The Secant method for nondifferentiable operators. Appl. Math. Lett. 15, 395–399 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hernandez, M.A., Rubio, M.J.: Semilocal convergence of the Secant method under mild convergence conditions of differentiability. Comput. Math. Appl. 44, 277–285 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Huang, Z.D.: The convergence ball of Newton’s method and the uniqueness ball of equations under Hölder-type continuous derivatives. Comput. Math. Appl. 25, 247–251 (2004)

    Article  Google Scholar 

  11. Pavaloiu, I.: On the convergence of a Steffensen-type method, “Babes-Bolyai” University Faculty of Mathematics Research Seminars: Seminar in Mathematical Analysis. Preprint Mr. 7, 121–126 (1991)

    MathSciNet  Google Scholar 

  12. Pavaloiu, I.: Remarks on the Secant method of nonlinear operational equations, “Babes-Bolyai” University Faculty of Mathematics Research Seminars: Seminar in Mathematical Analysis. Preprint Mr. 7, 105–120 (1991)

    Google Scholar 

  13. Ren, H., Wu, Q.: The convergence ball of the Secant method under Hölder continuous divided differences. J. Comput. Appl. Math. 194, 284–293 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ren, H., Wu, Q.: Mysovskii-type theorem for the Secant method under Hölder continuous Fréchet derivative. J. Math. Anal. Appl. 320, 415–424 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ren, H., Yang, S., Wu, Q.: A new semilocal convergence theorem for the Secant method under Hölder continuous divided differences. Appl. Math. Comput. 182, 41–48 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Rheinboldt, W.C.: An adaptive continuation process for solving systems of nonlinear equations. Banach Cent. Publ. 3, 129–142 (1977)

    MathSciNet  Google Scholar 

  17. Schmidt, J.W.: Regula-falsi verfahren mit konsistenter steigung und majoranten pinzip. Period. Math. Hung. 5, 187–193 (1974)

    Article  MATH  Google Scholar 

  18. Sergeev, A.: On the method of chords. Sib. Mat. Z. 2, 282–289 (1961)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Ioannis K. Argyros

  2. Department of Information and Electronics, Hangzhou Radio and TV University, Hangzhou, 310012, Zhejiang, People’s Republic of China

    Hongmin Ren

Authors
  1. Ioannis K. Argyros
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongmin Ren
    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., Ren, H. On an improved local convergence analysis for the Secant method. Numer Algor 52, 257–271 (2009). https://doi.org/10.1007/s11075-009-9271-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-009-9271-6

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation