Skip to main content
Springer Nature Link
Log in

On Ulm’s method using divided differences of order one

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

Abstract

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Tzv Akad Nauk Est SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones in Burmeister (Z Angew Math Mech 52:101–110, 1972), Kornstaedt (Aequ Math 13:21–45, 1975), Moser (1973), and Potra and Pták (Cas Pest Mat 108:333–341, 1983) do not. We also show that under the same hypotheses and computational cost, finer error bounds can be obtained. Some error bounds are also shown to be sharp. Numerical examples are also provided further validating the results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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 Theory and Application of Abstract Polynomial Equations. St. Lucie/CRC/Lewis, Boca Raton, Florida, USA (1998)

    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 Applications of Newton-Type Iterations. Springer, New York (2008)

    MATH  Google Scholar 

  4. Argyros, I.K., Chen, D.: An inverse-free Jarratt type approximation in a Banach space. Approx. Theory Appl. 12, 19–30 (1996)

    MATH  MathSciNet  Google Scholar 

  5. Burmeister, W.: Inversion freie verfahren zur lösung nichtlinearen operatorgleichungen. Z. Angew. Math. Mech. 52, 101–110 (1972)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  7. Hald, O.H.: On a Newton–Moser type method. Numer. Math. 23, 411–425 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hernánadez, M.A., Rubio, M.J., Ezquerro, J.A.: Secant-like methods for solving integral equations of the Hammerstein type. J. Comput. Appl. Math. 115, 245–254 (2001)

    Article  Google Scholar 

  9. Kornstaedt, H.J.: Funktionallongleichungen und iterations verfahren. Aequ. Math. 13, 21–45 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kurchatov, V.A.: Optimization, with respect to the number of arithmetic operations and entropy, of difference methods of linearization (Russian). Iz. VUZ 4, 33–37 (1990)

    MathSciNet  Google Scholar 

  11. Moser, J.: Stable and random motions in dynamical systems with special emphasis on celestial mechanics, Herman Weil Lectures. In: Annals of Mathematics Studies, vol. 77. Princeton Univ. Press, Princeton, NJ (1973)

    Google Scholar 

  12. Petzeltova, H.: Remark on a Newton–Moser type method. Comment. Math. Univ. Carol. 21, 719–725 (1980)

    MATH  MathSciNet  Google Scholar 

  13. Potra, F.A.: An application of the induction method of V. Pták to the study of regula falsi. Apl. Mat. 26, 111–120 (1981)

    MATH  MathSciNet  Google Scholar 

  14. Potra, F.A.: An error analysis for the secant method. Numer. Math. 38, 427–445 (1981/82)

    Article  MathSciNet  Google Scholar 

  15. Potra, F.A., Pták, V.: Nondiscrete induction and an inversion-free modification of Newton’s method. Cas. Pest. Mat. 108, 333–341 (1983)

    MATH  Google Scholar 

  16. Schmidt, J.W.: Überlinear konvergente Mehrschrittverfahren vom Regula falsi- und Newton-Typ (German). Z. Angew. Math. Mech. 53, 103–114 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  17. Schmidt, J.W.: Eine Übertragung der Regula Falsi auf Gleichungen in Banachräumen, I. Nichtlineare Gleichungssysteme (German). Z. Angew. Math. Mech. 43, 1–8 (1963)

    Article  MathSciNet  Google Scholar 

  18. Schmidt, J.W.: Eine Übertragung der Regula Falsi auf Gleichungen in Banachräumen, II. Nichtlineare Gleichungssysteme (German). Z. Angew. Math. Mech. 43, 97–110 (1963)

    Article  Google Scholar 

  19. Sergeev, A.S.: The method of chords (Russian). Sib. Mat. Z. 2, 282–289 (1961)

    MATH  MathSciNet  Google Scholar 

  20. Ulm, S.: On iterative methods with successive approximation of the inverse operator (in Russian). Tzv. Akad. Nauk Est. SSR 16, 403–411 (1967)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Ioannis K. Argyros

Authors
  1. Ioannis K. Argyros
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Ioannis K. Argyros.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Argyros, I.K. On Ulm’s method using divided differences of order one. Numer Algor 52, 295–320 (2009). https://doi.org/10.1007/s11075-009-9274-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-009-9274-3

Keywords

Mathematics Subject Classifications (2000)