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On the convergence radius of the modified Newton method for multiple roots under the center–Hölder condition

Abstract

It is very important to enlarge the convergence ball of an iterative method. Recently, the convergence radius of the modified Newton method for finding multiple roots of nonlinear equations has been presented by Ren and Argyros when the involved function is Hölder and center–Hölder continuous. Different from the technique and the hypothesis used by them, in this paper, we also investigate the convergence radius of the modified Newton method under the condition that the derivative \(f^{(m)}\) of function f satisfies the center–Hölder continuous condition. The radius given here is larger than that given by Ren and Argyros. The uniqueness ball of solution is also discussed. Some examples are given to show applications of our theorem.

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References

  1. 1.

    Schröder, E.: Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Obreshkov, N.: On the numerical solution of equations (Bulgarian). Annuaire Univ. Sofia Fac. Sci. Phys. Math. 56, 73–83 (1963)

    MATH  Google Scholar 

  3. 3.

    Hansen, E., Patrick, M.: A family of root finding methods. Numer. Math. 27, 257–269 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Traub, J.F.: Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York (1977)

    Google Scholar 

  5. 5.

    Osada, N.: An optimal multiple root-finding method of order three. J. Comput. Appl. Math. 51, 131–133 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Neta, B.: Numerical Methods for the Solution of Equations. Net-A-Sof, California (1983)

    Google Scholar 

  7. 7.

    Victory, H.D., Neta, B.: A higher order method for multiple zeros of nonlinear functions. Int. J. Comput. Math. 12, 329–335 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Dong, C.: A family of multiopoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math. 21, 363–367 (1987)

    Article  MATH  Google Scholar 

  9. 9.

    Neta, B.: New third order nonlinear solvers for multiple roots. Appl. Math. Comput 202, 162–170 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Neta, B., Jhonson, A.N.: High-order nonlinear solver for multiple roots. Comput. Math. Appl. 55, 2012–2017 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Chun, C., Neta, B.: A third-order modification of Newton’s method for multiple roots. Appl. Math. Comput. 211, 474–479 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Chun, C., Bae, H., Neta, B.: New families of nonlinear third-order solvers for finding multiple roots. Comput. Math. Appl. 57, 1574–1582 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Homeier, H.: On Newton-type methods for multiple roots with cubic convergence. J. Comput. Appl. Math. 231, 249–254 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Neta, B.: Extension of Murakami’s High order nonlinear solver to multiple roots. Int. J. Comput. Math. 8, 1023–103 (2010)

    Article  MathSciNet  Google Scholar 

  15. 15.

    Li, S.G., Cheng, L.Z., Neta, B.: Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput. Math. Appl. 59, 126–135 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Zhou, X., Chen, X., Song, Y.: Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. J. Comput. Appl. Math. 235, 4199–4206 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Ren, H.M., Argyros, I.K.: Convergence radius of the modified Newton method for multiple zeros under Hölder continuous derivative. Appl. Math. Comput. 217, 612–621 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Wang, X.H.: The convergence ball on Newton’s method, Chinese Sci. Bull. A Special Issue of Mathematics, Physics, Chemistry 25, 36–37 (1980)

    Google Scholar 

  19. 19.

    Traub, J., Wozniakowski, H.: Convergence and complexity of Newton iteration for operator equation. J. Assoc. Comput. Mech. 26, 250–258 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Argyros, I.K.: On the convergence and application of Newtons method under weak Hölder continuity assumptions. Int. J. Comput. Math. 80, 767–780 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. 21.

    Zhou, X., Song, Y.: Convergence radius of Osada’s method for multiple roots under Hölder and Center–Hölder continuous conditions. ICNAAM 2011 (Greece). AIP Conf. Proc. 1389, 1836–1839 (2011)

    Article  Google Scholar 

  22. 22.

    Bi, W.H., Ren, H.M., Wu, Q.B.: Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative. Numer. Algor. 58, 497–512 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    Abramowitz, M., Stegun, I.S.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. In: Applied Math Series, vol. 55., Washington, D.C. (1964)

  24. 24.

    Rall, L.B.: Computational Solution of Nonlinear Operator Equations, pp 124–125. John Wiley and Sons, Inc (1969)

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Correspondence to Yongzhong Song.

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Zhou, X., Chen, X. & Song, Y. On the convergence radius of the modified Newton method for multiple roots under the center–Hölder condition. Numer Algor 65, 221–232 (2014). https://doi.org/10.1007/s11075-013-9702-2

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Keywords

  • Nonlinear equations
  • Multiple roots
  • Convergence radius
  • The modified Newton method
  • Center–Hölder condition