Skip to main content
SpringerLink
Log in

Fifth-order iterative method for finding multiple roots of nonlinear equations

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

Abstract

This paper presents a fifth-order iterative method as a new modification of Newton’s method for finding multiple roots of nonlinear equations with unknown multiplicity m. Its convergence order is analyzed and proved. Moreover, several numerical examples demonstrate that the proposed iterative method is superior to the existing methods.

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Neta, B.: Extension of Murakami’s high order nonlinear solver to multiple roots. Int. J. Comput. Math. 87, 1023–1031 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, Englewood (1964)

    MATH  Google Scholar 

  13. King, R.F.: A secant method for multiple roots. BIT 17, 321–328 (1977)

    Article  MATH  Google Scholar 

  14. Wu, X.Y., Fu, D.S.: New higher-order convergence iteration methods without employing derivatives for solving nonlinear equations. Comput. Math. Appl. 41, 489–495 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wu, X.Y., Xia, J.L., Shao, R.: Quadratically convergent multiple roots finding method without derivatives. Comput. Math. Appl. 42, 115–119 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Steffensen, I.F.: Remark on Iteration, vol. 16, pp. 64–72. Skand, Aktuarietidskr (1933)

  17. Wu, X.Y.: A new continuation Newton-like method and its deformation. Appl. Math. Comput. 112, 75–78 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Parida, P.K., Gupta, D.K.: An improved method for finding multiple roots and it’s multiplicity of nonlinear equations in R. Appl. Math. Comput. 202, 498–503 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yun, B.I.: A derivative free iterative method for finding multiple roots of nonlinear equations. Appl. Math. Lett. 22, 1859–1863 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kioustelidis, J.B.: A derivative-free transformation preserving the order of convergence of iteration methods in case of multiple zeros. Numer. Math. 33, 385–389 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bi, W., Ren, H., Wu, Q.: New family of seventh-order methods for nonlinear equations. Appl. Math. Comput. 203, 408–412 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bi, W., Ren, H., Wu, Q.: Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 225, 105–112 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yun, B.I.: Transformation methods for finding multiple roots of nonlinear equations. Appl. Math. Comput. 217, 599–606 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Halley, E.: A new, exact and easy method of finding the roots of equations generally and that without any previous reduction. Phil. Trans. Roy. Soc. Lond. 18, 136–148 (1694)

    Google Scholar 

  25. Laguerre, E.N.: Sur une méthode pour obtener par approximation les racines d’une équation algébrique qui a toutes ses racines réelles. Nouvelles Ann. de Math. 2e séries 19, 88–103 (1880)

    Google Scholar 

  26. Dong, C.: A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation. Math. Numer. Sinica 11, 445–450 (1982)

    Google Scholar 

  27. Werner, W.: Iterationsverfahren höherer Ordnung zur Lösung nicht linearer Gleichungen. Z. Angew. Math. Mech. 61, T322–T324 (1981)

    Google Scholar 

  28. Gautschi, W.: Numerical Analysis: an Introduction. Birkhäuser (1997)

  29. Neta, B.: On a family of multipoint methods for non-linear equations. Int. J. Comput. Math. 9, 353–361 (1981)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 400044, People’s Republic of China

    Xiaowu Li & Chunlai Mu

  2. School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871, People’s Republic of China

    Jinwen Ma

  3. Center for Chinese Agricultural Policy, Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, 100101, People’s Republic of China

    Linke Hou

Authors
  1. Xiaowu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunlai Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jinwen Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Linke Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaowu Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, X., Mu, C., Ma, J. et al. Fifth-order iterative method for finding multiple roots of nonlinear equations. Numer Algor 57, 389–398 (2011). https://doi.org/10.1007/s11075-010-9434-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-010-9434-5

Keywords

Search

Navigation