Skip to main content

Convergence Analysis of the Modified Chebyshev’s Method for Finding Multiple Roots

Vietnam Journal of Mathematics volume 50, pages 59–68 (2022)Cite this article

Abstract

In this paper, an estimate of the radius of convergence ball of the modified Chebyshev’s method for finding multiple roots of nonlinear equations is provided under the hypotheses that the (m + 1)st derivative f(m+ 1) of function f is Hölder continuous and bounded. The unique ball of a solution is also established. Finally, some examples are provided to show the effectiveness of our results.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Amat, S., Busquier, S., Gutiérrez, J.M., Hernández, M.A.: On the global convergence of Chebyshev’s iterative method. J. Comput. Appl. Math. 220, 17–21 (2008)

    MathSciNet  Article  Google Scholar 

  2. Behl, R., González, D., Maroju, P., Motsa, S.S.: An optimal and efficient general eighth-order derivative free scheme for simple roots. J. Comput. Appl. Math. 330, 666–675 (2018)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  4. Candela, V.F., Marquina, A.: Recurrence relations of rational cubic method II: the Chebyshev method. Computing 45, 335–367 (1990)

    MathSciNet  Article  Google Scholar 

  5. Castro, R.A., Rodríguez, J.C., Sierra, W.W., Di Giorgi, G.L., Gómez, S.J.: Chebyshev–Halley method on Riemannian manifolds. J. Comput. Appl. Math. 336, 30–53 (2018)

    MathSciNet  Article  Google Scholar 

  6. Chun, C., Neta, B.: Comparative study of methods of various orders for finding repeated roots of nonlinear equations. J. Comput. Appl. Math. 340, 11–42 (2018)

    MathSciNet  Article  Google Scholar 

  7. 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)

    MathSciNet  Article  Google Scholar 

  8. Hansen, E., Patrick, M.: A family of root finding methods. Numer. Math. 27, 257–269 (1976)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  10. Jamaludin, N.A.A., Nik Long, N.M.A., Salimi, M., Sharifi, S.: Review of some iterative methods for solving nonlinear equations with multiple zeros. Afr. Mat. 30, 355–369 (2019)

    MathSciNet  Article  Google Scholar 

  11. Kumar, S., Kanwar, V., Singh, S.: On some modified families of multipoint iterative methods for multiple roots of nonlinear equations. Appl. Math. Comput. 218, 7382–7394 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Kumari, C., Parida, P.K.: Local convergence analysis for Chebyshev’s method. J. Appl. Math. Comput. 59, 405–421 (2019)

    MathSciNet  Article  Google Scholar 

  13. Luo, X.F., Ye, X.T.: Convergence of the family of the deformed Euler–Halley iterations with parameters under the hölder condition of the derivative. Numer. Math. J. Chin. Univ. 27, 85–192 (2005)

    MATH  Google Scholar 

  14. Margaris, A., Goulianas, K.: Finding all roots of 2 × 2 nonlinear algebraic systems using back-propagation neural networks. Neural Comput. Appl. 21, 891–904 (2012)

    Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  16. Otadi, M.: System of fully fuzzy nonlinear equations with fuzzy neural network. Neural Comput. Appl. 21, 369–376 (2012)

    Article  Google Scholar 

  17. Ren, H.M., Wu, Q.B.: The convergence ball of the secant method under hölder continuous divided differences. J. Comput. Appl. Math. 194, 284–293 (2006)

    MathSciNet  Article  Google Scholar 

  18. 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)

    MathSciNet  MATH  Google Scholar 

  19. Schröer, E.: Ueber unendlich viele Algorithmen zur auflöung der Gleichungen. Math. Ann. 2, 317–365 (1870)

    MathSciNet  Article  Google Scholar 

  20. Sharma, J.R., Kumar, D.: On a class of efficient higher order Newton-like methods. Math. Model. Anal. 24, 105–126 (2019)

    MathSciNet  Article  Google Scholar 

  21. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1980)

    Book  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  24. Wu, Q.B., Ren, H.M., Bi, W.H.: The convergence ball of Wang’s method for finding a zero of a derivative. Math. Comput. Model. 49, 740–744 (2009)

    MathSciNet  Article  Google Scholar 

  25. Xu, L., Wang, X.: Topics on Methods and Examples of Mathematical Analysis (in Chinese). High Education Press (1983)

  26. Zhanlav, T., Otgondorj, K.H., Chuluunbaatar, O.: Families of optimal derivative-free two- and three-point iterative methods for solving nonlinear equations. Comput. Math. Math. Phys. 59, 864–880 (2019)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant Nos. 11771393, 11632015), Scientific Research Fund of Zhejiang Provincial Education Department (Grant No. FX2016073) and the Science Foundation of Taizhou University (Grant No. 2017PY028).

Author information

Authors and Affiliations

  1. Department of Mathematics, Taizhou University, Linhai, Zhejiang, 317000, People’s Republic of China

    Rongfei Lin

  2. College of Information and Engineering, Hangzhou Polytechnic, Hangzhou, Zhejiang, 311402, People’s Republic of China

    Hongmin Ren

  3. Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, 310027, People’s Republic of China

    Qingbiao Wu

  4. Department of Mathematics, University of Hafr Al-Batin, Hafr Al-Batin, 31991, Saudi Arabia

    Yasir Khan

  5. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, People’s Republic of China

    Juelian Hu

Authors
  1. Rongfei Lin
    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

  3. Qingbiao Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yasir Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Juelian Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yasir Khan.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lin, R., Ren, H., Wu, Q. et al. Convergence Analysis of the Modified Chebyshev’s Method for Finding Multiple Roots. Vietnam J. Math. 50, 59–68 (2022). https://doi.org/10.1007/s10013-020-00470-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-020-00470-8

Keywords

  • Chebyshev’s method
  • Nonlinear equations
  • Multiple roots
  • Hölder condition
  • Local convergence
  • Radius of convergence

Mathematics Subject Classification (2010)

  • 65F10
  • 65F50
  • 65H10