Skip to main content
SpringerLink
Log in

Optimal Fourth-Order Weerakoon–Fernando-Type Methods for Multiple Roots and Their Dynamics

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we present optimal fourth-order methods for finding multiple roots of non-linear equations, where the multiplicity is known in advance. These methods are based on the third-order method given by Weerakoon and Fernando for simple roots. The dynamical behavior of these methods around multiple roots is studied using basin of attraction in complex plane. We also present numerical examples to confirm our theoretical results.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Alexander, D.S.: A History of Complex Dynamics: From Schroder to Fatou and Julia. Friedr & Sohn Veriage, Wiesbaden (1994)

    Book  Google Scholar 

  2. Beardon, A.F.: Iteration of Rational Functions. Springer, New York (1991)

    Book  Google Scholar 

  3. Blanchard, P.: The dynamics of Newton’s method. Proc. Sympos. Appl. Math. 49, 139–154 (1994)

    Article  MathSciNet  Google Scholar 

  4. Chicharro, F.I., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 780153, 1–11 (2013)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley publishing Company, New York (1989)

    MATH  Google Scholar 

  7. Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iterations. J. Assoc. Comput. Mach. 21, 643–651 (1974)

    Article  MathSciNet  Google Scholar 

  8. Liu, B., Zhou, X.: A new family of fourth order methods for multiple roots of non-linear equation. Nonlinear Anal. Model. Control 18, 143–152 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Petkovic, M.S., Neta, B., Petcovic, L.D., Dzflnic, J.: Multipoint Methods for Solving Nonlinear Equations. Elsevier, Waltham, MA (2013)

    Google Scholar 

  10. Sharma, J.R., Sharma, R.: Modified Jarratt method for computing multiple roots. Appl. Math. Comp. 217, 878–881 (2010)

    Article  MathSciNet  Google Scholar 

  11. Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intell. 24, 37–46 (2002)

    Article  MathSciNet  Google Scholar 

  12. Vrscay, E.R., Gilbert, W.J.: Extraneous fixed points, basin boundaries and chaotic dynamics for Schroder and Konig rational iteration functions. Num. Math. 52, 1–16 (1988)

    Article  Google Scholar 

  13. Weerakoon, S., Fernando, T.G.I.: A Variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett 17, 87–93 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, 110021, India

    Prem Bahadur Chand, Pankaj Jain & Kriti Sethi

  2. Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, Av. La Paz 137, 26006, Logroño, Spain

    Francisco I. Chicharro

Authors
  1. Prem Bahadur Chand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francisco I. Chicharro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pankaj Jain
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kriti Sethi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco I. Chicharro.

Additional information

Publisher's Note

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

This research was partially supported by both Ministerio de Ciencia, Innovación y Universidades and Generalitat Valenciana, Spain, under Grants PGC-2018-095896-B-C22 and PROMETEO/2016/089, respectively.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chand, P.B., Chicharro, F.I., Jain, P. et al. Optimal Fourth-Order Weerakoon–Fernando-Type Methods for Multiple Roots and Their Dynamics. Mediterr. J. Math. 16, 67 (2019). https://doi.org/10.1007/s00009-019-1350-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-019-1350-x

Mathematics Subject Classification

Keywords

Search

Navigation