Skip to main content
SpringerLink
Log in

Efficient methods of optimal eighth and sixteenth order convergence for solving nonlinear equations

  • Published:
SeMA Journal Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We present simple yet efficient three- and four-point iterative methods for solving nonlinear equations. The methodology is based on fourth order Kung–Traub method and further developed by using rational Hermite interpolation. Three-point method requires four function evaluations and has the order of convergence eight, whereas the four-point method requires the evaluation of five functions and has the order of convergence sixteen, that means, the methods are optimal in the sense of Kung–Traub hypothesis (Kung and Traub, J ACM 21:643–651, 1974). The methods are tested through numerical experimentation. Their performance is compared with already established methods in literature. It is observed that new algorithms are well-behaved and very effective in high precision computations. Moreover, the presented basins of attraction also confirm stable nature of the algorithms.

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

Similar content being viewed by others

References

  1. 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 

  2. Bi, W., Wu, Q., Ren, H.: A new family of eighth-order iterative methods for solving nonlinear equations. Appl. Math. Comput. 214, 236–245 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Chapra, S.C., Canale, R.P.: Numerical Methods for Engineers. McGraw-Hill Book Company, New York (1988)

    Google Scholar 

  4. Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Three-step iterative methods with optimal eighth order of convergence. J. Comput. Appl. Math. 23, 3189–3194 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Danby, J.M.A., Burkardt, T.M.: The solution of Kepler’s equation, I. Celest. Mech. 40, 95–107 (1983)

    Article  MATH  Google Scholar 

  6. Geum, Y.H., Kim, Y.I.: A multi-parameter family of three-step eighth-order iterative methods locating a simple root. Appl. Math. Comput. 215, 3375–3382 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Geum, Y.H., Kim, Y.I.: A family of optimal sixteenth-order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth-step weighting function. Comput. Math. Appl. 61, 3278–3287 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hoffman, J.D.: Numerical Methods for Engineers and Scientists. McGraw-Hill Book Company, New York (1992)

    MATH  Google Scholar 

  9. Jarratt, P., Nudds, D.: The use of rational functions in the iterative solution of equations on a digital computer. Comput. J. 8, 62–65 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jay, L.O.: A note on Q-order of convergence. BIT 41, 422–429 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Khan, Y., Fardi, M., Sayevand, K.: A new general eighth-order family of iterative methods for solving nonlinear equations. Appl. Math. Lett. 25, 2262–2266 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, L., Wang, X.: Eighth-order methods with high efficiency index for solving nonlinear equations. Appl. Math. Comput. 215, 3449–3454 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Lotfi, T., Sharifi, S., Salimi, M., Siegmund, S.: A new class of three-point methods with optimal convergence order eight and its dynamics. Numer. Algor. 68, 261–288 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ostrowski, A.M.: Solutions of Equations and System of Equations. Academic Press, New York (1960)

    MATH  Google Scholar 

  16. Petković, M.S., Neta, B., Petković, L.D., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Elsevier, Boston (2013)

    MATH  Google Scholar 

  17. Sharma, J.R., Arora, H.: A new family of optimal eighth order methods with dynamics for nonlinear equations. Appl. Math. Comput. 273, 924–933 (2016)

    MathSciNet  Google Scholar 

  18. Sharma, J.R., Guha, R.K., Sharma, R.: A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algor. 54, 445–458 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sharma, R., Sharma, J.R., Kalra, N.: A novel family of weighted-Newton optimal eighth order methods with dynamics. SeMa. doi:10.1007/s40324-017-0129-x (2017)

  20. Thukral, R.: A new eighth-order iterative method for solving nonlinear equations. Appl. Math. Comput. 217, 222–229 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Thukral, R., Petković, M.S.: A family of three-point methods of optimal order for solving nonlinear equations. J. Comput. Appl. Math. 233, 2278–2284 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tornheim, L.: Convergence of multipoint iterative methods. J. ACM 11, 210–220 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  23. Traub, J.F.: Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York (1982)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Vrscay, E.R., Gilbert, W.J.: Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions. Numer. Math. 52, 1–16 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, X., Liu, L.: Modified Ostrowski’s method with eighth-order convergence and high efficiency index. Appl. Math. Lett. 23, 549–554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, X., Qin, Y., Qian, W., Zhang, S., Fan, X.: A family of Newton type iterative method for solving nonlinear equations. Algorithms. 8, 786–798 (2015)

    Article  MathSciNet  Google Scholar 

  28. Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media (2003)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur, 148106, India

    Janak Raj Sharma & Sunil Kumar

Authors
  1. Janak Raj Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sunil Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Janak Raj Sharma.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sharma, J.R., Kumar, S. Efficient methods of optimal eighth and sixteenth order convergence for solving nonlinear equations. SeMA 75, 229–253 (2018). https://doi.org/10.1007/s40324-017-0131-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40324-017-0131-3

Keywords

Mathematics Subject Classification

Search

Navigation