Skip to main content
SpringerLink
Log in

Efficient derivative-free variants of Hansen-Patrick’s family with memory for solving nonlinear equations

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

Abstract

In this paper, we present a new tri-parametric derivative-free family of Hansen-Patrick type methods for solving nonlinear equations numerically. The proposed family requires only three functional evaluations to achieve optimal fourth order of convergence. In addition, acceleration of convergence speed is attained by suitable variation of free parameters in each iterative step. The self-accelerating parameters are estimated from the current and previous iteration. These self-accelerating parameters are calculated using Newton’s interpolation polynomials of third and fourth degrees. Consequently, the R-order of convergence is increased from 4 to 7, without any additional functional evaluation. Furthermore, the most striking feature of this contribution is that the proposed schemes can also determine the complex zeros without having to start from a complex initial guess as would be necessary with other methods. Numerical experiments and the comparison of the existing robust methods are included to confirm the theoretical results and high computational efficiency.

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  7. King, R.F.: A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jarratt, P.: Some efficient fourth-order multipoint methods for solving equations. BIT 9, 119–124 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sharma, J.R., Guha, R.K., Sharma, R.: Some variants of Hansen-Patrick method with third and fourth order convergence. Appl. Math. Comput. 214, 171–177 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Kansal, M., Kanwar, V., Bhatia, S.: New modifications of Hansen-Patrick’s family with optimal fourth and eighth orders of convergence. Appl. Math. Comput. 269, 507–519 (2015)

    MathSciNet  Google Scholar 

  11. Ortega, J.M., Rheinboldt, W.C.: Iterative Solutions of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  12. Petković, M.S., Dz̆unić, J., Petković, L.D.: A family of two-point methods with memory for solving nonlinear equations. Appl. Anal. Discrete Math. 5, 298–317 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sharifi, S., Siegmund, S., Salimi, M.: Solving nonlinear equations by a derivative-free form of the King’s family with memory. Calcolo. doi:10.1007/s10092-015-0144-1

  14. Sharma, J.R., Guha, R.K., Gupta, P.: Some efficient derivative free methods with memory for solving nonlinear equations. Appl. Math. Comput. 219, 699–707 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Dz̆unić, J., Petković, M.S., Petković, L.D.: Three-point methods with and without memory for solving nonlinear equations. Appl. Math. Comput. 218, 4917–4927 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Cordero, A., Fardi, M., Ghasemi, M., Torregrosa, J.R.: Accelerated iterative methods for ?nding solutions of nonlinear equations and their dynamical behavior. Calcolo 51, 17–30 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Three-step iterative methods with optimal eighth-order convergence. J. Comput. Appl. Math. 235, 3189–3194 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Andreu, C., Cambil, N., Cordero, A., Torregrosa, J.R.: A class of optimal eighth-order derivative free methods for solving the Danchick-Guass problem. Appl. Math. Comput. 232, 237–246 (2014)

    MathSciNet  Google Scholar 

  19. Dzunic, J., Petkovic, M.S.: On generalized multipoint root-solvers with memory. J. Comput. Appl. Math. 236, 2909–2920 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hazrat, R.: Mathematica: A Problem-Centered Approach. Springer, New York (2010)

    Book  MATH  Google Scholar 

  21. Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)

    MathSciNet  MATH  Google Scholar 

  22. Soleymani, F., Sharma, R., Li, X., Tohidi, E.: An optimized derivative-free form of the Potra-Pták method. Math. Comput. Modell. 56, 97–104 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cordero, A., Lotfi, T., Bakhtiari, P., Torregrosa, J.R.: An efficient two-parametric family with memory for nonlinear equations. Numer. Algoritm. 68, 323–335 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dzunic, J.: On efficient two-parameter methods for solving nonlinear equations. Numer. Algoritm. 63(3), 549–569 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jay, I.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Danby, J.M.A., Burkardt, T.M.: The solution of Kepler’s equation. I. Celest. Mech. 31, 95–107 (1983)

    Article  MATH  Google Scholar 

  27. Otolorin, O.: A new Newton-like iterative method for roots of analytic functions. Int. J. Math. Ed. Sci. Tech. 36, 539–572 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University Institute of Engineering and Technology, Panjab University, Chandigarh, 160 014, India

    Munish Kansal, V. Kanwar & Saurabh Bhatia

Authors
  1. Munish Kansal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Kanwar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Saurabh Bhatia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Kanwar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kansal, M., Kanwar, V. & Bhatia, S. Efficient derivative-free variants of Hansen-Patrick’s family with memory for solving nonlinear equations. Numer Algor 73, 1017–1036 (2016). https://doi.org/10.1007/s11075-016-0127-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-016-0127-6

Keywords

Mathematics Subject Classificaton (2010)

Search

Navigation