Skip to main content
SpringerLink
Log in

On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations

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

Abstract

We present derivative free methods with memory with increasing order of convergence for solving systems of nonlinear equations. These methods relied on the basic family of fourth order methods without memory proposed by Sharma et al. (Appl. Math. Comput. 235, 383–393, 2014). The order of convergence of new family is increased from 4 of the basic family to \(2+\sqrt {5} \approx 4.24\) by suitable variation of a free self-corrected parameter in each iterative step. In a particular case of the family even higher order of convergence \(2+\sqrt {6} \approx 4.45\) is achieved. It is shown that the new methods are more efficient in general. The presented numerical tests confirm the 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.

Similar content being viewed by others

References

  1. Ostrowski, A.M.: Solution of Equations and Systems of Equations. Academic Press, New York (1966)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Kelley, C.T.: Solving Nonlinear Equations with Newton’s. SIAM, Philadelphia (2003)

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

    MATH  Google Scholar 

  5. Steffensen, J.F.: Remarks on iteration, Skand. Aktuar Tidskr 16, 64–72 (1933)

    Google Scholar 

  6. Sharma, J.R., Arora, H., Petković, M.S.: An efficient derivative free family of fourth order methods for solving systems of nonlinear equations. Appl. Math. Comput 235, 383–393 (2014)

    Article  MathSciNet  Google Scholar 

  7. Petković, M.S., Džunić, 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 

  8. Argyros, I.K.: Advances in the Efficiency of Computational Methods and Applications. World Scientific Publishing Oxon, United Kingdom (2000)

    Book  MATH  Google Scholar 

  9. Grau-Sánchez, M., Grau, À., Noguera, M.: Ostrowski type methods for solving systems of nonlinear equations. Appl. Math. Comput 218, 2377–2385 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grau-Sánchez, M., Noguera, M.: A technique to choose the most efficient method between secant method and some variants. Appl Math. Comput 218, 6415–6426 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grau-Sánchez M., Grau, À., Noguera, M.: Frozen divided difference scheme for solving systems of nonlinear equations. J. Comput. Appl. Math. 235, 1739–1743 (2011)

    Article  MathSciNet  Google Scholar 

  12. Potra, F-A., Pták, V.: Nondiscrete Induction and Iterarive Processes. Pitman Publishing, Boston (1984)

    Google Scholar 

  13. Grau-Sánchez, M., Noguera, M., Amat, S.: On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods. J. Comput. Appl. Math 237, 363–372 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33(2) (2007). 15. Art. 13.

  17. http://www.mpfr.org/mpfr-2.1.0/timings.html.

  18. Grau-Sánchez, M., Peris, J.M., Gutiérrez, J.M.: Accelerated iterative methods for finding solutions of a system of nonlinear equations. Appl Math. Comput 190, 1815–1823 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Electronic Engineering, Department of Mathematics, University of Niš, 18000, Niš, Serbia

    Miodrag S. Petković

  2. Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, 148106, Punjab, India

    Janak Raj Sharma

Authors
  1. Miodrag S. Petković
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Janak Raj Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miodrag S. Petković.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Petković, M.S., Sharma, J.R. On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations. Numer Algor 71, 457–474 (2016). https://doi.org/10.1007/s11075-015-0003-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-015-0003-9

Keywords

Search

Navigation