Skip to main content
SpringerLink
Log in

A simple yet efficient two-step fifth-order weighted-Newton method for nonlinear models

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

Abstract

An iterative method with fifth order of convergence for solving nonlinear systems is formulated and analyzed. Primary goal of the development of method is to keep the both, convergence order and computational efficiency, as high as possible. Both of these factors are investigated thoroughly in theoretical as well as numerical manner. Comparison with the existing iterative methods of similar order is carried out to examine the performance of method. Numerical testing clearly indicates the high degree of precision and efficiency of the new method.

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

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Ostrowski, A. M.: Solution of Equation and Systems of Equations. Academic Press, New York (1960)

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

    MATH  Google Scholar 

  4. McNamee, J. M.: Numerical Methods for Roots of Polynomials Part, vol. I. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  5. Grau-Sánchez, M., Grau, À. , Noguera, M.: On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. J. Comput. Appl. Math. 236, 1259–1266 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cordero, A., Hueso, J. L., Martínez, E., Torregrosa, J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Xu, Z., Jieqing, T.: The fifth order of three-step iterative methods for solving systems of nonlinear equations. Math. Numer. Sin. 35, 297–304 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Sharma, J. R., Gupta, P.: An efficient fifth order method for solving systems of nonlinear equations. Comput. Math. Appl. 67, 591–601 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Xiao, X. Y., Yin, H.: A new class of methods with higher order of convergence for solving systems of nonlinear equations. Appl. Math. Comput. 264, 300–309 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Choubey, N., Jaiswal, J. P.: Improving the order of convergence and efficiency index of an iterative method for nonlinear systems. Proc. Natl. Acad. Sci. India A - Phys. Sci. 86, 221–227 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Sharma, J. R., Guha, R. K.: Simple yet efficient Newton-like method for systems of nonlinear equations. Calcolo 53, 451–473 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cordero, A., Gómez, E., Torregrosa, J. R.: Efficient high-order iterative methods for solving nonlinear systems and their application on heat conduction problems. Complexity 2017, 6457532 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Xiao, X. Y., Yin, H.: Accelerating the convergence speed of iterative methods for solving nonlinear systems. Appl. Math. Comput. 333, 8–19 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sharma, R., Sharma, J. R., Kalra, N.: A Modified Newton–\(\overset {..}{\text {O}}\)zban composition for solving nonlinear systems. Int. J. Comput. Methods 17, 1950047 (2020)

  15. Wu, S., Wang, H.: A modified Newton-like method for nonlinear equations. Comput. Appl. Math. 39, 1–18 (2020)

    Article  MathSciNet  Google Scholar 

  16. Solaiman, O. S., Hashim, I.: An iterative scheme of arbitrary odd order and its basins of attraction for nonlinear systems. Comput. Mater. Contin. 66, 1427–1444 (2021)

    Google Scholar 

  17. Kansal, M., Cordero, A., Bhalla, S., Torregrosa, J. R.: New fourth-and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability analysis. Numer. Algorithms 87, 1017–1060 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Behl, R., Bhalla, S., Magreñán, A. ́A., Kumar, S.: An efficient high order iterative scheme for large nonlinear systems with dynamics. J. Comput. Appl. Math. 404, 113249 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  19. Arroyo, V., Cordero, A., Torregrosa, J. R.: Approximation of artificial satellites’ preliminary orbits: The efficiency challenge. Math. Comput. Model. 54, 1802–1807 (2011)

    Article  MATH  Google Scholar 

  20. Brent, R. P.: Some efficient algorithms for solving systems of nonlinear equations. SIAM J. Numer. Anal. 10, 327–344 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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

  22. Chandrasekhar, S.: Radiative Transfers. Dover, New York (1960)

    Google Scholar 

  23. Hueso, J. L., Martínez, E., Teruel, C.: Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems. J. Comput. Appl. Math. 275, 412–420 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Singh, H., Sharma, J. R.: Reduced cost numerical methods of sixth-order convergence for systems of nonlinear models. Rev. Real Acad. Cienc. Exactas Fis. Nat. - A: Mat. 116, 1–24 (2022)

    MathSciNet  MATH  Google Scholar 

  25. Narang, M., Bhatia, S., Alshomrani, A. S., Kanwar, V.: General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations. J. Comput. Appl. Math. 352, 23–39 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Harmandeep Singh & Janak Raj Sharma

  2. Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Chennai, 601103, India

    Sunil Kumar

Authors
  1. Harmandeep Singh
    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

  3. Sunil Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Janak Raj Sharma.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher’s note

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

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, H., Sharma, J.R. & Kumar, S. A simple yet efficient two-step fifth-order weighted-Newton method for nonlinear models. Numer Algor 93, 203–225 (2023). https://doi.org/10.1007/s11075-022-01412-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-022-01412-w

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation