Skip to main content
SpringerLink
Log in

Extended iterative schemes based on decomposition for nonlinear models

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

We suggest the local analysis of a class of iterative schemes based on decomposition technique for solving Banach space valued nonlinear models. Earlier results used hypotheses up to the fourth derivative to establish convergence. But we only apply the first derivative in our convergence theorem. We also provide computable radius of convergence ball, error estimates and uniqueness of the solution results not studied in earlier works. Hence, we enhance the applicability of these schemes. Furthermore, we explore, using basin of attraction tool, the dynamics of the schemes when they are applied on various complex polynomials. This article is concluded with numerical experiments.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Amat, S., Argyros, I.K., Busquier, S., Hernández-Verón, M.A., Martínez, E.: On the local convergence study for an efficient k-step iterative method. J. Comput. Appl. Math. 343, 753–761 (2018)

    Article  MathSciNet  Google Scholar 

  2. Argyros, I.K.: Convergence and Application of Newton-Type Iterations. Springer, Berlin (2008)

    MATH  Google Scholar 

  3. Argyros, I.K., Cho, Y.J., Hilout, S.: Numerical Methods for Equations and Its Applications. Taylor & Francis, CRC Press, New York (2012)

    Book  Google Scholar 

  4. Argyros, I.K., Magreñán, Á.A.: On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252(1), 336–346 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Argyros, I.K., Magreñán, Á.A.: A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71(1), 1–23 (2015)

    Article  MathSciNet  Google Scholar 

  6. Argyros, I.K., Cho, Y.J., George, S.: Local convergence for some third order iterative methods under weak conditions. J. Korean Math. Soc. 53(4), 781–793 (2016)

    Article  MathSciNet  Google Scholar 

  7. Argyros, I.K., George, S.: Local convergence for an almost sixth order method for solving equations under weak conditions. SeMA J. 75(2), 163–171 (2017)

    Article  MathSciNet  Google Scholar 

  8. Argyros, I.K., Sharma, D., Parhi, S.K., Sunanda, S.K.: On the convergence, dynamics and applications of a new class of nonlinear system solvers. Int. J. Appl. Comput. Math. 6 (5), Article Number: 142 (2020). https://doi.org/10.1007/s40819-020-00893-4

  9. Argyros, I.K., George, S.: On the complexity of extending the convergence region for Traub’s method. J. Complex. (2020). https://doi.org/10.1016/j.jco.2019.101423

    Article  MathSciNet  MATH  Google Scholar 

  10. Argyros, I.K., Behl, R., González, D., Motsa, S.S.: Ball convergence for combined three-step methods under generalized conditions in Banach space. Stud. Univ. Babes-Bolyai Math. 65(1), 127–137 (2020)

    Article  MathSciNet  Google Scholar 

  11. Argyros, I. K., George, S.: Ball comparison between four fourth convergence order methods under the same set of hypotheses for solving equations. Int. J. Appl. Comput. Math. 7, Article Number: 9 (2021). https://doi.org/10.1007/s40819-020-00946-8

  12. Behl, R., Martínez, E.: A new higher-order and efficient family of iterative techniques for nonlinear models. Complexity 2020. Article ID 1706841, pp. 1–11 (2020)

  13. Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A modified Newton-Jarratt’s composition. Numer. Algorithms 55(1), 87–99 (2010)

    Article  MathSciNet  Google Scholar 

  14. Cordero, A., García-Maimó, J., Torregrosa, J.R., Vassileva, M.P.: Solving nonlinear problems by Ostrowski-Chun type parametric families. J. Math. Chem. 53(1), 430–449 (2014)

    Article  MathSciNet  Google Scholar 

  15. Cordero, A., Villalba, E.G., Torregrosa, J.R., Triguero-Navarro, P.: Convergence and stability of a parametric class of iterative schemes for solving nonlinear systems. Mathematics 9 (1), Article Number 86, pp. 1–19 (2021). https://doi.org/10.3390/math9010086

  16. Ezquerro, J., Hernández, M.A.: On Halley-type iteration with free second derivative. J. Comput. Appl. Math. 170, 455–459 (2004)

    Article  MathSciNet  Google Scholar 

  17. Ezquerro, J.A., González, D., Hernández, M.A.: Majorizing sequences for Newton’s method from initial value problems. J. Comput. Appl. Math. 236, 2246–2258 (2012)

    Article  MathSciNet  Google Scholar 

  18. 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(1), 363–372 (2013)

    Article  MathSciNet  Google Scholar 

  19. Kou, J., Li, Y., Wang, X.: A composite fourth-order iterative method for solving nonlinear equations. Appl. Math. Comput. 184(2), 471–475 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Kumar, D., Sharma, J.R., Jäntschi, L.: Convergence analysis and complex geometry of an efficient derivative-free iterative method. Mathematics 7 (10), Article Number 919, pp. 1–11 (2019)

  21. Magreñán, Á.A.: Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Maroju, P., Magreñán, Á.A., Sarría, Í., Kumar, A.: Local convergence of fourth and fifth order parametric family of iterative methods in Banach spaces. J. Math. Chem. 58, 686–705 (2020)

    Article  MathSciNet  Google Scholar 

  23. Neta, B., Scott, M., Chun, C.: Basins of attraction for several methods to find simple roots of nonlinear equations. Appl. Math. Comput. 218, 10548–10556 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Noor, M.A., Waseem, M., Noor, K.I., Ali, M.A.: New iterative technique for solving nonlinear equations. Appl. Math. Comput. 265, 1115–1125 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Petković, M.S., Neta, B., Petković, L., Dz̃unić, D.: Multipoint Methods for Solving Nonlinear Equations. Elsevier, Amsterdam (2013)

  26. Rall, L.B.: Computational Solution of Nonlinear Operator Equations. Robert E. Krieger, New York (1979)

    MATH  Google Scholar 

  27. Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Sharma, D., Parhi, S.K.: Extending the applicability of a Newton-simpson-like method. Int. J. Appl. Comput. Math. 6 (3), Article number: 79 (2020). https://doi.org/10.1007/s40819-020-00832-3

  29. Sharma, J.R., Guna, R.K., Sharma, R.: An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62(2), 307–323 (2013)

    Article  MathSciNet  Google Scholar 

  30. Sharma, J.R.: A composite third order Newton-Steffensen method for solving nonlinear equations. Appl. Math. Comput. 169(1), 242–246 (2005)

    MathSciNet  MATH  Google Scholar 

  31. Singh, S., Gupta, D.K., Badoni, R.P., Martínez, E., Hueso, J.L.: Local convergence of a parameter based iteration with Hölder continuous derivative in Banach spaces. Calcolo 54(2), 527–539 (2017)

    Article  MathSciNet  Google Scholar 

  32. Traub, J.F.: Iterative Methods for Solution of Equations. Prentice-Hall, Upper Saddle River (1964)

    MATH  Google Scholar 

Download references

Acknowledgements

Funding was provided by University Grants Commission (Grant No: NOV2017-402662).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

  2. Department of Mathematics, IIIT Bhubaneswar, Bhubaneswar, Odisha, 751003, India

    Debasis Sharma & Shanta Kumari Sunanda

  3. Department of Computer Science, University of Oklahoma, Norman, OK, 73071, USA

    Christopher I. Argyros

  4. Department of Mathematics, Fakir Mohan University, Balasore, Odisha, 756020, India

    Sanjaya Kumar Parhi

Authors
  1. Ioannis K. Argyros
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Debasis Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christopher I. Argyros
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sanjaya Kumar Parhi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Shanta Kumari Sunanda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Debasis Sharma.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Argyros, I.K., Sharma, D., Argyros, C.I. et al. Extended iterative schemes based on decomposition for nonlinear models. J. Appl. Math. Comput. 68, 1485–1504 (2022). https://doi.org/10.1007/s12190-021-01570-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01570-5

Keywords

Mathematics Subject Classification

Search

Navigation