Skip to main content
SpringerLink
Log in

Semilocal convergence of a computationally efficient iterative method in Banach spaces under weak condition

  • Proceedings: ICMAA 2016
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

In this article, we discuss the semilocal convergence of well established efficient eighth-order method for solving nonlinear equations in Banach spaces under relaxed condition. The semilocal convergence of this scheme is established by using recurrence relations. We derive a system of recurrence relations for the method and then prove the existence and uniqueness result that shows the R-order of the method. Finally, numerical example is presented to validate the theoretical discussions.

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. Kantorovich, L.V., and G.P. Akilov. 1982. Functional Analysis. Oxford: Pergamon Press.

    MATH  Google Scholar 

  2. Gutiérrez, J.M., and M.A. Hernández. 2001. An acceleration of Newton’s method: super-Halley method. Applied Mathematics and Computation 117: 223–239.

    Article  MathSciNet  Google Scholar 

  3. Wu, Q., and Y. Zhao. 2006. Third-order convergence theorem by using majorizing functions for a modified Newton’s method in Banach spaces. Applied Mathematics and Computation 175: 1515–1524.

    Article  MathSciNet  Google Scholar 

  4. Rall, L.B. 1979. Computational solution of nonlinear operator equations. New York: Robert E. Krieger.

    MATH  Google Scholar 

  5. Amat, S., M.A. Hernández, and N. Romero. 2012. Semilocal convergence of a sixth order iterative method for quadratic equations. Applied Numerical Mathematics 62: 833–841.

    Article  MathSciNet  Google Scholar 

  6. Hernández, M.A., and N. Romero. 2005. On a characterization of some Newton-like methods of R-order at least three. Journal of Computational and Applied Mathematics 183: 53–66.

    Article  MathSciNet  Google Scholar 

  7. Ezquerro, J.A., and M.A. Hernández. 2000. Recurrence relations for Chebyshev-type methods. Applied Mathematics and Optimization 41: 227–236.

    Article  MathSciNet  Google Scholar 

  8. Gutiérrez, J.M., and M.A. Hernández. 1997. Third-order iterative methods for operators with bounded second derivative. Journal of Computational and Applied Mathematics 82: 171–183.

    Article  MathSciNet  Google Scholar 

  9. Gutiérrez, J.M., and M.A. Hernández. 1998. Recurrence relations for the super-Halley method. Computers and Mathematics with Applications 7: 1–8.

    Article  MathSciNet  Google Scholar 

  10. Parida, P.K., and D.K. Gupta. 2007. Recurrence relations for a Newton-like method in Banach spaces. Journal of Computational and Applied Mathematics 206: 873–887.

    Article  MathSciNet  Google Scholar 

  11. Candela, V., and A. Marquina. 1990. Recurrence relations for rational cubic methods I: the Halley method. Computing 44: 169–184.

    Article  MathSciNet  Google Scholar 

  12. Candela, V., and A. Marquina. 1990. Recurrence relations for rational cubic methods II: the Chebyshev method. Computing 45: 355–367.

    Article  MathSciNet  Google Scholar 

  13. Hernández, M.A., and M.A. Salanova. 1999. Sufficient conditions for semilocal convergence of a fourth-order multipoint iterative method for solving equations in Banach spaces. Southwest Journal of Pure and Applied Mathematics 1: 29–40.

    MathSciNet  MATH  Google Scholar 

  14. Hueso, J.L., and E. Martínez. 2014. Semilocal convergence of a family of iterative methods in Banach spaces. Numerical Algorithms 67: 365–384.

    Article  MathSciNet  Google Scholar 

  15. Zheng, L., K. Zhang, and L. Chen. 2015. On the convergence of a modified Chebyshev-like’s method for solving nonlinear equations. Taiwanese Journal of Mathematics 19: 193–209.

    Article  MathSciNet  Google Scholar 

  16. Chen, L., C. Gu, and Y. Ma. 2011. Semilocal convergence for a fifth-order Newton’s method using recurrence relations in Banach spaces. Journal of Applied Mathematics 2011: 15. Article ID 786306.

    MathSciNet  MATH  Google Scholar 

  17. Cordero, A., M.A. Hernández-Verón, N. Romero, and J.R. Torregrosa. 2015. Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces. Journal of Computational and Applied Mathematics 273: 205–213.

    Article  MathSciNet  Google Scholar 

  18. Wang, X., J. Kou, and C. Gu. 2011. Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numerical Algorithms 57: 441–456.

    Article  MathSciNet  Google Scholar 

  19. Zheng, L., and C. Gu. 2012. Semilocal convergence of a sixth-order method in Banach spaces. Numerical Algorithms 61: 413–427.

    Article  MathSciNet  Google Scholar 

  20. Jaiswal, J.P. 2016. Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency. Numerical Algorithms 71: 933–951.

    Article  MathSciNet  Google Scholar 

  21. Grau-Sánchez, M., Á. Grau, and M. Noguera. 2011. On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics 236: 1259–1266.

    Article  MathSciNet  Google Scholar 

  22. Ganesh, M., and M.C. Joshi. 1991. Numerical solvability of Hammerstein integral equations of mixed type. IMA Journal of Numerical Analysis 11: 21–31.

    Article  MathSciNet  Google Scholar 

  23. Ezquerro, J.A., and M.A. Hernandez. 2009. New iterations of R-order four with reduced computational cost. BIT Numerical Mathematics 49: 325–342.

    Article  MathSciNet  Google Scholar 

  24. Bruns, D.D., and J.E. Bailey. 1977. Nonlinear feedback control for operating a non isothermal CSTR near an unstable steady state. Chemical Engineering Sciences 32: 257–264.

    Article  Google Scholar 

Download references

Acknowledgements

The author is thankful to the reviewer(s) and editor for their useful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, MP, 462051, India

    J. P. Jaiswal

  2. Faculty of Science, Barkatullah University, Bhopal, MP, 462026, India

    J. P. Jaiswal

  3. Regional Institute of Education, Bhopal, MP, 462013, India

    J. P. Jaiswal

Authors
  1. J. P. Jaiswal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. P. Jaiswal.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest for publishing this article.

Research involving human participants and/or animals

The author declares that there is no involvement of human participants and/or animals in this research.

Informed consent

Not applicable here.

Funding

This study is supported by Science and Engineering Research Board, New Delhi, India under the scheme Start Up Research Grant (Young Scientists) (Ref. No. YSS/2015/001507).

Additional information

The submission is specifically for the special volume of ICMAA 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jaiswal, J.P. Semilocal convergence of a computationally efficient iterative method in Banach spaces under weak condition. J Anal 28, 141–154 (2020). https://doi.org/10.1007/s41478-017-0050-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-017-0050-9

Keywords

Mathematics Subject Classification

Search

Navigation