Skip to main content
Log in

Semilocal Convergence and Its Computational Efficiency of a Seventh-Order Method in Banach Spaces

  • Research Article
  • Published:
Proceedings of the National Academy of Sciences, India Section A: Physical Sciences Aims and scope Submit manuscript

Abstract

The present paper describes the study of semilocal convergence of seventh-order iterative method, in Banach spaces for solving nonlinear equations. The existence and uniqueness results have been verified followed by error bounds. It is also discussed that the considered scheme has not only the higher convergence order but also improved computational efficiency, which is the significant issue when dealing the nonlinear system problems. The theoretical development has been justified by applying it to a numerical problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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

References

  1. Kantorovich LV, Akilov GP (1982) Functional analysis. Pergamon Press, Oxford

    MATH  Google Scholar 

  2. Rall LB (1979) Computational solution of nonlinear operator equations. Robert E Krieger, New York

    MATH  Google Scholar 

  3. Amat S, Bermúdez C, Busquier S, Plaza S (2010) On a third-order Newton-type method free of bilinear operators. Numer Linear Algebra Appl 17:639–653

    MathSciNet  MATH  Google Scholar 

  4. Amat S, Busquier S, Gutiérrez JM (2011) Third-order iterative methods with applications to Hammerstein equations: a unified approach. J Comput Appl Math 235:2936–2943

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Ezquerro JA, Grau-Sánchez M, Grau A, Hernández MA, Noguera M, Romero N (2011) On iterative methods with accelerated convergence for solving systems of nonlinear equations. J Optim Theory Appl 151:163–174

    Article  MathSciNet  Google Scholar 

  8. Gutiérrez JM, Hernández MA (1998) Recurrence relations for the super-Halley method. Comput Math Appl 36:1–8

    Article  MathSciNet  Google Scholar 

  9. Hernández MA, Romero N (2007) General study of iterative processes of R-order at least three under convergence conditions. J Optim Theory Appl 133:163–177

    Article  MathSciNet  Google Scholar 

  10. Parida PK, Gupta DK (2008) Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces. J Math Anal Appl 345:350–361

    Article  MathSciNet  Google Scholar 

  11. Ezquerro JA, Hernández MA (2009) Fourth-order iterations for solving Hammerstein integral equations. Appl Numer Math 59:1149–1158

    Article  MathSciNet  Google Scholar 

  12. Ezquerro JA, Gutiérrez JM, Hernández MA, Salanova MA (1998) The application of an inverse-free Jarratt-type approximation to nonlinear integral equations of Hammerstein-type. Comput Math Appl 36:9–20

    Article  MathSciNet  Google Scholar 

  13. Hernández MA, Salanova MA (1999) Sufficient conditions for semilocal convergence of a fourth-order multipoint iterative method for solving equations in Banach spaces. Southwest J Pure Appl Math 1:29–40

    MathSciNet  MATH  Google Scholar 

  14. Wang X, Gu C, Kou J (2011) Semilocal convergence of a multipoint fourth-order super Halley method in Banach spaces. Numer Algorithms 56:497–516

    Article  MathSciNet  Google Scholar 

  15. Zheng L, Zhang K, Chen L (2015) On the convergence of a modified Chebyshev-like’s method for solving nonlinear equations. Taiwan J Math 19:193–209

    Article  MathSciNet  Google Scholar 

  16. Cordero A, Hernández MA, Romero N, Torregrosa JR (2015) Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces. J Comput Appl Math 273:205–213

    Article  MathSciNet  Google Scholar 

  17. Chen L, Gu C, Ma Y (2011) Semilocal convergence for a fifth-order Newton’s method using recurrence relations in Banach spaces. J Appl Math 2011:15

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  21. Ostrowski AM (1966) Solutions of equations and systems of equations. Academic Press, New York

    MATH  Google Scholar 

  22. Traub JF (1982) Iterative methods for the solution of equations. Chelsea Publishing Company, New York

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Gautschi W (1997) Numerical analysis: an introduction. Birkhäuser, Boston

    MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jai Prakash Jaiswal.

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 and Its Computational Efficiency of a Seventh-Order Method in Banach Spaces. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 90, 271–279 (2020). https://doi.org/10.1007/s40010-018-0590-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40010-018-0590-7

Keywords

Mathematics Subject Classification

Navigation