Skip to main content
SpringerLink
Log in

Analysis of semilocal convergence in banach spaces under relaxed condition and computational efficiency

  • Published:
Numerical Analysis and Applications Aims and scope Submit manuscript

Abstract

The current paper is concerned with the study of semilocal convergence of a fifth-order method for solving nonlinear equations in Banach spaces under mild condition. The existence and uniqueness theorem has been proved followed by the error estimates. The computational efficiency of the considered scheme over the identical order methods is also examined, which endorses the nobility of the present scheme from computational point of view. Lastly, application of theoretical development has been made in nonlinear integral equation.

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

    MATH  Google Scholar 

  2. Amat, S. and Busquier, S., Third-Order Iterative Methods under Kantorovich Conditions, J. Math. An. Appl., 2007, vol. 336, no. 1, pp. 243–261.

    Article  MathSciNet  MATH  Google Scholar 

  3. Parida, P.K. and Gupta, D.K., Recurrence Relations for a Newton-Like Method in Banach Spaces, J. Comput. Appl.Math., 2007, vol. 206, no. 2, pp. 873–887.

    Article  MathSciNet  MATH  Google Scholar 

  4. Hernández, M.A. and Salanova, M.A., Sufficient Conditions for Semilocal Convergence of a Fourth-Order Multipoint IterativeMethod for Solving Equations in Banach Spaces, Southwest J. Pure Appl.Math., 1999, vol. 1, pp. 29–40.

    MATH  Google Scholar 

  5. Zheng, L. and Gu, C., Fourth-Order Convergence TheorembyUsingMajorizing Functions for Super-Halley Method in Banach Spaces, Int. J. Comp. Math., 2013, vol. 90, pp. 423–434.

    Article  MATH  Google Scholar 

  6. Cordero, A., Hernández-Verón, M.A., Romero, N., and Torregrosa, S., Semilocal Convergence by Using Recurrence Relations for Fifth-Order Method in Banach Spaces, J. Comput. Appl. Math., 2015, vol. 273, pp. 205–213.

    Article  MathSciNet  MATH  Google Scholar 

  7. Cordero, A., Ezquerro, J.A., Hernández-Verón, M.A., and Torregrosa, S., On the Local Convergence of a Fifth-Order IterativeMethod in Banach Spaces, J. Appl. Math. Comput., 2015, vol. 251, pp. 396–403.

    Article  MATH  Google Scholar 

  8. Singh, S., Gupta, D.K., Martínez, E., and Hueso, J.L., Semilocal and Local Convergence of a Fifth-Order Iteration with Fréchet Derivative Satisfying Hölder Condition, J. Appl. Math. Comput., 2016, vol. 276, pp. 266–277.

    Article  Google Scholar 

  9. Ganesh, M. and Joshi, M.C., Numerical Solvability of Hammerstein Integral Equations ofMixed Type, IMA. J. Num. An., 1991, vol. 11, pp. 21–31.

    Article  MATH  Google Scholar 

  10. Ezquerro, J.A. and Hernández, M.A., On the R-Order of the Halley Method, J. Math. An. Appl., 2005, vol. 303, no. 2, pp. 591–601.

    Article  Google Scholar 

  11. Ezquerro, J.A. and Hernández, M.A., New Iterations of R-Order Four with Reduced Computational Cost, BIT Num. Math., 2009, vol. 49, no. 2, pp. 325–342.

    Article  MathSciNet  MATH  Google Scholar 

  12. Bruns, D.D. and Bailey, J.E., Nonlinear Feedback Control for Operating a Nonisothermal CSTR near an Unstable Steady State, Chem. Eng. Sci., 1977, vol. 32, no. 3, pp. 257–264.

    Article  Google Scholar 

  13. Argyros, I.K., Remarks on the Convergence of Newton’s Method under Hölder Continuity Conditions, Tamkang J.Math., 1992, vol. 23, pp. 269–277.

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. Traub, J.F., Iterative Methods for the Solution of Equations, New York: Chelsea Publ., 1982.

    MATH  Google Scholar 

  17. Grau-Sánchez, M., Grau, Á., and Noguera, M., On the Computational Efficiency Index and Some Iterative Methods for Solving Systems of Nonlinear Equations, J. Comput. Appl. Math., 2011, vol. 236, pp. 1259–1266.

    Article  MathSciNet  MATH  Google Scholar 

  18. Gautschi, W., Numerical Analysis: An introduction, Boston: Birkhäuser, 1997.

    MATH  Google Scholar 

  19. Sharma, J.R. and Gupta, P., An Efficient Fifth-Order Method for Solving Systems of Nonlinear Equations, J. Comput. Math. Appl., 2014, vol. 67, no. 3, pp. 591–601.

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, L., Gu, C., and Ma, Y., Semilocal Convergence for a Fifth-Order Newton’s Method Using Recurrence Relations in Banach Spaces, J. Appl. Math., 2011, vol. 2011, art. ID 786306.

  21. Hernández-Verón, M.A. and Martínez, E., On the Semilocal Convergence of a Three Steps Newton-Type Iterative Process underMild Convergence Conditions, Num. Algor., 2015, vol. 70, no. 2, pp. 377–392.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    J. P. Jaiswal

  2. Faculty of Science, Barkatullah University, Bhopal, M.P., 462026, India

    J. P. Jaiswal

  3. Regional Institute of Education, Bhopal, M.P., 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.

Additional information

Original Russian Text © J.P. Jaiswal, 2017, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2017, Vol. 20, No. 2, pp. 157–168.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jaiswal, J.P. Analysis of semilocal convergence in banach spaces under relaxed condition and computational efficiency. Numer. Analys. Appl. 10, 129–139 (2017). https://doi.org/10.1134/S1995423917020045

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995423917020045

Keywords

Search

Navigation