Skip to main content
SpringerLink
Log in

Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators

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

Abstract

In this paper, the convergence and dynamics of improved Chebyshev-Secant-type iterative methods are studied for solving nonlinear equations in Banach space settings. Their semilocal convergence is established using recurrence relations under weaker continuity conditions on first-order divided differences. Convergence theorems are established for the existence-uniqueness of the solutions. Next, center-Lipschitz condition is defined on the first-order divided differences and its influence on the domain of starting iterates is compared with those corresponding to the domain of Lipschitz conditions. Several numerical examples including Automotive Steering problems and nonlinear mixed Hammerstein-type integral equations are analyzed, and the output results are compared with those obtained by some of similar existing iterative methods. It is found that improved results are obtained for all the numerical examples. Further, the dynamical analysis of the iterative method is carried out. It confirms that the proposed iterative method has better stability properties than its competitors.

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

Similar content being viewed by others

References

  1. Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41(3-4), 433–445 (2001)

    Article  MathSciNet  Google Scholar 

  2. Ezquerro, J.A., Grau-Sánchez, Miquel, Hernández, M.A.: Solving non-differentiable equations by a new one-point iterative method with memory. J. Complex. 28(1), 48–58 (2012)

    Article  MathSciNet  Google Scholar 

  3. Ioannis , K.A., Ezquerro, J.A., Gutiérrez, J.M., hernández, M.A., saïd Hilout: On the semilocal convergence of efficient Chebyshev-Secant-type methods. J. Comput. Appl. Math. 235(10), 3195–3206 (2011)

    Article  MathSciNet  Google Scholar 

  4. Hongmin, R., Ioannis, K.A.: Local convergence of efficient Secant-type methods for solving nonlinear equations. Appl. Math. comput. 218(14), 7655–7664 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Ioannis, Ioannis K.A., Hongmin, R.: On the semilocal convergence of derivative free methods for solving nonlinear equations. J. Numer. Anal. Approx. Theory 41 (1), 3–17 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Hongmin, R., Ioannis, K.A.: On the convergence of King-Werner-type methods of order \(1+\sqrt {2}\) free of derivatives. Appl. Math. Comput. 256, 148–159 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Kumar, A., Gupta, D.K., Martínez, E., Sukhjit, S.: Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces. J. Comput. Appl. Math. 330, 732–741 (2018)

    Article  MathSciNet  Google Scholar 

  8. Grau-Sánchez, M., Noguera, M., Gutiérrez, J.M.: Frozen iterative methods using divided differences “à la Schmidt–Schwetlick”. J. Optim. Theory Appl. 160 (3), 931–948 (2014)

    Article  MathSciNet  Google Scholar 

  9. Louis, B.R.: Computational Solution of Nonlinear Operator Equations. Wiley, New York (1969)

    Google Scholar 

  10. Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)

    Article  MathSciNet  Google Scholar 

  11. Parisa, B., Cordero, A., Taher, L., Kathayoun, M., Torregrosa, J.R.: Widening basins of attraction of optimal iterative methods. Nonlinear Dynamics 87 (2), 913–938 (2017)

    Article  MathSciNet  Google Scholar 

  12. Chun, C., Neta, B.: The basins of attraction of Murakami’s fifth order family of methods. Appl. Numer. Math. 110, 14–25 (2016)

    Article  MathSciNet  Google Scholar 

  13. Magreñán, Á. A.: A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Ramandeep, B., Cordero, A., Motsa, S.S., Torregrosa, J.R.: Stable high-order iterative methods for solving nonlinear models. Appl. Math. Comput. 303, 70–88 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Pramanik, S.: Kinematic synthesis of a six-member mechanism for automotive steering. Trans Ame Soc. Mech. Eng. J. Mech. Des. 124(4), 642–645 (2002)

    Google Scholar 

Download references

Funding

This research was partially supported by Ministerio de Economí a y Competitividad under grant PGC2018-095896-B-C22.

Author information

Authors and Affiliations

  1. Department of Mathematics, Lalit Narayan Mithila University, Darbhanga, 846004, India

    Abhimanyua Kumar

  2. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    D. K. Gupta

  3. Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, València, Spain

    Eulalia Martínez & José L. Hueso

Authors
  1. Abhimanyua Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. K. Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eulalia Martínez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. José L. Hueso
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abhimanyua Kumar.

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

Kumar, A., Gupta, D.K., Martínez, E. et al. Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators. Numer Algor 86, 1051–1070 (2021). https://doi.org/10.1007/s11075-020-00922-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-00922-9

Keywords

Mathematics subject classification (2010)

Search

Navigation