Skip to main content

Improved semilocal convergence analysis in Banach space with applications to chemistry

Abstract

We present a new semilocal convergence analysis for Secant methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost on the parameters involved our convergence criteria are weaker and the error bounds more precise than in earlier studies. A numerical example is also presented to illustrate the theoretical results obtained in this study.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. S. Amat, S. Busquier, On a higher order Secant method. Appl. Math. Comput. 141(2–3), 321–329 (2003)

    Google Scholar 

  2. S. Amat, S. Busquier, M. Negra, Adaptive approximation of nonlinear operators. Numer. Funct. Anal. Optim. 25, 397–405 (2004)

    Article  Google Scholar 

  3. S. Amat, J.A. Ezquerro, M.A. Hernández, Approximation of inverse operators by a new faily of high-order iterative methods. Numer. Linear Algebra Appl. https://doi.org/10.1002/nla.1917

  4. S. Amat, M.A. Hernández, M.J. Rubio, Improving the applicability of the Secant method to solve nonlinear systems of equations. To appear in Applied Mathematics and Computation

  5. I.K. Argyros, Computational theory of iterative methods, in Series: Studies in Computational Mathematics, vol. 15, ed. by C.K. Chui, L. Wuytack (Elsevier, New York, 2007)

    Google Scholar 

  6. I.K. Argyros, S. Hilout, Weaker conditions for the convergence of Newton’s method. J. Complex. AMS 28, 364–387 (2012)

    Article  Google Scholar 

  7. I.K. Argyros, Y.J. Cho, S. Hilout, Numerical Method for Equations and its Applications (CRC Press/Taylor and Francis, New York, 2012)

    Google Scholar 

  8. I.A. Argyros, D. González, A.A. Magreñán et al., A semilocal convergence for a uniparametric family of efficient Secant-like methods. J. Funct. Spaces 2014(467980), 10 (2014). https://doi.org/10.1155/2014/467980

    Article  Google Scholar 

  9. I.A. Argyros, Á.A. Magreñán, Iterative Methods and Their Dynamics with Applications: A Contemporary Study (CRC Press, Boca Raton, 2017)

    Book  Google Scholar 

  10. I.A. Argyros, D. González, Local convergence for an improved Jarratt-type method in Banach space. Int. J. Interact. Multimed. Artif. Intell. 3(4), 20–25 (2015)

    Google Scholar 

  11. E. Cătinaş, The inexact, inexact perturbed, and quasi-Newton methods are equivalent models. Math. Comp. 74(249), 291–301 (2005)

    Article  Google Scholar 

  12. S. Chandrasekhar, Radiative Transfer (Dover Publ, New York, 1960)

    Google Scholar 

  13. J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in Nonlinear Functional Analysis and Appications, ed. by L.B. Rall (Academic Press, New York, 1971), pp. 425–472

    Chapter  Google Scholar 

  14. J.A. Ezquerro, J.M. Hernández, M.J. Rubio et al., Secant-like methods for solving nonlinear integral equations of the Hammerstein type. J. Comput. Appl. Math. 115, 245–254 (2000)

    Article  Google Scholar 

  15. J.A. Ezquerro, J.M. Gutiérrez, M.A. Hernández, N. Romero, M.J. Rubio, The Newton method: from Newton to Kantorovich. Gac. R. Soc. Mat. Esp. 13(1), 53–76 (2010). (Spanish)

    Google Scholar 

  16. W.B. Gragg, R.A. Tapia, Optimal error bounds for the Newton–Kantorovich theorem. SIAM J. Numer. Anal. 11, 10–13 (1974)

    Article  Google Scholar 

  17. L.V. Kantorovich, G.P. Akilov, Functional Analysis (Pergamon Press, Oxford, 1982)

    Google Scholar 

  18. N. Lal, An effective approach for Mobile ad hoc Nextwork via I-Watchdog Protocol. Int. J. Interact. Multimed. Artif. Intell. 3(1), 36–43 (2014)

    Google Scholar 

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

    Google Scholar 

  20. L.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic press, New York, 1970)

    Google Scholar 

  21. F.A. Potra, V. Pták, Nondiscrete induction and iterative processes. Research Notes in Mathematics, 103. Pitman (Advanced Publishing Program), Boston, MA, (1984)

  22. P.D. Proinov, General local convergence theory for a class of iterative processes and its applications to Newton’s method. J. Complex 25, 38–62 (2009)

    Article  Google Scholar 

  23. W.C. Rheinboldt, An adaptative continuation process for solving systems of nonlinear equations. Banach ctz. Publ. 3, 129–142 (1975)

    Article  Google Scholar 

  24. J.F. Traub, Iterative Method for Solutions of Equations (Prentice-Hall, New Jersey, 1964)

    Google Scholar 

  25. J.W. Schmidt, Untere Fehlerschranken fun Regula-Falsi Verhafren. Period. Hungar. 9, 241–247 (1978)

    Article  Google Scholar 

  26. F. Silva, C. Analide, P. Novais, Assessing road traffic expression. Int. J. Artif. Intell. Interact. Multimed. 3(1), 20–27 (2014)

    Google Scholar 

  27. T. Yamamoto, A convergence theorem for Newton-like methods in Banach spaces. Numer. Math. 51, 545–557 (1987)

    Article  Google Scholar 

  28. P.P. Zabrejko, D.F. Nguen, The majorant method in the theory of Newton–Kantorovich approximations and the Pták error estimates. Numer. Funct. Anal. Optim. 9, 671–684 (1987)

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Plan Propio de Investigación, Desarrollo e Innovación [2015–2017]. Research group: Modelación matemática aplicada a la ingeniería(MOMAIN), by the the grant SENECA 19374/PI/14 and by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-\(\{01\}\)-P.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Á. A. Magreñán.

Additional information

This is one of several papers published together in Journal of Mathematical Chemistry on the “Special Issue: CMMSE 2017”.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Argyros, I.K., Giménez, E., Magreñán, Á.A. et al. Improved semilocal convergence analysis in Banach space with applications to chemistry. J Math Chem 56, 1958–1975 (2018). https://doi.org/10.1007/s10910-017-0823-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-017-0823-z

Keywords

  • Secant method
  • Banach space
  • Majorizing sequence
  • Divided difference
  • Local convergence
  • Semilocal convergence