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Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces

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Abstract

Semilocal convergence for an iteration of order five for solving nonlinear equations in Banach spaces is established under second-order Fréchet derivative satisfying the Lipschitz condition. It is done by deriving a number of recurrence relations. A theorem for the existence-uniqueness along with the estimation of error bounds of the solution is established. Its R-order is shown to be equal to five. Both efficiency and computational efficiency indices are given. A variety of examples are worked out to show its applicability. In comparison to existing methods having similar R-orders, improved results in terms of computational efficiency index and error bounds are found using our methodology.

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References

  1. Cordero A., Hueso J.L., Martinez E., Torregrosa J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, L., Gu, C., Ma Y.: Semilocal convergence for a fifth order Newton’s method using Recurrence relations in Banach spaces. J. Appl. Math. 2011, 1–15 (2011)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Zheng L., Gu C.: Recurrence relations for semilocal convergence of a fifth order method in Banach spaces. Numer. Algorithms 59, 623–638 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Proinov P.D., Ivanov S.I.: On the convergence of Halley’s method for multiple polynomial zeros. Mediterr. J. Math. 12, 555–572 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ezquerro, J.A., Hernández-Verón M.A.: On the domain of starting points of Newton’s method under center lipschitz conditions. Mediterr. J. Math. (2015). doi:10.1007/s00009-015-0596-1

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Parida P.K., Gupta D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206, 873–887 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hueso J.L., Martínez E.: Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms 67, 365–384 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Argyros, I.K., Hilout S.: Numerical methods in nonlinear analysis. World Scientific Publ. Comp., New Jersey (2013)

  12. Argyros, I.K., Hilout, S., Tabatabai, M.A.: Mathematical modelling with applications in biosciences and engineering. Nova Publishers, New York (2011)

  13. Argyros I.K., Khattri S.K.: Local convergence for a family of third order methods in Banach spaces. J. Math. 46, 53–62 (2004)

    MathSciNet  MATH  Google Scholar 

  14. Argyros I.K., Hilout A.S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kantorovich, L.V., Akilov G.P.: Functional analysis. Pergamon Press, Oxford (1982)

  16. Argyros I.K., George S., Magreñán A.A.: Local convergence for multi-point-parametric Chebyshev-Halley-type methods of higher convergence order. J. Comput. Appl. Math. 282, 215–224 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Argyros I.K., Magreñán A.A.: A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71, 1–23 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Amat S., Hernández M.A., Romero N.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Chun, C., St\({\breve{a}}\)nic\({\breve{a}}\), P., Neta, B.: Third-order family of methods in Banach spaces. Comput. Math. Appl. 61, 1665–1675 (2011)

  20. Ostrowski, A.M.: Solution of equations in Euclidean and Banach spaces, 3rd edn. Academic Press, New-York (1977)

  21. Jaiswal J.P.: Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency. Numer. Algorithms 71, 933–951 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)

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Authors and Affiliations

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

    Sukhjit Singh & D. K. Gupta

  2. Instituto de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, Spain

    E. Martínez

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

    José L. Hueso

Authors
  1. Sukhjit Singh
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  2. D. K. Gupta
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  3. E. Martínez
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  4. José L. Hueso
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Correspondence to Sukhjit Singh.

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Singh, S., Gupta, D.K., Martínez, E. et al. Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces. Mediterr. J. Math. 13, 4219–4235 (2016). https://doi.org/10.1007/s00009-016-0741-5

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  • DOI: https://doi.org/10.1007/s00009-016-0741-5

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