Convergence Analysis of a Fifth-Order Iterative Method Using Recurrence Relations and Conditions on the First Derivative


Using conditions on the second Fréchet derivative, fifth order of convergence was established in Singh et al. (Mediterr J Math 13:4219–4235, 2016) for an iterative method. In this paper, we establish fifth order of convergence of the method using assumptions only on the first Fréchet derivative of the involved operator.

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  1. 1.

    Amat, S., Hernandez, 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  Article  Google Scholar 

  2. 2.

    Argyros, I.K., George, S.: Semilocal convergence analysis of a fifth-order method using recurrence relations in Banach space under weak conditions. Appl. Math. 45(2), 223–231 (2018)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Argyros, I.K., George, S.: Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications, Volume-III. Nova Publishes, New York (2019)

    Google Scholar 

  4. 4.

    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)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Argyros, I.K., Magreñán, A.A.: Iterative Methods and Their Dynamics with Applications. CRC Press, New York (2017)

    Google Scholar 

  6. 6.

    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)

    MathSciNet  Article  Google Scholar 

  7. 7.

    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)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Chun, C., Stanica, P., Neta, B.: Third-order family of methods in Banach spaces. Comput. Math. Appl. 61, 1665–1675 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

    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)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Cordero, A., Hernandez-Veron, 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)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Ezquerro, J.A., Hernandez-Veron, M.A.: On the domain of starting points of Newton’s method under center lipschitz conditions. Mediterr. J. Math. (2015).

    Article  MATH  Google Scholar 

  12. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

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

    Google Scholar 

  15. 15.

    Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)

    Google Scholar 

  16. 16.

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

    Google Scholar 

  17. 17.

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

    Google Scholar 

  18. 18.

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

    MathSciNet  Article  Google Scholar 

  19. 19.

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

    MathSciNet  Article  Google Scholar 

  20. 20.

    Rudin, W.: Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1976)

    Google Scholar 

  21. 21.

    Singh, S., Gupta, D.K., Martinez, E., Hueso, J.L.: Semilocal convergence Analysis of an iteration of order five using recurrence relations in Banach spaces. Mediterr. J. Math. 13, 4219–4235 (2016)

    MathSciNet  Article  Google Scholar 

  22. 21.

    Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)

    Google Scholar 

  23. 22.

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

    MathSciNet  Article  Google Scholar 

  24. 23.

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

    MathSciNet  Article  Google Scholar 

  25. 24.

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

    MathSciNet  Article  Google Scholar 

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The works of Santhosh George and Jidesh P are supported by the Core Research Grant by SERB, Department of Science and Technology, Govt. of India, EMR/2017/001594.

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George, S., Argyros, I.K., Jidesh, P. et al. Convergence Analysis of a Fifth-Order Iterative Method Using Recurrence Relations and Conditions on the First Derivative. Mediterr. J. Math. 18, 57 (2021).

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  • Fréchet derivative
  • Order of convergence
  • Iterative method
  • Banach space

Mathematics Subject Classification

  • 65G49
  • 47H99