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Local convergence for some high convergence order Newton-like methods with frozen derivatives

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Abstract

We present a local convergence analysis of some families of Newton-like methods with frozen derivatives in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Amat et al. (Appl Math Lett. 25:2209–2217, 2012), Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam, 2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Xiao and Yin (Appl Math Comput, 2015) the local convergence was proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these methods. In this paper we expand the applicability of these methods using only hypotheses on the first derivative and Lipschitz constants. Numerical examples are also presented in this study.

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References

  1. Amat, S., Busquier, C., Bermudez, S.P.: On two families of high order Newton-type methods. Appl. Math. Lett. 25, 2209–2217 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Amat, S., Hernández, M.A., Romero, N.: Semilocal convergence of a sixth order iterative method for quadratic equations. Appl. Numer. Math. 62, 833–841 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Argyros, I.K.: Computational theory of iterative methods. In: Chui, C.K., Wuytack, L. (eds.) Series: Studies in Computational Mathematics, vol. 15. Elsevier Publ. Co., New York (2007)

  4. Argyros, I.K.: A semilocal convergence analysis for directional Newton methods. Math. Comput. 80, 327–343 (2011)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Argyros, I.K., Hilout, S.: Computational Methods in Nonlinear Analysis. Efficient Algorithms, Fixed Point Theory and Applications. World Scientific, Singapore (2013)

    Book  MATH  Google Scholar 

  7. Argyros, I.K., Ren, H.: Improved local analysis for certain class of iterative methods with cubic convergence. Numer. Algorithms 59, 505–521 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chun, C., Lee, M.Y., Neta, B., Dzunic, J.: On optimal fourth order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cordero, A., Torregrosa, J.R., Vasileva, M.P.: Increasing the order of convergence of iterative schemes for solving nonlinear systems. J. Comput. Appl. Math. 252, 86–94 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gutiérrez, J.M., Magren̄án, A.A., Romero, N.: On the semi-local convergence of Newton–Kantorovich method under center-Lipschitz conditions. Appl. Math. Comput. 221, 79–88 (2013)

    Article  MathSciNet  Google Scholar 

  11. Ezquerro, J.A., Gonzalez, D., Hernández, M.A.: A modification of the classic conditions of Newton–Kantorovich for Newtons method. Math. Comput. Model. 57, 584–594 (2013)

    Article  MATH  Google Scholar 

  12. Ezquerro, J.A., Gonzalez, D., Hernández, M.A.: A variant of the Newton–Kantorovich theorem for nonlinear integral equations of mixed Hammerstein type. Appl. Math. Comput. 218, 9536–9546 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hernández, M.A., Salanova, M.A.: Modification of the Kantorovich assumptions for semi-local convergence of the Chebyshev method. J. Comput. Appl. Math. 126, 131–143 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  15. Kou, J.S., Li, Y.T., Wang, X.H.: A modification of Newton method with third-order convergence. Appl. Math. Comput. 181, 1106–1111 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Magreñán, A.A.: Different anomalies in a Jarratt family of iterative root finding methods. Appl. Math. Comput. 233, 29–38 (2014)

    Article  MathSciNet  Google Scholar 

  17. Petkovic, M.S., Neta, B., Petkovic, L., Džunič, J.: Multipoint Methods for Solving Nonlinear Equations. Elsevier, Amsterdam (2013)

    MATH  Google Scholar 

  18. Potra, F.A., Pták, V.: Nondiscrete induction and iterative processes. In: Research Notes in Mathematics, vol. 103. Pitman, Boston (1984)

  19. Ren, H., Wu, Q.: Convergence ball and error analysis of a family of iterative methods with cubic convergence. Appl. Math. Comput. 209, 369–378 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rheinboldt, W.C.: An adaptive continuation process for solving systems of nonlinear equations. In: Tikhonov, A.N., et al. (eds.) Mathematical Models and Numerical Methods, pub. 3, pp. 129–142 Banach Center, Warsaw (1977)

  21. Traub, J.F.: Iterative Methods for the Solution of Equations. AMS Chelsea Publishing, Providence (1982)

    MATH  Google Scholar 

  22. Xiao, X., Yin, H.: A new class of methods with higher order of convergence for solving systems of nonlinear equations. Appl. Math. Comput. 264, 300–309 (2015)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

  2. Department of Mathematical and Computational Sciences, NIT Karnataka, Mangaluru, 575 025, India

    Santhosh George

Authors
  1. Ioannis K. Argyros
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  2. Santhosh George
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Correspondence to Ioannis K. Argyros.

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Argyros, I.K., George, S. Local convergence for some high convergence order Newton-like methods with frozen derivatives. SeMA 70, 47–59 (2015). https://doi.org/10.1007/s40324-015-0039-8

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  • DOI: https://doi.org/10.1007/s40324-015-0039-8

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