Convergence of Halley's method under centered Lipschitz condition on the second Fréchet derivative


  • Ioannis K. Argyros Cameron University
  • Hongmin Ren Hangzhou Polytechnic


Halley's method, Fréchet-derivative, Banach space, semi-local convergence, Frechet derivative, centered Lipschitz condition


We present a semi-local as well as a local convergence analysis of Halley's method for approximating a locally unique solution of a nonlinear equation in a Banach space setting.
We assume that the second Fréchet-derivative satisfies a centered Lipschitz condition.
Numerical examples are used to show that the new convergence criteria are satisfied but earlier ones are not satisfied.


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I.K. Argyros, The convergence of Halley-Chebyshev type method under Newton-Kantorovich hypotheses, Appl. Math. Lett., 6 (1993), pp. 71-74,

I.K. Argyros, Computational theory of iterative methods, Series: Studies in Computational Mathematics 15, Editors, C.K. Chui and L. Wuytack, Elservier Publ. Co. New York, USA, 2007.

I.K. Argyros, Y.J. Cho and S. Hilout, On the semilocal convergence of the Halley method using recurrent functions, J. Appl. Math. Computing., 37 (2011), pp. 221-246,

I.K. Argyros and H.M. Ren, Ball convergence theorems for Halley's method in Banach spaces, J. Appl. Math. Computing, 38 (2012), pp. 453-465,

I.K. Argyros and H.M. Ren, On the Halley method in Banach space, Applicationes Mathematicae, to appear 2012.

P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer-Verlag, Berlin, Heidelberg, 2004.

J.M. Gutiérrez and M.A. Hernández, Newton's method under weak Kantorovich conditions, IMA J. Numer. Anal., 20 (2000), pp. 521-532,

X.B. Xu and Y.H., Ling, Semilocal convergence for Halley's method under weak Lipschitz condition, Appl. Math. Comput., 215 (2009), pp. 3057-3067,




How to Cite

Argyros, I. K., & Ren, H. (2013). Convergence of Halley’s method under centered Lipschitz condition on the second Fréchet derivative. Rev. Anal. Numér. Théor. Approx., 42(1), 3–20. Retrieved from