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On the local convergence of a family of Euler-Halley type iterations with a parameter

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Abstract

This paper aims to study the local convergence of a family of Euler-Halley type methods with a parameter α for solving nonlinear operator equations under the second-order generalized Lipschitz assumption. The radius r α of the optimal convergence ball and the error estimation of the method corresponding to α are estimated for each α ∈ ( − ∞ , + ∞ ). For each α > 0, we get r α  ≥ r  − α and the upper bound of the error estimation of the method with α > 0 is not larger than the one with α < 0. For each α ≤ 0, we get the precise value of r α , which is closely linked to the dynamical property of the method applied to a real or a complex function, and the optimal error estimation, which decreases when α→0 − . Results show that the method corresponding to α is better than the one corresponding to − α for each α > 0 and the Chebyshev-Euler method is the best among all methods in the family with α ∈ ( − ∞ , 0] from the view of both safe choice of the initial point and error estimation.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Huang Zhengda & Ma Guochun

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  1. Huang Zhengda
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  2. Ma Guochun
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Correspondence to Huang Zhengda.

Additional information

This work was supported of NSFC(Grant No.10731060).

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Zhengda, H., Guochun, M. On the local convergence of a family of Euler-Halley type iterations with a parameter. Numer Algor 52, 419–433 (2009). https://doi.org/10.1007/s11075-009-9284-1

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  • DOI: https://doi.org/10.1007/s11075-009-9284-1

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