Convergence analysis for a family of improved super-Halley methods under general convergence condition

Authors:
Xiuhua Wang;Dongyang Shi;Jisheng Kou
Affiliations:
Department of Mathematics, Zhengzhou University, Zhengzhou, China 450052 and School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, China 432100;Department of Mathematics, Zhengzhou University, Zhengzhou, China 450052;School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, China 432100
Venue:
Numerical Algorithms
Year:
2014

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Abstract

In this paper, we focus on the semilocal convergence for a family of improved super-Halley methods for solving non-linear equations in Banach spaces. Different from the results in Wang et al. (J Optim Theory Appl 153:779---793, 2012), the condition of Hölder continuity of third-order Fréchet derivative is replaced by its general continuity condition, and the latter is weaker than former. Moreover, the R-order of the methods is also improved. By using the recurrence relations, we prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is analyzed with the third-order Fréchet derivative of the operator satisfies general continuity condition and Hölder continuity condition.