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.
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Wang, X., Shi, D. & Kou, J. Convergence analysis for a family of improved super-Halley methods under general convergence condition. Numer Algor 65, 339–354 (2014). https://doi.org/10.1007/s11075-013-9708-9
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DOI: https://doi.org/10.1007/s11075-013-9708-9
Keywords
- Recurrence relations
- Semilocal convergence
- Nonlinear equations in Banach spaces
- Super-Halley method
- R-order of convergence