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Extending the convergence domain of Newton’s method for twice Fréchet differentiable operators

Ioannis K. Argyros

Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA

E-mail Address: iargyros@cameron.edu

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and

Á. Alberto Magreñán

Departamento de TFG/TFM, Universidad Internacional de La Rioja, 26002 Logroño, La Rioja, Spain

E-mail Address: angel.magrenan@unir.net

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https://doi.org/10.1142/S0219530515500013Cited by:0 (Source: Crossref)

Abstract

We present a semi-local convergence analysis of Newton’s method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Using center-Lipschitz condition on the first and the second Fréchet derivatives, we provide under the same computational cost a new and more precise convergence analysis than in earlier studies by Huang [A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math.47 (1993) 211–217] and Gutiérrez [A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math.79 (1997) 131–145]. Numerical examples where the old convergence criteria cannot apply to solve nonlinear equations but the new convergence criteria are satisfied are also presented at the concluding section of this paper.

Keywords:

AMSC: 47H17, 49M15, 65G99, 65B05, 65H10

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