### A unifying semilocal convergence theorem for Newton-like methods in Banach space.

Argyros, Ioannis K.I. (1998)

Southwest Journal of Pure and Applied Mathematics [electronic only]

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Argyros, Ioannis K.I. (1998)

Southwest Journal of Pure and Applied Mathematics [electronic only]

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Ioannis K. Argyros (2002)

Applicationes Mathematicae

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We present a local and a semilocal analysis for Newton-like methods in a Banach space. Our hypotheses on the operators involved are very general. It turns out that by choosing special cases for the "majorizing" functions we obtain all previous results in the literature, but not vice versa. Since our results give a deeper insight into the structure of the functions involved, we can obtain semilocal convergence under weaker conditions and in the case of local convergence a larger convergence...

Argyros, Ioannis K. (2003)

Southwest Journal of Pure and Applied Mathematics [electronic only]

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Argyros, Ioannis K. (1998)

Southwest Journal of Pure and Applied Mathematics [electronic only]

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Argyros, Ioannis K. (1996)

Southwest Journal of Pure and Applied Mathematics [electronic only]

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Argyros, Ioannis K. (1995)

Southwest Journal of Pure and Applied Mathematics [electronic only]

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Ioannis K. Argyros (2002)

Applicationes Mathematicae

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We provide local convergence theorems for the convergence of Newton's method to a solution of an equation in a Banach space utilizing only information at one point. It turns out that for analytic operators the convergence radius for Newton's method is enlarged compared with earlier results. A numerical example is also provided that compares our results favorably with earlier ones.

Tetsuro Yamamoto (1987)

Numerische Mathematik

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Ioannis K. Argyros, Sanjay K. Khattri (2013)

Applicationes Mathematicae

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We develop local and semilocal convergence results for Newton's method in order to solve nonlinear equations in a Banach space setting. The results compare favorably to earlier ones utilizing Lipschitz conditions on the second Fréchet derivative of the operators involved. Numerical examples where our new convergence conditions are satisfied but earlier convergence conditions are not satisfied are also reported.

Ioannis K. Argyros (2001)

Applicationes Mathematicae

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We provide new local and semilocal convergence results for Newton's method. We introduce Lipschitz-type hypotheses on the mth-Frechet derivative. This way we manage to enlarge the radius of convergence of Newton's method. Numerical examples are also provided to show that our results guarantee convergence where others do not.

Ioannis K. Argyros, Saïd Hilout (2011)

Applicationes Mathematicae

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We provide a new semilocal result for the quadratic convergence of Newton's method under ω*-conditioned second Fréchet derivative on a Banach space. This way we can handle equations where the usual Lipschitz-type conditions are not verifiable. An application involving nonlinear integral equations and two boundary value problems is provided. It turns out that a similar result using ω-conditioned hypotheses can provide usable error estimates indicating only linear convergence for Newton's...

Ioannis K. Argyros (2006)

Applicationes Mathematicae

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The Newton-Mysovskikh theorem provides sufficient conditions for the semilocal convergence of Newton's method to a locally unique solution of an equation in a Banach space setting. It turns out that under weaker hypotheses and a more precise error analysis than before, weaker sufficient conditions can be obtained for the local as well as semilocal convergence of Newton's method. Error bounds on the distances involved as well as a larger radius of convergence are obtained. Some numerical...

Argyros, Ioannis K. (2001)

Southwest Journal of Pure and Applied Mathematics [electronic only]

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Ioannis K. Argyros, Saïd Hilout (2009)

Applicationes Mathematicae

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We introduce a new idea of recurrent functions to provide a new semilocal convergence analysis for two-step Newton-type methods of high efficiency index. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in many interesting cases. Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar type, and a differential equation containing a Green's kernel are also provided. ...

Ioannis K. Argyros, Saïd Hilout (2013)

Applicationes Mathematicae

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We use a two-point Newton-like method to approximate a locally unique solution of a nonlinear equation containing a non-differentiable term in a Banach space setting. Using more precise majorizing sequences than in earlier studies, we present a tighter semi-local and local convergence analysis and weaker convergence criteria. This way we expand the applicability of these methods. Numerical examples are provided where the old convergence criteria do not hold but the new convergence criteria...

Ioannis K. Argyros (2007)

Applicationes Mathematicae

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We answer a question posed by Cianciaruso and De Pascale: What is the exact size of the gap between the semilocal convergence domains of the Newton and the modified Newton method? In particular, is it possible to close it? Our answer is yes in some cases. Using some ideas of ours and more precise error estimates we provide a semilocal convergence analysis for both methods with the following advantages over earlier approaches: weaker hypotheses; finer error bounds on the distances involved,...