On the convergence of perturbed Newton-like methods in Banach space and applications.
Argyros, Ioannis K. (1996)
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.I. (1998)
Southwest Journal of Pure and Applied Mathematics [electronic only]
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Ioannis K. Argyros, Hongmin Ren (2012)
Applicationes Mathematicae
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We provide a semilocal convergence analysis for Halley's method using convex majorants in order to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. Our results reduce and improve earlier ones in special cases.
Argyros, Ioannis K. (1995)
Southwest Journal of Pure and Applied Mathematics [electronic only]
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Ioannis K. Argyros, Hongmin Ren (2012)
Applicationes Mathematicae
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We present ball convergence results for Newton's method in order to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. Our hypotheses involve very general majorants on the Fréchet derivatives of the operators involved. In the special case of convex majorants our results, compared with earlier ones, have at least as large radius of convergence, no less tight error bounds on the distances involved, and no less precise information on the uniqueness...
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...
Argyros, Ioannis K. (2003)
Southwest Journal of Pure and Applied Mathematics [electronic only]
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Argyros, Ioannis K. (2001)
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...
Tetsuro Yamamoto (1987)
Numerische Mathematik
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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, 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.
Argyros, Ioannis K. (1998)
Southwest Journal of Pure and Applied Mathematics [electronic only]
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Ioannis K. Argyros (2009)
Applicationes Mathematicae
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We provide a local as well as a semilocal convergence analysis for Newton's method to approximate a locally unique solution of an equation in a Banach space setting. Using a combination of center-gamma with a gamma-condition, we obtain an upper bound on the inverses of the operators involved which can be more precise than those given in the elegant works by Smale, Wang, and Zhao and Wang. This observation leads (under the same or less computational cost) to a convergence analysis with...
Ioannis K. Argyros, Santhosh George (2015)
Applicationes Mathematicae
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We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations. Using more precise majorant conditions than in earlier studies, we provide: a larger radius of convergence; tighter error estimates on the distances involved; and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.
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.