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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Argyros, Ioannis K. (1996)
Southwest Journal of Pure and Applied Mathematics [electronic only]
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Ioannis K. Argyros (2005)
Applicationes Mathematicae
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The Newton-Kantorovich hypothesis (15) has been used for a long time as a sufficient condition for convergence of Newton's method to a locally unique solution of a nonlinear equation in a Banach space setting. Recently in [3], [4] we showed that this hypothesis can always be replaced by a condition weaker in general (see (18), (19) or (20)) whose verification requires the same computational cost. Moreover, finer error bounds and at least as precise information on the location of the...
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 (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...
Argyros, Ioannis K. (1995)
Southwest Journal of Pure and Applied Mathematics [electronic only]
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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, 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. (2003)
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...
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.
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,...
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, 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.
Argyros, Ioannis K. (2003)
Southwest Journal of Pure and Applied Mathematics [electronic only]
Similarity:
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.