Nondiscrete induction and an inversion-free modification of Newton's method
Florian-Alexandru Potra, Vlastimil Pták (1983)
Časopis pro pěstování matematiky
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Florian-Alexandru Potra, Vlastimil Pták (1983)
Časopis pro pěstování matematiky
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Miel, George J. (1981)
International Journal of Mathematics and Mathematical Sciences
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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...
Florian-Alexandru Potra (1981)
Aplikace matematiky
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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, 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...
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. ...