Error bounds for Newton's method under a weak Kantorovich-type hypothesis.
Argyros, Ioannis K. (2002)
Southwest Journal of Pure and Applied Mathematics [electronic only]
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Argyros, Ioannis K. (2002)
Southwest Journal of Pure and Applied Mathematics [electronic only]
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Tetsuro Yamamoto (1986)
Numerische Mathematik
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Marek Kuczma, Halina Światak (1991)
Annales Polonici Mathematici
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Ioannis K. Argyros (2005)
Applicationes Mathematicae
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The Newton-Kantorovich approach and the majorant principle are used to provide new local and semilocal convergence results for Newton-like methods using outer or generalized inverses in a Banach space setting. Using the same conditions as before, we provide more precise information on the location of the solution and on the error bounds on the distances involved. Moreover since our Newton-Kantorovich-type hypothesis is weaker than before, we can cover cases where the original Newton-Kantorovich...
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...
Rafał Ziobro (2015)
Formalized Mathematics
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Solving equations in integers is an important part of the number theory [29]. In many cases it can be conducted by the factorization of equation’s elements, such as the Newton’s binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books [28], [14]. In the second section of the article, Fermat’s Little Theorem is proved in a classical way, on the basis of divisibility of Newton’s binomial. Although slightly redundant...
Argyros, Ioannis K. (1997)
Southwest Journal of Pure and Applied Mathematics [electronic only]
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Sahari, M.L., Djellit, I. (2006)
Discrete Dynamics in Nature and Society
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Tetsuro Yamamoto (1986)
Numerische Mathematik
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Henry Brougham, Edward John Routh
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F.A. Potra, V. Pták (1980)
Numerische Mathematik
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Ioannis K. Argyros (2000)
Czechoslovak Mathematical Journal
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We provide local convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our convergence balls differ from earlier ones. In fact we show that with a simple numerical example that our convergence ball contains earlier ones. This way we have a wider choice of initial guesses than before. Our results can be used to solve undetermined systems,...