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http://dx.doi.org/10.4134/JKMS.2014.51.1.137

CONVERGENCE OF THE RELAXED NEWTON'S METHOD  

Argyros, Ioannis Konstantinos (Department of Mathematics Sciences Cameron University)
Gutierrez, Jose Manuel (Department of Mathematics and Computation University of La Rioja)
Magrenan, Angel Alberto (Department of Mathematics and Computation University of La Rioja)
Romero, Natalia (Department of Mathematics and Computation University of La Rioja)
Publication Information
Journal of the Korean Mathematical Society / v.51, no.1, 2014 , pp. 137-162 More about this Journal
Abstract
In this work we study the local and semilocal convergence of the relaxed Newton's method, that is Newton's method with a relaxation parameter 0 < ${\lambda}$ < 2. We give a Kantorovich-like theorem that can be applied for operators defined between two Banach spaces. In fact, we obtain the recurrent sequence that majorizes the one given by the method and we characterize its convergence by a result that involves the relaxation parameter ${\lambda}$. We use a new technique that allows us on the one hand to generalize and on the other hand to extend the applicability of the result given initially by Kantorovich for ${\lambda}=1$.
Keywords
relaxed Newton's method; Banach space; Kantorovich hypothesis; majorizing sequence; local convergence; semilocal convergence;
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