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In this book the authors use a technique based on recurrence
relations to study the convergence of the Newton method under mild
differentiability conditions on the first derivative of the
operator involved. The authors' technique relies on the
construction of a scalar sequence, not majorizing, that satisfies a
system of recurrence relations, and guarantees the convergence of
the method. The application is user-friendly and has certain
advantages over Kantorovich's majorant principle. First, it allows
generalizations to be made of the results obtained under conditions
of Newton-Kantorovich type and, second, it improves the results
obtained through majorizing sequences. In addition, the authors
extend the application of Newton's method in Banach spaces from the
modification of the domain of starting points. As a result, the
scope of Kantorovich's theory for Newton's method is substantially
broadened. Moreover, this technique can be applied to any iterative
method. This book is chiefly intended for researchers and
(postgraduate) students working on nonlinear equations, as well as
scientists in general with an interest in numerical analysis.
This book shows the importance of studying semilocal convergence in
iterative methods through Newton's method and addresses the most
important aspects of the Kantorovich's theory including implicated
studies. Kantorovich's theory for Newton's method used techniques
of functional analysis to prove the semilocal convergence of the
method by means of the well-known majorant principle. To gain a
deeper understanding of these techniques the authors return to the
beginning and present a deep-detailed approach of Kantorovich's
theory for Newton's method, where they include old results, for a
historical perspective and for comparisons with new results, refine
old results, and prove their most relevant results, where
alternative approaches leading to new sufficient semilocal
convergence criteria for Newton's method are given. The book
contains many numerical examples involving nonlinear integral
equations, two boundary value problems and systems of nonlinear
equations related to numerous physical phenomena. The book is
addressed to researchers in computational sciences, in general, and
in approximation of solutions of nonlinear problems, in particular.
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