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Book
Newton’s method: an updated approach of Kantorovich’s theory
Authors: ---
ISBN: 3319559761 3319559753 Year: 2017 Publisher: Cham : Springer International Publishing : Imprint: Birkhäuser,

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Abstract

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.

Convergence and applications of Newton-type iterations
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ISBN: 9780387727417 0387727418 9780387727431 1441924922 9786611491727 1281491721 0387727434 Year: 2008 Publisher: New York : Springer,

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Recent results in local convergence and semi-local convergence analysis constitute a natural framework for the theoretical study of iterative methods. This monograph provides a comprehensive study of both basic theory and new results in the area. Each chapter contains new theoretical results and important applications in engineering, modeling dynamic economic systems, input-output systems, optimization problems, and nonlinear and linear differential equations. Several classes of operators are considered, including operators without Lipschitz continuous derivatives, operators with high order derivatives, and analytic operators. Each section is self-contained. Examples are used to illustrate the theory and exercises are included at the end of each chapter. The book assumes a basic background in linear algebra and numerical functional analysis. Graduate students and researchers will find this book useful. It may be used as a self-study reference or as a supplementary text for an advanced course in numerical functional analysis.

Points fixes, zéros et la méthode de Newton
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ISSN: 1154483X ISBN: 3540376607 3540309950 Year: 2006 Volume: v. 54 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer,

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Cet ouvrage est consacré aux points fixes d'applications différentiables, aux zéros de systèmes non-linéaires et à la méthode de Newton. Il s'adresse à des étudiants de mastère ou préparant l'agrégation de mathématique et à des chercheurs confirmés. La première partie est consacrée à la méthode des approximations successives et confronte un point de vue «systèmes dynamiques» (théorèmes de Grobman-Hartman, de la variété stable) à des exemples issus de l'analyse numérique. La seconde partie de cet ouvrage expose la méthode de Newton et ses développements les plus récents (théorie alpha de Smale, systèmes sous ou sur-déterminés). Elle présente une nouvelle approche de ce sujet et un ensemble de résultats originaux publiés pour la première fois dans un ouvrage de langue française. This is an advanced text on fixed points, zeros of nonlinear systems and the Newton method. Its first part, devoted to fixed points, includes the Grobman-Hartman and the stable manifold theorems. The second part describes the Newton method from a modern point of view: Smale's alpha theory, underdetermined and overdetermined systems of equations. These results are illustrated by various examples from numerical analysis.

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