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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.
List of contents
Operators and Equations.- The Newton Kantorovich (NK) Method.- Applications of the Weaker Version of the NK Theorem.- Special Methods.- Newton-like Methods.- Analytic Computational Complexity We Are Concerned with the Choice of Initial Approximations.- Variational Inequalities.- Convergence Involving Operators with Outer or Generalized Inverses.- Convergence on Generalized Banach Spaces: Improving Error Bounds and Weakening of Convergence Conditions.- Point to Set Mappings.- The Newton Kantorovich Theorem and Mathematical Programming.
Summary
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.
Additional text
From the reviews:
"The book is devoted to iterative methods of approximative solving nonlinear operator equations … . The main part of the book deals with the classical Newton-Kantorovich method … . Undoubtedly, it can be used for an advanced study of the Newton-Kantorovich method and other iterative methods of approximate solving nonlinear operator equations. The reviewer recommends this book to all who deal with nonlinear operator equations and their approximate solutions." (Peter Zabreiko, Zentrablatt MATH, Vol. 1153, 2009)
“This monograph deals with Netwon-Kantorovich (N-K) type methods for solving equations in Banach spaces, presenting fundamental results as well as many of the author’s own developments. It is addressed to graduate students and researchers with background in linear algebra and numerical functional analysis. … it also as ‘a reference book for an advanced numerical-functional analysis course’, one would expect more details on applications in order to illustrate how such powerful numerical tools can be used.” (Elena Resmerita, Mathematical Reviews, Issue 2010 c)
Report
From the reviews:
"The book is devoted to iterative methods of approximative solving nonlinear operator equations ... . The main part of the book deals with the classical Newton-Kantorovich method ... . Undoubtedly, it can be used for an advanced study of the Newton-Kantorovich method and other iterative methods of approximate solving nonlinear operator equations. The reviewer recommends this book to all who deal with nonlinear operator equations and their approximate solutions." (Peter Zabreiko, Zentrablatt MATH, Vol. 1153, 2009)
"This monograph deals with Netwon-Kantorovich (N-K) type methods for solving equations in Banach spaces, presenting fundamental results as well as many of the author's own developments. It is addressed to graduate students and researchers with background in linear algebra and numerical functional analysis. ... it also as 'a reference book for an advanced numerical-functional analysis course', one would expect more details on applications in order to illustrate how such powerful numerical tools can be used." (Elena Resmerita, Mathematical Reviews, Issue 2010 c)
