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Convergence and applications of Newton-type iterations. (English) Zbl 1153.65057
New York, NY: Springer (ISBN 978-0-387-72741-7/hbk). xv, 506 p. (2008).
The book is devoted to iterative methods of approximative solving nonlinear operator equations in Banach and Hilbert spaces as well as equations on manifolds and in spaces with more complicated structure. The main part of the book deals with the classical Newton-Kantorovich method, however, the author gives much attention to different modifications of this method; in particular, to the secant method, Gauss-Newton method, Broyden method, Stirling method, Steffensen method, tangent hyperbolic method, King-Werner method and so on. In addition, some modifications of the Newton-Kantorovich method for operator equation with a parameter are considered. At last, the author discusses applications of iterative methods to solving variational inequalities, equations with multi-valued operators, also to optimization problems.
Here is the contents of this book: Introduction, 1. Operators and equations, 2. The Newton-Kantorovich (NK) method, 3. Applications of the weaker version of the NK theorem, 4. Special methods, 5. Newton-like methods, 6. Analytic computational complexity: We are concerned with the choice of initial approximations, 7. Variational inequalities, 8. Convergence involving operators with outer or generalized inverses, 9. Convergence on generalized Banach spaces: Improving error bounds and weakening of convergence conditions, 10. Point-to-set mappings, 11. The Newton-Kantorovich theorem and mathematical programming, References, Glossary of symbols, Index.
Many among the results presented in the book are illustrated with numerical examples. Each chapter ends with exercises. It must be admitted that the book is not a systematic analysis of the Newton-Kantorovich method and its modifications. I think that the aim of this book is to give the sufficiently complete presentation about numerous results obtained by the author and his collaborators. So, among 220 references cited in the book, 38 are written by the author and his coauthors. Moreover, in the book there are no references on fundamental articles by B. A. Vertgeim and I. P. Mysovskikh (the first proved the analogue of fundamental Kantorovich theorem in the case of Hölder continuous derivatives, the second proved the similar analogue in the case when the derivatives have uniformly bounded inverses).
There are other annoying omissions of such type. One can find other deficiencies in the book: the absence of a system for notations unified for a book as a whole (probably, the system of notations in each section is borrowed from the corresponding articles); in chapter 3 the author compares the Kantorovich theorem with closed theorems by Moore, Miranda and so on, however, he omits their exact formulations and, as a result, the part of the author’s arguments are vague; the logic of partitioning onto chapters of results collected in the book is incomprehensible (so, chapter 2, devoted to the Newton-Kantorovich method, contains sections about the weak Newton-Kantorovich method, the Gauss-Kantorovich method and so on, although in the book there are chapters 4, 5, and 8 devoted to modifications of the Newton-Kantorovich method and Newton-like methods).
In general, one can remark that the computational aspect of the book prevails its methodological one. However, it is easy to scold books but it is difficult to write them. However, despite these remarks, the book presents numerous results in Numerical Functional Analysis with their complete proofs that one can find earlier only in special mathematical journals. 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.

65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65K10 Numerical optimization and variational techniques
47J25 Iterative procedures involving nonlinear operators
49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H04 Set-valued operators
49M15 Newton-type methods
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