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Iterative solution of nonlinear equations in several variables. (English) Zbl 0949.65053
Classics in Applied Mathematics. 30. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xxvi, 572 p. (2000).
The book provides an excellent survey of theoretical results on finite-dimensional functional analysis. The Newton method for solving nonlinear systems surves as a basis where a variety of its modifications is presented, problems related to local and global convergence are studied, iterative methods for numerical solutions are considered.
The book includes five parts, subdivided in fourteen chapters, a bibliography involving about 850 papers and books, an Index and a Glossary of symbols. Each chapter is subdivided in sections, most of which are supplemented with “Notes and Remarks” giving some additional results to the problems discussed as well as a specified list of references. There are also exercises included which make the book very useful for graduate students and others working in this field.
Part I (Chapters 1 to 3) has an introductory character. It collects background material from analysis and linear algebra used in following parts. Chapter 1 presents several typical mathematical problems leading to solutions of nonlinear systems. Chapter 2 is devoted to linear algebra problems such as eigenvalue problems, matrix norms, invertibility of linear operators in \(\mathbb{R}^n\), partial order relations. Chapter 3 includes \(n\)-dimensional calculus, Gâteaux and Fréchet derivatives, convex functionals and their properties.
Part II (Chapters 4 to 6) presents various existence and uniqueness theorems with nonconstructive proofs. Chapter 4 studies gradient operators, rising from minimization problems as well as convex functionals. In Chapter 5, contraction mapping theorems for inverse and implicit function are proved and continuations of local homeomorphisms to global ones are considered. Chapter 6 is devoted to the theory of the degree of a mapping, which allows simple proofs of known fixed point theorems and is used to study monotone and coercive mappings.
Part III (Chapters 7 and 8) describes the most common iterative methods for solving systems of nonlinear equations. In Chapter 7, the Newton’s method, the secant and Steffensen methods as well as their variations are discussed. Iterative methods for linear systems are generalized to and applied to nonlinear problems. Continuation methods for the study of existence problems of operator equations are also discussed. Chapter 8 is devoted to minimization problems, including the gradient and conjugate gradient iterations.
Part IV (Chapters 9 to 11) discusses local convergence and convergence order of iterative processes. In Chapter 9 two different measures the \(Q\)- and \(R\)-factors of the asymptotic rate of convergence for sequences and iterative processes are introduced. Chapter 10 presents results on local convergence and rate of convergence for one-step stationary iterations; for the Newton method, for the \(m\)-step Newton-SOR and \(m\)-step SOR-Newton and some of the minimization methods from Chapter 8. Chapter 11 considers multistep iteration methods as well as one-step nonstationary iterations to which the techniques from previous chapter do not apply. The main examples are secant-type methods, modified Newton iterations and variable-step generalized linear iterations. Results on point of attraction and rate of convergence are derived as well.
Part V (Chapter 12 to 14) discusses problems on about existence of a solution \(x^\ast \) to \(Fx=0\) in the case of convergence of the iterations, about the choice of initial approximations for \(x^\ast\) and on estimations of the error after stopping the iterations. Chapter 12 is devoted to generalizations of contraction mapping theorems, approximate contractions and sequences of contractions, nonlinear majorants of contractions. The results are applied to Newton’s method. While Chapter 12 is based on the usage of norms in \(\mathbb{R}^n\), in Chapter 13 \(\mathbb{R}^n\) is considered to be a partially ordered linear space and all results show monotonic convergence of the iterates. The last Chapter 14 presents minimization methods for real functionals in \(\mathbb{R}^n\). Considered are gradient and gradient-related methods, Newton-type methods, conjugate direction methods, univariate relaxation processes.
The first edition of the book was published in 1970 (see Zbl 0241.65046). The present new edition is equipped with a Preface, where the authors indicate “some of the principle developments and changes” and give some recent books and papers related to topics considered.

65H10 Numerical computation of solutions to systems of equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
01A75 Collected or selected works; reprintings or translations of classics
47J25 Iterative procedures involving nonlinear operators
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