Iterative methods without inversion (Q5891013)

The book deals with iterative methods for solving the operator equation \({\mathsf f}(x) = 0\) in Banach spaces; the methods under consideration do not require inversion of exact or approximate ``linearization'' of the original equation. Convergence analyses of these methods are based on the classical Kantorovich majorization principle. However, the corresponding analysis is realized under the assumption about regular smoothness of the derivative \({\mathsf f}'(x)\) or its approximations. The condition of the regular smoothness was introduced by A. Galperin and Z. Waksman in the eighties of the XX century and allowed the authors to obtain a series of new and interesting results about the convergence for the classical Newton-Kantorovich method. The results of this book demonstrate that the condition of the regular smoothness is successfully applied also for the analysis of iterative methods without inversion.NEWLINENEWLINEThe book consists of an introduction, 7 chapters, references, and index. Each (exclusive the first) chapter starts with a motivation and is completed with research projects. Chapter 1 \textsc{Tools of the trade} is auxiliary; it presents Banach's lemma on perturbation, the Sherman-Morrison formula, generalized inverses in Hilbert spaces, and so on; all these facts are useful in the basic chapters. Chapter 2 \textsc{Ulm's method} describes Ulm's method with the scheme NEWLINE\[NEWLINEx_+: = x - {\mathsf A}{\mathsf f}(x), \qquad {\mathsf A}_+: = 2{\mathsf A} - {\mathsf A}{\mathsf f}'(x_+){\mathsf A}NEWLINE\]NEWLINE and, further, presents, under the condition of the regular smoothness, theorems about the convergence of this method, the rate of the convergence, and a posteriori error bounds. The condition of the regular smoothness is used in the form NEWLINE\[NEWLINE\omega^{-1}(\min\, \{\|{\mathsf f}'(x)\|,\|{\mathsf f}'(x')\|\} - \underline{h} + \|{\mathsf f}'(x') - {\mathsf f}'(x)\|) - \omega^{-1}(\min\, \{\|{\mathsf f}'(x)\|,\|{\mathsf f}'(x')\|\} - \underline{h}) \leq \|x' - x\|NEWLINE\]NEWLINE NEWLINE\[NEWLINE(0< \underline{h} \leq \underline{h}({\mathsf f}'), \quad \underline{h}({\mathsf f}'): = \inf_{x \in D}\;\|{\mathsf f}'(x)\|).NEWLINE\]NEWLINE As application, the Chandrasekhar integral equation is considered. Chapter 3 \textsc{Ulm's method without deri\-vatives} repeats arguments of Chapter 3 for the case when \({\mathsf f}(x)\) is not differentiable; of course, the derivative \({\mathsf f}'(x)\) in this chapter is changed into the appropriate difference operator. Chapter 4 \textsc{Broyden's method} deals with Broyden's approximations that are described by the scheme NEWLINE\[NEWLINEx_+: = x - {\mathsf A}{\mathsf f}(x), \qquad {\mathsf A}_+: = \frac{{\mathsf A}{\mathsf f}(x_+)}{\langle {\mathsf A}^*{\mathsf A}{\mathsf f}(x),{\mathsf f}(x_+) - {\mathsf f}(x),\rangle} \, \langle {\mathsf A}^*{\mathsf A}{\mathsf f}(x),\cdot \rangle.NEWLINE\]NEWLINE In the chapter, theorems about the convergence of the method are also obtained, the rate of this convergence, and so on. The applications to the complementarity problem and functional and integral equations are presented. Chapter 5 \textsc{Optimal secant updates of low rank} deals with iterative schemes of the type NEWLINE\[NEWLINEx_+ := x - {\mathsf A}{\mathsf f}(x), \quad {\mathsf A}_+ := {\mathsf A} + {\mathsf B}NEWLINE\]NEWLINE where the update \({\mathsf B}\) is a linear operator of a finite rank (most often \(1\) or \(2\)) such that \({\mathsf A}_+\) is invertible and satisfies the secant equation \({\mathsf A}^{-1}(x_+ - x) = {\mathsf f}(x_+) - {\mathsf f}(x)\). In particular, the modified Newton-Kantorovich, the modified secant method are studied. Chapter 6 \textsc{Optimal secant-type methods} is devoted to secant-type methods, which are described by the scheme NEWLINE\[NEWLINEz_+ = F(z,{\mathsf f}(z))NEWLINE\]NEWLINE (Ulm's and Broyden's methods can be considered partial cases). The main part of this chapter is devoted to the scalar equations under different assumptions about continuity and smoothness of \({\mathsf f}(x)\). The last Chapter 7 \textsc{Majorant generators and their convergence domains} presents a curious analysis of the convergence for successive approximations, described by the scheme NEWLINE\[NEWLINEu_+ := F(u,v), \quad v_+ := G(u,v), \qquad u \in {\mathbb R}, \;v \in {\mathbb R}^m.NEWLINE\]NEWLINE Such a scheme (really, a difference equation) is generated, as a result of the use of Kantorovich's majorization technique, in the previous Chapters 2--6. Here the author presents some general results about the convergence domain of the scheme. \textsc{References} contains 59 items. \textsc{Index} covers the content of the book.NEWLINENEWLINEOn the whole, the book will be useful to all specialists (teachers and researchers) in analysis: classical, functional, and numerical.