Iterative Methods without Inversion presents the iterative methods for solving operator equations f(x) = 0 in Banach and/or Hilbert spaces. It covers methods that do not require inversions of f (or solving linearized subproblems). The typical representatives of the class of methods discussed are Ulm’s and Broyden’s methods. Convergence analyses of the methods considered are based on Kantorovich’s majorization principle which avoids unnecessary simplifying assumptions like differentiability of the operator or solvability of the equation. These analyses are carried out under a more general assumption about degree of continuity of the operator than traditional Lipschitz continuity: regular continuity.
- The methods discussed are analyzed under the assumption of regular continuity of divided difference operator, which is more general and more flexible than the traditional Lipschitz continuity.
- An attention is given to criterions for comparison of merits of various methods and to the related concept of optimality of a method of certain class.
- Many publications on methods for solving nonlinear operator equations discuss methods that involve inversion of linearization of the operator, which task is highly problematic in infinite dimensions.
- Accessible for anyone with minimal exposure to nonlinear functional analysis.
Table of Contents
Introduction. Some useful tools of the trade. Ulm’s method. Ulm’s method without derivatives. Broyden’s method. Optimal secantupdates
of low rank. Optimal secant-type methods. Majorant generators and their convergence domains. Bibliography
"The book is well organised and clearly written and presents a limited but illustrative number of computational examples that are intended to provide results that can be used to validate the reader's own implementations and to give a sense of how the algorithms perform. It will be accessible to anyone who has reasonable knowledge of basic nonlinear functional analysis.
Among many other positive features of the book, I especially appreciate the fact that Chapters 2-7 begin with a motivation, give numerical examples and end by stating research project(s), thus enabling and challenging interested readers to pursue further developments. The book will be very useful to graduate students and young researchers beginning their scientific careers in the field of computational mathematics and to anyone else interested in numerical analysis."
- Vasile Berinde, Mathematical Reviews, August 2017