Topics in Validated Computations: Proceedings of Imacs-Gamm International Workshop on Validated Computations

This volume is the result of the IMACS-GAMM International Workshop on Validated Computations held at the University of Oldenburg (Germany) from August 30 to September 3, 1993. The intention of this workshop, in which 29 scientists from 10 countries took part, was to bring together experts in some selected areas of the application of interval arithmetic in the development of algorithms with result verification. Most of the talks delivered during this meeting were supposed to be survey talks or expository contributions on the fields represented by selected speakers. Among them were presented applications in numerical linear algebra, solving nonlinear equations, eigenvalue problems, ODE's and PDE's, and optimization. Since it is now 20 years ago that the monograph of G. Alefeld and J. Herzberger "Einfiihrung in die Intervallrechnung" appeared, the choice covered by this volume was according to the following idea: some of the papers should give an updated survey of how things have developed during the years after the appearance of the book - for example, the chapters on interval Newton methods and iterative methods of including the inverse of a matrix - but the main portion of the contributions should deal with areas of applications which were not included in the monograph, mainly because of lack of convincing results at that early stage of interval arithmetic. Just to mention a few of them: optimization, applications in PDE's and eigenvalue problems. In this sense, this volume is thought to be a kind of continuation of the old monograph. In order to make this volume easily readable, the notations are unified and introduced in a first introductory chapter with some basic facts about interval arithmetic. Here, the main ideas and properties of the interval operations are summarized with some notations differing from those chosen in the old monograph taking into account the development of the last two decades in interval arithmetic literature. Again in this volume, there exists a kind of appendix, although not explicitly announced as such. Firstly, there is a chapter with instructions how to implement the algorithms nowadays described in this volume by reviewing the enormous progress having been made in the design and creation of programming environments for this purpose (compared with the old-fashioned ALGOL 60 procedures in the monograph mentioned). Secondly, there is a short critical report on the complexity of some common problems in interval analysis where we want to calculate op- optimal solutions. Last, but not least, as a reference to the numerous practical applications of interval arithmetic, we want to mention the chapter on selected examples for solving nontrivial problems in engineering.
The purpose of this volume is to provide the interval analysis community with surveys of some important recent developments in the creation of validated numerical algorithms in the form of a book with non-overlapping chapters written by several experts in their fields. At the same time we want to inform the numerical analysts and the appliers of numerical software about the enormous variety of problem-solving algorithms now available, even for sophisticated problems which were out of reach at the beginning of research in this direction some two decades ago. This volume is self-contained and offers the interested reader applications m nearly all main branches of numerical analysis, and it can be used as an up-to-minate textbook for studying advanced interval algorithms. We want to express our sincere gratitude to the sponsors of the international workshop, Volkswagen-Stiftung, Ministerium fur Wissenschaft und Kultur des Landcs Niedersachsen and the Universitatsgesellschaft Oldenburg e.V., and to all participants for their encouraging discussions during the presentation of these papers.