Preface

Summary

The present book gives a presentation of the interval arithmetical tools and methods for the solution of linear and nonlinear systems of equations in the presence of uncertainties in parameters (data errors) and computer arithmetic (rounding errors). It is based on lectures which I gave repeatedly at Freiburg University. A standard background in linear algebra, analysis and numerical analysis is required.

Since there are now over 2000 publications on interval arithmetic (Garloff, 1985, 1987) I have been rather selective in the choice of the material. The major restrictions are:

1. Only finite-dimensional problems are treated; some reasons for this limitation are discussed below. Thus we also do not touch recent applications of interval arithmetic to computer-assisted proofs in analysis (cf. Eckmann & Wittwer, 1985; Lanford, 1986; Matsumoto, Chua & Ayaki, 1988).

2. Eigenvalue problems are not discussed; this omission is mainly due to a lack of time on my part. (For some references see Remarks to Chapter 5.)

3. Range enclosure problems are treated only to the extent needed for an understanding of the basic principles, and to allow applications to implicit functions; the subject essentially belongs to the field of global optimization and a systematic presentation of the state of the art might well fill another book of this size. (For some references see Remarks to Chapter 2.)

In writing this book l tried to achieve several objectives. My first goal was to develop the tools which are necessary to solve the basic problems of finite-dimensional numerical analysis by interval methods.