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Computer Arithmetic and Validity: Theory, Implementation, and Applications

Ulrich Kulisch
de Gruyter
Publication Date: 
Number of Pages: 
Studies in Mathematics 33
[Reviewed by
Luiz Henrique de Figueiredo
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This is a treatise on the theory and implementation of computer arithmetic by one of the leading experts in the subject. It is the product of work done in the last 30 years whose evolution is explained in three previous books by Kulisch. The goal of the present book, stated in its preface, is to help "improve the accuracy of numerical computing and control the quality of the computed results".

The book has three parts. The first part lays out an algebraic theory of computer arithmetic, including interval arithmetic. The second part discusses in detail the implementation of computer arithmetic, including floating-point arithmetic and hardware support for interval arithmetic, a feature that Kulisch (with others) argues is long overdue: the potential resulting improvement in the reliability of numerical computation would easily offset any (small) extra costs. The third part is an introduction to the principles of verified computing, including a brief discussion of interval methods for global optimization, automatic differentiation, verified integration, and the interval Newton method.

The book will be useful mostly as a reference work for researchers in computer arithmetic, but its third part can also be used as an introduction to verified computing for users and students of applied mathematics.

Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.