Ulrich Kulisch - Böcker

This is the revised and extended second edition of the successful basic book on computer arithmetic. It is consistent with the newest recent standard developments in the field. The book shows how the arithmetic and mathematical capability of the digital computer can be enhanced in a quite natural way. The work is motivated by the desire and the need to improve the accuracy of numerical computing and to control the quality of the computed results (validity). The accuracy requirements for the elementary floating-point operations are extended to the customary product spaces of computations including interval spaces. The mathematical properties of these models are extracted into an axiomatic approach which leads to a general theory of computer arithmetic. Detailed methods and circuits for the implementation of this advanced computer arithmetic on digital computers are developed in part two of the book. Part three then illustrates by a number of sample applications how this extended computer arithmetic can be used to compute highly accurate and mathematically verified results. The book can be used as a high-level undergraduate textbook but also as reference work for research in computer arithmetic and applied mathematics.

This computing supplementum collects a number of original contributions which all aim to compute rigorous and reliable error bounds for the solution of numerical problems. An introductory article by the editors about the meaning and diverse methods of automatic result verification is followed by 16 original contributions. The first chapter deals with automatic result verification for standard mathematical problems, such as enclosing the solution of ordinary boundary value problems, linear programming problems, linear systems of equations and eigenvalue problems. The second chapter deals with applications of result verification methods to problems of the technical sciences. The contributions consider critical bending vibrations stability tests for periodic differential equations, geometric algorithms in the plane, and the periodic solution of the oregonator, a mathematical model in chemical kinetics. The contributions of the third chapter are concerned with extending and developing the tools required in scientific computation with automatic result verification.

Enclosure methods and their applications have been developed to a high standard during the last decades. These methods guarantee the validity of the computed results. This means they are of the same standard as the rest of mathematics. The book deals with a wide variety of aspects of enclosure methods. All contributions follow the common goal to push the limits of enclosure methods forward. Topics that are treated include basic questions of arithmetic, proving conjectures, bounds for Krylow type linear system solvers, bounds for eigenvalues, the wrapping effect, algorithmic differencing, differential equations, finite element methods, application in robotics, and nonsmooth global optimization.

PASCAL-XSC (PASCAL-eXtension for Scientific Computation) ist eine Erweiterung der weitverbreiteten Programmiersprache PASCAL. Sie verfolgt das Ziel, das Programmieren, insbesondere im Bereich technisch-wissenschaftlicher Anwendungen, durch zusätzliche Sprachkonzepte wie Module, dynamische Felder und Operatoren erheblich zu vereinfachen. Eine optimale Arithmetik mit hochgenauen Standardfunktionen und exakter Ausdrucksauswertung ermöglicht eine automatisierte, zuverlässige Kontrolle der berechneten Ergebnisse. Damit unterstützt PASCAL-XSC die Entwicklung von Routinen mit automatischer Ergebnisverifikation. Mittels der Compiler-Implementierung in C kann PASCAL-XSC auf PC's, Workstations, Großrechnern und Supercomputern gleichermaßen eingesetzt werden. In dem vorliegenden Lehr- und Handbuch wird PASCAL-XSC vollständig beschrieben. Zur praktischen Verwendung und zum leichteren Kennenlernen und Vertrautwerden mit den über PASCAL hinausgehenden neuen Sprachelementen ist ein ausführlicher Abschnitt mit Übungsaufgaben und Lösungen aufgenommen worden. Ein kompletter Satz von Syntaxdiagrammen, sowie ausführliche Register und Verzeichnisse schließen dieses Buch ab.

This manual describes a PASCAL extension for scientific computation with the short title PASCAL-XSC (PASCAL eXtension for Scientific Computation). The language is the result of a long term effort of members of the Institute for Applied Mathematics of Karlsruhe University and several associated scientists. PASCAL­ XSC is intended to make the computer more powerful arithmetically than usual. It makes the computer look like a vector processor to the programmer by providing the vector/matrix operations in a natural form with array data types and the usual operator symbols. Programming of algorithms is thus brought considerably closer to the usual mathematical notation. As an additional feature in PASCAL-XSC, all predefined operators for real and complex numbers and intervals, vectors, matrices, and so on, deliver an answer that differs from the exact result by at most one rounding. Numerical mathematics has devised algorithms that deliver highly accurate and automatically verified results by applying mathematical fixed point theorems. That is, these computations carry their own accuracy control. However, their imple­ mentation requires arithmetic and programming tools that have not been available previously. The development of PASCAL-XSC has been aimed at providing these tools within the PASCAL setting. Work on the subject began during the 1960's with the development of a general theory of computer arithmetic. At first, new algorithms for the realization of the arithmetic operations had to be developed and implemented.

C-XSC is a tool for the development of numerical algorithms delivering highly accurate and automatically verified results. It provides a large number of predefined numerical data types and operators. These types are implemented as C++ classes. Thus, C-XSC allows high-level programming of numerical applications in C and C++. The most important features of C-XSC are: real, complex, interval, and complex interval arithmetic; dynamic vectors and matrices; subarrays of vectors and matrices; dot precision data types, predefined arithmetic operators with maximum accuracy; standard functions of high accuracy; multiple precision arithmetic and standard functions; rounding control for I/O data; error handling and a library of problem solving routines with automatic result verification. Thus, C-XSC makes the computer more powerful concerning the arithmetic. C-XSC is immediately usable by C programmers, easy to learn, user-extendable, and may also be combined with other tools. The book can be used as a textbook and as a reference manual.It consists of an introduction to advanced computer arithmetic, a chapter describing the programming languages C and C++, the major chapter "C-XSC Reference", sample programmes and indices.

As suggested by the title of this book Numerical Toolbox for Verified Computing, we present an extensive set of sophisticated tools to solve basic numerical problems with a verification of the results. We use the features of the scientific computer language PASCAL-XSC to offer modules that can be combined by the reader to his/her individual needs. Our overriding concern is reliability - the automatic verification of the result a computer returns for a given problem. All algorithms we present are influenced by this central concern. We must point out that there is no relationship between our methods of numerical result verification and the methods of program verification to prove the correctness of an imple~entation for a given algorithm. This book is the first to offer a general discussion on • arithmetic and computational reliability, • analytical mathematics and verification techniques, • algorithms, and • (most importantly) actual implementations in the form of working computer routines. Our task has been to find the right balance among these ingredients for each topic. For some topics, we have placed a little more emphasis on the algorithms. For other topics, where the mathematical prerequisites are universally held, we have tended towards more in-depth discussion of the nature of the computational algorithms, or towards practical questions of implementation. For all topics, we present exam­ ples, exercises, and numerical results demonstrating the application of the routines presented.