Our aim in writing this book was to provide an extensive set of C++ programs for solving basic numerical problems with verification of the results. This C++ Toolbox for Verified Computing I is the C++ edition of the Numerical Toolbox for Verified Computing l. The programs of the original edition were written in PASCAL-XSC, a PASCAL eXtension for Scientific Computation. Since we published the first edition we have received many requests from readers and users of our tools for a version in C++. We take the view that C++ is growing in importance in the field of numeri cal computing. C++ includes C, but as a typed language and due to its modern concepts, it is superior to C. To obtain the degree of efficiency that PASCAL-XSC provides, we used the C-XSC library. C-XSC is a C++ class library for eXtended Scientific Computing. C++ and the C-XSC library are an adequate alternative to special XSC-Ianguages such as PASCAL-XSC or ACRITH-XSC. A shareware version of the C-XSC library and the sources of the toolbox programs are freely available via anonymous ftp or can be ordered against reimbursement of expenses. The programs of this book do not require a great deal of insight into the features of C++. Particularly, object oriented programming techniques are not required."

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.

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. This 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.

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; dotprecision 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 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 programs, and indices.

The programming language PASCAL-XSC (PASCAL eXtension for Scientific Computation) significantly simplifies programming in the area of scientific and technical computing. PASCAL-XSC provides a large number of predefined data types with arithmetic operators and predefined functions of highestaccuracy for real and complex numbers, for real and complex intervals, and for the corresponding vectors and matrices. Thus PASCAL-XSC makes the computer more powerful concerning the arithmetic. Through an implementation in C, compilers for PASCAL-XSC are available for a large variety of computers such as personal computers, workstations, mainframes, and supercomputers. PASCAL-XSC provides a module concept, an operator concept, functions and operators with general result type, overloading of functions, procedures, and operators, dynamic arrays, access to subarrays, rounding control by the user, and accurate evaluation of expressions. The language is particularly suited for the development of numerical algorithms that deliver highly accurate and automatically verified results. A number of problem-solving routines with automatic resultverification have already been implemented. PASCAL-XSC contains Standard PASCAL. It is immediately usable by PASCAL programmers. PASCAL-XSC is easy to learn and ideal for programming education. The book can be used as a textbook for lectures on computer programming. It contains a major chapter with sample programs, exercises, and solutions. A complete set of syntax diagrams, detailed tables, and indices complete the book.

Scientific Computation with Result Verification has been a persevering research topic at the Institute for Applied Mathematics of Karlsruhe University for many years. A good number of meetings have been devoted to this area. The latest of these meetings was held from 30 September to 2 October, 1987, in Karlsruhe; it was co-sponsored by the GAMM Committee on "Computer Arithmetic and Scientific Computation." - - This volume combines edited versions of selected papers presented at this confer ence, including a few which were presented at a similar meeting one year earlier. The selection was made on the basis of relevance to the topic chosen for this volume. All papers are original contributions. In an appendix, we have supplied a short account of the Fortran-SC language which permits the programming of algorithms with result verification in a natural manner. The editors hope that the publication of this material as a Supplementum of Computing will further stimulate the interest of the scientific community in this important tool for Scientific Computation. In particular, we would like to make application scientists aware of its potential. The papers in the second chapter of this volume should convince them that automatic result verification may help them to design more reliable software for their particular tasks. We wish to thank all contributors for adapting their manuscripts to the goals of this volume. We are also grateful to the Publisher, Springer-Verlag of Vienna, for an efficient and quick production."

Was ist los in der Mathematik? Das fragen sich viele Mathematiker, aber auch alle, die fruher einmal Mathematik gelernt haben und jetzt in anderen Feldern der Wissenschaft und Wirtschaft tatig sind. Sie wollen einen profunden, aber leicht lesbaren UEberblick uber das, was sich in diesem Zweig der Wissenschaft tut, wollen aber auch etwas uber Menschen erfahren, die dieses Feld pragen. "UEberblicke Mahematik" bietet dies alles und noch viel mehr, z. B. auch, was Politiker von Mathematik halten. Die Herausgeber - allesamt ausgewiesene Fachleute auf ihrem Gebiet - zeigen, dass man bei der Lekture eines Mathematikbuches nicht verzweifeln muss, sondern auch "einfach so" viel uber neue Gebiete und Entwicklungen erfahren kann.

PASCAL-XSC (PASCAL-e"X"tension for "S"cientific "C"omputation) ist eine Erweiterung der weitverbreiteten Programmiersprache PASCAL. Sie verfolgt das Ziel, das Programmieren, insbesondere im Bereich technisch-wissenschaftlicher Anwendungen, durch zusAtzliche Sprachkonzepte wie Module, dynamische Felder und Operatoren erheblich zu vereinfachen. Eine optimale Arithmetik mit hochgenauen Standardfunktionen und exakter Ausdrucksauswertung ermAglicht eine automatisierte, zuverlAssige Kontrolle der berechneten Ergebnisse. Damit unterstA1/4tzt PASCAL-XSC die Entwicklung von Routinen mit automatischer Ergebnisverifikation. Mittels der Compiler-Implementierung in C kann PASCAL-XSC auf PC's, Workstations, GroArechnern und Supercomputern gleichermaAen eingesetzt werden. In dem vorliegenden Lehr- und Handbuch wird PASCAL-XSC vollstAndig beschrieben. Zur praktischen Verwendung und zum leichteren Kennenlernen und Vertrautwerden mit den A1/4ber PASCAL hinausgehenden neuen Sprachelementen ist ein ausfA1/4hrlicher Abschnitt mit Aoebungsaufgaben und LAsungen aufgenommen worden. Ein kompletter Satz von Syntaxdiagrammen, sowie ausfA1/4hrliche Register und Verzeichnisse schlieAen dieses Buch ab.

Computer. Algebra fdr den Ingenieur (B. Buchberger I B. Kutzler) Das Gebiet der Computer-Algebra stent dem Ingenieur ein neues Arsenal von computer-unterstutzten Methoden zur Losung von Problemen des technichlwis senschaftlichen Rechnens zur Verfugung. In diesem Kapitel wird zunachst das neue Gebiet der Computer-Algebra charakterisiert und insbesondere vom Gebiet der Numerik abgegrenzt bzw. aufgezeigt, wie Computer-Algebra im Verein mit Numerik die ProblemlOsepotenz um eine wesentliche Qualitat erweitert. Dann werden die typischen Grundrechenoperationen, die in Computer-Algebra-Software systemen moglich sind, und die Verbindung dieser Operationen zu Programmen an Hand von Beispielen, insbesondere von konkreten Anwendungen aus der Ingenieur mathematik, demonstriert. 1m nachsten Abschnitt werden dann die wichtigsten Computer-Algebra-Softwaresysteme und ihre Verfugbarkeit fo. r den Benutzer be sprochen. Schlie13lich wird im Abschnitt Computer-Algebra-Algorithmen auf die der Computer-Algebra zugrundeliegende Mathematik eingegangen, indem fur einige typische Problemstellungen die zur Losung fuhrenden matbematisch/algorith mischen Ideen skizziert werden. Algorithmen zur Methode der finiten Elemente fdr Vektorrechner (M. Kratz) Eine neuartige Klasse sehr leistungsfahiger Computer hat die Moglichkeit der numerischen Datenverarbeitung wesentlich erweitert: die sog. Vektorrechner. Ihre V erarbei tungsgesch windigkei t kann diejenige gro13er U ni versalrechner um Zehnerpotenzen ubertreffen - jedoch nur mit neuen, der speziellen Maschinen architektur angepa13ten Algorithmen und Programmen. Das Besondere ist die Funktionsweise ihrer Prozessoren, die lange Folgen von Daten nach dem Flie13band prinzip verknupfen. Aus dem Flie13bandverfahren folgt, da13 sich die volle Leistung 6 der Maschinen erst bei genugend langen Operndenstromen einstellt."