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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."
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