Table Of ContentR. Klatte U. Kulisch A. Wiethoff
C. Lawo M. Rauch
c-xsc
A C++ Class Library for
Extended Scientific Computing
Translated by G. F. Corliss C. Lawo R. Klatte
A. Wiethoff C. Wolff
Springer-Verlag Berlin Heidelberg GmbH
Prof. Dr. Ulrich Kulisch
Dr. Rudi Klatte
Dipl.-Math. Andreas Wiethoff
Dipl.-Übers. Carola Wolff
Institut für Angewandte Mathematik
Universität Karlsruhe
Kaiserstrasse 12, W-7500 Karlsruhe 1, Deutschland
Prof. George F. Corliss
Department of Mathematics, Statistics and Computer Science
MU Marquette University
William Wehr Physics Building
Milwaukee, WI 53233, USA
Dr. Christian Lawo
Wichernstrasse 12, W -7 528 Karlsdorf, Deutschland
Dipl.-Math. Michael Rauch
Albert-Einstein-Strasse 9, W-7518 Bretten, Deutschland
Mathematics Subject Classification (1991): 65-XX, 65-04, 65FXX,
65GXX, 65G05, 65GlO, 65HXX, 65LXX, 68-XX, 68NXX
ISBN 978-3-540-56328-0 ISBN 978-3-642-58058-1 (eBook)
DOI 10.1007/978-3-642-58058-1
Library of Congress Cataloging-in-Publication Data available.
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© Springer-Verlag Berlin Heidelberg 1993
Originaliy published by Springer-Verlag Berlin Heidelberg New Y ork in 1993
UNIX is a registered trademark of AT &T Bell LaboratorieS.
IBM and PS/2 are registered trademarks of International Business Machines Corporation.
Borland C++ is a registered trademark of Borland International, Inc.
Typesetting: Camera ready by author/editor using TEX
41/3140 -5 4 3 2 I 0 -Printed on acid-free paper
Preface
The programming environment C - XSC (C for eXtended Scientif
ic Qomputing) is a powerful and easy to use programming tool, es
pecially for scientific and engineering applications. C - XSC makes
the computer more powerful arithmetically and significantly sim
plifies programming. C - XSC is implemented as a numerical class
library in the programming language C++, a superset of the well
known language C. C - XSC may be interpreted as an arithmetic
module extending the properties of the language C.
The speed of digital computers is ever increasing. Within the
last 40 years, the speed of electronic digital computers has been
increased roughly from about 100 to 100 billion floating-point
operations per second. This increased computing power has made
it feasible to address ever larger problems. A significant question
is whether floating-point arithmetic, which may fail already in
simple calculations, is still adequate to be used in computers of
such gigantic speeds for huge problems.
Numerical mathematics has contrived algorithms which deliver
highly accurate and automatically verified results by applying
mathematical fixed-point theorems. This means that these com
putations carry their own accuracy control. However, their im
plementation requires suitable arithmetic support and powerful
programming tools which were not previously available. The de
velopment of C - XSC has aimed at providing these tools within C
and c++. C-XSC is particularly suited for the development of
numerical algorithms that deliver highly accurate and automati
cally verified results, which are essential, for example, in simula
tion runs where the user has to distinguish between computational
artifacts and genuine reactions of the model. Problem-solving
functions with automatic result verification have been developed
in C - XSC for several standard problems of numerical analysis.
VI Preface
C - XSC consists of a run time system written in C including an
optimal dot product and many predefined data types for elements
of the most commonly used vector spaces such as real and complex
numbers, vectors, and matrices. Operat.ors for elements of these
types are predefined and can be called by their usual operator
symbols. Thus, arithmetic expressions and numerical algorithms
are expressed in a notation that is very close to the usual math
ematical notation. Using C-XSC, many programs can be read
like a technical report. All predefined numerical operators are of
highest accuracy. That is, the computed result differs from the
correct result by at most one rounding.
We provide 24 mathematical standard functions for real and com
plex arguments and for real and complex interval arguments. The
computed values are accurate to within one or two bits in the last
place. While the emphasis in computing is traditionally on speed,
in C - XSC, the emphasis is more on accuracy and reliability of
results. The total time for solving 'a problem is the sum of the
programming effort, the processing time, and the time for the
interpretation of results. We contend that C - XSC reduces this
sum considerably.
C programmers should be able to use and write programs in C
XSC immediately. C - XSC simplifies programming by providing
many predefined data types and arithmetic operators. Programs
are much easier to read, to write, and to debug. The program
ming environment C - XSC is easily portable to any computer
that supports a standard-conforming C++ compiler.
Work on a general theory of computer arithmetic began during
the 1960's at the Institute of Applied Mathematics at the Univer
sity of Karlsruhe, Germany. New algorithms for the realization of
the arithmetic operations were developed. The design of appropri
ate programming languages began around 1975 with preliminary
implementation studies based on PASCAL and on FORTRAN.
Compilers for the extended languages were implemented. Since
about 1980, algorithms for standard problems of numerical anal
ysis with automatic result verification and for many applications
have been developed. Based on this work, the implementation of
C - XSC started in 1990. Thus, many colleagues and scientists
Preface
VII
have directly and indirectly contributed to the realization of C
XSC. The authors would like to thank each one of them for his
or her cooperation.
This reference manual contains the complete description of the C -
XSC language extension, a chapter with sample programs, and a
library of problem-solving functions and programs with automatic
result verification for standard problems of numerical analysis. It
also includes a complete set of syntax diagrams, detailed tables,
and an index. The material in this book should be easily accessible
for C or C++ programmers.
This book is a translation of an unpublished German manuscript.
The authors are very grateful to George Corliss who helped to
polish the text and the contents, to Carola Wolff who translated
parts of the book, and to Dietrich Cordes who implemented large
parts of the C run time system.
Karlsruhe, October 1992 The Authors
The Realization of this Book
This manual was completely written in the text system L\.TE)X.
Co-Author Andreas Wiethofl' gathered the text, designed the nec
essary macros and environments (with the help of Dietmar Ratz),
developed the syntax diagrams, carried out corrections, and drew
up the final version including the appendices and indices. He also
designed the final layout of this book.
The Authors
Contents
1 Introd uction 1
1.1 Typography 2
1.2 C - XSC: A Class Library in the Programming Language C++ 2
1.3 C - XSC: A Programming Environment for Scientific Computing
with Result Verification 4
1.4 SurveyofC-XSC 11
2 The Programming Languages C and C++ 21
2.1 A Short Introduction to C . . . . . . . . . 21
2.1.1 Overview .............. 21
2.1.2 Data Types, Operators, and Expressions 23
2.1.3 Control Flow .. . . . . . . . . 29
2.1.4 Functions ........... . 33
2.1.5 The Structure of a C Program . 35
2.1.6 External Variables ...... . 35
2.1. 7 The Scope of Variables and Functions. 36
2.1.8 The C Preprocessor . . . . . . . . 37
2.1.9 Pointers ............... . 39
2.1.10 Pointers and Function Arguments .. 40
2.1.11 Arithmetic with Pointers and Arrays 41
2.1.12 Structures ....... . 43
2.1.13 The C Standard Library 45
2.2 Additional Features in C++ 46
2.2.1 Overview ..... 46
2.2.2 A Sample Program 47
2.2.3 Comments..... 52
2.2.4 Classes....... 52
2.2.5 Member Functions 55
2.2.6 Friend Functions . 56
2.2.7 Reference Variables 57
x Contents
2.2.8 Constructors and Destructors .. 57
2.2.9 The Structure of a C++ Program 59
2.2.10 Inline Functions ......... . 60
2.2.11 Overloaded Operators and Functions 61
2.2.12 Input and Output with Streams. 62
2.2.13 Memory Management Operators. 63
2.2.14 Type Casting ....... . 64
2.2.15 Additional Features of C++ 65
3 C - XSC Reference 67
3.1 Constants, Data Types, and Variables. 68
3.1.1 Constants..... 68
3.1.2 Variables ..... . 68
3.1.3 Scalar Data Types 69
3.1.4 Vector Data Types 79
3.1.5 Matrix Data Types 84
3.1.6 Dotprecision Data Types . 91
3.1.7 Multiple-Precision Data Types. 95
3.1.8 User-Defined Data Types. · 101
3.2 Expressions ......... . .102
3.2.1 Implicit Type Casting .102
3.2.2 Explicit Type Casting .102
3.2.3 Arithmetic Operators . .116
3.2.4 Relational Operators · 132
3.2.5 Standard Functions. .140
3.3 Statements.......... .142
3.3.1 Assignments ..... .142
3.3.2 Manipulation of Index Bounds . · 155
3.3.3 Resize of Vectors and Matrices. · 157
3.3.4 Addition of a Product to a Dotprecision Accumulator . · 160
3.3.5 Rounding of Dotprecision Accumulators · 162
3.3.6 Input and Output. . . . . . . . · 163
3.4 Error Handling . . . . . . . . . . . . . · 174
3.5 Pitfalls for Programming with C - XSC · 176
Contents Xl
Appendix
A Syntax Diagrams 181
A.1 Data Types . . . . . . . . . . . . . . · 184
A.2 Management of Vectors and Matrices · 185
A.3 Definition of Variables · 186
AA Expressions . . . . · 187
A.5 Logical Expressions · 190
A.6 Type Castings ... · 190
A.7 Assignments . . . . · 196
A.8 Arithmetic Standard Functions · 198
A.9 Other Functions ........ . · 200
A.10 Input and Output ....... . · 201
A.ll Extension of a Syntax Diagram of C++ . · 205
B The Structure of the C - XSC Package 207
B.1 Header Files . . . .207
B.2 Module Libraries .208
C Error List 209
D Sample Programs 213
D.1 Rounding Control of Arithmetic Operations · 213
D.2 Rounding Control of Input and Output · 216
D.3 Scalar Product ..... . · 217
D.4 Transpose of a Matrix .. · 219
D.5 Trace of a Product Matrix .221
D.6 Inverse of a Matrix .... · 222
D.7 Multiple-Precision Arithmetic .225
D.8 Interval Newton Method .. . .226
D.9 Runge-Kutta Method .... . · 228
D.10 Complex Polynomial Arithmetic. · 231
D.ll Automatic Differentiation .... .235
E Scientific Computation with Verified Results 241
E.1 Evaluation of Polynomials .242
E.2 Matrix Inversion ......... . .243
E.3 Linear Systems of Equations . . . . .243
EA Eigenvalues of Symmetric Matrices .244
XlI Contents
E.5 Fast Fourier Transform ..... .245
E.6 Zeros of a Nonlinear Equation . .246
E.7 System of Nonlinear Equations .247
E.8 Ordinary Differential Equations .248
Bibliography 253
Index 261
