c-xsc R. Klatte U. Kulisch A. Wiethoff C. Lawo M. Rauch A C++ Class Library for Extended Scientific Computing Springer-Verlag Berlin Heidelberg GmbH

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2 R. 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

3 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 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 ISBN (ebook) DOI / Library of Congress Cataloging-in-Publication Data available. This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 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/ I 0 - Printed on acid-free paper

4 Preface The programming environment C - XSC (C for extended Scientific Qomputing) is a powerful and easy to use programming tool, especially for scientific and engineering applications. C - XSC makes the computer more powerful arithmetically and significantly simplifies programming. C - XSC is implemented as a numerical class library in the programming language C++, a superset of the wellknown 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 computations carry their own accuracy control. However, their implementation requires suitable arithmetic support and powerful programming tools which were not previously available. The development 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 automatically verified results, which are essential, for example, in simulation 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.

5 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 mathematical 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 complex 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 programming 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 University of Karlsruhe, Germany. New algorithms for the realization of the arithmetic operations were developed. The design of appropriate 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 analysis with automatic result verification and for many applications have been developed. Based on this work, the implementation of C - XSC started in Thus, many colleagues and scientists

6 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 necessary 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

7 Contents 1 Introd uction Typography C - XSC: A Class Library in the Programming Language C C - XSC: A Programming Environment for Scientific Computing with Result Verification SurveyofC-XSC 11 2 The Programming Languages C and C A Short Introduction to C Overview Data Types, Operators, and Expressions Control Flow Functions The Structure of a C Program External Variables The Scope of Variables and Functions The C Preprocessor Pointers Pointers and Function Arguments Arithmetic with Pointers and Arrays Structures The C Standard Library Additional Features in C Overview A Sample Program Comments Classes Member Functions Friend Functions. Reference Variables

8 x Contents Constructors and Destructors The Structure of a C++ Program Inline Functions Overloaded Operators and Functions Input and Output with Streams Memory Management Operators Type Casting Additional Features of C C - XSC Reference 3.1 Constants, Data Types, and Variables Constants Variables Scalar Data Types Vector Data Types Matrix Data Types Dotprecision Data Types Multiple-Precision Data Types User-Defined Data Types. 3.2 Expressions Implicit Type Casting Explicit Type Casting Arithmetic Operators Relational Operators Standard Functions. 3.3 Statements Assignments Manipulation of Index Bounds Resize of Vectors and Matrices Addition of a Product to a Dotprecision Accumulator Rounding of Dotprecision Accumulators Input and Output Error Handling Pitfalls for Programming with C - XSC

9 Contents Xl Appendix A Syntax Diagrams A.1 Data Types A.2 Management of Vectors and Matrices A.3 Definition of Variables AA Expressions.... A.5 Logical Expressions A.6 Type Castings... A.7 Assignments.... A.8 Arithmetic Standard Functions A.9 Other Functions.... A.10 Input and Output.... A.ll Extension of a Syntax Diagram of C++. B The Structure of the C - XSC Package B.1 Header Files... B.2 Module Libraries C Error List D Sample Programs D.1 Rounding Control of Arithmetic Operations D.2 Rounding Control of Input and Output D.3 Scalar Product.... D.4 Transpose of a Matrix.. D.5 Trace of a Product Matrix D.6 Inverse of a Matrix... D.7 Multiple-Precision Arithmetic D.8 Interval Newton Method... D.9 Runge-Kutta Method.... D.10 Complex Polynomial Arithmetic. D.ll Automatic Differentiation... E Scientific Computation with Verified Results E.1 Evaluation of Polynomials E.2 Matrix Inversion.... E.3 Linear Systems of Equations.... EA Eigenvalues of Symmetric Matrices

10 XlI Contents E.5 Fast Fourier Transform... E.6 Zeros of a Nonlinear Equation. E.7 System of Nonlinear Equations E.8 Ordinary Differential Equations Bibliography Index

11 List of Figures 1.1 Subset Relation of Numerical Basic Spaces Structure of the c++ Sample Program Public and Private Members of a C++ Class The real Floating-Point Format A Complex Interval Structure of a Dotprecision Accumulator Screen of Multiple-Precision Numbers Hierarchy of Scalar Types.116 B.1 Hierarchy of the Data Types.208

12 List of Tables 1.1 Basic Spaces of Scientific Computing Precedence and Associativity of the C Operators Functions of the C++ Sample Program Standard Input and Standard Output Streams in C Special real Constants and real Values Arithmetic Operators for Scalars Definition of the Arithmetic Vector and Matrix Operators Structure of the Result Type for Matrix/Vector Operations Arithmetic Operators for Vectors and Matrices Arithmetic Operators for Dotprecision Terms Arithmetic Operators for Multiple-Precision Numbers Relational Operators for Scalars Definition of the Relational Vector and Matrix Operators Relational Operators for Vectors and Matrices Relational Operators for Dotprecision Terms Relational Operators for Multiple-Precision Data Types Standard Functions Functions accumulate() for Scalar Operands Functions accumulate() for Vector Terms I/O Environment Attributes and Default Values C.1 Possible Modules Detecting Errors. C.2 Possible Types of Errors in C - XSC

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