Ch. Ullrich

FORTRAN for contemporary numerical computation

In addition to the integers, the real and complex numbers, the real segments (intervals) and complex segments as well as vectors and matrices over all of these comprise the fundamental data types in computation. We extendFORTRAN so that it accepts operands and operators for all of these types as primitives in expressions.We briefly review the spaces corresponding to these data types and the definitions of the arithmetic operations in their computer representable subsets. Then we give a general description of the language extension including the additional basic external functions and intrinsic functions for the new data types. Following this we give the syntax for the extended language in the form of easily traceable syntax diagrams. Comments on the semantics are also included.ZusammenfassungIn numerischen Rechnungen treten neben den ganzen Zahlen häufig auch reelle und komplexe Zahlen, reelle und komplexe Intervalle sowie Vektoren und Matrizen über diesen Mengen auf. In der vorliegenden Arbeit erweitern wir die ProgrammierspracheFORTRAN so, daß Ausdrücke mit Operanden und Operatoren für all diese Typen (Datenmengen) akzeptiert werden.Wir beginnen mit einer kurzen Zusammenstellung dieser Räume und der arithmetischen Verknüpfungen in den Teilmengen, welche auf einem Rechner darstellbar sind. Es folgt dann eine allgemeine Beschreibung der Spracherweiterung sowie der neuen Standardfunktionen und Standardformelfunktionen für die zusätzlichen Datentypen. Im zweiten Teil der Arbeit geben wir dann die vollständige Syntax für die erweiterte Sprache in Form von leicht lesbaren Syntaxdiagrammen an. Wir erläutern auch die Semantik der Spracherweiterung.

New developments in Pascal-SC

PASCAL-SC is an efficient programming language for scientific computation which was developed at the beginning of the eighties as an extension of standard PASCAL. It provides arithmetical operations with controlled rounding and an optimal scalar product. Functions with arbitrary result type and an operator concept permit the realization of higher arithmetical operations, such as matrix-vector operations. The application of PASCAL-SC for the development of reliable numerical software raised several wishes for further extension. These extensions have been incorporated in the language.A new data type dotprecision together with arithmetic operations is introduced permitting scalar product expressions to be evaluated with maximum accuracy in a convenient manner. Dynamic arrays and a module concept support the definition of numeric libraries. Over loading of functions and procedures improve the readability of scientific programs. A program for the verified solution of linear systems illustrates the strength of the new language.

PASCAL-SC and Numerical and Non Numerical Applications

Today a lot of lectures are unthinkable without accompanying practical course for proving the acquired knowledge directly on the computer. This especially applies to the present lectures and practical courses which were held during the time of the HECTOR-project by the staff of the Institute for Applied Mathematics: “Introduction to PASCAL-SC”, “Higher Numerical Mathematics” and “Algorithms, Data Structures and Programming Methods”.

Iterative Methods in the Spaces of Rounded Computations

Numerical algorithms are usually executed in a space over the finite set of floating-point numbers. Numerous properties are missing in such spaces in contrast to mathematical spaces in which we are used to study algorithms. For this reason numerical algorithms show another behaviour than we would expect based on theoretical investigations. This paper summarizes some results for iterative methods, which can be derived directly by algebraic and order properties of the spaces of rounded computations.