Rudi Klatte

Complex sector arithmetic

In the past interval analysis in the complex plane has used nearly exclusively the rectangular arithmeticRℂ resp. the circular arithmeticKℂ. Both arithmetics satisfy the inclusion property where the equality does not hold in general. To introduce new interval arithmetics it is therefore interesting to consider other subsets of the powerset ofℂ. This paper describes different alternatives of sector arithmetics, for which the equality in (*) is valid.ZusammenfassungFür die Intervallrechnung in der komplexen Zahlenebeneℂ wurde bisher fast ausschließlich die RechteckarithmetikRℂ bzw. die KreisarithmetikKℂ verwendet. Beide Arithmetiken erfüllen bezüglich der Multiplikation die Einschließungseigenschaft wobei die Gleichheit im allgemeinen nicht erfüllt ist. Zur Einführung neuer Intervallarithmetiken ist daher die Betrachtung anderer Teilmengen der Potenzmenge ℙℂ überℂ interessant. Die Arbeit gibt eine Beschreibung verschiedener Möglichkeiten für Kreisringsektorarithmetiken, die in (*) jeweils das Gleichheitszeichen erfüllen.

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.

Consequences of a properly implemented computer arithmetic for periodicities of iterative methods

In ordered sets it is possible to show under certain assumptions two basic theorems concerning the cycle length of sequences of iterates generated by monotone operators. These results are applied to different iterative methods, where the conclusions are valid for the sequences of iterates produced by the numerical computations only, if the used computer arithmetic is properly implemented.

Interval Methods for Global Optimization Using the Boxing Method

The global optimization problem with simple bounds which is the scope of this work can be defined in general as min x∈X f(x) where X is a — possibly multidimensional — interval. The original problem can be solved with verified accuracy with the aid of interval subdivision methods. These algorithms are based on the well-known branch-and-bound principle. The methods pruning the search tree of these algorithms are the so-called accelerating devices. One of the most effective of these is the interval Newton step, however, its time complexity is relatively high as compared with other accelerating devices. Therefor it should only be deployed if there are no other possibilities to effectively bound the search tree. Methods like the boxing method can decrease the number of the applied Newton steps. The present paper discusses some of these methods and shows the numerical effects of their implementation.

On conversions between screens

The well known concepts of screens and roundings are not always sufficient if one intends to derive statements in connection with the generation of arithmetics in spaces which are more general than those considered in [5]. In this paper the concept of the screen therefore is generalized to the concept of the (Si)-screen and instead of roundings the conversions are introduced as mappings between (Si)-screens. The properties of (Si)-screens and conversions are studied.ZusammenfassungDie bisher bekannten Begriffe des Rasters und der Rundung erlauben, wie neuere Untersuchungen zeigen, nicht immer die Beschreibung des Sachverhalts bzw. die Herleitung von Aussagen bei der Erzeugung von Arithmetiken. In der vorliegenden Arbeit wird daher bei der Definition des (Si)-Rasters von der Forderung einer Inklusionsbeziehung abgesehen. An die Stelle der bisher verwendeten Rundungen treten Konvertierungen, die den Übergang zwischen den (Si)-Rastern ermöglichen. Es werden die Eigenschaften der (Si)-Raster und der Konvertierungen selbst betrachtet.

C - XSC Reference

In the previous chapter The Programming Languages C and C++, we briefly introduced those aspects and features of the programming languages C and C++ that are needed to write programs for scientific computation.

The Programming Languages C and C

C - XSC is implemented as a set of classes in the programming language C++. We present here a brief review of the programming languages C and C++ for the benefit of readers unfamiliar with either C or C++. These reviews are no substitute for complete textbooks for C and C++. For example, good introductions to both languages can be found in [Ker90, Mas91] and [Str91, Wei90].

Problem-Solving Routines

Routines for solving common numerical problems have been developed in PASCAL-XSC. They are supplied by means of an additional module library. The methods used compute a highly accurate inclusion of the true solution of the problem and verify the existence and uniqueness of the solution in the given interval. The advantages of these new routines are: The solution is computed with high accuracy, even for many ill-conditioned cases. The accuracy of the computed solution is always controlled. The correctness of the result is automatically verified, i.e. an inclusion set is computed which guarantees the existence and uniqueness of the exact solution within the bounds computed. If no solution exists, or if the problem is extremely ill-conditioned, an error message is returned.

The Arithmetic Modules

Numerical methods require computations not only in the space of real numbers, but also with complex numbers, and vectors and matrices over these numbers (see [1], [2], [19], or [33]). To fulfill all these requirements, PASCAl-XSC provides the corresponding types with the necessary operators and functions.

Exercises with Solutions

Here are some exercises with which the reader can practice the language PASCAL-XSC by solving various exercises and applying the new language elements to the development of complete programs.