Edgar W. Kaucher

Interval Analysis in the Extended Interval Space IR

This paper shows, how the extended Interval Space IR can be used to write formulas, theorems, and proofs in a closed form, i.e. without using the left and right interval bounds. So a basic generalization and moreover a simplification and improvement of the theorems and proofs is achieved.

E-methods for fixed point equations f(x)=x

This paper provides newly implemented [11], [13] and widely applicable methods for, computing inclusion (i. e. a containing interval) (Einschließung) of the solution of a fixed point equationf(x)=x as well as autmatic verification the existence (Existenz) and uniqueness (Eindeutigkeit) of the solution. These methods make essential use of a new computer arithmetic defined by semimorphisms as developed in [7] and [8]. We call such methods E-Methods in correspondance to the three German words. A priori estimations such as a bound for a Lipschitz constant etc. are not required by the new algorithm. So the algorithm including the a posteriori proof of existence and uniqueness of the fixed point is programmable on computers for linear as well as for nonlinear problems. This is a key feature of our results. The computations produced by E-methods deliver answers the components of which have accuracy better than 10−t+1 (wheret denotes the mantissa length employed in the computer).ZusammenfassungEs werden neuartige sehr allgemeine Methoden vorgestellt, die sowohl eine Einschließung der Lösung von Fixpunktgleichungenf(x)=x als auch automatisch dieExistenz und gegebenenfallsEindeutigkeit der Lösung nachweisen. Diese Methoden machen wesentlichen Gebrauch von neuen Rechnerarithmetiken, die charakterisiert sind wie in [2], [7] und [8] entwickelt. Wir nennen solche Methoden E-Methoden in Übereinstimmung mit den drei Anfangsbuchstaben. A-priori-Abschätzungen wie z. B. für Schranken von Lipschitzkonstanten sind nicht mehr notwendig. Daher ist es in eleganter Weise möglich, Algorithmen zu implementieren, die einen automatischen Existenz- und Eindeutigkeitsnachweis für den Fixpunkt von linearen und nichtlinearen Fixpunktgleichungen ermöglichen. Die mit E-Methoden berechneten Lösungen haben i. a. eine relative Genauigkeit, die besser als 10−t+1 ist (wobeit die Mantissenlänge des verwendeten Rechners bezeichnet).

Small Bounds for the Solution of Systems of Linear Equations

An algorithm is presented to solve a system of linear equations Ax = b of high order. There are no restrictions for A; A may be a floating-point or interval matrix. The algorithm leads to small, guaranteed bounds for the Solution even for ill-conditioned matrices. It takes about six times the Computing time needs for the usual floating-point Gaussian algorithm with comparable accuracy.

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.

Generalized iteration methods for bounds of the solution of fixed point operator-equations

Some general fixed point theorems are obtained describing methods for preservation of a relation between fixed points or between fixed points and iteratives of related iteration operators. If the relation is especially an order relation or an inclusion relation some useful iteration methods to compute bounds for fixed points are stated. Applications and some practical results are discussed.ZusammenfassungIn der vorliegenden Arbeit werden Fixpunktsätze angegeben, die insbesondere Relationen zwischen Fixpunkten oder zwischen Fixpunkten und Iterierten von Iterationsoperatoren erhalten. Für den Fall, daß die Relationen speziell eine Ordnungs- oder Inklusionsrelation ist, werden brauchbare Iterationsmethoden zur Berechnung von Schranken für Fixpunkte angegeben. Es werden Anwendungen und praktische Ergebnisse diskutiert.

Solving function space problems with guaranteed close bounds

Publisher Summary This chapter focuses on solving function space problems with guaranteed close bounds. Using arithmetical routines and methods, it becomes possible to compute very close and reliable bounds economically for the solutions of functional, differential, and integral equations. These methods are called E-methods corresponding to their properties. The methods are suitable for software packages. They provide automatically error control for the first time. Without any additional demand on the part of the user, an incomparably high level of software security and reliability can be achieved. Based on these methods, it is at present economically possible to contain the solutions of functional equations within guaranteed functional bounds. The computation of a reliable error estimate for the solution involves the determination of a set of vector functions X ( t ) containing the solution x ( t ) for all t . The computation of X ( t ) requires new computer arithmetic, such as segment arithmetic and functional ultra arithmetic. All these must be executed automatically and intrinsically on a computer without intervention by the users. Consequently, such methods are called error controlling methods. In most cases, an error controlling method necessarily proves the existence of the solution x ( t ) in X ( t ) automatically.

Validating computation in a function space

A methodology for the numerical solution of differential equations and integral equations which furnishes computer generated bounds of high quality is developed. Functions and operators on functions are implemented by means of computer representable counterparts, the latter being the constituents of ultra-arithmetic. An interval ultra-arithmetic is also developed. This furnishes the basis for fixed point iteration techniques which deliver the solution bounds. High quality of these bounds results from the method of iterative residual correction in a function space.

ITERATIVE RESIDUAL CORRECTION

This chapter presents the iterative residual correction (IRC) method for functional spaces, in particular, in the computational framework of functoids. The method of IRC is a well-known computational technique for improving the accuracy of an approximation to the solution of equations, especially linear equations. Until recently, this method was applied in the context of problems cast in modular number systems, such as floating-point representation systems. Functoids and their corresponding roundings have a great similarity to floating-point number structures and their roundings. The chapter reviews the IRC process in a floating-point system with particular emphasis on two arithmetic features: (1) the need for increasing accuracy in the computation of residuals during the process and (2) the propagation of information among the digits in a floating-point system that the IRC process engenders in a floating-point system. The chapter describes the latter feature in a context for achieving annihilation of digits in the residuals, and it is this feature that motivates the subsequent treatment of IRC in function spaces.

ULTRA-ARITHMETIC AND ROUNDINGS

This chapter focuses on ultra-arithmetic and roundings. It presents the constructs of function-space arithmetic, that is, spaces, bases, roundings, and approximate operations. It also presents the analogous constructs that are to be employed when computation that supplies inclusion is in question. In ultra-arithmetic, structures, data types, and operations corresponding to functions are developed for direct digital implementation. A digital computer equipped with ultra-arithmetic is a highly congenial tool for computation. Problems associated with functions would be solvable on computers just as one now solves algebraic problems. Moreover, the considerably enlarged set of structures, data types, and operations would make for the generation of far-reaching concepts of computer architecture.

COMMENTS ON PROGRAMMING LANGUAGE

This chapter discusses the requirements of a programming language. The methodology introduced in computer arithmetic and numerical analysis, composed as it is of many higher-level concepts and constructs, requires that a complementary higher-level programming language be furnished. Such a programming language will enable the computer user to employ these many constructs in an effective and congenial manner. Examples of programming languages that have been devised with this principle of accommodation for other classes of computer-arithmetic constructs exist and have demonstrated their effectiveness. For the methodology, the requirements of the higher programming language may be succinctly summarized as follows: the language should make an efficient use of functoids and their algebraic structure possible. The chapter also presents a list some key features that a prospective programming language should have.