We discuss two closely related interval arithmetic systems: i) the system of directed (generalized) intervals studied by E. Kaucher, and ii) the system of normal intervals together with the outer and inner interval operations. A relation between the two systems becomes feasible due to introduction of special notations and a so-called normal form of directed intervals. As an application, it has been shown that both interval systems can be used for the computation of tight inner and outer inclusions of ranges of functions and consequently for the development of software for automatic computation of ranges of functions.

This report presents an experimental Mathematica

: This paper generalizes the specification of Basic Interval Arithmetic Subroutines (BIAS) to support interval arithmetic on directed (i.e. proper and improper) intervals. This is due to our understanding that the arithmetic involving improper intervals will be increasingly used in future applications and the corresponding interval arithmetic implementations require no additional cost. We extend BIAS specification to be sufficiently precise and complete, to include everything a user needs, such as subroutine's purpose, name, method of invocation and details of its behaviour and communication with the environment. The specified interval arithmetic subroutines for directed intervals are consistent with conventional interval arithmetic and IEEE floating-point arithmetic. Key Words: specification, interval arithmetic Category: D.2.1, D.3., K.6.3 1 Introduction Interval arithmetic [Alefeld and Herzberger 1974], [Moore 1966] is widely recognized as a valuable computing technique. Numerou...

This paper describes an implementation of a general interval arithmetic extension, which comprises the following extensions of the conventional interval arithmetic: (1) extension of the set of normal intervals by improper intervals; (2) extension of the set of arithmetic operations for normal intervals by nonstandard operations; (3) extension by infinite intervals. We give a possible realization scheme of such an universal interval arithmetic in any programming environment supporting IEEE floating-point arithmetic. A PASCAL-XSC module is reported which allows easy programming of numerical algorithms formulated in terms of conventional interval arithmetic or of any of the enlisted extended interval spaces, and provides a common base for comparison of such numerical algorithms. 1

Abstract: This paper generalizes the speci cation of Basic Interval Arithmetic Subroutines (BIAS) to support interval arithmetic on directed (i.e. proper and improper) intervals. This is due to our understanding that the arithmetic involving improper intervals will be increasingly used in future applications and the corresponding interval arithmetic implementations require no additional cost. We extend BIAS speci cation to be su ciently precise and complete, to include everything a user needs, such as subroutine's purpose, name, method of invocation and details of its behaviour and communication with the environment. The speci ed interval arithmetic subroutines for directed intervals are consistent with conventional interval arithmetic and IEEE oating-point arithmetic. Key Words: speci cation, interval arithmetic Category: D.2.1, D.3., K.6.3

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