The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus efficient solution of some interval algebraic problems. In this paper we summarize and present all distributive relations, known by now, on multiplication and addition of generalized (proper and improper) intervals.

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...

Abstract. This document is an assessment of the value of optimal linear interpolation enclosures and of nonstandard intervals, especially with respect to applications in computer graphics, and of the extent a future IEEE interval standard should support these. It turns out that essentially all present applications of nonstandard intervals to practical problems can be matched by similarly efficient approaches based on standard intervals only. On the other hand, a number of applications were inspired by the use of nonstandard arithmetic. This suggests the requirement of a minimal support for nonstandard intervals, allowing implementations of nonstandard interval arithmetic to be compatible with the standard, while a full support by making one of the conflicting variants required seems not appropriate.

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

We discuss the solution to the interval algebraic system A x = b involving interval n n matrix A and interval vector b in directed interval arithmetic involving improper intervals. We givesome new relations for directed intervals, which form the basis for a directed interval matrix algebra. Using such relations we prove convergence of an iterative method, formulated by L. Kupriyanova, under simple explicit conditions on the interval matrix A. We propose an iterative numerical algorithm for the solution to a class of interval algebraic systems A x = b. Cramer-type formula for a special case of real matrices and interval right-hand side are used for the computation of an initial approximation for the iteration method. A Mathematica function performing the proposed algorithm is described.

The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus e cient solution of interval algebraic problems. This paper generalizes the distributive relations, known by now, on multiplication and addition of proper and improper intervals. A complete proof of the main results is presented, demonstrating an original technique based on functional notations and transition formulae between different interval structures. A variety of equivalent forms and different representations are discussed together with some examples. This paper is an extraction from [19] and will be updated permanently to include current improvements, generalizations and applications of the conditionally distributive relations. The second part of the paper is scheduled for the end of 2000 and will include several directions for the application of the generalized distributive relations. Keywords:

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

Abstract: Discrete mechanics deals with discrete mechanical systems, such as cellular automata, in which time proceeds in integer steps and the configuration space is discrete. Directly modeling discrete mechanical systems is a well known alternative to starting from a continuous setting, discretizing the model, and finally force the model to the finite alphabet of a computer. The time evolution of discrete dynamical systems, however, can be calculated exactly. In order to take into account imprecision in the input data and the need to accommodate a finite alphabet, extended interval analysis is introduced in the discrete mechanical systems formulation developed by Baez and Gilliam. It is shown how the Euler-Lagrange equation must be modified when working with interval input.

In this paper we present the theoretical base for some modifications in interval multiplication algorithms. A diversity of proposed implementation approaches is summarized along with a discussion on their costefficiency. It is shown that some improvements can be achieved by utilizing some properties of interval multiplication formulae and no special hardware support. Both conventional and extended interval multiplication operations are considered.