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.

Abstract. Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of ‘Idempotent Mathematics ’ with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.

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.

Although interval arithmetic is increasingly used in combination with computer algebra and other methods, both approaches --- symbolic-algebraic and interval-arithmetic --- are used separately. Implementing symbolic interval arithmetic seems not suitable due to the exponential growth in the "size" of the end-points. In this paper we propose a methodology for "true" symbolic-algebraic manipulations on symbolic-numerical interval expressions involving interval variables instead of symbolic intervals. Due to the better algebraic properties, resembling to classical analysis, and the containment of classical interval arithmetic as a special case, we consider the algebraic extension of conventional interval arithmetic as an appropriate environment for solving interval algebraic problems. Based on the distributivity relations, a general framework for simplification of symbolicnumerical expressions involving intervals is given and some of the wider implications of the theory pertaining to inte...

The arithmetic on the extended set of proper and improper intervals is an algebraic completion of the conventional interval arithmetic and thus facilitates the explicit solution of certain interval algebraic problems. Due to the existence of inverse elements with respect to addition and multiplication operations certain interval algebraic equations can be solved by elementary algebraic transformations. The conditionally distributive relations between extended intervals allow that complicated interval algebraic equations, multi-incident on the unknown variable, be reduced to simpler ones. In this paper we give the general type of "pseudo-linear" interval equations in the extended interval arithmetic. The algebraic solutions to a pseudo-linear interval equation in one variable are studied. All numeric and parametric algebraic solutions, as well as the conditions for nonexistence of the algebraic solution to some basic types pseudo-linear interval equations in one variable are found. Some examples leading to algebraic solution of the equations under consideration and the extra functionalities for performing true symbolic-algebraic manipulations on interval formulae in a Mathematica package are discussed.

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...

In this report we deal with the correct formulation of a special extended interval arithmetic in the context of interval Newton like methods. We first demonstrate some of the problems arising from selected older definitions. Then we investigate the basic aim, concept, and properties important for defining a correct extended interval division. Finally, we give a proper way for defining the extended interval operations needed in our special context, and we prove their inclusion isotonicity. Additionally, we give some sample applications. We conclude with two updated implementations of our extended interval operations in the toolbox environments [2] and [3].

The paper presents a diagrammatic representation of a standard interval space (the so-called MR-diagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the IS-diagram representation devised earlier by the author to represent interval relations. First, the MR-diagram is defined together with appropriate graphical notions and con-structions for basic interval relations and operations. Second, diagrammatic constructions for all standard arithme-tic operations are presented. Several examples of the use of these constructions to aid reasoning about various simple, though nontrivial, properties of interval arithmetic are included in order to show how the representation facilitates both deeper understanding of the subject matter and reasoning about its properties.

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.