Is an interval the right result of arithmetic operations on intervals?

Abstract

For many scientists interval arithmetic (IA, I arithmetic) seems to be easy and simple. However, this is not true. Interval arithmetic is complicated. This is confirmed by the fact that, for years, new, alternative versions of this arithmetic have been created and published. These new versions tried to remove shortcomings and weaknesses of previously proposed options of the arithmetic, which decreased the prestige not only of interval arithmetic itself, but also of fuzzy arithmetic, which, to a great extent, is based on it. In our opinion, the main reason for the observed shortcomings of the present IA is the assumption that the direct result of arithmetic operations on intervals is also an interval. However, the interval is not a direct result but only a simplified representative (indicator) of the result. This hypothesis seems surprising, but investigations prove that it is true. The paper shows what conditions should be satisfied by the result of interval arithmetic operations to call it a "result", how great its dimensionality is, how to perform arithmetic operations and solve equations. Examples illustrate the proposed method of interval computations.

Inconsistencies in Numerical Simulations of Dynamical Systems Using Interval Arithmetic

“…Regarding the theoretical investigations, many versions of interval arithmetic have been suggested in literature. One of the most respected versions is widely recognized as the standard arithmetic interval (SIA) [Moore et al, 2009;Piegat & Landowski, 2017]. SIA has a long history, but it is usually attributed to Moore's development to a mature stage [Chalco-Cano et al, 2014].…”

“…Although, the success of interval arithmetic is widely accepted, some concerns have also been reported in literature. The authors in [Piegat & Landowski, 2017, 2015 have presented a considerable list of works in which the SIA has had some faults. The main worries are related to the overestimation of results, the dependence on the form of the mathematical expression used in problem solving, the calculation of interval borders and the lack of important properties, such as distributivity and multiplicative cancellation.…”

“…The main worries are related to the overestimation of results, the dependence on the form of the mathematical expression used in problem solving, the calculation of interval borders and the lack of important properties, such as distributivity and multiplicative cancellation. According to Piegat and Landowski [2017], these problems lead to controversial results.…”

A discussion on “On the solution of a class of fuzzy system of linear equations”

“…To introduce the horizontal concept of the fuzzy number, first an interval notation in Relative Distance Measure interval arithmetic (RDMIA) should be described. The detailed concept of RDMIA and differences between RDMIA and standard interval arithmetic (SIA) are presented in [5,[10][11][12]. In the RDMIA approach, an element x of interval X ¼ ½x; x is described using RDM variable a x 2 ½0; 1 as x ¼ x þ a x ðx À xÞ.…”

Solving different practical granular problems under the same system of equations

“…They give seven examples of different solution versions and show that the solution notion can have subjectively individual character resulting from interests of a person. However, it should be taken into account that also objective, mathematical interpretations and definitions of solutions (results) can be found (Popova 1998;Piegat and Landowski 2017b). Difficulty degree of a solved problem depends also on the uncertainty degree of information granules occurring in the problem.…”