On the solution set of a linear equation with the right-hand side and operator given by intervals
While solving problems in various areas of knowledge by mathematical methods, the problem is often encountered of processing approximate data. The approximate character can be expressed, in particular, by interval representation of the results of the corresponding measurements. For solving problems of the type, research is intensively conducted in recent years within the so-called interval mathematics. One of the problems, which can be ranked as classical, is that of solving a system of linear equations with the coefficients and the right-hand side given by intervals; i.e., finding the set of solutions \(x\) to an equation (1) \(\mathbb{A} x = \mathbb{B}\), where \(\mathbb{A} \in I (\mathbb{R}^{m \times n})\) is a real interval \(m \times n\)-matrix, \(\mathbb{B} \in I (\mathbb{R}^m)\) is an \(m\)-dimensional interval vector, and \(x \in \mathbb{R}^n\). The solution set of system (1) can be defined in different ways depending on the quantifiers that bound the coefficients of the matrix and the right-hand side. We consider the following solution sets of equation (1), which are most frequent in the literature: \({\mathcal R}_1 = \{x \in \mathbb{R}^n \mid \exists C \in \mathbb{A} \;\exists c \in \mathbb{B}\;Cx = c\}\) is the total solution set, \({\mathcal R}_2 = \{x \in \mathbb{R}^n \mid \forall C \in \mathbb{A} \;\exists c \in \mathbb{B} \;Cx = c\}\) is the admissible solution set, \({\mathcal R}_3 = \{x \in \mathbb{R}^n \mid \forall c \in \mathbb{B}\;\exists C \in \mathbb{A}\;Cx = c\}\) is the set first considered while solving an interval modal control problem, \({\mathcal R}_4 = \{(x \in \mathbb{R}^n |\forall C \in \mathbb{A}\;\exists c \in \mathbb{B}\;Cx = c) \& (\forall d \in \mathbb{B}\;\exists D \in \mathbb{A}\;Dx = d)\}\) is the set of all pointwise algebraic interval solutions. The purpose of the present article is to describe these sets in terms of the endpoints of the intervals \(\mathbb{A}\), and \(\mathbb{B}\); moreover, we consider an analog of equation (1) in arbitrary ordered vector spaces.
set of all pointwise algebraic interval solutions
scientific article; zbMATH DE number 917116
scientific article