## 2 Problem Formulation  
The algorithm presented in the present section allows to determine a solution set of any structure described by a system of algebraic equations. This type of structure includes all bar structures in terms of linear mechanics and some plate and shell structures described by a set of ordinary differential equations, i.e., a circular plate and a shells in a rotational symmetry state. This section presents a mathematical description of the problem using hermetic mathematical language. This is necessary to keep the wording precise and to avoid misinterpretation. Additionally, flowchart illustrating the algorithm is presented in Section 3. Let the system of linear equations be given in the form  
\[\mathbf{A}\mathbf{x}=\mathbf{b}, \tag{1}\]  
in which \(\mathbf{A}\in\mathbb{R}^{s\times s}\) and \(\mathbf{b}\in\mathbb{R}^{s}\). Let this system depend on parameters, which are collected in vector \(\mathbf{p}\in\mathbb{R}^{c}\) in a way that \(\mathbf{A}=\mathbf{A}(\mathbf{p})\). We assume that parameters \(\mathbf{p}=(p_{1},\ldots,p_{e})\) are not precisely known, but their bounds are known. Hence, the parameters are described by the intervals  
\[\mathbf{p}\in[\mathbf{p}]=([p_{1}]\,,\ldots,[p_{e}]), \tag{2}\]  
which have positive length. Then the system (1) can be treated as family of linear systems  
\[\mathbf{A}(\mathbf{p})\mathbf{x}=\mathbf{b},\quad\mathbf{p}\in[\mathbf{p}]\,. \tag{3}\]  
In problems with precisely defined parameters, a given set of material, geometrical and load parameters implies one solution that may consist of many data, such as the displacements of all nodes of a truss structure. The presence of uncertain parameters in the formulation causes the possibility of obtaining many results depending on the values of these parameters. The set of solutions to a task with uncertain parameters is the set of all possible solutions to this task. Hence, the set of solutions to the family (3), called a united solution set, is  
\[\Xi_{\text{uni}}=\left\{\mathbf{x}\in\mathbb{R}^{s}:\exists\mathbf{p}\in[ \mathbf{p}]\,,\mathbf{A}(\mathbf{p})\mathbf{x}=\mathbf{b}\right\}. \tag{4}\]  
Let the parametric linear system (3) be decomposable, that is possible to express as sum  
\[\left(\mathbf{A}^{(0)}+\sum_{i=1}^{e}\mathbf{A}^{(i)}p_{i}\right)\mathbf{x}= \mathbf{b},\quad\mathbf{p}\in[\mathbf{p}]\,, \tag{5}\]in which matrices \(\mathbf{A}^{(0)}\), \(\mathbf{A}^{(i)}\) (\(i=1,\ldots,e\)), vector \(\mathbf{b}\) and intervals \([p_{i}]\) (\(i=1,\ldots,e\)) are given. For this class of systems the characterization  
\[\mathbf{x}\in\Xi_{\mathrm{uni}}\ \ \Leftrightarrow\ \ (\exists\mathbf{p}\in[ \mathbf{p}])\ \ \mathbf{0}=\left(\mathbf{b}-\mathbf{A}^{(0)}\mathbf{x}\right)-\sum_{i=1}^{e} \left(\mathbf{A}^{(i)}\mathbf{x}\right)p_{i}\ \ \Leftrightarrow\ \ \{\mathbf{0}\}\subset\mathcal{V}(\mathbf{x}), \tag{6}\]  
directly follows the form of solution set \(\Xi_{\mathrm{uni}}\), where  
\[ \begin{split}&\mathcal{V}(\mathbf{x})=\mathcal{V}^{(0)}(\mathbf{x })+\sum_{i=1}^{e}\mathcal{V}^{(i)}(\mathbf{x}),\\ &\mathcal{V}^{(0)}(\mathbf{x})=\mathbf{b}-\mathbf{A}^{(0)} \mathbf{x},\quad\mathcal{V}^{(i)}(\mathbf{x})=-\left(\mathbf{A}^{(i)}\mathbf{ x}\right)*\left[p_{i}\right].\end{split}  \tag{7}\]  
It is possible to express the relation \(\mathbf{x}\in\Xi_{\mathrm{uni}}\) with the use of inclusion \(\{\mathbf{0}\}\subset\mathcal{V}(\mathbf{x})\). Sets \(\mathcal{V}^{(i)}(\mathbf{x})\subset\mathbb{R}^{s}\) (\(i=1,\ldots,e\)) are line segments, therefore \(\mathcal{V}(\mathbf{x})\subset\mathbb{R}^{s}\) is a zonotope-the Minkowski sum of point \(\mathcal{V}^{(0)}(\mathbf{x})\) and those line segments. A zonotope is a convex polyhedron that has a point symmetry with respect to its center. It is possible to graphically interpret the inclusion \(\{\mathbf{0}\}\subset\mathcal{V}(\mathbf{x})\), which was presented in the paper [23]. Let \(\mathcal{X},\mathcal{Y}\subset\mathbb{R}^{s}\) be nonempty closed convex sets. Then  
\[