On a class of parameterized solutions to interval parametric linear systems

TL;DR

This paper introduces a new method for obtaining parameterized solutions to interval parametric linear systems, compares its effectiveness with existing methods, and discusses its properties and limitations.

Contribution

The paper presents a novel approach for solving interval parametric linear systems and analyzes its properties and advantages over previous methods.

Findings

The new method provides effective solution enclosures.

It compares favorably with existing methods in certain cases.

Some previously proposed applications are not feasible in the general case.

Abstract

Presented is a new method yielding parameterized solution to an interval parametric linear system. Some properties of this method are discussed. The solution enclosure it provides is compared to the enclosures by other methods. It is shown that an application, proposed by other authors, cannot be done in the general case.

Tables

Table 1. Table 1: Bounds for □ ​ Σ □ Σ \square\Sigma in Example 1 , δ = 0.01 𝛿 0.01 \delta=0.01 , obtained by pKRank1 and PDM.

$𝐱$	outer		inner
	pKRank1	PDM	pKRank1	PDM
$x_1$	$[7.01522,7.16659]$	$[7.01480,7.16702]$	$[7.01736,7.16446]$	$[7.01777,7.16405]$
$x_2$	$[4.11780,4.24583]$	$[4.11736,4.24628]$	$[4.11987,4.24377]$	$[4.12030,4.24333]$
$x_3$	$[5.39374,5.51535]$	$[5.39331,5.51578]$	$[5.39567,5.51342]$	$[5.39609,5.51300]$
$x_4$	$[2.13805,2.22558]$	$[2.13770,2.22594]$	$[2.13962,2.22401]$	$[2.13997,2.22367]$
$x_5$	$[1.06046,1.12136]$	$[1.06017,1.12165]$	$[1.06171,1.12011]$	$[1.06200,1.11982]$

Table 2. Table 4: Monotonic dependence of the solution components on the interval parameters. Left: sign ( U ) 𝑈 (U) in the parameterized solution; Right: true dependence.

	$p_3$	$p_1$	$p_{12}$	$p_2$	$p_{22}$
$x_1$	-1	1	1	-1	-1
$x_2$	1	-1	-1	1	1
$x_3$	1	1	1	-1	-1

Equations

$$\begin{array}{rl}& A(p)x=a(p),p\in \mathbf{p}\in {\mathbb{IR}}^K,\\ & A(p):=A_0+\sum _{k=1}^K{p}_k{A}_k,a(p):=a_0+\sum _{k=1}^K{p}_k{a}_k,\end{array}$$

$${\Sigma }_{\mathrm{uni}}^p={\Sigma }_{\mathrm{uni}}(A(p),a(p),\mathbf{p}):=\{x\in {\mathbb{R}}^n\mid (\exists p\in \mathbf{p})(A(p)x=a(p))\}.$$

$$x(p,r)=\tilde{x}+V(\check{p}-p)+r,p\in \mathbf{p},r\in \mathbf{r}=[-\hat{r},\hat{r}],$$

$$\left(A_0+L{D}_{g(p_{{\pi }^{\prime }})}R\right)x=a_0+L{D}_{g(p_{{\pi }^{\prime }})}t+F{p}_{{\pi }^{\prime \prime }},p\in \mathbf{p},$$

$$\varrho \left(\left|(RCL)D_{g({\check{p}}_{{\pi }^{\prime }}-{\mathbf{p}}_{{\pi }^{\prime }})}\right|\right)<1,$$

$$\left(I-RCL{D}_{g(p_{{\pi }^{\prime }})}\right)y=R\check{x}-RCF{p}_{{\pi }^{\prime \prime }}-RCL{D}_{g(p_{{\pi }^{\prime }})}t,p\in [-\hat{p},\hat{p}],$$

$$\varrho (\sum _{i=1}^K\left|{\left(A(\check{p})\right)}^{-1}{A}_i\right|{\hat{p}}_i)<1,$$

$$x\in \check{x}-(CF)[-{\hat{p}}_{{\pi }^{\prime \prime }},{\hat{p}}_{{\pi }^{\prime \prime }}]+(CL)\left(D_{g([-{\hat{p}}_{{\pi }^{\prime }},{\hat{p}}_{{\pi }^{\prime }}])}|\mathbf{y}-t|\right).$$

$$x(p)=\check{x}-(CF)\left({\check{p}}_{{\pi }^{\prime \prime }}-p_{{\pi }^{\prime \prime }}\right)+\left(CL{D}_{|\mathbf{y}-t|}\right)g\left({\check{p}}_{{\pi }^{\prime }}-p_{{\pi }^{\prime }}\right),p\in \mathbf{p},$$

$$x(p,r)=\check{x}-(CF)({\check{p}}_{{\pi }^{\prime \prime }}-p_{{\pi }^{\prime \prime }})+(CL{D}_{\check{y}-t})g({\check{p}}_{{\pi }^{\prime }}-p_{{\pi }^{\prime }})+r,p\in \mathbf{p},r\in \mathbf{r}=[-\hat{r},\hat{r}],$$

$$x\in \mathbf{x}$$

$$=$$

$$x(p,y)$$

$$x(\mathbf{p},\mathbf{y})$$

$$\begin{gathered} \left(CL{D}_{|\check{y}-t|}\right)g({\check{p}}_{{\pi }^{\prime }}-{\mathbf{p}}_{{\pi }^{\prime }})-\left(CL{D}_{|\mathbf{y}-\check{y}|}\right)g({\check{p}}_{{\pi }^{\prime }}-{\mathbf{p}}_{{\pi }^{\prime }})\le \left(CL{D}_{|\mathbf{y}-t|}\right)g({\check{p}}_{{\pi }^{\prime }}-{\mathbf{p}}_{{\pi }^{\prime }})\le \\ \left(CL{D}_{|\check{y}-t|}\right)g({\check{p}}_{{\pi }^{\prime }}-{\mathbf{p}}_{{\pi }^{\prime }})+\left(CL{D}_{|\mathbf{y}-\check{y}|}\right)g({\check{p}}_{{\pi }^{\prime }}-{\mathbf{p}}_{{\pi }^{\prime }}). \end{gathered}$$

$$x(\mathbf{p},\mathbf{r})=x(\mathbf{p})=\mathbf{x},$$

$${\mathbf{x}}_{R1}=\check{x}-(CF)g({\check{p}}_{{\pi }^{\prime \prime }}-{\mathbf{p}}_{{\pi }^{\prime \prime }})+(CL)\left(D_{g({\check{p}}_{{\pi }^{\prime }}-{\mathbf{p}}_{{\pi }^{\prime }})}(\mathbf{y}-t)\right),$$

$${\mathbf{x}}_{NP}=C{a}_0+(CF){\mathbf{p}}_{{\pi }^{\prime \prime }}+(CL)\left(D_{g({\check{p}}_{{\pi }^{\prime }})}t+D_{g({\check{p}}_{{\pi }^{\prime }}-{\mathbf{p}}_{{\pi }^{\prime }})}(\mathbf{y}-t)\right),$$

$${\mathbf{v}}^{\mathrm{Low}}:=\check{x}+{\left(U\mathbf{q}\right)}^{-}+\mathbf{r},$$

$$x_{\ast }^{-}\in {\mathbf{v}}^{\mathrm{Low}},x_{\ast }^{+}\in {\mathbf{v}}^{\mathrm{Up}},$$

$${\left({\mathbf{x}}_{\mathrm{in}}(\mathbf{q})\right)}^{-}=\check{x}+U\mathbf{q}+\hat{r},{\left({\mathbf{x}}_{\mathrm{in}}(\mathbf{q})\right)}^{+}=\check{x}+U\mathbf{q}-\hat{r}.$$

$$\left(\begin{array}{ccccc}{p}_1+p_6& -p_6& 0& 0& 0\\ -p_6& p_2+p_6+p_7& -p_7& 0& 0\\ 0& -p_7& p_3+p_7+p_8& -p_8& 0\\ 0& 0& -p_8& p_4+p_8+p_9& -p_9\\ 0& 0& 0& -p_9& p_5+p_9\end{array}\right)x=\left(\begin{array}{c}10\\ 0\\ 10\\ 0\\ 0\end{array}\right)$$

$${\left(\square {\Sigma }_{uni}((\text{ref}pls))\right)}_i=[x_{\ast ,i}^{-},x_{\ast ,i}^{+}]=\left[x_i(p^{-s_i}),x_i(p^{s_i})\right]$$

$$x(p)$$

$$x_i(p^{-s})$$

$$x_i(p^s)$$

$$\check{p}-{\mathbf{p}}^{-s}=\check{p}-(\check{p}-s\hat{p})=s\hat{p},$$

$$x_i(p^{-s})$$

$$x_i(p^s)$$

$${\check{x}}_i-{\lambda }^{\prime \prime }({\mathbf{p}}_{{\pi }^{\prime \prime }}^{\prime }{)}_{-s^{\prime \prime }}+{\lambda }^{\prime }({\mathbf{p}}_{{\pi }^{\prime }}^{\prime }{)}_{-s^{\prime }}+[r_{\ast ,i}^{-},r_{\ast ,i}^{+}],$$

$${\check{x}}_i-|{\lambda }^{\prime \prime }|({\mathbf{p}}_{{\pi }^{\prime \prime }}^{\prime }{)}_{-s^{\prime \prime }{s}_{{\lambda }^{\prime \prime }}}+|{\lambda }^{\prime }|({\mathbf{p}}_{{\pi }^{\prime }}^{\prime }{)}_{-s^{\prime }{s}_{{\lambda }^{\prime }}}+[r_{\ast ,i}^{-},r_{\ast ,i}^{+}].$$

$$s_{{\lambda }^{\prime \prime }}:=\mathrm{sign}({\lambda }^{\prime \prime })=-s^{\prime \prime }$$

Full text

[TABLE]

• (i)

$\Sigma_{uni}\left(A(p),a(p),\mathbf{p}\right)$* and the solution set $\Sigma_{\rm uni}((\ref{eqY}))$ of system (5) are bounded*

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• (ii)

$\mathbf{y}\supseteq\Sigma_{\rm uni}((\ref{eqY}))$* is computable by methods that require (6) (cf.[9])*

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• (iii)

every $x\in\Sigma_{uni}\left(A(p),a(p),\mathbf{p}\right)$ satisfies

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The condition (6) is weaker and holds true when the condition (4) is satisfied, cf. [9]. The interval vector $\mathbf{y}$ in Theorem 1 (ii) can be obtained by a variety of numerical methods, many of them are discussed in [9].

• i)

there exists an united parameterized solution of the system (1), (3)

[TABLE]

where $\mathbf{y}\supseteq\Sigma_{\rm uni}((\ref{eqY}))$,

• ii)

with the same $\mathbf{y}$ used in (7) and in (8), interval evaluation $x\left(\mathbf{p}\right)$ of $x(p)$ is equal to the interval vector $\mathbf{x}$ obtained by Theorem 1.

3. Another method for parameterized solution. In [4] Kolev proposes a parameterized solution based: (a) on a generalized method of Neumaier and Pownuk [8] (abbreviated here as iGNP), and (b) on affine arithmetic. It is reported in [4] that the implementation of the proposed there parameterized method is eight times slower than the interval method iGNP from [8]. We suppose that the considerable slow down is due to the affine arithmetic which is used in both the implementation of iGNP and the parameterized solution derivation. It is discussed in [9] that the proposed there interval method (Theorem 1), abbreviated as iGRank1, is applicable to the same expanded class of parametric systems as the method iGNP and provides interval solution enclosure of the same (sometimes better) quality while overcoming some specific features that have to be considered in the implementation of iGNP. In what follows (Theorem 3) we propose a new parameterized solution, abbreviated as pKRank1, which is based on Theorem 1 and does not require affine arithmetic. It will be shown (Corollary 1) that the interval solution enclosures based on pKRank1 and pPRank1 (Theorem 2) are the same in exact arithmetic. Also, the parameterized solutions pKRank1 and pPRank1 are applicable to the interval method iGNP with $\mathbf{y}$ obtained by the latter (Proposition 1).

• (i)

There exists a parameterized solution enclosure of $\Sigma_{uni}((\ref{pls}))$

[TABLE]

where $\check{y}=R\check{x}$, $\mathbf{y}$ is that of Theorem 1 (ii), and $\hat{r}=|CL|D_{|\mathbf{y}-\check{y}|}g(\hat{p}_{\pi^{\prime}})$.

• (ii)

The interval evaluation $x(\mathbf{p},\mathbf{r})$ of the function in (i) is equal to the interval vector $\mathbf{x}$, obtained by Theorem 1, provided that both vectors are based on the same $\mathbf{y}$ of Theorem 1 (ii).

[TABLE]

Consider the right-hand side in (9) as an interval function $x(p,y)$ of $p\in\mathbf{p}$, $y\in\mathbf{y}$ and rearrange it as follows.

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The interval evaluation $x(\mathbf{p},\mathbf{y})$ of the last expression for $x(p,y)$ is

[TABLE]

the latter implying the representation (i). In order to prove (ii) we need to prove that $x(\mathbf{p},\mathbf{y})=(\ref{x2(p,y)})$. Since $|\check{y}-t|-|\mathbf{y}-\check{y}|\leq|\mathbf{y}-t|\leq|\check{y}-t|+|\mathbf{y}-\check{y}|$,

[TABLE]

Since $g(\check{p}_{\pi^{\prime}}-\mathbf{p}_{\pi^{\prime}})$ and $\left(CLD_{|\mathbf{y}-\check{y}|}\right)g(\check{p}_{\pi^{\prime}}-\mathbf{p}_{\pi^{\prime}})$ are symmetric interval vectors, $\left(CLD_{|\mathbf{y}-\check{y}|}\right)g(\check{p}_{\pi^{\prime}}-\mathbf{p}_{\pi^{\prime}})=-\left(CLD_{|\mathbf{y}-\check{y}|}\right)g(\check{p}_{\pi^{\prime}}-\mathbf{p}_{\pi^{\prime}})$, which implies the required assertion and (ii). ∎

[TABLE]

where $x(p,r)$ is that of Theorem 3, $x(p)$ is that of Theorem 2 and $\mathbf{x}$ is that of Theorem 1, provided that all computations are in exact arithmetic and both parameterized solutions use the same $\mathbf{y}$ of Theorem 1 (ii).

[TABLE]

obtained by Theorem 1, and the interval vector

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obtained by the implementation of [8, Theorem 4], are equal.

Proposition 1 implies that the two kinds of parameterized solutions, obtained by Theorem 2 and Theorem 3, are applicable to the generalized method of Neumaier and Pownuk [8] (iGNP) with $\mathbf{y}$ obtained by the latter method. Since [9] reports for better solution enclosures provided by Theorem 1 compared to iGNP for some problems, as well as for a better performance in a computing environment, it is expected that these advantages will be attributable to the above two kinds of parameterized solutions, obtained by Theorem 2 and Theorem 3. One advantage of the parameterized solutions involving the remainder term $r\in[-\hat{r},\hat{r}]$ is that they allow obtaining an inner estimate of the hull solution, presented in the next proposition.

[TABLE]

With $\mathbf{x}_{*}=\left[x_{*}^{-},x_{*}^{+}\right]=\square\Sigma^{p}_{\rm uni}((\ref{pls}))$,

[TABLE]

Define $\mathbf{x}_{\rm in}(\mathbf{q}):=\check{x}+U\mathbf{q}+\mathbf{r}_{-}$, where the interval evaluation is in Kaucher interval arithmetic [11] and $\mathbf{r}_{-}$ denotes $\text{\rm dual}(\mathbf{r})$. In classical interval arithmetic

[TABLE]

For every $i$, $1\leq i\leq n$, such that $\left(\mathbf{x}_{\rm in}(\mathbf{q})\right)^{-}_{i}>\left(\mathbf{x}_{\rm in}(\mathbf{q})\right)^{+}_{i}$, substitute $\left(\mathbf{x}_{\rm in}(\mathbf{q})\right)_{i}=\emptyset$. Then, it holds $\mathbf{x}_{\rm in}(\mathbf{q})\subseteq\square\Sigma^{p}_{\rm uni}((\ref{pls}))\subseteq x(\mathbf{q})$.

The methodology in [9] has an expanded scope of applications for systems involving rank one uncertainty structure. Next example demonstrates the advantage of the proposed here parameterized solution (Theorem 3) to a variety of parameterized solutions based on affine arithmetic and compared in [5].

[TABLE]

after [12], where the parameters vary within given intervals $p_{i}\in[1-\delta,1+\delta]$. This example is considered in [5, Example 5] and the outer solution enclosures obtained by six parameterized solutions based on affine arithmetic are compared to the direct parameterized method (abbreviated PDM) of [3]. Here we compare the parameterized inner and outer bounds for the solution set, obtained by the method of Theorem 3 and Proposition 2, and the corresponding bounds obtained by [3], thus comparing to the other six parameterized solutions considered in [5]. We present the results for the smallest uncertainty $\delta=0.01$ and the largest uncertainty $\delta=0.25$, considered in [5].

Table 1 presets inner and outer bounds obtained by us for $\delta=0.01$. These bounds are much sharper than, and can be compared to, the bounds obtained by three other parameterized solutions reported in [5, Table 4]. For the results in Table 1, Table 2 presents two measures of the quality of a solution enclosure: sharpness $O_{s}$ of the solution enclosure $\mathbf{x}_{\rm out}$ defined by $Q_{s}(\mathbf{x}_{\rm in},\mathbf{x}_{\rm out}):=\left\{0\text{ if }\mathbf{x}_{\rm in}=\emptyset,{\rm rad}(\mathbf{x}_{\rm in})/{\rm rad}(\mathbf{x}_{\rm out})\text{ otherwise}\right\}$, and percentage $O_{w}$ by which an interval $\mathbf{y}$ overestimates the interval $\mathbf{x}$, $\mathbf{x}\subseteq\mathbf{y}$, defined by $O_{w}(\mathbf{x},\mathbf{y}):=\left(1-{\rm rad}(\mathbf{x})/{\rm rad}(\mathbf{y})\right)100$. It is seen from Table 2 that the range of the sharpness measure is very close for the two methods pKRank1 and PDM. On the other hand, the percentage by which ${\rm PDM}_{\rm out}$ overestimates ${\rm pKRank1}_{\rm out}$ is between $0.55$% and $0.96$%. Table 3 presents the two measures of the quality of a solution enclosure for the case of large parameter uncertainties $\delta=0.25$ in Example 1. Although the percentage by which ${\rm PDM}_{\rm out}$ overestimates ${\rm pKRank1}_{\rm out}$ is more pronounced in this case, the ranges of sharpness is very close for these two methods and the methods compared in [5]. The first conclusion from Example 1 is that the methods based on condition (4) provide sharper solution enclosure than the methods based on condition (6) for systems with rank one uncertainty structure. The second important conclusion from this example is that the sharpness measure is not quite informative when comparing the solution enclosure of different methods in contrast to the percentage of overestimation.

4. On an application of pKRank1. In [4, Section 3] Kolev proposes to apply the parameterized solution of type Theorem 3 for determining $\square\Sigma^{p}$ of parametric systems involving rank one interval parameters. In this section we consider such an application in more details and demonstrate that this might be dangerous.

Let (3) be the equivalent, optimal rank one, representation of (1), which involves only rank one interval parameters. Let $A(\check{p})$ be nonsingular. Denote $C=A^{-1}(\check{p})$, $\check{x}=Ca(\check{p})$, $\check{y}=R\check{x}$, and let (4) hold true. Let $i$ be arbitrary, $1\leq i\leq n$, and let $\left(\Sigma_{uni}((\ref{pls}))\right)_{i}$ be monotone with respect to each parameter $p_{k}$, $k\in\pi=((\pi^{\prime\prime})^{\top},(\pi^{\prime})^{\top})^{\top}$, so that

[TABLE]

for an $s_{i}\in\{-1,1\}^{K}$ ($|s_{i}|=1\in\mathbb{R}^{K}$), where $-1$ means decreasing and $1$ – increasing. In order to simplify the notation, in what follows we will omit the subscript in $s_{i}$. Denote $s_{i}^{\top}=s^{\top}=(s^{\prime\prime},s^{\prime})$. We consider $\mathbf{x}$ in Theorem 1 as an interval evaluation of the function

[TABLE]

where $y(p)$ is the solution of the system (5). Replacing in this function the two endpoint vectors $p^{-s}$, respectively $p^{s}$, we obtain

[TABLE]

In order to simplify the presentation, we denote $\lambda^{\prime\prime}=C_{i\bullet}F$, $\lambda^{\prime}=C_{i\bullet}LD_{\check{y}-t}$. For $\mathbf{p}\in\mathbb{IR}$ and $s\in\{-1,1\}$, we have $\mathbf{p}^{-s}=\check{p}-s\hat{p}$, $\mathbf{p}^{s}=\check{p}+s\hat{p}$ and

[TABLE]

Thus, we have

[TABLE]

Now, in order to operate simultaneously with both $p^{-s}$, $p^{s}$, as well as simultaneously with both $x_{i}(p^{-s})$, $x_{i}(p^{s})$, we use Kaucher interval arithmetic and the relations between proper and improper intervals. Consider the following interval expression in $\mathbb{KR}$

[TABLE]

where $[r_{*,i}^{-},r_{*,i}^{+}]=\left[C_{i\bullet}LD_{y(s\hat{p}_{\pi})-\check{y}}\,(s^{\prime}\hat{p}_{\pi^{\prime}}),\;C_{i\bullet}LD_{y(-s\hat{p}_{\pi})-\check{y}}\,(-s^{\prime}\hat{p}_{\pi^{\prime}})\right]$ and $\mathbf{p}^{\prime}_{\pi}=[-\hat{p}_{\pi},\hat{p}_{\pi}]$. (11) is equivalent to

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If

[TABLE]

then

[TABLE]

similarly $\lambda^{\prime}(\mathbf{p}^{\prime}_{\pi^{\prime}})_{-s^{\prime}}=[\lambda^{\prime}s^{\prime}\hat{p}_{\pi^{\prime}},-\lambda^{\prime}s^{\prime}\hat{p}_{\pi^{\prime}}]$. Thus, (11), (12), become equivalently

[TABLE]

Thus, by (13), (12) is equivalent to  $\check{x}_{i}-\lambda^{\prime\prime}\mathbf{p}^{\prime}_{\pi^{\prime\prime}}+\lambda^{\prime}\mathbf{p}^{\prime}_{\pi^{\prime}}+[r_{*,i}^{-},r_{*,i}^{+}]$.

Now, we compare (12) to $x_{i}(\mathbf{p},\mathbf{r})$, where $x(p,r)$ is the parameterized solution from Theorem 3. The coefficients $\lambda^{\prime\prime}$, $\lambda^{\prime}$ are the same in both expressions. Consider three cases.

• •

Obviously, under (13), the first three terms in the two expressions are equivalent.

• •

If for some $k\in\pi$, $\lambda_{k}=0$, then the equality relation (14) is preserved and the equivalence between the first three terms in (12) and $x_{i}(\mathbf{p},\mathbf{r})$ is also preserved. However, $s_{k}\neq s_{\lambda_{k}}$ and $s_{k}$ cannot be inferred from $\lambda_{k}$.

• •

If for some $k\in\pi$, $0\neq s_{\lambda_{k}}\neq-s_{k}$, the equality relation (14) turns into inclusion (due to $|\lambda_{k}|\left(\mathbf{p}^{\prime}_{k}\right)_{-}\subseteq|\lambda_{k}|\left(\mathbf{p}^{\prime}_{k}\right)$), which contradicts to the initial assumption. In this case, the first three terms in (12) and $x_{i}(\mathbf{p},\mathbf{r})$ are equivalent but $s_{k}$ also cannot be inferred from $\lambda_{k}$.

Thus, we have proven the following theorem.

It follows from (14) that with given $s_{i}$, $\square\left(\Sigma_{\rm uni}((\ref{pls}))\right)_{i}$ can be obtained by solving (1) for $\mathbf{p}^{-s_{i}}$, respectively, for $\mathbf{p}^{s_{i}}$, $\mathbf{p}\in\mathbb{IR}^{K}$, or by solving the equivalent centered system

[TABLE]

for $p^{\prime}=s_{i}\hat{p}$, respectively, for $p^{\prime}=-s_{i}\hat{p}$, that is $(\mathbf{p}^{\prime})^{s_{i}}$, resp., $(\mathbf{p}^{\prime})^{-s_{i}}$.

Note, that the matrix $\left(CF,\;CLD_{\check{y}-t}\right)$ is different from the matrix $\left(-CF,\right.$ $\left.CLD_{\check{y}-t}\right)$ in the parameterized solution of Theorem 3. Note also, that the interval $[r_{*,i}^{-},r_{*,i}^{+}]\in\mathbb{IR}$ is not symmetric in general and differs from the symmetric interval $[-r,r]$ in Theorem 3.

By Proposition 1, it follows that (for rank one uncertainty structure of the system) the parameterized solution obtained by the method of [4] (based on affine arithmetic) will have the same signs of the parameter coefficients as the parameterized solution of Theorem 3. The example, considered in [4], illustrates the first case ($s_{\lambda}=s_{i}$) in the proof of Theorem 4. By the following example we illustrate the last case ($0\neq s_{\lambda_{k}}\neq-s_{k}$) in the proof of Theorem 4, which implies that the parameterized solution of type Theorem 3 cannot be used in general for determining the hull solution to interval parametric linear systems involving rank one parameters.

[TABLE]

The coefficient matrix for everyone of the parameters has rank one. Therefore, by [8, Corollary 3] the parametric united solution set has linear boundary and its interval hull is obtained for particular endpoints of the parameter intervals. Table 4 (right) presents global monotonicity (single entry) or local monotonicity (two entries respectively for the lower and the upper bounds) type of the parametric solution set with respect to interval parameters. An equivalent representation of the system is defined by

[TABLE]

Applying Theorem 3 we obtain a parameterized solution with reminder term $x(p,r)=\check{x}+U(p_{3},p_{1},p_{12},p_{2},p_{22})^{\top}+r$, where $r\in[-\hat{r},\hat{r}]$ and

[TABLE]

The sign of matrix $U^{\prime}=\left(CF,\;CLD_{\check{y}-t}\right)$ is presented in Table 4 (left). Comparing the left and right tables of monotonicity, it is clear that the sign of matrix $U^{\prime}$ does not represent the true monotonic dependence. Furthermore, no one of the elements of $U,U^{\prime}$ is zero. Applying the monotonicity defined by sign$(U^{\prime})$ we obtain an interval vector, which is contained in $\square\Sigma^{p}$. Therefore, using sign$(U^{\prime})$ is dangerous.

Acknowledgements This work is partly supported by the National Scientific Program “Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES)”, contract No DO1-205/23.11.2018, financed by the Ministry of Education and Science in Bulgaria.