Global error bounds of the extended vertical linear complementarity problems for Dashnic–Zusmanovich matrices and Dashnic–Zusmanovich-B matrices

Abstract

Global error bounds of the extended vertical linear complementarity problems for Dashnic–Zusmanovich (DZ) matrices and Dashnic–Zusmanovich-B (DZ-B) matrices are presented, respectively. The obtained error bounds are sharper than those of Zhang et al. (Comput. Optim. Appl. 42(3):335–352, 2009) in some cases. Some numerical examples are given to illustrate the obtained results.

1 Introduction

The extended vertical linear complementarity problem (EVLCP) is to find a vector $x\in R^{n}$ such that

$$\begin{aligned} r(x): = \min ( {{M_{0}}x + {q_{0}},{M_{1}}x + {q_{1}}, \ldots ,{M_{k}}x + {q_{k}}} ), \end{aligned}$$

or to prove that there is no such vector x, where the min operator works componentwise for both vectors and matrices. It is denoted by $\operatorname{EVLCP}(M, q)$, where

$$ q = ( {{q_{0}},{q_{1}}, \ldots , {q_{k}}} ),\qquad {q_{l}} \in{R^{n}},\quad l = 0,1, \ldots ,k,$$

is a block vector and

$$ M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} ),\qquad {M_{l}} \in{R^{n \times n}},\quad l = 0,1, \ldots ,k,$$

is a block matrix. When $k = 1$, ${M_{0}} = I$, ${q_{0}} = 0$, the $\operatorname{EVLCP}(M, q)$ comes back to linear complementarity problems (LCP), and when ${M_{0}} = I$, ${q_{0}} = 0$, the $\operatorname{EVLCP}(M, q)$ reduces to vertical linear complementarity problems (VLCP) [2]. The extended vertical linear complementarity problems are widely used in optimization theory, control theory, neural network model, convergence analysis, sensitive analysis, verification of the solutions, and so on, see [1, 3–7].

Many scholars are interested in the research on the error of the solution for the $\operatorname{EVLCP}(M, q)$ including the LCP case. Various results on the solution and its error bounds for the $\operatorname{EVLCP}(M, q)$ have appeared recently, see [8–13]. For example, Gowda and Sznajder [14] extended the sufficient and necessary condition for the existence and uniqueness of the solution from the LCP to the EVLCP. Afterwards, Sznajder and Gowda [15] provided some equivalent forms for the condition above. Xiu and Zhang [16] extended the error bound for the LCP given by [6] to the EVLCP. Zhang et al. [1] extended the error bound of the LCP given by Chen and Xiang [8] to the general EVLCP by the row rearrangement technique and provided some computable error bounds for two types of special block matrices. However, these error bounds generally can not be calculated accurately because they involve computing the inverse of matrices. In order to overcome this shortcoming, in this paper, we continue to explore the extended vertical linear complementarity problems, and we propose new error bounds for the other types of special block matrices, named DZ matrices [17, 18] and DZ-B matrices [19], only relying on the elements of such matrices. The obtained results extend the corresponding results in [1]. The validity of new error bounds is theoretically guaranteed, and numerical examples show the validity of the new results.

The remainder of this paper is organized as follows. In Sect. 2, we recall some related definitions, theorems, lemmas, and notations, which will be used in the proof of this paper. In Sect. 3, we prove that each block in any row rearrangement of the block matrix $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ is a DZ matrix if each matrix ${M_{l}}$ ($l = 0,1, \ldots ,k$) is a DZ matrix, propose a computable error bound of the $\operatorname{EVLCP}(M, q)$ with each matrix in M being a DZ matrix, and present numerical examples to show the effectiveness of the new error bound. In Sect. 4, we prove that each block in any row rearrangement of the block matrix $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ is a DZ-B matrix if each matrix ${M_{l}}$ ($l = 0,1, \ldots ,k$) is a DZ-B matrix, provide a calculable error bound of the $\operatorname{EVLCP}(M, q)$ with each matrix in M being a DZ-B matrix, and use numerical examples to indicate the validity of the new error bound. Some conclusions are summarized in Sect. 5.

2 Preliminaries

In this section, we recall some theorems, definitions, lemmas, and notations. Given a matrix $A = ( {{a_{ij}}} ) \in{R^{n \times n}}$, $i,j \in N = \{ {1,2,\ldots ,n} \}$, denote

$$ {r_{i}} ( A ) = \sum_{j = 1,j \ne i}^{n} { \vert {{a_{ij}}} \vert },\quad \forall i \in N.$$

The first one is the existence and uniqueness condition of the solution for the $\operatorname{EVLCP}(M, q)$ given by Gowda and Sznajder [14].

Theorem 1

([14])

For any block vector q, the $\operatorname{EVLCP}(M, q)$ has a unique solution if and only if the block matrix $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ has the W-property, i.e.,

$$\begin{aligned} \min ( {{M_{0}}x,{M_{1}}x, \ldots ,{M_{k}}x} ) \le 0 \le \max ( {{M_{0}}x,{M_{1}}x, \ldots ,{M_{k}}x} ) \quad \Rightarrow \quad x = 0, \end{aligned}$$

(1)

where the min and max operators work componentwise for both vectors and matrices.

Using the row rearrangement technique, Zhang et al. [1] presented a sufficient and necessary condition for the block matrix M with the row W-property and proposed a global error bound for the $EVLCP(M, q)$.

Definition 1

([1])

The block matrix $M'= (M_{0}',M_{1}',\ldots ,M_{k}')$ is called a row rearrangement of $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ if, for any $i \in N$,

$$\begin{aligned} { \bigl( {M_{j}^{'}} \bigr)_{i.}} = { ( {{M_{{j_{i}}}}} )_{i.}} \in \bigl\{ {{{ ( {{M_{0}}} )}_{i.}},{{ ( {{M_{1}}} )}_{i.}},\ldots ,{{ ( {{M_{k}}} )}_{i.}}} \bigr\} = \bigl\{ {{{ \bigl( {M_{0}^{'}} \bigr)}_{i.}},{{ \bigl( {M_{1}^{'}} \bigr)}_{i.}},\ldots ,{{ \bigl( {M_{k}^{ '}} \bigr)}_{i.}}} \bigr\} , \end{aligned}$$

(2)

where $A_{i.}$ means the ith row of a given matrix A. This is also true for the block vectors q and $q^{'}$. Denote by $R ( M )$ and $R ( q )$ the set of all row rearrangements of M and q, respectively.

Lemma 1

([1])

The block matrix $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ has the row W-property if and only if $( {I - D} )M_{j}^{'} +D M_{l}^{'}$ is nonsingular for any two blocks $M_{j}^{'}$ and $M_{l}^{'}$ of ${M^{'}} \in R ( M )$ and for any $D = \operatorname{diag} ( {{d_{i}}} )$ with ${d_{i}} \in [ {0,1} ]$ ($i \in N$).

Theorem 2

([1])

Let ${x^{*} }$ be the solution of the $\operatorname{EVLCP}(M, q)$. If the block matrix $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ has the row W-property, then for any $x \in{R^{n}}$,

$$ \bigl\Vert {x - {x^{*}}} \bigr\Vert \le \alpha ( M ) \cdot \bigl\Vert {r ( x )} \bigr\Vert , $$

(3)

where

$$ \alpha ( M ): = \max_{{M^{'}} \in R ( M )} \max _{j < l \in \{ {0,1, \ldots ,k} \}} \max_{d \in{{ [ {0,1} ]}^{n}}} \bigl\Vert {{{ \bigl[ { ( {I - D} )M_{j}^{ '} + DM_{l}^{'}} \bigr]}^{ - 1}}} \bigr\Vert ,$$

$D = \operatorname{diag} ( {{d_{i}}} )$ is defined as Lemma 1, and ${M_{j}^{'}}$, ${M_{l}^{'}}$ are any two blocks in ${M^{'}} \in R ( M )$.

Obviously, the upper bound in (3) for $\Vert {x - {x^{*} }} \Vert$ cannot be calculated easily because it is difficult to compute $\Vert {{{ [ { ( {I - D} )M_{j}^{'} + DM_{l}^{ '}} ]}^{ - 1}}} \Vert$ precisely in general. Hence, some computable upper bounds for $\alpha ( M )$ are provided under various matrix norms by using the structural properties of matrices ${{M_{j}}}$, $j \in \{ {1,2,\ldots,k} \}$. When all ${M_{j}}$ are strictly diagonally dominant ($SDD$) matrices, Zhang et al. [1] gave a calculable upper bound for $\alpha (M)$ under the infinity norm (denoted by ${\alpha _{\infty }}(M)$) as follows. Here a matrix $A = ( {{a_{ij}}} ) \in{C^{n \times n}}$ is called an $SDD$ matrix if $\vert {{a_{ii}}} \vert > \sum_{j \ne i}^{n} { \vert {{a_{ij}}} \vert }$ for each $i \in N$.

Theorem 3

([1])

If ${M_{0}},{M_{1}}, \ldots ,{M_{k}}$ are $SDD$, and for each $i \in N$, ${ ( {{M_{j}}} )_{ii}}{ ( {{M_{l}}} )_{ii}} > 0$ for any $j < l \in \{ {0,1,\ldots ,k} \}$, then $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ has the row W-property and

$$ {\alpha _{\infty }}(M) \le \frac{1}{{{{\min }_{i \in N}} \{ {{{ ( {\min ( {{{\tilde{M}}_{0}}e,{{\tilde{M}}_{1}}e,\ldots ,{{\tilde{M}}_{k}}e} )} )}_{i}}} \}}},$$

where ${{\tilde{M}}_{i}}$ is the comparison matrix of ${M_{i}}$, i.e., ${ ( {{{\tilde{M}}_{i}}} )_{\tau \tau }} = \vert {{{ ( {{M_{i}}} )}_{\tau \tau }}} \vert$, ${ ( {{{ \tilde{M}}_{i}}} )_{\tau j}} = - \vert {{{ ( {{M_{i}}} )}_{\tau j}}} \vert$ for $\tau \ne j$, ${ ( {{M_{i}}} )_{\tau j}}$ is the element in the τth row and the jth column of ${{M_{i}}}$, and ${ ( {{{\tilde{M}}_{i}}} )_{\tau j}}$ is the element in the τth row and the jth column of ${{{\tilde{M}}_{i}}}$.

Theorem 4

([1])

If ${M_{0}},{M_{1}}, \ldots ,{M_{k}}$ are matrices with positive diagonal entries, and the spectral radius $\rho ( {\max ( {\Lambda _{0}^{ - 1} \vert {{B_{0}}} \vert ,\Lambda _{1}^{ - 1} \vert {{B_{1}}} \vert ,\ldots , \Lambda _{k}^{ - 1} \vert {{B_{k}}} \vert } )} ) < 1$, then $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ has the row W-property and

$$ {\alpha _{\infty }}(M) \le{ \Bigl\Vert {{{ \Bigl[ {I - \max _{i = 0,1,\ldots ,k} \bigl( {\Lambda _{i}^{ - 1} \vert {{B_{i}}} \vert } \bigr)} \Bigr]}^{ - 1}}\max _{i = 0,1, \ldots ,k} \bigl( {\Lambda _{i}^{ - 1}} \bigr)} \Bigr\Vert _{\infty }},$$

where ${\Lambda _{i}}$ is the diagonal part of ${M_{i}}$, ${B_{i}} = {\Lambda _{i}} - {M_{i}}$ for ${i = 0,1,\ldots ,k}$.

3 A global error bound for the EVLCP of Dashnic–Zusmanovich matrices

Dashnic–Zusmanovich matrix, as a subclass of the class of nonsingular P-matrix, was introduced by Dashnic and Zusmanovich [17] to upper bound for the infinity norm of its inverse matrix, whose definition and related conclusion are listed as follows.

Definition 2

([17])

A matrix $A = ( {{a_{ij}}} ) \in{C^{n \times n}}$ is called a Dashnic–Zusmanovich (DZ) matrix if there exists an index $i \in N$ such that for any $j \in N$, $j \ne i$,

$$ \vert {{a_{ii}}} \vert \bigl( { \vert {{a_{jj}}} \vert - {r_{j}} ( A ) + \vert {{a_{ji}}} \vert } \bigr) > {r_{i}} ( A ) \vert {{a_{ji}}} \vert .$$

Theorem 5

([18])

Let $A = ( {{a_{ij}}} ) \in{C^{n \times n}}$ be a DZ matrix. Then

$$ { \bigl\Vert {{A^{ - 1}}} \bigr\Vert _{\infty }} \le \max \Bigl\{ { \max_{j \in N,j \ne i} {\alpha _{1}} ( A ), \max _{j \in N,j \ne i} {\alpha _{2}} ( A )} \Bigr\} ,$$

where

$$\begin{aligned}& {\alpha _{1}} ( A ) = \frac{{ \vert {{a_{ji}}} \vert + \vert {{a_{ii}}} \vert }}{{ ( { \vert {{a_{jj}}} \vert - {r_{j}} ( A ) + \vert {{a_{ji}}} \vert } ) \vert {{a_{ii}}} \vert - \vert {{a_{ji}}} \vert {r_{i}} ( A )}}, \\& {\alpha _{2}} ( A ) = \frac{{ \vert {{a_{jj}}} \vert - {r_{j}} ( A ) + \vert {{a_{ji}}} \vert + {r_{i}} ( A )}}{{ ( { \vert {{a_{jj}}} \vert - {r_{j}} ( A ) + \vert {{a_{ji}}} \vert } ) \vert {{a_{ii}}} \vert - \vert {{a_{ji}}} \vert {r_{i}} ( A )}}. \end{aligned}$$

Next, we will propose an upper bound for ${\alpha _{\infty }} ( M )$ with each ${M_{l}}$ ($l = 0,1, \ldots ,k$) being a DZ matrix. Before that, some useful propositions are provided below.

Proposition 1

Let $A = ( {{a_{ij}}} ) \in{C^{n \times n}}$ and $B = ( {{b_{ij}}} ) \in{C^{n \times n}}$ be all DZ matrices with positive diagonal elements. If there exists an index $i\in N$ such that, for any $j\in N$, $j\neq i$, ${a_{ij}}{b_{ij}} \geq 0$ (or ${a_{ji}}{b_{ji}} \geq 0$), and

$$\begin{aligned}& \vert {{a_{ii}}} \vert \bigl( { \vert {{a_{jj}}} \vert - {r_{j}} ( A ) + \vert {{a_{ji}}} \vert } \bigr) > {r_{i}} ( A ) \vert {{a_{ji}}} \vert ,\qquad \vert {{b_{ii}}} \vert \bigl( { \vert {{b_{jj}}} \vert - {r_{j}} ( B ) + \vert {{b_{ji}}} \vert } \bigr) > {r_{i}} ( B ) \vert {{b_{ji}}} \vert ,\\& \vert {{a_{ii}}} \vert \bigl( { \vert {{b_{jj}}} \vert - {r_{j}} ( B ) + \vert {{b_{ji}}} \vert } \bigr) > {r_{i}} ( A ) \vert {{b_{ji}}} \vert ,\qquad \vert {{b_{ii}}} \vert \bigl( { \vert {{a_{jj}}} \vert - {r_{j}} ( A ) + \vert {{a_{ji}}} \vert } \bigr) > {r_{i}} ( B ) \vert {{a_{ji}}} \vert , \end{aligned}$$

then $( {I - D} )A + DB$ is also a DZ matrix for any $D = \operatorname{diag} ( {{d_{t}}} )$ with ${d_{t}} \in [ {0,1} ]$ ($t \in N$).

Proof

Both A and B are DZ matrices, and there exists $i\in N$ such that, for any $j\in N$, $j\neq i$,

$$ \vert {{a_{ii}}} \vert \bigl( { \vert {{a_{jj}}} \vert - {r_{j}} ( A ) + \vert {{a_{ji}}} \vert } \bigr) > {r_{i}} ( A ) \vert {{a_{ji}}} \vert ,\qquad \vert {{b_{ii}}} \vert \bigl( { \vert {{b_{jj}}} \vert - {r_{j}} ( B ) + \vert {{b_{ji}}} \vert } \bigr) > {r_{i}} ( B ) \vert {{b_{ji}}} \vert .$$

Note that ${d_{i}} \in [ {0,1} ]$, then $1 - {d_{i}} \ge 0$ and ${d_{i}} \ge 0$, and they are not equal to 0 at the same time. Let $( {I - D} )A + DB = C = { ( {{c_{ij}}} )}$, then

$$\begin{aligned}& \vert {{c_{ii}}} \vert = ( {1 - {d_{i}}} ) \vert {{a_{ii}}} \vert + {d_{i}} \vert {{b_{ii}}} \vert ,\qquad \vert {{c_{ij}}} \vert = ( {1 - {d_{i}}} ) \vert {{a_{ij}}} \vert + {d_{i}} \vert {{b_{ij}}} \vert ,\\& {r_{i}} ( C ) = \sum_{j \ne i}^{n} { \bigl( { ( {1 - {d_{i}}} ) \vert {{a_{ij}}} \vert + {d_{i}} \vert {{b_{ij}}} \vert } \bigr)} = ( {1 - {d_{i}}} ){r_{i}} ( A ) + {d_{i}} {r_{i}} ( B ). \end{aligned}$$

Hence, we get

$$\begin{aligned}& \vert {{c_{jj}}} \vert - {r_{j}} ( C ) + \vert {{c_{ji}}} \vert \\& \quad = ( {1 - {d_{j}}} ) \vert {{a_{jj}}} \vert + {d_{j}} \vert {{b_{jj}}} \vert - \bigl( { ( {1 - {d_{j}}} ){r_{j}} ( A ) + {d_{j}} {r_{j}} ( B )} \bigr) + ( {1 - {d_{j}}} ) \vert {{a_{ji}}} \vert + {d_{j}} \vert {{b_{ji}}} \vert \\& \quad = ( {1 - {d_{j}}} ) \bigl[ { \vert {{a_{jj}}} \vert - {r_{j}} ( A ) \vert {{a_{ji}}} \vert } \bigr] + {d_{j}} \bigl[ { \vert {{b_{jj}}} \vert - {r_{j}} ( B ) + \vert {{b_{ji}}} \vert } \bigr], \end{aligned}$$

and

$$\begin{aligned}& \vert {{c_{ii}}} \vert \bigl( { \vert {{c_{jj}}} \vert - {r_{j}} ( C ) + \vert {{c_{ji}}} \vert } \bigr) \\& \quad = \bigl[ { ( {1 - {d_{i}}} ) \vert {{a_{ii}}} \vert + {d_{i}} \vert {{b_{ii}}} \vert } \bigr]\times \bigl[ { ( {1 - {d_{j}}} ) \bigl( { \vert {{a_{jj}}} \vert - {r_{j}} ( A ) \vert {{a_{ji}}} \vert } \bigr) + {d_{j}} \bigl( { \vert {{b_{jj}}} \vert - {r_{j}} ( B ) + \vert {{b_{ji}}} \vert } \bigr)} \bigr] \\& \quad = ( {1 - {d_{i}}} ) ( {1 - {d_{j}}} ) \bigl( { \vert {{a_{jj}}} \vert - {r_{j}} ( A ) \vert {{a_{ji}}} \vert } \bigr) \vert {{a_{ii}}} \vert + ( {1 - {d_{i}}} ){d_{j}} \bigl( { \vert {{b_{jj}}} \vert - {r_{j}} ( B ) + \vert {{b_{ji}}} \vert } \bigr) \vert {{a_{ii}}} \vert \\& \qquad {} + {d_{i}} ( {1 - {d_{j}}} ) \bigl( { \vert {{a_{jj}}} \vert - {r_{j}} ( A ) \vert {{a_{ji}}} \vert } \bigr) \vert {{b_{ii}}} \vert + {d_{i}} {d_{j}} \bigl( { \vert {{b_{jj}}} \vert - {r_{j}} ( B ) + \vert {{b_{ji}}} \vert } \bigr) \vert {{b_{ii}}} \vert \\& \quad > ( {1 - {d_{i}}} ) ( {1 - {d_{j}}} ){r_{i}} ( A ) \vert {{a_{ji}}} \vert + ( {1 - {d_{i}}} ){d_{j}} {r_{i}} ( A ) \vert {{b_{ji}}} \vert \\& \qquad {} + {d_{i}} ( {1 - {d_{j}}} ){r_{i}} ( B ) \vert {{a_{ji}}} \vert + {d_{i}} {d_{j}} {r_{i}} ( B ) \vert {{b_{ji}}} \vert \\& \quad = \bigl[ { ( {1 - {d_{i}}} ){r_{i}} ( A ) + {d_{i}} {r_{i}} ( B )} \bigr] \bigl[ { ( {1 - {d_{j}}} ) \vert {{a_{ji}}} \vert + {d_{j}} \vert {{b_{ji}}} \vert } \bigr] \\& \quad = {r_{i}} ( C ) \vert {{c_{ji}}} \vert , \end{aligned}$$

that is,

$$ \vert {{c_{ii}}} \vert \bigl( { \vert {{c_{jj}}} \vert - {r_{j}} ( C ) + \vert {{c_{ji}}} \vert } \bigr) > {r_{i}} ( C ) \vert {{c_{ji}}} \vert .$$

From Definition 2, the conclusion follows. □

Based on Proposition 1, Lemma 1, and the fact that a DZ matrix is nonsingular, the block matrix composed of DZ matrices has the row W-property.

Proposition 2

If ${{M_{0}},{M_{1}}, \ldots ,{M_{k}}}$ are DZ matrices and each ${{{M_{{l_{i}}}}}}$ (${l_{i}} = 0,1,\ldots ,k$) satisfies the hypotheses of Proposition 1, then each $M_{l}^{'}$ in ${M^{'}} \in R ( M )$ is a DZ matrix, and consequently, $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ has the row W-property.

Proof

By Definition 1, for the ith row ${ ( {M_{l}^{'}} )_{i}}$ of ${M_{l}^{'}}$, $i \in N$, there exists ${l_{i}} \in \{ {0,1,\ldots ,k} \}$ such that ${ ( {M_{l}^{'}} )_{i}} = { ( {{M_{{l_{i}}}}} )_{i}}$, i.e., ${ ( {{M_{{l_{i}}}}} )_{ii}} = { ( {M_{l}^{'}} )_{ii}}$, ${r_{i}} ( {{M_{{l_{i}}}}} ) = {r_{i}} ( {M_{l}^{'}} )$. Since ${{M_{{l_{i}}}}}$ is a DZ matrix, then

$$ \bigl\vert {{{ ( {{M_{{l_{i}}}}} )}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ ( {{M_{{l_{i}}}}} )}_{jj}}} \bigr\vert - {r_{j}} ( {{M_{{l_{i}}}}} ) + \bigl\vert {{{ ( {{M_{{l_{i}}}}} )}_{ji}}} \bigr\vert } \bigr) > {r_{i}} ( {{M_{{l_{i}}}}} ) \bigl\vert {{{ ( {{M_{{l_{i}}}}} )}_{ji}}} \bigr\vert ,\quad j \ne i, j\in N,$$

that is,

$$ \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{jj}}} \bigr\vert - {r_{j}} \bigl( {M_{l}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ji}}} \bigr\vert } \bigr) > {r_{i}} \bigl( {M_{l}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ji}}} \bigr\vert , \quad j \ne i, j\in N.$$

By Definition 2, ${M_{l}^{'}}$ is a DZ matrix.

Let $M_{j}^{'}$, $M_{l}^{'}$ be any two blocks in ${M^{'}} \in R ( M )$, then $M_{j}^{'}$, $M_{l}^{'}$ are all DZ matrices. Based on Proposition 1, $( {I - D} )M_{j}^{'} + DM_{l}^{'}$ is a DZ matrix for any $D = \operatorname{diag} ( {{d_{i}}} )$, ${d_{i}} \in [ {0,1} ]$ ($i \in N$). Thus $( {I - D} )M_{j}^{'} + DM_{l}^{'}$ is nonsingular. By Lemma 1, the block matrix $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ has the row W-property. □

Theorem 6

Let $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$, and let each ${M_{p}}$ ($p = 0,1,2,\ldots ,k$) be a DZ matrix with positive diagonal elements and satisfy the hypotheses of Proposition 1. Then

$$ \alpha{(M)_{\infty }} \le 2\max \biggl\{ { \frac{{\max \{ {\varphi _{\max }^{p}} \}}}{{\min \{ {\beta _{\min }^{p}} \}}}, \frac{{\max \{ {\omega _{\max }^{p}} \}}}{{\min \{ {\beta _{\min }^{p}} \}}}} \biggr\} ,$$

where

$$\begin{aligned}& \varphi _{\max }^{p} = \max_{\tau \ne i \in N} \bigl\{ { \varphi _{\tau i}^{p}} \bigr\} ,\qquad \varphi _{\tau i}^{p} = \bigl\vert {{{ ( {{M_{p}}} )}_{\tau i}}} \bigr\vert + \bigl\vert {{{ ( {{M_{p}}} )}_{ii}}} \bigr\vert ,\qquad \beta _{\min }^{p} = \min _{\tau \ne i \in N} \bigl\{ {\beta _{i\tau }^{p}} \bigr\} ,\\& \omega _{\max }^{p} = \max_{\tau \ne i \in N} \bigl\{ { \omega _{i\tau }^{p}} \bigr\} , \qquad \omega _{i\tau }^{p} = \bigl\vert {{{ ( {{M_{p}}} )}_{\tau \tau }}} \bigr\vert - {r_{ \tau }} ( {{M_{p}}} ) + \bigl\vert {{{ ( {{M_{p}}} )}_{\tau i}}} \bigr\vert + {r_{i}} ( {{M_{p}}} ),\\& \beta _{i\tau }^{p} = \bigl\vert {{{ ( {{M_{p}}} )}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ ( {{M_{p}}} )}_{\tau \tau }}} \bigr\vert - {r_{\tau }} ( {{M_{p}}} ) + \bigl\vert {{{ ( {{M_{p}}} )}_{\tau i}}} \bigr\vert } \bigr) - {r_{i}} ( {{M_{p}}} ) \bigl\vert {{{ ( {{M_{p}}} )}_{\tau i}}} \bigr\vert . \end{aligned}$$

Proof

For any two blocks $M_{j}^{'}$, $M_{l}^{'}$ in ${M^{'}} \in R ( M )$ and any $D = \operatorname{diag} ( {{d_{i}}} )$ with ${d_{i}} \in [ {0,1} ]$ ($i \in N$), let ${M_{D}} = { ( {{m_{ij}}} )} = ( {I - D} )M_{j}^{ '} + DM_{l}^{'}$. By Propositions 1 and 2, it holds that $M_{j}^{'}$, $M_{l}^{'}$, and ${M_{D}}$ are all DZ matrices with positive diagonal elements. By Theorem 5, we have that

$$ { \bigl\Vert {M_{D}^{ - 1}} \bigr\Vert _{\infty }}\le \max \Bigl\{ { \max_{\tau \in N,\tau \ne i} {\alpha _{1}} ( {{M_{D}}} ),\max_{\tau \in N,\tau \ne i} {\alpha _{2}} ( {{M_{D}}} )} \Bigr\} $$

holds for each matrix $D = \operatorname{diag} ( {{d_{i}}} )$ with ${d_{i}} \in [ {0,1} ]$ ($i \in N$), where

$$\begin{aligned}& {\alpha _{1}} ( {{M_{D}}} ) = \frac{{ \vert {{m_{\tau i}}} \vert + \vert {{m_{ii}}} \vert }}{{ ( { \vert {{m_{\tau \tau }}} \vert - {r_{\tau }} ( {{M_{D}}} ) + \vert {{m_{\tau i}}} \vert } ) \vert {{m_{ii}}} \vert - \vert {{m_{\tau i}}} \vert {r_{i}} ( {{M_{D}}} )}},\\& {\alpha _{2}} ( A ) = \frac{{ \vert {{m_{\tau \tau }}} \vert - {r_{\tau }} ( {{M_{D}}} ) + \vert {{m_{\tau i}}} \vert + {r_{i}} ( {{M_{D}}} )}}{{ ( { \vert {{m_{\tau \tau }}} \vert - {r_{\tau }} ( {{M_{D}}} ) + \vert {{m_{\tau i}}} \vert } ) \vert {{m_{ii}}} \vert - \vert {{m_{\tau i}}} \vert {r_{i}} ( {{M_{D}}} )}}. \end{aligned}$$

Hence, it holds that

$$\begin{aligned} \vert {{m_{\tau i}}} \vert + \vert {{m_{ii}}} \vert =& ( {1 - {d_{\tau }}} ) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert + {d_{\tau }} \bigl\vert {{{ \bigl( {M_{l}^{ '}} \bigr)}_{\tau i}}} \bigr\vert + ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{ii}}} \bigr\vert + {d_{i}} \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \\ < & \bigl[ { \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert + \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{ii}}} \bigr\vert } \bigr] + \bigl[ { \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert + \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ii}}} \bigr\vert } \bigr] \\ =& { \bigl( {\varphi _{\tau i}^{j}} \bigr)^{'}} + { \bigl( { \varphi _{\tau i}^{l}} \bigr)^{'}} \\ =& 2\max_{t = j,l} { \bigl( {\varphi _{\max }^{t}} \bigr)^{'}}, \end{aligned}$$

where

$$ { \bigl( {\varphi _{\max }^{t}} \bigr)^{'}} = \max _{\tau \ne i \in N} \bigl\{ {{{ \bigl( {\varphi _{\tau i}^{l}} \bigr)}^{'}}} \bigr\} ,\qquad { \bigl( {\varphi _{\tau i}^{l}} \bigr)^{'}} = \bigl\vert {{{ \bigl( {M_{t}^{'}} \bigr)}_{ \tau i}}} \bigr\vert + \bigl\vert {{{ \bigl( {M_{t}^{'}} \bigr)}_{ii}}} \bigr\vert ,\quad t = j, l.$$

Further, we get

$$\begin{aligned}& \bigl( { \vert {{m_{\tau \tau }}} \vert - {r_{\tau }} ( {{M_{D}}} ) + \vert {{m_{\tau i}}} \vert } \bigr) \vert {{m_{ii}}} \vert - \vert {{m_{\tau i}}} \vert {r_{i}} ( {{M_{D}}} ) \\& \quad = \bigl[ { ( {1 - {d_{\tau }}} ) \bigl( { \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{ \tau }} \bigl( {M_{j}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) + {d_{\tau }} \bigl( { \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{l}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{l}^{ '}} \bigr)}_{\tau i}}} \bigr\vert } \bigr)} \bigr] \\& \qquad {} \times \bigl[ { ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{ii}}} \bigr\vert + {d_{i}} \bigl\vert {{{ \bigl( {M_{l}^{ '}} \bigr)}_{ii}}} \bigr\vert } \bigr] \\& \qquad {} - \bigl[ { ( {1 - {d_{\tau }}} ) \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert + {d_{\tau }} \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] \bigl[ { ( {1 - {d_{i}}} ){r_{i}} \bigl( {M_{j}^{'}} \bigr) + {d_{i}} {r_{i}} \bigl( {M_{l}^{'}} \bigr)} \bigr] \\& \quad = ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{j}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} + ( {1 - {d_{\tau }}} ){d_{i}} \bigl\vert {{{ \bigl( {M_{l}^{ '}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{j}^{ '}} \bigr) + \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{ \tau i}}} \bigr\vert } \bigr) \\& \qquad {} + {d_{\tau }} ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{l}^{ '}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{l}^{ '}} \bigr) + \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ \tau i}}} \bigr\vert } \bigr) \\& \qquad {} + {d_{\tau }} {d_{i}} \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{l}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} - ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){r_{i}} \bigl( {M_{j}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert - ( {1 - {d_{\tau }}} ){d_{i}} {r_{i}} \bigl( {M_{l}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert \\& \qquad {} - {d_{\tau }} ( {1 - {d_{i}}} ){r_{i}} \bigl( {M_{j}^{ '}} \bigr) \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert - {d_{\tau }} {d_{i}} {r_{i}} \bigl( {M_{l}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert \\& \quad > ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{j}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {}+ ( {1 - {d_{\tau }}} ){d_{i}} {r_{i}} \bigl( {M_{l}^{ '}} \bigr) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert + {d_{\tau }} ( {1 - {d_{i}}} ){r_{i}} \bigl( {M_{j}^{ '}} \bigr) \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert \\& \qquad {} + {d_{\tau }} {d_{i}} \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{l}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} - ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){r_{i}} \bigl( {M_{j}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert - ( {1 - {d_{\tau }}} ){d_{i}} {r_{i}} \bigl( {M_{l}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert \\& \qquad {} - {d_{\tau }} ( {1 - {d_{i}}} ){r_{i}} \bigl( {M_{j}^{ '}} \bigr) \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert - {d_{\tau }} {d_{i}} {r_{i}} \bigl( {M_{l}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert \\& \quad = ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{j}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} + {d_{\tau }} {d_{i}} \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{l}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {}- ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){r_{i}} \bigl( {M_{j}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert - {d_{\tau }} {d_{i}} {r_{i}} \bigl( {M_{l}^{ '}} \bigr) \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert \\& \quad = ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ) \bigl[ { \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{j}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) - {r_{i}} \bigl( {M_{j}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] \\& \qquad {} + {d_{\tau }} {d_{i}} \bigl[ { \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{l}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) - {r_{i}} \bigl( {M_{l}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] \\& \quad = ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){ \bigl( {\beta _{i\tau }^{j}} \bigr)^{'}} + {d_{\tau }} {d_{i}} { \bigl( {\beta _{i\tau }^{l}} \bigr)^{'}} \\& \quad \ge ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){ \bigl( {\beta _{\min }^{j}} \bigr)^{'}} + {d_{\tau }} {d_{i}} { \bigl( {\beta _{\min }^{l}} \bigr)^{'}}, \end{aligned}$$

where

$$\begin{aligned}& { \bigl( {\beta _{\min }^{j}} \bigr)^{'}} = \mathop{ \min } _{\tau \ne i \in N} { \bigl( {\beta _{i\tau }^{t}} \bigr)^{ '}},\\& { \bigl( {\beta _{i\tau }^{t}} \bigr)^{'}} = \bigl\vert {{{ \bigl( {M_{t}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {M_{t}^{ '}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{t}^{ '}} \bigr) + \bigl\vert {{{ \bigl( {M_{t}^{'}} \bigr)}_{ \tau i}}} \bigr\vert } \bigr) - {r_{i}} \bigl( {M_{t}^{'}} \bigr) \bigl\vert {{{ \bigl( {M_{t}^{'}} \bigr)}_{\tau i}}} \bigr\vert ,\quad t = j, l. \end{aligned}$$

Therefore, we have

$$\begin{aligned} {\alpha _{1}} ( {{M_{D}}} ) =& \frac{{ \vert {{m_{\tau i}}} \vert + \vert {{m_{ii}}} \vert }}{{ ( { \vert {{m_{\tau \tau }}} \vert - {r_{\tau }} ( {{M_{D}}} ) + \vert {{m_{\tau i}}} \vert } ) \vert {{m_{ii}}} \vert - \vert {{m_{\tau i}}} \vert {r_{i}} ( {{M_{D}}} )}} \\ \ge & \frac{{2\max_{t = j,l} {{ ( {\varphi _{\max }^{t}} )}^{'}}}}{{ ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){{ ( {\beta _{\min }^{j}} )}^{'}} + {d_{\tau }}{d_{i}}{{ ( {\beta _{\min }^{l}} )}^{'}}}} \\ \ge & \frac{{2\max_{t = j,l} {{ ( {\varphi _{\max }^{t}} )}^{'}}}}{{\min \{ {{{ ( {\beta _{\min }^{j}} )}^{'}},{{ ( {\beta _{\min }^{l}} )}^{'}}} \}}} \end{aligned}$$

and

$$\begin{aligned}& \vert {{m_{\tau \tau }}} \vert - {r_{\tau }} ( {{M_{D}}} ) + \vert {{m_{\tau i}}} \vert + {r_{i}} ( {{M_{D}}} ) \\& \quad = \bigl[ { ( {1 - {d_{\tau }}} ) \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{\tau \tau }}} \bigr\vert + {d_{\tau }} \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert } \bigr] - \bigl[ { ( {1 - {d_{\tau }}} ){r_{\tau }} \bigl( {M_{j}^{ '}} \bigr) + {d_{\tau }} {r_{\tau }} \bigl( {M_{l}^{'}} \bigr)} \bigr] \\& \qquad {} + \bigl[ { ( {1 - {d_{\tau }}} ) \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert + {d_{\tau }} \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] + \bigl[ { ( {1 - {d_{i}}} ){r_{i}} \bigl( {M_{j}^{'}} \bigr) + {d_{i}} {r_{i}} \bigl( {M_{l}^{'}} \bigr)} \bigr] \\& \quad = ( {1 - {d_{\tau }}} ) \bigl[ { \bigl\vert {{{ \bigl( {M_{j}^{ '}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{j}^{ '}} \bigr) + \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{ \tau i}}} \bigr\vert } \bigr] \\& \qquad {} + {d_{\tau }} \bigl[ { \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{ \tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{l}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] + \bigl[ { ( {1 - {d_{i}}} ){r_{i}} \bigl( {M_{j}^{ '}} \bigr) + {d_{i}} {r_{i}} \bigl( {M_{l}^{'}} \bigr)} \bigr] \\& \quad < \bigl[ { \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{j}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert + {r_{i}} \bigl( {M_{j}^{'}} \bigr)} \bigr] \\& \qquad {} + \bigl[ { \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} \bigl( {M_{l}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert + {r_{i}} \bigl( {M_{l}^{'}} \bigr)} \bigr] \\& \quad = { \bigl( {\omega _{i\tau }^{j}} \bigr)^{'}} + { \bigl( { \omega _{i\tau }^{l}} \bigr)^{'}} \\& \quad \le 2\max_{t = j,l} { \bigl( {\omega _{\max }^{t}} \bigr)^{'}}, \end{aligned}$$

where

$$\begin{aligned}& { \bigl( {\omega _{\max }^{t}} \bigr)^{'}} = \max _{\tau \ne i \in N} { \bigl( {\omega _{i\tau }^{t}} \bigr)^{ '}},\\& { \bigl( {\omega _{i\tau }^{t}} \bigr)^{'}} = \bigl\vert {{{ \bigl( {M_{t}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{ \tau }} \bigl( {M_{t}^{'}} \bigr) + \bigl\vert {{{ \bigl( {M_{t}^{ '}} \bigr)}_{\tau i}}} \bigr\vert + {r_{i}} \bigl( {M_{t}^{'}} \bigr),\quad t = j, l. \end{aligned}$$

So, it holds that

$$\begin{aligned} {\alpha _{2}} ( A ) =& \frac{{ \vert {{m_{\tau \tau }}} \vert - {r_{\tau }} ( {{M_{D}}} ) + \vert {{m_{\tau i}}} \vert + {r_{i}} ( {{M_{D}}} )}}{{ ( { \vert {{m_{\tau \tau }}} \vert - {r_{\tau }} ( {{M_{D}}} ) + \vert {{m_{\tau i}}} \vert } ) \vert {{m_{ii}}} \vert - \vert {{m_{\tau i}}} \vert {r_{i}} ( {{M_{D}}} )}} \\ \le & \frac{{2\max_{t = j,l} {{ ( {\omega _{\max }^{t}} )}^{'}}}}{{ ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){{ ( {\beta _{\min }^{j}} )}^{'}} + {d_{\tau }}{d_{i}}{{ ( {\beta _{\min }^{l}} )}^{'}}}} \\ \le & \frac{{2\max_{t = j,l} {{ ( {\omega _{\max }^{t}} )}^{'}}}}{{\min \{ {{{ ( {\beta _{\min }^{j}} )}^{'}},{{ ( {\beta _{\min }^{l}} )}^{'}}} \}}} \end{aligned}$$

and

$$ { \bigl\Vert {{M_{D}}^{ - 1}} \bigr\Vert _{\infty }} \le 2\max \biggl\{ { \frac{{\max_{t = j,l} {{ ( {\varphi _{\max }^{t}} )}^{'}}}}{{\min \{ {{{ ( {\beta _{\min }^{j}} )}^{'}},{{ ( {\beta _{\min }^{l}} )}^{'}}} \}}}, \frac{{\max_{t = j,l} {{ ( {\omega _{\max }^{t}} )}^{'}}}}{{\min \{ {{{ ( {\beta _{\min }^{j}} )}^{'}},{{ ( {\beta _{\min }^{l}} )}^{'}}} \}}}} \biggr\} .$$

By Definition 1, we can regard ${M_{j}^{'}}$, ${M_{l}^{'}}$ as two blocks in a row rearrangement of $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$, and thus for $t = j$ or $t = l$ and for ${i \in N}$, there exists ${{t_{i}} \in \{ {0,1,\ldots ,k} \}}$ such that

$$ { \bigl( {\varphi _{\tau i}^{t}} \bigr)^{'}} = \varphi _{\tau i}^{{t_{i}}},\qquad { \bigl( {\omega _{i\tau }^{t}} \bigr)^{'}} = \omega _{i \tau }^{{t_{i}}},\qquad { \bigl( {\beta _{i\tau }^{t}} \bigr)^{'}} = \beta _{i\tau }^{{t_{i}}},$$

this implies that

$$\begin{aligned} \max_{t = j,l} { \bigl( {\varphi _{\max }^{t}} \bigr)^{'}} =& \max_{t = j,l} \Bigl\{ { \max _{\tau \ne i \in N} \bigl\{ {{{ \bigl( { \varphi _{\tau i}^{t}} \bigr)}^{'}}} \bigr\} } \Bigr\} = \max_{t = j,l} \Bigl\{ { \max_{ \tau \ne i \in N} \bigl\{ { \bigl( {\varphi _{\tau i}^{{t_{i}}}} \bigr)} \bigr\} } \Bigr\} \\ =& \max_{\tau \ne i \in N} \bigl\{ {\mathop{ \max } _{t = j,l} \bigl\{ { \bigl( {\varphi _{\tau i}^{{t_{i}}}} \bigr)} \bigr\} } \bigr\} \leq \max_{\tau \ne i \in N} \Bigl\{ {\max_{p = 0,1,\ldots ,k} \bigl\{ { \bigl( {\varphi _{\tau i}^{p}} \bigr)} \bigr\} } \Bigr\} \\ =& \max_{p = 0,1,\ldots ,k} \bigl\{ {\mathop{ \max } _{\tau \ne i \in N} \bigl\{ { \bigl( {\varphi _{\tau i}^{p}} \bigr)} \bigr\} } \bigr\} = \max_{p = 0,1, \ldots ,k} \bigl\{ {\varphi _{\max }^{p}} \bigr\} . \end{aligned}$$

Similarly, we get

$$\begin{aligned} \max_{t = j,l} { \bigl( {\omega _{\max }^{t}} \bigr)^{'}} =& \max_{t = j,l} \Bigl\{ { \max _{\tau \ne i \in N} \bigl\{ {{{ \bigl( { \omega _{i\tau }^{t}} \bigr)}^{'}}} \bigr\} } \Bigr\} = \max_{t = j,l} \Bigl\{ { \max_{ \tau \ne i \in N} \bigl\{ { \bigl( {\omega _{i\tau }^{{t_{i}}}} \bigr)} \bigr\} } \Bigr\} \\ =& \max_{\tau \ne i \in N} \Bigl\{ {\max_{t = j,l} \bigl\{ { \bigl( {\omega _{i\tau }^{{t_{i}}}} \bigr)} \bigr\} } \Bigr\} \le \max _{\tau \ne i \in N} \Bigl\{ {\max_{p = 0,1,\ldots ,k} \bigl\{ { \bigl( { \omega _{i\tau }^{p}} \bigr)} \bigr\} } \Bigr\} \\ =& \max_{p = 0,1,\ldots ,k} \Bigl\{ { \max_{\tau \ne i \in N} \bigl\{ { \bigl( {\omega _{i \tau }^{p}} \bigr)} \bigr\} } \Bigr\} = \max _{p = 0,1,\ldots ,k} \bigl\{ {\omega _{\max }^{p}} \bigr\} \end{aligned}$$

and

$$\begin{aligned} \min _{\tau \ne i \in N} \bigl\{ {{{ \bigl( {\beta _{ \min }^{j}} \bigr)}^{'}},{{ \bigl( {\beta _{\min }^{l}} \bigr)}^{'}}} \bigr\} =& \min _{t = j,l} \bigl\{ { \min _{\tau \ne i \in N} \bigl\{ {\beta _{ \min }^{t}} \bigr\} } \bigr\} = \min _{t = j,l} \bigl\{ {\min _{\tau \ne i \in N} \bigl\{ {\beta _{i \tau }^{{t_{i}}}} \bigr\} } \bigr\} \\ =& \min _{\tau \ne i \in N} \bigl\{ {\min _{t = j,l} \bigl\{ {\beta _{i\tau }^{{t_{i}}}} \bigr\} } \bigr\} \ge \min _{\tau \ne i \in N} \bigl\{ { \min _{p = 0,1,\ldots,k} \bigl\{ {\beta _{i\tau }^{p}} \bigr\} } \bigr\} \\ =& \min _{p = 0,1,\ldots,k} \bigl\{ {\min _{\tau \ne i \in N} \bigl\{ {\beta _{i\tau }^{p}} \bigr\} } \bigr\} = \min _{p = 0,1,\ldots ,k} \bigl\{ { \beta _{\min }^{p}} \bigr\} . \end{aligned}$$

Hence, for any two blocks ${M_{j}^{'}}$, ${M_{l}^{'}}$ in ${M^{'}} \in R ( M )$,

$$ { \bigl\Vert {{M_{D}}^{ - 1}} \bigr\Vert _{\infty }} \le 2\max \biggl\{ { \frac{{\max \{ {\varphi _{\max }^{p}} \}}}{{\min \{ {\beta _{\min }^{p}} \}}}, \frac{{\max \{ {\omega _{\max }^{p}} \}}}{{\min \{ {\beta _{\min }^{p}} \}}}} \biggr\} , \quad p = 0,1, \ldots ,k.$$

By the arbitrariness of ${M_{j}^{'}}$ and ${M_{l}^{'}}$, the conclusion follows. □

We illustrate our results with the following two examples.

Example 1

Let $M = ( {{M_{0}},{M_{1}},{M_{2}}} )$, where

$$\begin{array}{c}{M}_0=\left[\begin{array}{cccc}2& 0& 0.2& 0.2\\ 0.3& 3& 1.2& 1\\ 0.5& 0.1& 3& 2\\ 0.3& 0.1& 1& 2\end{array}\right],M_1=\left[\begin{array}{cccc}2& 0& 0.8& 0.1\\ 0.5& 2& 0.3& 0.2\\ 1& 0.5& 3& 0.5\\ 0.2& 0.5& 1& 3\end{array}\right],\hfill \\ M_2=\left[\begin{array}{cccc}2& 0& 0.2& 0.8\\ 1.7& 3& 0.85& 0.39\\ 0.12& 0.5& 3& 0.1\\ 0.2& 0.3& 0.1& 3\end{array}\right]\hfill \end{array}$$

are DZ matrices and $SDD$ matrices. Thus $M = ( {{M_{0}},{M_{1}},{M_{2}}} )$ has the row W-property. By Theorem 6, it holds that

$$ {\alpha _{\infty }}(M) \le 10.2509.$$

By Theorem 3 (Theorem 4.4 of [1]), we have

$$ {\alpha _{\infty }}(M) \le 16.6667.$$

Since

$$ \rho \bigl( {\max \bigl( {\Lambda _{0}^{ - 1} \vert {{B_{0}}} \vert ,\Lambda _{1}^{ - 1} \vert {{B_{1}}} \vert ,\Lambda _{2}^{ - 1} \vert {{B_{2}}} \vert } \bigr)} \bigr) = 0.9778 < 1,$$

then by Theorem 4 (Theorem 4.3 of [1]), we get

$$ {\alpha _{\infty }}(M) \le 25.2523.$$

Example 2

Let $M = ( {{M_{0}},{M_{1}},{M_{2}}} )$, where

$$\begin{array}{c}{M}_0=\left[\begin{array}{cccc}2& 0& 0.2& 0.2\\ 0.3& 3& 1.2& 1\\ 0.5& 0.1& 3& 2\\ 1& 0.1& 1& 2\end{array}\right],M_1=\left[\begin{array}{cccc}2& 0& 0.8& 0.1\\ 0.5& 2& 0.3& 0.2\\ 1& 0.5& 3& 0.5\\ 0.2& 0.5& 1& 3\end{array}\right],\hfill \\ M_2=\left[\begin{array}{cccc}2& 0& 0.2& 0.8\\ 1.7& 3& 0.85& 0.39\\ 0.12& 0.5& 3& 0.1\\ 0.2& 0.3& 0.1& 3\end{array}\right]\hfill \end{array}$$

are DZ matrices, but ${M_{0}}$ is not an $SDD$ matrix, and

$$ \rho \bigl( {\max \bigl( {\Lambda _{0}^{ - 1} \vert {{B_{0}}} \vert ,\Lambda _{1}^{ - 1} \vert {{B_{1}}} \vert ,\Lambda _{2}^{ - 1} \vert {{B_{2}}} \vert } \bigr)} \bigr) = 1.0728 > 1.$$

Thus Theorems 3 and 4 cannot work for this case. However, by Theorem 6, it holds that

$$ {\alpha _{\infty }}(M) \le 16.3429.$$

Examples 1 and 2 show that the bound in Theorem 6 is sharper than that in Theorems 3 and 4 in some cases.

4 A global error bound for the EVLCP of Dashnic–Zusmanovich-B matrices

In 2020, Zhou et al. [19] introduced error bounds of the linear complementarity problems of Dashnic–Zusmanovich-B matrices, whose definition is listed below.

Definition 3

([19])

A matrix $M = ( {{m_{ij}}} ) \in{R^{n \times n}}$ is called a Dashnic–Zusmanovich-B (DZ-B) matrix if ${B^{+} }$ is a DZ matrix, where $M = {B^{+} } + C$,

$$B^{+}=(b_{ij})=\left(\begin{array}{ccc}{m}_{11}-r_1^{+}& \cdots & m_{1n}-r_1^{+}\\ ⋮& \ddots & ⋮\\ m_{n1}-r_n^{+}& \cdots & m_{nn}-r_n^{+}\end{array}\right),C=\left(\begin{array}{ccc}{r}_1^{+}& \cdots & r_1^{+}\\ ⋮& \ddots & ⋮\\ r_n^{+}& \cdots & r_n^{+}\end{array}\right),$$

(4)

and $r_{i}^{+} = \max \{ {0,{m_{ij}}|i \ne j} \}$.

Next, we will present an upper bound for ${\alpha _{\infty }} ( M )$ with each ${M_{l}} \in{R^{n \times n}}$ ($l = 0,1, \ldots ,k$) being a DZ-B matrix. Before that, some useful results are presented as follows.

Proposition 3

Let $A = ( {{a_{ij}}} ) \in{R^{n \times n}}$ and $B = ( {{b_{ij}}} ) \in{R^{n \times n}}$ be all DZ-B matrices of the form

$$ A = B_{A}^{+} + {C_{A}}, \qquad B = B_{B}^{+} + {C_{B}},$$

where $B_{A}^{+}$, $B_{B}^{+}$, ${C_{A}}$, ${C_{B}}$ are as (4), and $B_{A}^{+}$, $B_{B}^{+}$ are all DZ matrices and satisfy the hypotheses of Proposition 1. Then $( {I - D} )A + DB$ is also a DZ-B matrix for any $D = \operatorname{diag} ( {{d_{i}}} )$ with ${d_{i}} \in [ {0,1} ]$ ($i \in N$).

Proof

Since both A and B are DZ-B matrices, then A and B can be split separately into

$$\begin{aligned} ( {I - D} )A + DB =& ( {I - D} ) \bigl( {B_{A}^{+} + {C_{A}}} \bigr) + D \bigl( { B_{B}^{+} + {C_{B}}} \bigr) \\ =& \bigl[ { ( {I - D} )B_{A}^{+} + D B_{B}^{+} } \bigr] + \bigl[ { ( {I - D} ){C_{A}} + D{C_{B}}} \bigr]. \end{aligned}$$

By Proposition 1, we have that ${ ( {I - D} )B_{A}^{+} + D B_{B}^{+} }$ is a DZ matrix, and ${ ( {I - D} )B_{A}^{+} + D B_{B}^{+} }$ and ${ ( {I - D} ){C_{A}} + D{C_{B}}}$ satisfy formula (4). So, by Definition 3, $( {I - D} )A + DB$ is a DZ-B matrix. □

Based on Proposition 3, Lemma 1, and the fact that a DZ-B matrix is nonsingular, we can prove the following result.

Proposition 4

If ${{M_{0}},{M_{1}}, \ldots ,{M_{k}}}$ are all DZ-B matrices and each ${{{M_{{l_{i}}}}}}$ (${l_{i}} = 0,1,\ldots k$) satisfies the hypotheses of Proposition 3, then each $M_{l}^{'}$ in ${M^{'}} \in R ( M )$ is a DZ-B matrix, and consequently, $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ has the row W-property.

Proof

By Definition 1, for the ith row ${ ( {M_{l}^{'}} )_{i}}$ of ${M_{l}^{'}}$ ($i \in N$), there exists ${l_{i}} \in \{ {0,1,\ldots,k} \}$ such that ${ ( {M_{l}^{'}} )_{i}} = { ( {{M_{{l_{i}}}}} )_{i}}$ and ${{M_{{l_{i}}}}}$ is a DZ-B matrix. So, it holds that

$$ {M_{{l_{i}}}} = B_{{l_{i}}}^{+} + {C_{{l_{i}}}}, \qquad { ( {{M_{{l_{i}}}}} )_{i}} = { \bigl( {B_{{l_{i}}}^{+} } \bigr)_{i}} + { ( {{C_{{l_{i}}}}} )_{i}},$$

that is,

$$ { \bigl( {M_{l}^{'}} \bigr)_{i}} = \bigl( {B_{l}^{+} } \bigr)_{i}^{'} + ( {{C_{l}}} )_{i}^{'},\qquad M_{l}^{'} = { \bigl( {B_{l}^{+} } \bigr)^{'}} + { ( {{C_{l}}} )^{'}}.$$

By Proposition 2, ${ ( {B_{l}^{+} } )^{'}}$ is a DZ matrix, and ${ ( {B_{l}^{+} } )^{'}}$ and ${ ( {{C_{l}}} )^{'}}$ satisfy formula (4). So, $M_{l}^{'}$ is a DZ-B matrix.

Let $M_{j}^{'}$, $M_{l}^{'}$ be any two blocks in ${M^{'}} \in R ( M )$, then $M_{j}^{'}$ and $M_{l}^{'}$ are all DZ-B matrices. By Proposition 3, $( {I - D} )M_{j}^{'} + DM_{l}^{'}$ is a DZ-B matrix for any $D = \operatorname{diag} ( {{d_{i}}} )$, ${d_{i}} \in [ {0,1} ]$ ($i \in N$). Thus $( {I - D} )M_{j}^{'} + DM_{l}^{'}$ is nonsingular. By Lemma 1, the block matrix $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$ has the row W-property. □

Theorem 7

Let $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$, and let each ${M_{p}}$ be a DZ-B matrix of the form ${M_{p}} = B_{p}^{+} + {C_{p}}$ as (4), and let $B_{p}^{+} = ( {{{ ( {{b^{p}}} )}_{i\tau }}} )$ satisfy the hypotheses of Proposition 3for $p = 0,1,\ldots ,k$. Then

$$ \alpha{(M)_{\infty }} \le 2(n - 1)\max \biggl\{ { \frac{{\max \{ {\chi _{\max }^{p}} \}}}{{\min \{ {\lambda _{\min }^{p}} \}}}, \frac{{\max \{ \gamma _{\max }^{p} \}}}{{\min \{ {\lambda _{\min }^{p}} \}}}} \biggr\} ,$$

where

$$\begin{aligned}& \chi _{\max }^{p} = \max_{\tau \ne i \in N} \bigl\{ {\chi _{\tau i}^{p}} \bigr\} ,\qquad \chi _{\tau i}^{p} = \bigl\vert {{{ \bigl( {{b^{p}}} \bigr)}_{\tau i}}} \bigr\vert + \bigl\vert {{{ \bigl( {{b^{p}}} \bigr)}_{ii}}} \bigr\vert ,\\& \gamma _{\max }^{p} = \max_{\tau \ne i \in N} \bigl\{ \gamma _{i\tau }^{p} \bigr\} ,\qquad \gamma _{i\tau }^{p} = \bigl\vert {{{ \bigl( {{b^{p}}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{ \tau }} \bigl( {B_{p}^{+} } \bigr) + \bigl\vert {{{ \bigl( {{b^{p}}} \bigr)}_{\tau i}}} \bigr\vert + {r_{i}} \bigl( {B_{p}^{+} } \bigr),\\& \lambda _{\min }^{p} = \min _{\tau \ne i \in N} \bigl\{ { \lambda _{i\tau }^{p}} \bigr\} ,\qquad \lambda _{i\tau }^{p} = \bigl\vert {{{ \bigl( {{b^{p}}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {{b^{p}}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{ \tau }} \bigl( {B_{p}^{+} } \bigr) + \bigl\vert {{{ \bigl( {{b^{p}}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) - {r_{i}} \bigl( {B_{p}^{+} } \bigr) \bigl\vert {{{ \bigl( {{b^{p}}} \bigr)}_{\tau i}}} \bigr\vert . \end{aligned}$$

Proof

For any two blocks $M_{j}^{'}$, $M_{l}^{'}$ in ${M^{'}} \in R ( M )$ and any $D = \operatorname{diag} ( {{d_{i}}} )$ with ${d_{i}} \in [ {0,1} ]$ ($i \in N$), denote ${M_{D}} = { ( {{m_{i\tau}}} )} = ( {I - D} )M_{j}^{ '} + DM_{l}^{'}$. By Propositions 3 and 4, it holds that $M_{j}^{'}$, $M_{l}^{'}$, and ${M_{D}}$ are all DZ-B matrices, then we can split $M_{j}^{'} = { ( {B_{j}^{+} } )^{'}} + { ( {{C_{j}}} )^{'}}$ and $M_{l}^{'} = { ( {B_{l}^{+} } )^{'}} + { ( {{C_{l}}} )^{'}}$ as (4). Let ${ ( {B_{t}^{+} } )^{'}} = ( {{{ ( {b_{t}^{ '}} )}_{i\tau }}} )$ for $t = j,l$. Since

$$\begin{aligned} {M_{D}} =& ( {I - D} )M_{j}^{'} + DM_{l}^{'} \\ =& ( {I - D} ) \bigl[ {{{ \bigl( {B_{j}^{+} } \bigr)}^{ '}} + {{ ( {{C_{j}}} )}^{'}}} \bigr] + D \bigl[ {{{ \bigl( {B_{l}^{+} } \bigr)}^{'}} + {{ ( {{C_{l}}} )}^{'}}} \bigr] \\ =& \bigl[ { ( {I - D} ){{ \bigl( {B_{j}^{+} } \bigr)}^{ '}} + D{{ \bigl( {B_{l}^{+} } \bigr)}^{'}}} \bigr] \bigl[ { ( {I - D} ){{ ( {{C_{j}}} )}^{'}} + D{{ ( {{C_{l}}} )}^{'}}} \bigr], \end{aligned}$$

then ${{M_{D}} = B_{D}^{+} + {C_{D}}}$, where

$$ {B_{D}^{+} = ( {{b_{i\tau }}} ) = ( {I - D} ){{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + D{{ \bigl( {B_{l}^{+} } \bigr)}^{'}}},\qquad {{C_{D}} = ( {I - D} ){{ ( {{C_{j}}} )}^{'}} + D{{ ( {{C_{l}}} )}^{'}}}.$$

So, both ${{{ ( {B_{j}^{+} } )}^{'}}}$ and ${{{ ( {B_{l}^{+} } )}^{'}}}$ are DZ matrices with positive diagonal elements, and ${B_{D}^{+} }$ is also a DZ matrix with positive diagonal entries by Proposition 1. Therefore, ${B_{D}^{+} }$ is a nonsingular matrix, and

$$\begin{aligned} M_{D}^{ - 1} = { \bigl( {B_{D}^{+} + {C_{D}}} \bigr)^{ - 1}}= { \bigl( {B_{D}^{+} \bigl( {I + {{ \bigl( {B_{D}^{+} } \bigr)}^{ - 1}} {C_{D}}} \bigr)} \bigr)^{ - 1}} ={ \bigl( {I + {{ \bigl( {B_{D}^{+} } \bigr)}^{ - 1}} {C_{D}}} \bigr)^{ - 1}} { \bigl( {B_{D}^{+} } \bigr)^{ - 1}}, \end{aligned}$$

that is,

$$ { \bigl\Vert {M_{D}^{ - 1}} \bigr\Vert _{\infty }} \le{ \bigl\Vert {{{ \bigl( {I + {{ \bigl( {B_{D}^{+} } \bigr)}^{ - 1}} {C_{D}}} \bigr)}^{ - 1}}} \bigr\Vert _{\infty }} \cdot{ \bigl\Vert {{{ \bigl( {B_{D}^{+} } \bigr)}^{ - 1}}} \bigr\Vert _{\infty }} \le ( {n - 1} ){ \bigl\Vert {{{ \bigl( {B_{D}^{+} } \bigr)}^{ - 1}}} \bigr\Vert _{\infty }},$$

where the last equality holds because ${ \Vert {{{ ( {I + {{ ( {B_{D}^{+} } )}^{ - 1}}{C_{D}}} )}^{ - 1}}} \Vert _{\infty }} \le ( {n - 1} )$, see Theorem 2.2 in [9].

In fact, since ${ ( {B_{j}^{+} } )^{'}}$, ${ ( {B_{l}^{+} } )^{'}}$, and ${B_{D}^{+} }$ are all DZ Z-matrices with positive diagonal elements, by Theorem 5, we have that

$$ { \bigl\Vert {{{ \bigl( {B_{D}^{+} } \bigr)}^{ - 1}}} \bigr\Vert _{ \infty }} \le \max \Bigl\{ {\max _{\tau \in N,\tau \ne i} {\alpha _{1}} \bigl( {B_{D}^{+} } \bigr),\max_{\tau \in N,\tau \ne i} {\alpha _{2}} \bigl( {B_{D}^{+} } \bigr)} \Bigr\} $$

holds for each matrix $D = \operatorname{diag} ( {{d_{i}}} )$ with ${d_{i}} \in [ {0,1} ]$ ($i \in N$), where

$$\begin{aligned}& {\alpha _{1}} \bigl( {B_{D}^{+} } \bigr) = \frac{{ \vert {{b_{\tau i}}} \vert + \vert {{b_{ii}}} \vert }}{{ ( { \vert {{b_{\tau \tau }}} \vert - {r_{\tau }} ( {B_{D}^{+} } ) + \vert {{b_{\tau i}}} \vert } ) \vert {{b_{ii}}} \vert - \vert {{b_{\tau i}}} \vert {r_{i}} ( {B_{D}^{+} } )}},\\& {\alpha _{2}} \bigl( {B_{D}^{+} } \bigr) = \frac{{ \vert {{b_{\tau \tau }}} \vert - {r_{\tau }} ( {B_{D}^{+} } ) + \vert {{b_{\tau i}}} \vert + {r_{i}} ( {B_{D}^{+} } )}}{{ ( { \vert {{b_{\tau \tau }}} \vert - {r_{\tau }} ( {B_{D}^{+} } ) + \vert {{b_{\tau i}}} \vert } ) \vert {{b_{ii}}} \vert - \vert {{b_{\tau i}}} \vert {r_{i}} ( {B_{D}^{+} } )}}. \end{aligned}$$

Hence, we get

$$\begin{aligned} \vert {{b_{\tau i}}} \vert + \vert {{b_{ii}}} \vert =& ( {1 - {d_{\tau }}} ) \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert + {d_{\tau }} \bigl\vert {{{ \bigl( {b_{l}^{ '}} \bigr)}_{\tau i}}} \bigr\vert + ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{ii}}} \bigr\vert + {d_{i}} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \\ < & \bigl[ { \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert + \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{ii}}} \bigr\vert } \bigr] + \bigl[ { \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert + \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ii}}} \bigr\vert } \bigr] \\ =& { \bigl( {\chi _{\tau i}^{j}} \bigr)^{'}} + { \bigl( { \chi _{\tau i}^{l}} \bigr)^{'}}= 2\max _{t = j,l} { \bigl( {\chi _{\max }^{t}} \bigr)^{'}}, \end{aligned}$$

where

$$ { \bigl( {\chi _{\max }^{t}} \bigr)^{'}} = \max _{\tau \ne i \in N} \bigl\{ {{{ \bigl( {\chi _{\tau i}^{l}} \bigr)}^{'}}} \bigr\} ,\qquad { \bigl( {\chi _{\tau i}^{l}} \bigr)^{'}} = \bigl\vert {{{ \bigl( {b_{t}^{'}} \bigr)}_{ \tau i}}} \bigr\vert + \bigl\vert {{{ \bigl( {b_{t}^{'}} \bigr)}_{ii}}} \bigr\vert ,\quad t = j, l.$$

Therefore, we have

$$\begin{aligned}& \bigl( { \vert {{b_{\tau \tau }}} \vert - {r_{\tau }} \bigl( {B_{D}^{+} } \bigr) + \vert {{b_{\tau i}}} \vert } \bigr) \vert {{b_{ii}}} \vert - \vert {{b_{\tau i}}} \vert {r_{i}} \bigl( {B_{D}^{+} } \bigr) \\& \quad = \bigl[ { ( {1 - {d_{\tau }}} ) \bigl( { \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{ \tau }} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) + {d_{ \tau }} \bigl( { \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr)} \bigr] \\& \qquad {} \times \bigl[ { ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {b_{j}^{ '}} \bigr)}_{ii}}} \bigr\vert + {d_{i}} \bigl\vert {{{ \bigl( {b_{l}^{ '}} \bigr)}_{ii}}} \bigr\vert } \bigr] \\& \qquad {} - \bigl[ { ( {1 - {d_{\tau }}} ) \bigl\vert {{{ \bigl( {b_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert + {d_{\tau }} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] \bigl[ { ( {1 - {d_{i}}} ){r_{i}} {{ \bigl( {B_{j}^{+} } \bigr)}^{ '}} + {d_{i}} {r_{i}} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}}} \bigr] \\& \quad = ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} + ( {1 - {d_{\tau }}} ){d_{i}} \bigl\vert {{{ \bigl( {b_{l}^{ '}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{j}^{ '}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} + {d_{\tau }} ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {b_{j}^{ '}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{l}^{ '}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} + {d_{\tau }} {d_{i}} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} - ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){r_{i}} { \bigl( {B_{j}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert - ( {1 - {d_{\tau }}} ){d_{i}} {r_{i}} { \bigl( {B_{l}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert \\& \qquad {}- {d_{\tau }} ( {1 - {d_{i}}} ){r_{i}} { \bigl( {B_{j}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ \tau i}}} \bigr\vert - {d_{\tau }} {d_{i}} {r_{i}} { \bigl( {B_{l}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ \tau i}}} \bigr\vert \\& \quad > ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} + ( {1 - {d_{\tau }}} ){d_{i}} {r_{i}} { \bigl( {B_{l}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{ \tau i}}} \bigr\vert + {d_{\tau }} ( {1 - {d_{i}}} ){r_{i}} { \bigl( {B_{j}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{l}^{ '}} \bigr)}_{\tau i}}} \bigr\vert \\& \qquad {} + {d_{\tau }} {d_{i}} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} - ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){r_{i}} { \bigl( {B_{j}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert - ( {1 - {d_{\tau }}} ){d_{i}} {r_{i}} { \bigl( {B_{l}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert \\& \qquad {} - {d_{\tau }} ( {1 - {d_{i}}} ){r_{i}} { \bigl( {B_{j}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ \tau i}}} \bigr\vert - {d_{\tau }} {d_{i}} {r_{i}} { \bigl( {B_{l}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ \tau i}}} \bigr\vert \\& \quad = ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ) \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} + {d_{\tau }} {d_{i}} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) \\& \qquad {} - ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){r_{i}} { \bigl( {B_{j}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert - {d_{\tau }} {d_{i}} {r_{i}} { \bigl( {B_{l}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{l}^{ '}} \bigr)}_{\tau i}}} \bigr\vert \\& \quad = ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ) \bigl[ { \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) - {r_{i}} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] \\& \qquad {} + {d_{\tau }} {d_{i}} \bigl[ { \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ \tau i}}} \bigr\vert } \bigr) - {r_{i}} {{ \bigl( {B_{l}^{+} } \bigr)}^{ '}} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] \\& \quad = ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){ \bigl( {\lambda _{i\tau }^{j}} \bigr)^{'}} + {d_{\tau }} {d_{i}} { \bigl( {\lambda _{i\tau }^{l}} \bigr)^{'}} \\& \quad \ge ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){ \bigl( { \lambda _{\min }^{j}} \bigr)^{'}} + {d_{ \tau }} {d_{i}} { \bigl( {\lambda _{\min }^{l}} \bigr)^{'}}, \end{aligned}$$

where

$$\begin{aligned}& { \bigl( {\lambda _{\min }^{j}} \bigr)^{'}} = \min _{\tau \ne i \in N} { \bigl( {\lambda _{i\tau }^{t}} \bigr)^{ '}},\\& { \bigl( {\lambda _{i\tau }^{t}} \bigr)^{'}} = \bigl\vert {{{ \bigl( {b_{t}^{'}} \bigr)}_{ii}}} \bigr\vert \bigl( { \bigl\vert {{{ \bigl( {b_{t}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{ \tau }} {{ \bigl( {B_{t}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{t}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr) - {r_{i}} { \bigl( {B_{j}^{+} } \bigr)^{'}} \bigl\vert {{{ \bigl( {b_{t}^{ '}} \bigr)}_{\tau i}}} \bigr\vert ,\quad t = j, l. \end{aligned}$$

Further, we get

$$\begin{aligned} {\alpha _{1}} \bigl( {B_{D}^{+} } \bigr) =& \frac{{ \vert {{b_{\tau i}}} \vert + \vert {{b_{ii}}} \vert }}{{ ( { \vert {{b_{\tau \tau }}} \vert - {r_{\tau }} ( {B_{D}^{+} } ) + \vert {{b_{\tau i}}} \vert } ) \vert {{b_{ii}}} \vert - \vert {{b_{\tau i}}} \vert {r_{i}} ( {B_{D}^{+} } )}} \\ \ge & \frac{{2\max_{t = j,l} {{ ( {\chi _{\max }^{t}} )}^{'}}}}{{ ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){{ ( {\lambda _{\min }^{j}} )}^{'}} + {d_{\tau }}{d_{i}}{{ ( {\lambda _{\min }^{l}} )}^{'}}}} \\ \ge & \frac{{2\max_{t = j,l} {{ ( {\chi _{\max }^{t}} )}^{'}}}}{{\min \{ {{{ ( {\lambda _{\min }^{j}} )}^{'}},{{ ( {\lambda _{\min }^{l}} )}^{'}}} \}}} \end{aligned}$$

and

$$\begin{aligned}& \vert {{b_{\tau \tau }}} \vert - {r_{\tau }} \bigl( {B_{D}^{+} } \bigr) + \vert {{b_{\tau i}}} \vert + {r_{i}} \bigl( {B_{D}^{+} } \bigr) \\& \quad = \bigl[ { ( {1 - {d_{\tau }}} ) \bigl\vert {{{ \bigl( {b_{j}^{ '}} \bigr)}_{\tau \tau }}} \bigr\vert + {d_{\tau }} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert } \bigr] - \bigl[ { ( {1 - {d_{\tau }}} ){r_{\tau }} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + {d_{\tau }} {r_{\tau }} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}}} \bigr] \\& \qquad {} + \bigl[ { ( {1 - {d_{\tau }}} ) \bigl\vert {{{ \bigl( {b_{j}^{ '}} \bigr)}_{\tau i}}} \bigr\vert + {d_{\tau }} \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] + \bigl[ { ( {1 - {d_{i}}} ){r_{i}} {{ \bigl( {B_{j}^{+} } \bigr)}^{ '}} + {d_{i}} {r_{i}} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}}} \bigr] \\& \quad = ( {1 - {d_{\tau }}} ) \bigl[ { \bigl\vert {{{ \bigl( {b_{j}^{ '}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] \\& \qquad {} + {d_{\tau }} \bigl[ { \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{ \tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{l}^{+} } \bigr)}^{ '}} + \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert } \bigr] + \bigl[ { ( {1 - {d_{i}}} ){r_{i}} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + {d_{i}} {r_{i}} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}}} \bigr] \\& \quad < \bigl[ { \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{j}^{'}} \bigr)}_{\tau i}}} \bigr\vert + {r_{i}} {{ \bigl( {B_{j}^{+} } \bigr)}^{'}}} \bigr] \\& \qquad {} + \bigl[ { \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{\tau }} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}} + \bigl\vert {{{ \bigl( {b_{l}^{'}} \bigr)}_{\tau i}}} \bigr\vert + {r_{i}} {{ \bigl( {B_{l}^{+} } \bigr)}^{'}}} \bigr] \\& \quad = { \bigl( {\gamma _{i\tau }^{j}} \bigr)^{'}} + { \bigl( { \gamma _{i\tau }^{l}} \bigr)^{'}} \\& \quad \le 2\max_{t = j,l} { \bigl( {\gamma _{\max }^{t}} \bigr)^{'}}, \end{aligned}$$

where

$$\begin{aligned}& { \bigl( {\gamma _{\max }^{t}} \bigr)^{'}} = \max _{\tau \ne i \in N} { \bigl( {\gamma _{i\tau }^{t}} \bigr)^{ '}},\\& { \bigl( {\gamma _{i\tau }^{t}} \bigr)^{'}} = \bigl\vert {{{ \bigl( {b_{t}^{'}} \bigr)}_{\tau \tau }}} \bigr\vert - {r_{ \tau }} { \bigl( {B_{t}^{+} } \bigr)^{'}} + \bigl\vert {{{ \bigl( {b_{t}^{ '}} \bigr)}_{\tau i}}} \bigr\vert + {r_{i}} { \bigl( {B_{t}^{+} } \bigr)^{'}},\quad t = j, l, \end{aligned}$$

and

$$\begin{aligned} {\alpha _{2}} \bigl( {B_{D}^{+} } \bigr) =& \frac{{ \vert {{b_{\tau \tau }}} \vert - {r_{\tau }} ( {B_{D}^{+} } ) + \vert {{b_{\tau i}}} \vert + {r_{i}} ( {B_{D}^{+} } )}}{{ ( { \vert {{b_{\tau \tau }}} \vert - {r_{\tau }} ( {B_{D}^{+} } ) + \vert {{b_{\tau i}}} \vert } ) \vert {{b_{ii}}} \vert - \vert {{b_{\tau i}}} \vert {r_{i}} ( {B_{D}^{+} } )}} \\ \le & \frac{{2\max_{t = j,l} {{ ( {\gamma _{\max }^{t}} )}^{'}}}}{{ ( {1 - {d_{\tau }}} ) ( {1 - {d_{i}}} ){{ ( {\lambda _{\min }^{j}} )}^{'}} + {d_{\tau }}{d_{i}}{{ ( {\lambda _{\min }^{l}} )}^{'}}}} \\ \le & \frac{{2\max_{t = j,l} {{ ( {\gamma _{\max }^{t}} )}^{'}}}}{{\min \{ {{{ ( {\lambda _{\min }^{j}} )}^{'}},{{ ( {\lambda _{\min }^{l}} )}^{'}}} \}}}. \end{aligned}$$

So, it holds that

$$ { \bigl\Vert {{M_{D}}^{ - 1}} \bigr\Vert _{\infty }} \le 2(n - 1)\max \biggl\{ { \frac{{\max_{t = j,l} {{ ( {\chi _{\max }^{t}} )}^{'}}}}{{\min \{ {{{ ( {\lambda _{\min }^{j}} )}^{'}},{{ ( {\lambda _{\min }^{l}} )}^{'}}} \}}}, \frac{{\max_{t = j,l} {{ ( {\gamma _{\max }^{t}} )}^{'}}}}{{\min \{ {{{ ( {\lambda _{\min }^{j}} )}^{'}},{{ ( {\lambda _{\min }^{l}} )}^{'}}} \}}}} \biggr\} .$$

By Definition 1, we can regard ${M_{j}^{'}}$, ${M_{l}^{'}}$ as two blocks in a row rearrangement of $M = ( {{M_{0}},{M_{1}}, \ldots ,{M_{k}}} )$, and thus for $t = j$ or $t = l$ and for ${i \in N}$, there exists ${{t_{i}} \in \{ {0,1,\ldots,k} \}}$ such that

$$ { \bigl( {\chi _{\tau i}^{t}} \bigr)^{'}} = \varphi _{\tau i}^{{t_{i}}},\qquad { \bigl( {\gamma _{i\tau }^{t}} \bigr)^{'}} = \gamma _{i \tau }^{{t_{i}}}, \qquad { \bigl( {\lambda _{i\tau }^{t}} \bigr)^{ '}} = \lambda _{i\tau }^{{t_{i}}},$$

this implies that

$$\begin{aligned} \max_{t = j,l} { \bigl( {\chi _{\max }^{t}} \bigr)^{ '}} =& \max_{t = j,l} \Bigl\{ {\max _{\tau \ne i \in N} \bigl\{ {{{ \bigl( {\chi _{\tau i}^{t}} \bigr)}^{'}}} \bigr\} } \Bigr\} = \max_{t = j,l} \Bigl\{ { \max_{\tau \ne i \in N} \bigl\{ { \bigl( {\chi _{\tau i}^{{t_{i}}}} \bigr)} \bigr\} } \Bigr\} \\ =& \max_{\tau \ne i \in N} \bigl\{ {\mathop{ \max } _{t = j,l} \bigl\{ { \bigl( {\chi _{\tau i}^{{t_{i}}}} \bigr)} \bigr\} } \bigr\} \ge \max_{\tau \ne i \in N} \Bigl\{ {\max_{p = 0,1,\ldots,k} \bigl\{ { \bigl( {\chi _{\tau i}^{p}} \bigr)} \bigr\} } \Bigr\} \\ =& \max_{p = 0,1,\ldots,k} \Bigl\{ {\max_{\tau \ne i \in N} \bigl\{ { \bigl( {\chi _{\tau i}^{p}} \bigr)} \bigr\} } \Bigr\} = \max _{p = 0,1, \ldots ,k} \bigl\{ {\chi _{\max }^{p}} \bigr\} . \end{aligned}$$

Similarly, we get

$$\begin{aligned} \max_{t = j,l} { \bigl( {\gamma _{\max }^{t}} \bigr)^{'}} =& \max_{t = j,l} \Bigl\{ { \max _{\tau \ne i \in N} \bigl\{ {{{ \bigl( { \gamma _{i\tau }^{t}} \bigr)}^{'}}} \bigr\} } \Bigr\} = \max_{t = j,l} \Bigl\{ { \max_{ \tau \ne i \in N} \bigl\{ { \bigl( {\gamma _{i\tau }^{{t_{i}}}} \bigr)} \bigr\} } \Bigr\} \\ =& \max_{\tau \ne i \in N} \Bigl\{ {\max_{t = j,l} \bigl\{ { \bigl( {\gamma _{i\tau }^{{t_{i}}}} \bigr)} \bigr\} } \Bigr\} \le \max _{\tau \ne i \in N} \Bigl\{ {\max_{p = 0,1,\ldots,k} \bigl\{ { \bigl( { \gamma _{i\tau }^{p}} \bigr)} \bigr\} } \Bigr\} \\ =& \max_{p = 0,1,\ldots,k} \Bigl\{ {\max_{\tau \ne i \in N} \bigl\{ { \bigl( {\gamma _{i\tau }^{p}} \bigr)} \bigr\} } \Bigr\} = \max _{p = 0,1,\ldots,k} \bigl\{ {\gamma _{\max }^{p}} \bigr\} \end{aligned}$$

and

$$\begin{aligned} \min _{\tau \ne i \in N} \bigl\{ {{{ \bigl( { \lambda _{\min }^{j}} \bigr)}^{'}},{{ \bigl( {\lambda _{\min }^{l}} \bigr)}^{'}}} \bigr\} =& \min _{t = j,l} \bigl\{ { \min _{\tau \ne i \in N} \bigl\{ { \lambda _{\min }^{t}} \bigr\} } \bigr\} = \min _{t = j,l} \bigl\{ {\min _{\tau \ne i \in N} \bigl\{ { \lambda _{i\tau }^{{t_{i}}}} \bigr\} } \bigr\} \\ = &\min _{\tau \ne i \in N} \bigl\{ {\min _{t = j,l} \bigl\{ {\lambda _{i\tau }^{{t_{i}}}} \bigr\} } \bigr\} \ge \min _{\tau \ne i \in N} \bigl\{ { \min _{p = 0,1,\ldots,k} \bigl\{ {\lambda _{i\tau }^{p}} \bigr\} } \bigr\} \\ =& \min _{p = 0,1,\ldots,k} \bigl\{ {\min _{\tau \ne i \in N} \bigl\{ {\lambda _{i\tau }^{p}} \bigr\} } \bigr\} = \min _{p = 0,1,\ldots,k} \bigl\{ { \lambda _{\min }^{p}} \bigr\} . \end{aligned}$$

Thus, for any two blocks ${M_{j}^{'}}$,${M_{l}^{'}}$ in ${M^{'}} \in R ( M )$, we have

$$ \alpha{(M)_{\infty }} \le 2(n - 1)\max \biggl\{ { \frac{{\max \{ {\chi _{\max }^{p}} \}}}{{\min \{ {\lambda _{\min }^{p}} \}}}, \frac{{\max \{ {\gamma _{\max }^{p}} \}}}{{\min \{ {\lambda _{\min }^{p}} \}}}} \biggr\} ,\quad p = 0,1,\ldots ,k.$$

By the arbitrariness of ${M_{j}^{'}}$ and ${M_{l}^{'}}$, the conclusion follows. □

We illustrate our results with the following two examples.

Example 3

Let $M = ( {{M_{0}},{M_{1}},{M_{2}}} )$, and let each ${M_{p}}$ be DZ-B matrices and $SDD$ matrices of the form ${M_{p}} = B_{p}^{+} + {C_{p}}$ ($p = 0,1,2$) as (4), where

$$\begin{array}{c}{M}_0=\left[\begin{array}{ccc}3.5& 1.5& 0.5\\ 1& 4& -0.5\\ -0.86& 1.64& 4.64\end{array}\right],M_1=\left[\begin{array}{ccc}3.2& 0.2& -2\\ 0.1& 2.1& 0.9\\ -1.84& 0.16& 3.16\end{array}\right],\hfill \\ M_2=\left[\begin{array}{ccc}2.78& 0.78& -0.22\\ 1.74& 3.5& 1.7\\ -0.12& 0.5& 3.5\end{array}\right],B_0^{+}=\left[\begin{array}{ccc}2& 0& -1\\ 0& 3& -1.5\\ -2.5& 0& 3\end{array}\right],\hfill \\ B_1^{+}=\left[\begin{array}{ccc}3& 0& -2.2\\ 0& 2& -1\\ -2& 0& 3\end{array}\right],B_2^{+}=\left[\begin{array}{ccc}2& 0& -1\\ 0& 1.76& -0.04\\ -0.62& 0& 3\end{array}\right].\hfill \end{array}$$

Thus $M = ( {{M_{0}},{M_{1}},{M_{2}}} )$ has the row W-property. By Theorem 7, it holds that

$$ {\alpha _{\infty }}(M) \le 14.6667.$$

By Theorem 3 (Theorem 4.4 of [1]), we have

$$ {\alpha _{\infty }}(M) \le 16.6667.$$

Since

$$ \rho \bigl( {\max \bigl( {\Lambda _{0}^{ - 1} \vert {{B_{0}}} \vert ,\Lambda _{1}^{ - 1} \vert {{B_{1}}} \vert ,\Lambda _{2}^{ - 1} \vert {{B_{2}}} \vert } \bigr)} \bigr) = 0.9909 < 1,$$

then by Theorem 4 (Theorem 4.3 of [1]), we get

$$ {\alpha _{\infty }}(M) \le 42.8184.$$

Example 4

Let $M = ( {{M_{0}},{M_{1}},{M_{2}}} )$, and let ${M_{p}}$ be DZ-B matrices of the form ${M_{p}} = B_{p}^{+} + {C_{p}}$ ($p = 0,1,2$) as (4), where

$$\begin{array}{c}{M}_0=\left[\begin{array}{ccc}3.5& 1.5& 0.5\\ 1& 4& -0.5\\ -0.86& 1.64& 4.64\end{array}\right],M_1=\left[\begin{array}{ccc}3.2& 0.2& -2\\ 0.1& 2.1& 0.9\\ -1.84& 0.16& 3.16\end{array}\right],\hfill \\ M_2=\left[\begin{array}{ccc}2.78& 0.78& -0.22\\ 2& 3.5& 1.7\\ -0.12& 0.5& 3.5\end{array}\right],B_0^{+}=\left[\begin{array}{ccc}2& 0& -1\\ 0& 3& -1.5\\ -2.5& 0& 3\end{array}\right],\hfill \\ B_1^{+}=\left[\begin{array}{ccc}3& 0& -2.2\\ 0& 2& -1\\ -2& 0& 3\end{array}\right],B_2^{+}=\left[\begin{array}{ccc}2& 0& -1\\ 0& 1.5& -0.3\\ -0.62& 0& 3\end{array}\right].\hfill \end{array}$$

Thus $M = ( {{M_{0}},{M_{1}},{M_{2}}} )$ has the row W-property. It is easy to check that ${M_{2}}$ is not $SDD$ and

$$ \rho \bigl( {\max \bigl( {\Lambda _{0}^{ - 1} \vert {{B_{0}}} \vert ,\Lambda _{1}^{ - 1} \vert {{B_{1}}} \vert ,\Lambda _{2}^{ - 1} \vert {{B_{2}}} \vert } \bigr)} \bigr) = 1.0347> 1.$$

Hence, we cannot use these bounds in Theorems 3 and 4 to estimate ${\alpha _{\infty }}(M)$. But, by Theorem 7, we get

$$ {\alpha _{\infty }}(M) \le 14.6667.$$

Examples 3 and 4 show that the bound in Theorem 7 is sharper than that in Theorems 3 and 4 in some cases.

5 Conclusions

In this paper, we present global error bounds for the extended vertical linear complementarity problems of DZ matrices and DZ-B matrices. These bounds are expressed in terms of elements of the matrices, so they can be checked easily. Numerical examples show the feasibility of new results. Finding computable global error bounds for the extended vertical linear complementarity problems of other matrices (S-$SOB$ matrices, S-$SOB$-B matrices, weakly chained diagonally dominant B-matrices, SB-matrices, etc.) under some additional conditions is an interesting problem. It is worth studying in the future.