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Error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices

Journal of Inequalities and Applications20172017:33

https://doi.org/10.1186/s13660-017-1303-5

Received: 1 October 2016

Accepted: 16 January 2017

Published: 2 February 2017

Abstract

In this paper, new error bounds for the linear complementarity problem are obtained when the involved matrix is a weakly chained diagonally dominant B-matrix. The proposed error bounds are better than some existing results. The advantages of the results obtained are illustrated by numerical examples.

Keywords

error boundlinear complementarity problemweakly chained diagonally dominant matrix B-matrix

MSC

90C3360G5065F35

1 Introduction

A linear complementarity problem (LCP) is to find a vector \(x\in \mathbb{R}^{n\times1}\) such that
$$(Mx+q)^{T}x= 0, \qquad Mx+q\geq0, \quad x\geq0, $$
where \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) and \(q\in \mathbb{R}^{n\times1}\). The LCP has various applications in the free boundary problems for journal bearing, the contact problem, and the Nash equilibrium point of a bimatrix game [13].
The LCP has a unique solution for any \(q\in \mathbb{R}^{n\times1}\) if and only if M is a P-matrix [4]. In [5], Chen et al. gave the following error bound for the LCP when M is a P-matrix:
$$\bigl\Vert x-x^{*}\bigr\Vert _{\infty}\leq\max _{d\in[0,1]^{n}} \bigl\Vert (I-D+DM)^{-1}\bigr\Vert _{\infty} \bigl\Vert r(x)\bigr\Vert _{\infty}, $$
where \(x^{*}\) is the solution of the LCP, \(r(x)=\min\{x, Mx+q\}\), \(D=\operatorname{diag}(d_{i})\) with \(0\leq d_{i}\leq1\), and the min operator \(r(x)\) denotes the componentwise minimum of two vectors. If M satisfies special structures, then some bounds of \(\max_{d\in [0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty}\) can be derived [611].

Definition 1

[4]

A matrix \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) is called a B-matrix if for any \(i, j\in\mathbb{N}=\{1,2,\ldots,n\}\),
$$\sum_{k\in N} m_{ik}>0, \qquad \frac{1}{n} \biggl(\sum_{k\in N} m_{ik} \biggr)>m_{ij}, \quad j\neq i. $$

Definition 2

[12]

A matrix \(A=[a_{ij}]\in\mathbb{R}^{n\times n}\) is called a weakly chained diagonally dominant (wcdd) matrix if A is diagonally dominant, i.e.,
$$\vert a_{ii}\vert \geq r_{i}(A)= \sum _{j=1, \neq i}^{n} \vert a_{ij}\vert , \quad \forall i\in\mathbb{N}, $$
and for each \(i\notin J(A)=\{ i\in\mathbb{N}: \vert a_{ii}\vert >r_{i}(A)\}\neq\emptyset\), there is a sequence of nonzero elements of A of the form \(a_{ii_{1}}, a_{i_{1}i_{2}}, \ldots, a_{i_{r} j}\) with \(j\in J(A)\).

Definition 3

[13]

A matrix \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) is called a weakly chained diagonally dominant (wcdd) B-matrix if it can be written in the form \(M=B^{+}+C\) with \(B^{+}\) a wcdd matrix whose diagonal entries are all positive.

García-Esnaola et al. [8] gave the upper bound for \(\max_{d\in [0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty}\) when M is a B-matrix: Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a B-matrix with the form
$$M=B^{+}+C, $$
where
$$ B^{+}=[b_{ij}]=\left [ \begin{matrix} m_{11}-r_{1}^{+} &\cdots &m_{1n}-r_{1}^{+} \\ \vdots & &\vdots \\ m_{n1}-r_{n}^{+} &\cdots &m_{nn}-r_{n}^{+} \end{matrix} \right ], $$
(1)
and \(r_{i}^{+}=\max\{0, m_{ij}\vert j\neq i\}\). Then
$$ \max_{d\in[0,1]^{n}} \bigl\Vert (I-D+DM)^{-1}\bigr\Vert _{\infty }\leq \frac{n-1}{\min\{\beta,1\}}, $$
(2)
where \(\beta= \min_{i\in\mathbb{N}}\{\beta_{i}\}\) and \(\beta_{i}=b_{ii}-\sum_{j\neq i} \vert b_{ij}\vert \).
To improve the bound in (2), Li et al. [14] presented the following result: Let \(M=[m_{ij}]\in \mathbb{R}^{n\times n}\) be a B-matrix with the form \(M=B^{+}+C\), where \(B^{+}=[b_{ij}]\) is defined as (1). Then
$$ \max_{d\in[0,1]^{n}} \bigl\Vert (I-D+DM)^{-1}\bigr\Vert _{\infty} \leq \sum_{i=1}^{n} \frac{n-1}{\min\{\bar{\beta}_{i},1\}} \prod_{j=1}^{i-1} \Biggl(1+ \frac{1}{\bar{\beta}_{j}} \sum_{k=j+1}^{n} \vert b_{jk}\vert \Biggr), $$
(3)
where \(\bar{\beta}_{i}=b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert l_{i}(B^{+})\), \(l_{k}(B^{+})=\max_{k\leq i\leq n} \{ \frac{1}{\vert b_{ii}\vert } \sum_{j=k, \neq i}^{n} \vert b_{ij}\vert \}\) and
$$\prod_{j=1}^{i-1} \Biggl(1+\frac{1}{\bar{\beta}_{j}} \sum_{k=j+1}^{n} \vert b_{jk} \vert \Biggr)=1, \quad\mbox{if } i=1. $$
Recently, when M is a weakly chained diagonally dominant (wcdd) B-matrix, Li et al. [13] gave a bound for \(\max_{d\in[0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty }\): Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a wcdd B-matrix with the form \(M=B^{+}+C\), where \(B^{+}=[b_{ij}]\) is defined as (1). Then
$$ \max_{d\in[0,1]^{n}} \bigl\Vert (I-D+DM)^{-1}\bigr\Vert _{\infty }\leq \sum_{i=1}^{n} \Biggl( \frac{n-1}{\min\{\tilde{\beta}_{i},1\}} \prod_{j=1}^{i-1} \frac{b_{jj}}{\tilde{\beta}_{j}} \Biggr), $$
(4)
where \(\tilde{\beta}_{i}=b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert >0\) and \(\prod_{j=1}^{i-1}\frac{b_{jj}}{\tilde{\beta}_{j}}=1\) if \(i=1\).

This bound in (4) holds when M is a B-matrix since a B-matrix is a weakly chained diagonally dominant B-matrix [13].

Now, some notation is given, which will be used in the sequel. Let \(A=[a_{ij}]\in\mathbb{R}^{n\times n}\). For \(i, j, k \in \mathbb{N}\), denote
$$\begin{aligned} &u_{i}(A)=\frac{1}{\vert a_{ii}\vert } \sum _{j= i+1}^{n} \vert a_{ij}\vert , \qquad u_{n}(A)=0, \\ &b_{k}(A)= \max_{k+1\leq i\leq n} \biggl\{ \frac{\sum_{j= k, \neq i}^{n} \vert a_{ij}\vert }{\vert a_{ii}\vert } \biggr\} , \qquad b_{n}(A)=1, \\ &p_{k}(A)= \max_{k+1\leq i\leq n} \biggl\{ \frac{ \vert a_{ik}\vert +\sum_{j= k+1, \neq i}^{n} \vert a_{ij}\vert b_{k}(A) }{\vert a_{ii}\vert } \biggr\} , \qquad p_{n}(A)=1. \end{aligned} $$

The rest of this paper is organized as follows: In Section 2, we present some new bounds for \(\max_{d\in[0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert_{\infty}\) when M is a wcdd B-matrix. Numerical examples are given to verify the corresponding results in Section 3.

2 Main results

In this section, some new upper bounds for \(\max_{d\in [0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty}\) are provided when M is a wcdd B-matrix. Firstly, several lemmas, which will be used later, are given.

Lemma 1

[13]

Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a wcdd B-matrix with the form \(M=B^{+}+C\), where \(B^{+}\) is defined as (1). Then
$$\bigl\Vert \bigl(I + \bigl(B^{+}_{D}\bigr)^{-1}C_{D} \bigr)^{-1}\bigr\Vert _{\infty} \leq n-1, $$
where \(B^{+}_{D}=I-D+DB^{+}\) and \(C_{D}=DC\).

Lemma 2

[15]

Let \(A=[a_{ij}]\in\mathbb{R}^{n\times n}\) be a wcdd M-matrix with \(u_{k}(A)p_{k}(A)<1\) (\(\forall k\in\mathbb{N}\)). Then
$$\begin{aligned} \bigl\Vert A^{-1}\bigr\Vert _{\infty} &\leq \max\Biggl\{ \sum_{i=1}^{n} \Biggl( \frac{1}{ a_{ii} (1-u_{i}(A)p_{i}(A) ) } \prod_{j=1}^{i-1} \frac{u_{j}(A)}{1-u_{j}(A)p_{j}(A)} \Biggr), \\ &\quad \sum_{i=1}^{n} \Biggl( \frac{p_{i}(A)}{ a_{ii} (1-u_{i}(A)p_{i}(A) ) } \prod_{j=1}^{i-1} \frac{1}{1-u_{j}(A)p_{j}(A)} \Biggr) \Biggr\} , \end{aligned} $$
where
$$\prod_{j=1}^{i-1} \frac{u_{j}(A)}{1-u_{j}(A)p_{j}(A)} =1, \qquad \prod_{j=1}^{i-1} \frac{1}{1-u_{j}(A)p_{j}(A)}=1, \quad\textit{if } i=1. $$

Lemma 3

[14]

Let \(\gamma>0\) and \(\eta\geq0\). Then, for any \(x\in[0,1]\),
$$\frac{1}{1-x+\gamma x}\leq \frac{1}{ \min\{\gamma, 1 \}}, \qquad \frac{\eta x}{1-x+\gamma x}\leq \frac{\eta}{\gamma}. $$

Theorem 1

Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a wcdd B-matrix with the form \(M=B^{+}+C\), where \(B^{+}=[b_{ij}]\) is defined as (1). If, for each \(i\in\mathbb{N}\),
$$\hat{\beta}_{i}=b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert p_{i}\bigl(B^{+}\bigr)>0, $$
then
$$ \begin{aligned}[b] &\max_{d\in[0,1]^{n}} \bigl\Vert (I-D+DM)^{-1}\bigr\Vert _{\infty } \\ &\quad\leq\max \Biggl\{ \sum_{i=1}^{n} \frac{n-1}{ \min \{\hat{\beta}_{i}, 1\} } \prod_{j=1}^{i-1} \Biggl( \frac{1}{\hat{\beta}_{j} }\sum_{k= j+1}^{n} \vert b_{jk}\vert \Biggr), \sum_{i=1}^{n} \frac{(n-1)p_{i}(B^{+})}{ \min \{\hat{\beta}_{i}, 1\} } \prod_{j=1}^{i-1} \frac{ b_{jj} }{\hat{\beta}_{j} } \Biggr\} , \end{aligned} $$
(5)
where
$$\prod_{j=1}^{i-1} \Biggl(\frac{1}{\hat{\beta}_{j} } \sum_{k= j+1}^{n} \vert b_{jk} \vert \Biggr) =1, \qquad \prod_{j=1}^{i-1} \frac{ b_{jj} }{\hat{\beta}_{j} }=1, \quad \textit{if } i=1. $$

Proof

Let \(M_{D}=I-D+DM\). Then
$$M_{D}=I-D+DM=I-D+D\bigl(B^{+}+C\bigr)=B_{D}^{+}+C_{D}, $$
where \(B_{D}^{+}=I-D+DB^{+}\). Similar to the proof of Theorem 2 in [13], we see that \(B_{D}^{+}\) is a wcdd M-matrix with positive diagonal elements and \(C_{D}=DC\), and, by Lemma 1,
$$ \bigl\Vert M_{D}^{-1}\bigr\Vert _{\infty}\leq \bigl\Vert \bigl(I+\bigl(B_{D}^{+}\bigr)^{-1}C_{D} \bigr)^{-1}\bigr\Vert _{\infty} \bigl\Vert \bigl(B_{D}^{+} \bigr)^{-1}\bigr\Vert _{\infty} \leq (n-1)\bigl\Vert \bigl(B_{D}^{+}\bigr)^{-1}\bigr\Vert _{\infty}. $$
(6)
By Lemma 2, we have
$$\begin{aligned} \bigl\Vert \bigl(B_{D}^{+} \bigr)^{-1}\bigr\Vert _{\infty} &\leq \max\Biggl\{ \sum_{i=1}^{n} \frac{1}{ (1-d_{i}+b_{ii}d_{i}) (1-u_{i}(B_{D}^{+})p_{i}(B_{D}^{+}) ) } \prod_{j=1}^{i-1} \frac{u_{j}((B_{D}^{+}))}{1-u_{j}((B_{D}^{+}))p_{j}(B_{D}^{+})}, \\ &\quad \sum_{i=1}^{n} \frac{p_{i}(B_{D}^{+})}{ (1-d_{i}+b_{ii}d_{i}) (1-u_{i}((B_{D}^{+}))p_{i}(B_{D}^{+}) ) } \prod_{j=1}^{i-1} \frac{1}{1-u_{j}(B_{D}^{+})p_{j}(B_{D}^{+})} \Biggr\} . \end{aligned} $$
By Lemma 3, we can easily get the following results: for each \(i , j, k\in\mathbb{N}\),
$$\begin{aligned}& b_{k}\bigl(B_{D}^{+} \bigr)= \max_{k+1\leq i\leq n} \biggl\{ \frac{\sum_{j= k, \neq i}^{n} \vert b_{ij}\vert d_{i} }{1-d_{i}+b_{ii}d_{i}} \biggr\} \leq \max _{k+1\leq i\leq n} \biggl\{ \frac{\sum_{j= k, \neq i}^{n} \vert b_{ij}\vert }{b_{ii}} \biggr\} =b_{k} \bigl(B^{+}\bigr), \\& p_{k}\bigl(B_{D}^{+}\bigr)= \max _{k+1\leq i\leq n} \biggl\{ \frac{ \vert b_{ik}\vert d_{i}+\sum_{j= k+1, \neq i}^{n} \vert b_{ij}\vert d_{i}b_{k}(B_{D}^{+}) }{1-d_{i}+b_{ii}d_{i}} \biggr\} \\& \hphantom{p_{k}(B_{D}^{+})}\leq\max_{k+1\leq i\leq n} \biggl\{ \frac{ \vert b_{ik}\vert +\sum_{j= k+1, \neq i}^{n} \vert b_{ij}\vert b_{k}(B_{D}^{+}) }{b_{ii}} \biggr\} \\& \hphantom{p_{k}(B_{D}^{+})} \leq\max_{k+1\leq i\leq n} \biggl\{ \frac{ \vert b_{ik}\vert +\sum_{j= k+1, \neq i}^{n} \vert b_{ij}\vert b_{k}(B^{+}) }{b_{ii}} \biggr\} \\& \hphantom{p_{k}(B_{D}^{+})}=p_{k}\bigl(B^{+}\bigr), \end{aligned}$$
and
$$ \begin{aligned}[b] \frac{1}{ (1-d_{i}+b_{ii}d_{i})(1-u_{i}(B_{D}^{+})p_{i}(B_{D}^{+}))} &= \frac{1}{ 1-d_{i}+b_{ii}d_{i}-\sum_{j= i+1}^{n} \vert b_{ij}\vert d_{i} p_{i}(B_{D}^{+})} \\ &\leq\frac{1}{ \min \{ b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert p_{i}(B^{+}), 1 \} } \\ &=\frac{1}{ \min \{ \hat{\beta}_{i}, 1 \} }. \end{aligned} $$
(7)
Furthermore, by Lemma 3, we have
$$ \begin{aligned}[b] \frac{u_{i}(B_{D}^{+})}{ 1-u_{i}(B_{D}^{+})p_{i}(B_{D}^{+})} &= \frac{\sum_{j= i+1}^{n} \vert b_{ij}\vert d_{i} }{ 1-d_{i}+b_{ii}d_{i} -\sum_{j=i+1}^{n} \vert b_{ij}\vert d_{i} p_{i}(B_{D}^{+})} \\ &\leq\frac{\sum_{j= i+1}^{n} \vert b_{ij}\vert }{ b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert p_{i}(B^{+}) } \\ &=\frac{1}{\hat{\beta}_{i} }\sum_{j= i+1}^{n} \vert b_{ij}\vert \end{aligned} $$
(8)
and
$$ \begin{aligned}[b] \frac{1}{ 1-u_{i}(B_{D}^{+})p_{i}(B_{D}^{+})} &= \frac{1-d_{i}+b_{ii}d_{i} }{ 1-d_{i}+b_{ii}d_{i} -\sum_{j=i+1}^{n} \vert b_{ij}\vert d_{i} p_{i}(B_{D}^{+})} \\ &\leq\frac{1-d_{i}+b_{ii}d_{i} }{ b_{ii}-\sum_{j= i+1}^{n} \vert b_{ij}\vert p_{i}(B^{+}) } \\ &=\frac{b_{ii}}{\hat{\beta}_{i} }. \end{aligned} $$
(9)
By (7), (8), and (9), we obtain
$$ \bigl\Vert \bigl(B_{D}^{+} \bigr)^{-1}\bigr\Vert _{\infty} \leq\max \Biggl\{ \sum _{i=1}^{n} \frac{1}{ \min \{\hat{\beta}_{i}, 1\} } \prod _{j=1}^{i-1} \Biggl(\frac{1}{\hat{\beta}_{j} }\sum _{k= j+1}^{n} \vert b_{jk}\vert \Biggr), \sum_{i=1}^{n} \frac {p_{i}(B^{+})}{ \min \{\hat{\beta}_{i}, 1\} } \prod _{j=1}^{i-1} \frac{ b_{jj} }{\hat{\beta}_{j} } \Biggr\} . $$
(10)
Therefore, the result in (5) holds from (6) and (10). □

Since a B-matrix is also a wcdd B-matrix, then by Theorem 1, we find the following result.

Corollary 1

Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a B-matrix with the form \(M=B^{+}+C\), where \(B^{+}=[b_{ij}]\) is defined as (1). Then
$$ \begin{aligned}[b] & \max_{d\in[0,1]^{n}} \bigl\Vert (I-D+DM)^{-1}\bigr\Vert _{\infty} \\ &\quad\leq \max \Biggl\{ \sum_{i=1}^{n} \frac{n-1}{ \min \{\hat{\beta}_{i}, 1\} } \prod_{j=1}^{i-1} \Biggl( \frac{1}{\hat{\beta}_{j} }\sum_{k= j+1}^{n} \vert b_{jk}\vert \Biggr), \sum_{i=1}^{n} \frac{(n-1)p_{i}(B^{+})}{ \min \{\hat{\beta}_{i}, 1\} } \prod_{j=1}^{i-1} \frac{ b_{jj} }{\hat{\beta}_{j} } \Biggr\} , \end{aligned} $$
(11)
where \(\hat{\beta}_{i}\) is defined as in Theorem  1.

We next give a comparison of the bounds in (4) and (5) as follows.

Theorem 2

Let \(M=[m_{ij}]\in\mathbb{R}^{n\times n}\) be a wcdd B-matrix with the form \(M=B^{+}+C\), where \(B^{+}=[b_{ij}]\) is defined as (1). Let \(\bar{\beta}_{i}\), \(\tilde{\beta}_{i}\), and \(\hat{\beta}_{i}\) be defined as in (3), (4), and (5), respectively. Then
$$ \begin{aligned}[b] &\max \Biggl\{ \sum _{i=1}^{n} \frac{n-1}{ \min \{\hat{\beta}_{i}, 1\} } \prod _{j=1}^{i-1} \Biggl(\frac{1}{\hat{\beta}_{j} }\sum _{k= j+1}^{n} \vert b_{jk}\vert \Biggr), \sum_{i=1}^{n} \frac{(n-1)p_{i}(B^{+})}{ \min \{\hat{\beta}_{i}, 1\} } \prod _{j=1}^{i-1} \frac{ b_{jj} }{\hat{\beta}_{j} } \Biggr\} \\ &\quad \leq \sum_{i=1}^{n} \Biggl( \frac{n-1}{\min\{\tilde{\beta}_{i},1\}} \prod_{j=1}^{i-1} \frac{b_{jj}}{\tilde{\beta}_{j}} \Biggr). \end{aligned} $$
(12)

Proof

Since \(B^{+}\) is a wcdd matrix with positive diagonal elements, for any \(i\in\mathbb{N}\),
$$ 0\leq p_{i}\bigl(B^{+}\bigr)\leq1, \qquad \tilde{\beta}_{i}\leq \hat{\beta}_{i}. $$
(13)
By (13), for each \(i\in\mathbb{N}\),
$$ \frac{1}{ \hat{\beta}_{i} } \leq\frac{1}{ \tilde{\beta}_{i} }, \qquad \frac{1}{ \min\{ \hat{\beta}_{i}, 1\} } \leq\frac{1}{ \min \{ \tilde{\beta}_{i}, 1\} }. $$
(14)
The result in (12) follows by (13) and (14). □

Remark 1

  1. (i)

    Theorem 2 shows that the bound in (5) is better than that in (4).

     
  2. (ii)

    When n is very large, one needs more computations to obtain these upper bounds by (5) than by (4).

     

3 Numerical examples

In this section, we present numerical examples to illustrate the advantages of our derived results.

Example 1

Consider the family of B-matrices in [14]:
$$M_{k}=\left [ \begin{matrix} 1.5 &0.5 &0.4 &0.5 \\ -0.1 &1.7 &0.7 &0.6 \\ 0.8 &-0.1 \frac{k}{k+1} &1.8 &0.7\\ 0 &0.7 &0.8 &1.8 \end{matrix} \right ], $$
where \(k\geq1\). Then \(M_{k}=B_{k}^{+}+C_{k}\), where
$$B_{k}^{+}=\left [ \begin{matrix} 1 &0 &-0.1 &0 \\ -0.8 &1 &0 &-0.1 \\ 0 &-0.1 \frac{k}{k+1}-0.8 &1 &-0.1\\ -0.8 &-0.1 &0 &1 \end{matrix} \right ]. $$
By (2), we have
$$\max_{d\in[0,1]^{4}} \bigl\Vert (I-D+DM_{k})^{-1} \bigr\Vert _{\infty }\leq \frac{4-1}{\min\{\beta,1\}}=30(k+1). $$
It is obvious that
$$30(k+1)\rightarrow+\infty, \quad \mbox{if } k\rightarrow+\infty. $$
By (3), we get
$$\max_{d\in[0,1]^{4}} \bigl\Vert (I-D+DM_{k})^{-1} \bigr\Vert _{\infty }\leq 15.2675. $$
By Theorem 7 of [11], we have
$$\max_{d\in[0,1]^{4}} \bigl\Vert (I-D+DM_{k})^{-1} \bigr\Vert _{\infty }\leq 13.6777. $$
By Corollary 1 of [13], we have
$$\max_{d\in[0,1]^{4}} \bigl\Vert (I-D+DM_{k})^{-1} \bigr\Vert _{\infty }\leq \sum_{i=1}^{4} \Biggl(\frac{3}{\min\{\tilde{\beta}_{i},1\}} \prod_{j=1}^{i-1} \frac{b_{jj}}{\tilde{\beta}_{j}} \Biggr)\approx15.2675. $$
By (11), we obtain
$$\max_{d\in[0,1]^{4}} \bigl\Vert (I-D+DM_{k})^{-1} \bigr\Vert _{\infty }\leq 9.9683. $$
In these two cases, the bounds in (2) are equal to 60 (\(k=1\)) and 90 (\(k=2\)), respectively.

Example 2

Consider the wcdd B-matrix in [13]:
$$M=\left [ \begin{matrix} 1.5 &0.2 &0.4 &0.5 \\ -0.1 &1.5 &0.5 &0.1 \\ 0.5 &-0.1 &1.5 &0.1\\ 0.4 &0.4 &0.8 &1.8 \end{matrix} \right ]. $$
Then \(M=B^{+}+C\), where
$$B^{+}=\left [ \begin{matrix} 1 &-0.3 &-0.1 &0 \\ -0.6 &1 &0 &-0.4 \\ 0 &-0.6 &1 &-0.4\\ -0.4 &-0.4 &0 &1 \end{matrix} \right ]. $$
By (4), we get
$$\max_{d\in[0,1]^{4}} \bigl\Vert (I-D+DM)^{-1}\bigr\Vert _{\infty }\leq 41.1111. $$
By (5), we have
$$\max_{d\in[0,1]^{4}} \bigl\Vert (I-D+DM)^{-1}\bigr\Vert _{\infty }\leq 21.6667. $$

4 Conclusions

In this paper, we present some new upper bounds for \(\max_{d\in[0,1]^{n}} \Vert (I-D+DM)^{-1}\Vert _{\infty}\) when M is a weakly chained diagonally dominant B-matrix, which improve some existing results. A numerical example shows that the given bounds are efficient.

Declarations

Acknowledgements

The author is grateful to the referees for their useful and constructive suggestions. This work is supported by the National Natural Science Foundation of China (11361074, 11501141), the Foundation of Science and Technology Department of Guizhou Province ([2015]7206), the Natural Science Programs of Education Department of Guizhou Province ([2015]420), and the Research Foundation of Guizhou Minzu University (16yjsxm002, 16yjsxm040).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Science, Guizhou Minzu University, Guiyang, P.R. China

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