# New error bounds for the extended vertical LCP††thanks: This research was
supported by National Natural Science Foundation of China (No.11961082).

Shiliang Wu, Hehui Wang
†School of Mathematics, Yunnan Normal University,
Kunming, Yunnan, 650500, P.R. China
‡School of Mathematics and Statistics, Yunnan University,
Kunming, Yunnan, 650091, P.R. China
Corresponding author: slwuynnu<EMAIL_ADDRESS>
###### Abstract

In this paper, by making use of this fact that for $a_{j},b_{j}\in\mathbb{R}$,
$j=1,2,\ldots,n$, there are $\lambda_{j}\in[0,1]$ with
$\sum_{j=1}^{n}\lambda_{j}=1$ such that

$\min_{1\leq j\leq n}\\{a_{j}\\}-\min_{1\leq j\leq
n}\\{b_{j}\\}=\sum_{j=1}^{n}\lambda_{j}(a_{j}-b_{j}),$

some new error bounds of the extended vertical LCP under the row
$\mathcal{W}$-property are obtained, which cover the error bounds in [Math.
Program., 106 (2006) 513-525] and [Comput. Optim. Appl., 42 (2009) 335-352].
Not only that, these new error bounds skillfully avoid the inconvenience
caused by the row rearrangement technique for error bounds to achieve the goal
of reducing the computation workload, which was introduced in the latter paper
mentioned above. Besides, with respect to the row $\mathcal{W}$-property, two
new sufficient and necessary conditions are obtained.

Keywords: The extended vertical LCP; row $\mathcal{W}$-property; error bound

AMS classification: 90C33, 65F10, 65F50, 65G40

##  1 Introduction

For $A_{j}\in\mathbb{R}^{n\times n}$ and $q_{j}\in\mathbb{R}^{n}$
($j=0,1,2,\ldots,k$) being given known matrices and the source terms, the
extended vertical linear complementarity problem is to find
$x\in\mathbb{R}^{n}$ such that

$r(x):=\min\\{A_{0}x+q_{0},A_{1}x+q_{1},\ldots,A_{k}x+q_{k}\\}=0,$ (1.1)

where $\min$ is the component minimum operator. Here, Eq. (1.1) is denoted by
EVLCP($\mathbf{A},\mathbf{q}$) for short, where

$\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})\ \mbox{and}\
\mathbf{q}=(q_{0},q_{1},\ldots,q_{k}).$

When $A_{0}=I$ and $q_{0}=0$ in (1.1), where $I$ denotes the identity matrix,
the EVLCP ($\mathbf{A},\mathbf{q}$) reduces to the vertical LCP, which was
introduced by Cottle and Dantzig [1], also see [2]. Further, when $A_{0}=I$
and $q_{0}=0$ and $k=1$ in (1.1), the EVLCP ($\mathbf{A},\mathbf{q}$) comes
back to the standard LCP ($A_{1},q_{1}$), see [3, 4].

So far, it has been found that the EVLCP ($\mathbf{A},\mathbf{q}$) has more
and more applications in the many fields, like, such as nonlinear networks
[5], control theory [6], the mixed lubrication problem [7], stochastic impulse
control games [8], the boundary value problem [9], the generalized bimatrix
games [10], the generalized Leontief input-output model [11], the discrete HJB
equations [12], volterra ecosystem [13] and so on. There exist many
literatures to pay attention to the existence of solutions and algorithms for
the EVLCP ($\mathbf{A},\mathbf{q}$), see [2, 15, 6, 14, 16, 8].

For the EVLCP ($\mathbf{A},\mathbf{q}$), another important and interested
topic in theory is error bound, which has drawn widespread attention because
of an important tool in theoretical analysis, including convergence analysis,
sensitive analysis and verification of the solutions. At present, there have
some results about error bounds in the literature. For instance, assume that
$x^{\ast}$ is the unique solution of the LCP ($A_{1},q_{1}$), for $A_{1}$
being $P$-matrix, a well-know global error bound was given in [17] by Mathias
and Pang, and described below

$\|x-x^{\ast}\|_{\infty}\leq\frac{1+\|A_{1}\|_{\infty}}{\alpha(A_{1})}\|r(x)\|_{\infty},\
\mbox{for any }\ x\in\mathbb{R}^{n},$ (1.2)

where

$\alpha(A_{1}):=\min_{\|x\|_{\infty}=1}\big{\\{}\max_{1\leq i\leq
n}x_{i}(A_{1}x)_{i}\big{\\}}.$

By the equivalent form of the minimum function, Chen and Xiang in [18]
obtained the following error bound in $\|\cdot\|_{p}\ (p\geq 1,\mbox{or}\
p=\infty)$ norms,

$\|x-x^{\ast}\|_{p}\leq\max_{d\in[0,1]^{n}}\|(I-D+DA_{1})^{-1}\|_{p}\|r(x)\|_{p},\
\mbox{for any }\ x\in\mathbb{R}^{n},$ (1.3)

where $D=\mbox{diag}(d)$ with $d\in[0,1]^{n}$, which is sharper than (1.2) in
$\|\cdot\|_{\infty}$, see [18]. Moreover, for $A_{1}$ being an $H_{+}$-matrix,
Chen and Xiang confirmed

$\max_{d\in[0,1]^{n}}\|(I-D+DA_{1})^{-1}\|_{p}\leq\|\langle
A_{1}\rangle^{-1}\max(\wedge_{1},I)\rangle\|_{p},$ (1.4)

where $\wedge_{1}$ is the diagonal part of $A_{1}$ and $\langle A_{1}\rangle$
is its comparison matrix (i.e., $\langle
A_{1}\rangle_{ii}=|(A_{1})_{ii}|,\langle A_{1}\rangle_{ij}=-|(A_{1})_{ij}|$
for $i\neq j$).

Recently, by the row rearrangement technique introduced by Zhang et al. in
[19], i.e.,
$\textbf{A}^{\prime}=(A^{\prime}_{0},A^{\prime}_{1},\ldots,A^{\prime}_{k})$ is
called a row rearrangement of $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ if for
any $i\in N:=\\{1,2,\ldots,n\\}$,

$(\textbf{A}^{\prime}_{j})_{i\cdot}=(A_{ji})_{i\cdot}\in\\{(A_{0})_{i\cdot},(A_{1})_{i\cdot},\ldots,(A_{k})_{i\cdot}\\}=\\{(A^{\prime}_{0})_{i\cdot},(A^{\prime}_{1})_{i\cdot},\ldots,(A^{\prime}_{k})_{i\cdot}\\},$

where $(\cdot)_{i\cdot}$ means the i-th row of a given matrix, for the EVLCP
($\mathbf{A},\mathbf{q}$), assume that $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$
has the row $\mathcal{W}$-property (see the following definition), Zhang et
al. in [19] presented the following result,

$\|x-x^{\ast}\|\leq\alpha(\textbf{A})\|r(x)\|,\ \mbox{for any }\
x\in\mathbb{R}^{n},$ (1.5)

where

$\alpha(\textbf{A})=\max_{\textbf{A}^{\prime}\in\mathcal{R}(\mathbf{A})}\max_{j<l\in\\{0,1,\ldots,k\\}}\max_{d\in[0,1]^{n}}\|((I-D)A_{j}^{\prime}+A^{\prime}_{l})^{-1}\|,$
(1.6)

$A_{j}^{\prime},A^{\prime}_{l}\in\mathbb{R}^{n\times n}$ are any two blocks in
$\textbf{A}^{\prime}\in\mathcal{R}(\mathbf{A})$ with $\mathcal{R}(\mathbf{A})$
being the set of all row rearrangements of $\mathbf{A}$. When the norm in
(1.5) is taken as $\|\cdot\|_{\infty}$, we denote

$\alpha_{\infty}(\textbf{A})=\max_{\textbf{A}^{\prime}\in\mathcal{R}(\mathbf{A})}\max_{j<l\in\\{0,1,\ldots,k\\}}\max_{d\in[0,1]^{n}}\|((I-D)A_{j}^{\prime}+A^{\prime}_{l})^{-1}\|_{\infty}.$

There is no doubt that Eq. (1.5) presents a general result for the upper
global error bound of the EVLCP ($\mathbf{A},\mathbf{q}$) under the row
$\mathcal{W}$-property. By investigating Eq. (1.5), clearly, it is easy to
know that before obtaining Eq. (1.5), we have to calculate and obtain the
exact value of (1.6). Whereas, in the implementations, calculating the exact
value of (1.6) is a very difficult task because there is a rearrangement for
the row of matrix $\mathbf{A}$, in particular, when the order of matrix
$\mathbf{A}$ is large. It is a main motivation of this present paper. In this
paper, to overcome this disadvantage caused by the row rearrangement technique
in essence, we have to carve out a new approach to obtain the error bound for
the EVLCP ($\mathbf{A},\mathbf{q}$). Our approach inspired by the work in
[18], we first develop a general equivalent form of the minimum function.
Then, based on this, some new error bounds for the EVLCP
($\mathbf{A},\mathbf{q}$) are obtained. Not only that, these new error bounds
cover some existing results in [18] and [19] as well. Meanwhile, it avoids the
row rearrangement of the system matrix $\mathbf{A}$. Incidentally, for the row
$\mathcal{W}$-property, two new sufficient and necessary conditions are given.

The rest of the article expands as follows. First, from the view of the
calculation time, section 2 further discusses the result in (1.5). Secondly,
section 3 provides some error bounds of the EVLCP ($\mathbf{A},\mathbf{q}$)
under the row $\mathcal{W}$-property by a general equivalent form of the
minimum function, which is entirely different from Eq. (1.5). Thirdly, section
4 presents some numerical examples to show the feasibility of the error bound.
Finally, section 5 is a brief conclusion.

By the way, the following notations, definitions and results will be used
throughout the paper, which can be founded in [20, 2]. Let $A=(a_{ij})$ and
$B=(b_{ij})\in\mathbb{R}^{n\times n}$. Then $A\geq(>)B$ means
$a_{ij}\geq(>)b_{ij}$ for $i,j=1,2,\ldots,n$. We indicate $|A|=(|a_{ij}|)$.
Matrix $A=(a_{ij})$ is called a strictly diagonal dominant matrix if
$|a_{ii}|>\sum_{j\neq i}|a_{ij}|,\ i\in N:=\\{1,2,\ldots,n\\}$. $\rho(\cdot)$
indicates the spectral radius of the matrix. A block matrix
$\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ has the row $\mathcal{W}$-property if

$\min(A_{0}x,A_{1}x,\ldots,A_{k}x)\leq
0\leq\max(A_{0}x,A_{1}x,\ldots,A_{k}x)\Rightarrow x=0.$

The EVLCP ($\mathbf{A},\mathbf{q}$) has a unique solution for any $\mathbf{q}$
if and only if $\mathbf{A}$ has the row $\mathcal{W}$-property.

##  2 Old error bound

In this section, we will discuss the result in (1.5) from the angle of the
calculation time.

To calculate $\alpha(\textbf{A})$, a natural question is how many times you
need to calculate $\|((I-D)A_{j}^{\prime}+A^{\prime}_{l})^{-1}\|$. Further,
when confronting Eq. (1.6), the first thought is to calculate the number of
elements in $\mathcal{R}(\mathbf{A})$, that is, we need to know that how many
the row rearrangements of $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ there are in
$\mathcal{R}(\mathbf{A})$. To answer this question, we get Proposition 2.1.

######  Proposition 2.1.

Let $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ with $A_{j}\in\mathbb{R}^{n\times
n}$ $(j=0,1,\ldots,k)$. Then the cardinality of $\mathcal{R}(\mathbf{A})$ is
$[(k+1)!]^{n}$.

Proof. If only one row is rearranged, noting that a total number of matrices
are $k+1$, there are a total of $(k+1)!$ different sorting methods. Because
there are a total of $n$ rows and the arrangement of different rows can be
arbitrarily combined, there are a total of $[(k+1)!]^{n}$ different sorting
methods. Therefore, matrix $\mathbf{A}$ has $[(k+1)!]^{n}$ different row
rearrangements, that is, the cardinality of $\mathcal{R}(\mathbf{A})$ is
$[(k+1)!]^{n}$. $\hfill{}\Box$

In fact, once a certain $\textbf{A}^{\prime}$ is selected, there exist the
total $C_{k+1}^{2}=\frac{k(k+1)}{2}$ kinds of options for choosing
$A_{j}^{\prime},A^{\prime}_{l}$. From Proposition 2.1, a total of elements in
$\mathcal{R}(\mathbf{A})$ are $[(k+1)!]^{n}$, so the calculation times of
$\|((I-D)A_{j}^{\prime}+A^{\prime}_{l})^{-1}\|$ are at most
$\frac{k(k+1)}{2}[(k+1)!]^{n}$ times to obtain $\alpha(\textbf{A})$.
Therefore, the calculation times of Theorem 4.1 in [19] for $k=1$ achieve at
most $2^{n}$ times. Whereas, the calculation time of Theorem 3.1 in [19] is 1.
The next question is whether Theorem 4.1 in [19] for $k=1$ is in contradiction
with Theorem 3.1 [19] because the form of Theorem 4.1 in [19] for $k=1$ is
different from in Theorem 3.1 in [19] as well. To answer this question, we
require Proposition 2.2.

######  Proposition 2.2.

Let $\mathbf{A}=(A_{0},A_{1})$ with $A_{0},A_{1}\in\mathbb{R}^{n\times n}$.
Then

$\max_{\textbf{A}^{\prime}\in\mathcal{R}(\mathbf{A})}\max_{d\in[0,1]^{n}}\|((I-D)A^{\prime}_{0}+A^{\prime}_{1})^{-1}\|=\max_{d\in[0,1]^{n}}\|((I-D)A_{0}+DA_{1})^{-1}\|.$

Proof. For any $\textbf{A}^{\prime}=(A_{0}^{\prime},A^{\prime}_{1})$,
$D=\mbox{diag}(d)$, $d\in[0,1]^{n}$, $i=1,2,\cdots,n$, we have

$((I-D)A^{\prime}_{0}+DA^{\prime}_{1})_{i.}=(1-d_{i})(A^{\prime}_{0})_{i.}+d_{i}(A^{\prime}_{1})_{i.}.$

According to the definition of row rearrangement, we get

$\\{(A^{\prime}_{0})_{i.},(A^{\prime}_{1})_{i.}\\}=\\{(A_{0})_{i.},(A_{1})_{i.}\\}.$

When $(A^{\prime}_{0})_{i.}=(A_{0})_{i.}$,
$(A^{\prime}_{1})_{i.}=(A_{1})_{i.}$, we take $\hat{d}_{i}=d_{i}$; when
$(A^{\prime}_{0})_{i.}=(A_{1})_{i.}$, $(A^{\prime}_{1})_{i.}=(A_{0})_{i.}$, we
take $\hat{d}_{i}=1-d_{i}$, $\hat{d}_{i}\in[0,1]$. Further, we have

$(1-d_{i})(A^{\prime}_{0})_{i.}+d_{i}(A^{\prime}_{1})_{i.}=(1-\hat{d}_{i})(A_{0})_{i.}+\hat{d}_{i}(A_{1})_{i.}.$

Let $\hat{D}=\mbox{diag}(\hat{d})$. Then

$(I-D)A^{\prime}_{0}+DA^{\prime}_{1}=(I-\hat{D})A_{0}+\hat{D}A_{1}.$

Therefore, from the arbitrariness of $\textbf{A}^{\prime}$, we obtain

$\max_{\textbf{A}^{\prime}\in\mathcal{R}(\mathbf{A})}\max_{d\in[0,1]^{n}}\|((I-D)A^{\prime}_{0}+A^{\prime}_{1})^{-1}\|=\max_{d\in[0,1]^{n}}\|((I-D)A_{0}+DA_{1})^{-1}\|.$

This completes the proof. $\hfill{}\Box$

Based on Proposition 2.2, in essence, Theorem 4.1 in [19] for $k=1$ is in line
with Theorem 3.1 [19]. The calculation times of both are the same, 1 time.

According to the process of the proof of Proposition 2.2, we can see that when
calculating $\|((I-D)A_{j}^{\prime}+DA^{\prime}_{l})^{-1}\|$, as long as two
of the $k+1$ elements in each row are chosen, no matter how many
$A_{j},A^{\prime}_{l}$ they can be combined into, ultimately, their
calculation results are the same, so there is no need to reconsider.
Therefore, calculating $\alpha(\textbf{A})$ only needs to calculate
$(C_{k+1}^{2})^{n}$ times $\|((I-D)A^{\prime}_{j}+A^{\prime}_{l})^{-1}\|$.
From this, we present Proposition 2.3.

######  Proposition 2.3.

Let $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$. Then there needs to calculate
$\alpha(\emph{{A}})$ at most $(C_{k+1}^{2})^{n}$ times
$\|((I-D)A_{j}^{\prime}+DA^{\prime}_{l})^{-1}\|$, i.e.,

$\alpha(\emph{{A}})=\max_{\textbf{A}^{\prime}\in\mathcal{R}(\mathbf{A})}\max_{B_{1},B_{2}}\max_{d\in[0,1]^{n}}\|((I-D)B_{1}+DB_{2})^{-1}\|,$
(2.1)

where $(B_{1})_{i.}=(A_{j})_{i.}$, $(B_{2})_{i.}=(A_{l})_{i.}$,
$j<l\in\\{0,1,\ldots,k\\}$.

To further explain Proposition 2.3, Example 2.1 is provided.

Example 2.1 Let $\mathbf{A}=(A_{0},A_{1},A_{2})$, where

$A_{0}=\left[\begin{array}[]{cc}1&1\\\ -1&1\end{array}\right],\
A_{1}=\left[\begin{array}[]{cc}1&0\\\ -2&1\end{array}\right],\
A_{2}=\left[\begin{array}[]{cc}2&1\\\ 0&1\end{array}\right].$

When using Eq. (2.1), from Proposition 2.3, its calculation times are 9.
Specifically as follows:

Let

$B_{1}=\left[\begin{array}[]{cc}1&1\\\
-1&1\end{array}\right],B_{2}=\left[\begin{array}[]{cc}1&0\\\
-2&1\end{array}\right].$

Then

$\displaystyle\mu_{1}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)B_{1}+DB_{2})^{-1}\|_{\infty}=3;$

let

$B_{1}=\left[\begin{array}[]{cc}1&1\\\
-1&1\end{array}\right],B_{2}=\left[\begin{array}[]{cc}2&1\\\
0&1\end{array}\right].$

Then

$\displaystyle\mu_{2}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)B_{1}+DB_{2})^{-1}\|_{\infty}=2;$

let

$B_{1}=\left[\begin{array}[]{cc}1&0\\\
-2&1\end{array}\right],B_{2}=\left[\begin{array}[]{cc}2&1\\\
0&1\end{array}\right].$

Then

$\displaystyle\mu_{3}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)B_{1}+DB_{2})^{-1}\|_{\infty}=3;$

let

$B_{1}=\left[\begin{array}[]{cc}1&1\\\
-1&1\end{array}\right],B_{2}=\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right].$

Then

$\displaystyle\mu_{4}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)B_{1}+DB_{2})^{-1}\|_{\infty}=2;$

let

$B_{1}=\left[\begin{array}[]{cc}1&1\\\
-2&1\end{array}\right],B_{2}=\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right].$

Then

$\displaystyle\mu_{5}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)B_{1}+DB_{2})^{-1}\|_{\infty}=3;$

let

$B_{1}=\left[\begin{array}[]{cc}1&1\\\
-1&1\end{array}\right],B_{2}=\left[\begin{array}[]{cc}2&1\\\
-2&1\end{array}\right].$

Then

$\displaystyle\mu_{6}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)B_{1}+DB_{2})^{-1}\|_{\infty}=\frac{3}{2};$

let

$B_{1}=\left[\begin{array}[]{cc}1&1\\\
-2&1\end{array}\right],B_{2}=\left[\begin{array}[]{cc}2&1\\\
0&1\end{array}\right].$

Then

$\displaystyle\mu_{7}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)B_{1}+DB_{2})^{-1}\|_{\infty}=2;$

let

$B_{1}=\left[\begin{array}[]{cc}1&0\\\
-1&1\end{array}\right],B_{2}=\left[\begin{array}[]{cc}2&1\\\
-2&1\end{array}\right].$

Then

$\displaystyle\mu_{8}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)B_{1}+DB_{2})^{-1}\|_{\infty}=3;$

let

$B_{1}=\left[\begin{array}[]{cc}1&0\\\
-1&1\end{array}\right],B_{2}=\left[\begin{array}[]{cc}2&1\\\
0&1\end{array}\right].$

Then

$\displaystyle\mu_{9}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)B_{1}+DB_{2})^{-1}\|_{\infty}=2.$

Based on the above computational results, we obtain

$\alpha_{\infty}(\textbf{A})=\max\\{\mu_{1},\mu_{2},\cdots,\mu_{9}\\}=3.$

However, directly using Eq. (1.5), its calculation times are
$\frac{k(k+1)}{2}[(k+1)!]^{n}=108$, this is because a lot of repetitive work
are done. For example, let,

$A^{\prime}_{0}=\left[\begin{array}[]{cc}1&1\\\
-2&1\end{array}\right],A^{\prime}_{1}=\left[\begin{array}[]{cc}1&0\\\
-1&1\end{array}\right],A^{\prime}_{2}=\left[\begin{array}[]{cc}2&1\\\
0&1\end{array}\right],$

then
$A^{\prime}=(A^{\prime}_{0},A^{\prime}_{1},A^{\prime}_{2})\in\mathcal{R}(\mathbf{A})$.
Similarly, let,

$A^{\prime\prime}_{0}=\left[\begin{array}[]{cc}1&0\\\
-2&1\end{array}\right],A^{\prime\prime}_{1}=\left[\begin{array}[]{cc}1&1\\\
-1&1\end{array}\right],A^{\prime\prime}_{2}=\left[\begin{array}[]{cc}2&1\\\
0&1\end{array}\right],$

then
$A^{\prime\prime}=(A^{\prime\prime}_{0},A^{\prime\prime}_{1},A^{\prime\prime}_{2})\in\mathcal{R}(\mathbf{A})$.
Of course, $A=(A_{0},A_{1},A_{2})\in\mathcal{R}(\mathbf{A})$. When using Eq.
(1.5), we have to calculate

$\max_{d\in[0,1]^{n}}\|((I-D)A_{0}+DA_{1})^{-1}\|_{\infty},\max_{d\in[0,1]^{n}}\|((I-D)A^{\prime}_{0}+DA^{\prime}_{1})^{-1}\|_{\infty}$

and

$\max_{d\in[0,1]^{n}}\|((I-D)A^{\prime\prime}_{0}+DA^{\prime\prime}_{1})^{-1}\|_{\infty}.$

Whereas, from the proof of Proposition 2.2, the above three formulas are
equivalent, so they belong to the repeated calculation. Proposition 2.3 just
avoids this.

Example 2.1 is simple, but it tells us that for any block matrix $\mathbf{A}$
containing three two-by-two matrices, i.e., $k=2$ and $n=2$, once the row
rearrangement of $\mathbf{A}$ is required, its calculation times are at most 9
to obtain the corresponding error bound. It’s easy to imagine that for the
block matrix $\mathbf{A}$ containing $k+1$ $n$-by-$n$ matrices, making use of
Proposition 2.3 to compute the error bound of the EVLCP
($\mathbf{A},\mathbf{q}$), once the row rearrangement occurs, the
corresponding computational cost should be highly expensive and unacceptable
for the sufficiently large $k$ or $n$! Therefore, to face this headwind, we
have to exploit a new and effective tool to obtain the error bound of the
EVLCP ($\mathbf{A},\mathbf{q}$).

##  3 New error bound

In this section, to avoid Proposition 2.3, we will give some new error bounds
for the EVLCP ($\mathbf{A},\mathbf{q}$). For this goal, we first present some
requisite lemmas.

######  Lemma 3.1.

Let all $a_{j},b_{j}\in\mathbb{R}$, $j=1,2,\ldots,n$. Then there are
$\lambda_{j}$ with $\lambda_{j}\in[0,1]$ such that

$\min_{1\leq j\leq n}\\{a_{j}\\}-\min_{1\leq j\leq
n}\\{b_{j}\\}=\sum_{j=1}^{n}\lambda_{j}(a_{j}-b_{j}).$ (3.1)

Proof. The result in Lemma 3.1 is given directly from the mean value theorem
of Lipschitz functions with the generalized gradient. $\hfill{}\Box$

######  Lemma 3.2.

Let $a_{j},b_{j},t_{j}\in\mathbb{R}$ with $a_{j}>0$, $t_{j}\in[0,1]$,
$(j=1,2,\ldots,n)$ and $\sum_{j=1}^{n}t_{j}=1$. Then

$\frac{\sum_{j=1}^{n}t_{j}b_{j}}{\sum_{j=1}^{n}t_{j}a_{j}}\leq\max_{1\leq
j\leq n}\bigg{\\{}\frac{|b_{j}|}{a_{j}}\bigg{\\}}.$ (3.2)

In addition, if $b_{j}=1$, then the inequality (3.3) simplifies as

$\frac{1}{\sum_{j=1}^{n}t_{j}a_{j}}\leq\max_{1\leq j\leq
n}\bigg{\\{}\frac{1}{a_{j}}\bigg{\\}}.$

Proof. The proof is straightforward. $\hfill{}\Box$

In addition, it is noted that the following inequality is still true, i.e.,

$\sum_{j=2}^{n}t_{j}b_{j}\leq\max_{2\leq j\leq n}\\{|b_{j}|\\},$

where $t_{j}\in[0,1]$ and $\sum_{j=2}^{n}t_{j}\leq 1$. In fact, by the simple
calculation, we have

$\displaystyle\sum_{j=2}^{n}t_{j}b_{j}=$
$\displaystyle\frac{\sum_{j=2}^{n}t_{j}b_{j}}{1-\sum_{j=2}^{n}t_{j}+\sum_{j=2}^{n}t_{j}}$
$\displaystyle=$
$\displaystyle\frac{\sum_{j=2}^{n}t_{j}b_{j}}{\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}(n-2)+\sum_{j=2}^{n}t_{j}}$
$\displaystyle=$
$\displaystyle\frac{t_{2}b_{2}+t_{3}b_{3}+\ldots+t_{n}b_{n}}{(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{2})+(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{3})+\ldots+(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{n})}$
$\displaystyle\leq$
$\displaystyle\frac{t_{2}|b_{2}|+t_{3}|b_{3}|+\ldots+t_{n}|b_{n}|}{(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{2})+(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{3})+\ldots+(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{n})}$
$\displaystyle\leq$
$\displaystyle\frac{(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{2})|b_{2}|+(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{3})|b_{3}|+\ldots+(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{n})|b_{n}|}{(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{2})+(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{3})+\ldots+(\frac{1-\sum_{j=2}^{n}t_{j}}{n-2}+t_{n})}$
$\displaystyle\leq$ $\displaystyle\max_{2\leq j\leq n}\\{|b_{j}|\\}.$

######  Lemma 3.3.

_[21]_ Matrix $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ has the row
$\mathcal{W}$-property if and only if for arbitrary nonnegative diagonal
matrix $X_{0},X_{1},\ldots,X_{k}$ with
$\mbox{diag}(X_{0}+X_{1}+\ldots+X_{k})>0$,

$\det(X_{0}A_{0}+X_{1}A_{1}+\ldots+X_{k}A_{k})\neq 0.$

To obtain the error bound for the EVLCP ($\mathbf{A},\mathbf{q}$), we require
a new sufficient and necessary for the row $\mathcal{W}$-property, see Lemma
3.4.

######  Lemma 3.4.

Matrix $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ has the row
$\mathcal{W}$-property if and only if
$D_{0}A_{0}+D_{1}A_{1}+\ldots+D_{k}A_{k}$ is nonsingular for arbitrary
nonnegative diagonal matrices $D_{j}=\mbox{diag}(d_{j})$ with
$d_{j}\in[0,1]^{n}$, $(j=0,1,\ldots,k)$ and $\sum_{j=0}^{k}D_{j}=I$.

Proof. The proof is straightforward by making use of Lemma 3.3. $\hfill{}\Box$

Remark 3.1 When $k=1$ in Lemma 3.4, the result in Lemma 3.4 goes back to Lemma
2.2 in [19]. Clearly, Lemma 3.4 is a generalization of their result.

Next, we discuss the error bound for the EVLCP ($\mathbf{A},\mathbf{q}$).

Assume that $x^{\ast}$ is the unique solution of the EVLCP
($\mathbf{A},\mathbf{q}$). Based on Lemma 3.1, we set

$a_{j}=A_{j}x+q_{j},b_{j}=A_{j}x^{\ast}+q_{j},j=0,1,\ldots,k.$ (3.3)

Substituting (3.3) into (3.1) yields

$r(x)=\min_{0\leq j\leq k}\\{A_{j}x+q_{j}\\}-\min_{0\leq j\leq
k}\\{A_{j}x^{\ast}+q_{j}\\}=(D_{0}A_{0}+D_{1}A_{1}+\ldots+D_{k}A_{k})(x-x^{\ast}),$
(3.4)

where $D_{j}=\mbox{diag}(d_{j})$ with $d_{j}\in[0,1]^{n}$ $(j=0,1,\ldots,k)$
are nonnegative diagonal matrices and $\sum_{j=0}^{k}D_{j}=I$.

Combining (3.4) with Lemma 3.4, we immediately gain the upper global error
bound of the EVLCP ($\mathbf{A},\mathbf{q}$) under the row
$\mathcal{W}$-property.

######  Theorem 3.1.

Let $D_{0},D_{1},\ldots,D_{k}$ satisfy the conditions of Lemma 3.4. If
$\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ has the row $\mathcal{W}$-property,
then for any $x\in\mathbb{R}^{n}$,

$\|x-x^{\ast}\|\leq\max\|(D_{0}A_{0}+D_{1}A_{1}+\ldots+D_{k}A_{k})^{-1}\|\|r(x)\|.$
(3.5)

Remark 3.2 When $A_{0}=I$ and $k=1$ in (3.5), Theorem 3.1 reduces to (1.3),
see Eq. (2.3) on page 516 in [18] as well. Further, when $k=1$ in (3.5),
Theorem 3.1 reduces to Theorem 3.1 in [19]. In addition, comparing (3.5) with
(1.5), the former advantage over the latter is that the former no longer
requires the row rearrangement of the matrix $\mathbf{A}$. That is to say,
(3.5) successfully avoids the row rearrangement of the matrix $\mathbf{A}$.
From the view of the calculation time, in general, Eq. (1.5) requires
$(\frac{(k+1)k}{2})^{n}$ times, Eq. (3.5) requires only 1 time. Compared with
the error bound (1.5) by the row rearrangement technique, our new error bound
(3.5) greatly reduces the computation workload in a way.

In addition, we also obtain the lower global error bound for the EVLCP
($\mathbf{A},\mathbf{q}$) under the row $\mathcal{W}$-property. By making use
of (3.4), we easily find that

$\|x-x^{\ast}\|\geq\frac{\|r(x)\|}{\max\|D_{0}A_{0}+D_{1}A_{1}+\ldots+D_{k}A_{k}\|}.$

Theorem 3.1 provides a new result for the upper global error bound of the
EVLCP ($\mathbf{A},\mathbf{q}$) under the row $\mathcal{W}$-property, whereas,
it contains arbitrary nonnegative diagonal matrices $D_{0},D_{1},\ldots,D_{k}$
such that it seems be an impossible task to find
$\max\|(D_{0}A_{0}+D_{1}A_{1}+\ldots+D_{k}A_{k})^{-1}\|$ under certain
conditions. To turn around this negative situation, an effective approach is
to limit the range of $\mathbf{A}$. That is to say, once we are able to choose
the suitable $\mathbf{A}$, Theorem 3.1 is able to get rid of this unfavorable
situation, on condition that the error can be permitted.

In the following, we consider the upper global error bound of the EVLCP
($\mathbf{A},\mathbf{q}$) from two aspects: (I)
$\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ with the diagonal part of every matrix
$A_{j}$ being positive; (II) $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ with
every matrix $A_{j}$ being strictly diagonally dominant. Under these two
cases, interestingly, we obtain the upper global error bound the same as these
in [19], see Theorem 3.2 and Theorem 3.3. Not only that, The proof of Theorem
3.2 is simpler than the proof of Theorem 4.3 in [19].

######  Theorem 3.2.

Let $A_{j}=\wedge_{j}-C_{j}$ in $\mathbf{A}$ with $\wedge_{j}>0$, where
$\wedge_{j}$ is the diagonal part of $A_{j}$, j=0,1,…,k. If
$A_{j}=\wedge_{j}-C_{j}$ satisfy

$\rho(\max_{0\leq j\leq k}\\{\wedge^{-1}_{j}|C_{j}|\\})<1,$ (3.6)

then $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ has the row
$\mathcal{W}$-property and

$\max\|(D_{0}A_{0}+D_{1}A_{1}+\ldots+D_{k}A_{k})^{-1}\|\leq\|[I-\max_{0\leq
j\leq k}\\{\wedge^{-1}_{j}|C_{j}|\\}]^{-1}\max_{0\leq j\leq
k}\\{\wedge^{-1}_{j}\\}\|.$ (3.7)

Proof. Let $V=\sum_{j=0}^{k}D_{j}\wedge_{j}$ and $U=\sum_{j=0}^{k}D_{j}C_{j}$.
Then

$(D_{0}A_{0}+D_{1}A_{1}+\ldots+D_{k}A_{k})^{-1}=(V-U)^{-1}=(I-V^{-1}U)^{-1}V^{-1}.$

Under the condition (3.6), together with Lemma 3.2, we have

$V^{-1}\leq\max_{0\leq j\leq k}\\{\wedge^{-1}_{j}\\},\
V^{-1}|U|\leq\max_{0\leq j\leq k}\\{\wedge^{-1}_{j}|C_{j}|\\},$

and

$\displaystyle|(I-V^{-1}U)^{-1}|=$
$\displaystyle|I+(V^{-1}U)+(V^{-1}U)^{2}+...|$ $\displaystyle\leq$
$\displaystyle I+(V^{-1}|U|)+(V^{-1}|U|)^{2}+...$ $\displaystyle\leq$
$\displaystyle I+(\max_{0\leq j\leq
k}\\{\wedge^{-1}_{j}|C_{j}|\\})+(\max_{0\leq j\leq
k}\\{\wedge^{-1}_{j}|C_{j}|\\})^{2}+...$ $\displaystyle=$
$\displaystyle[I-\max_{0\leq j\leq k}\\{\wedge^{-1}_{j}|C_{j}|\\}]^{-1}.$

So,

$|(I-V^{-1}U)^{-1}V^{-1}|\leq[I-\max_{0\leq j\leq
k}\\{\wedge^{-1}_{j}|C_{j}|\\}]^{-1}\max_{0\leq j\leq
k}\\{\wedge^{-1}_{j}\\}.$

Hence,

$\displaystyle\|(I-V^{-1}U)^{-1}V^{-1}\|\leq$
$\displaystyle\||(I-V^{-1}U)^{-1}V^{-1}|\|$ $\displaystyle\leq$
$\displaystyle\|[I-\max_{0\leq j\leq
k}\\{\wedge^{-1}_{j}|C_{j}|\\}]^{-1}\max_{0\leq j\leq
k}\\{\wedge^{-1}_{j}\\}\|,$

which implies that (3.7) is true. By Lemma 3.4, it is easy to see that
$\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ has the row $\mathcal{W}$-property.
$\hfill{}\Box$

In fact, the process of the proof of Theorem 3.2 is also suitable for the
proof of Theorem 2.1 in [18] and Theorem 3.2 in [19].

######  Theorem 3.3.

Let $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ with every matrix $A_{j}$ being
strictly row diagonally dominant and sign($\wedge_{0}$)=sign($\wedge_{j}$),
where $\wedge_{j}$ is a diagonal part of $A_{j}$, $j=0,1,\ldots,k$. Then
$\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ has the row $\mathcal{W}$-property and

$\max\|(D_{0}A_{0}+D_{1}A_{1}+\ldots+D_{k}A_{k})^{-1}\|_{\infty}\leq\frac{1}{\min_{i\in
N}\min((\langle A_{0}\rangle e)_{i},(\langle A_{1}\rangle
e)_{i},\ldots,(\langle A_{k}\rangle e)_{i})}.$ (3.8)

Here, the proof of Theorem 3.3 is omitted, one can see the proof of Theorem
3.3 in [19] for more details.

Comparing Theorem 4.4 in [19] with Theorem 3.3, the latter no longer requires
“each $i\in N$, $(A_{j})_{ii}(A_{l})_{ii}>0$, for any
$j<l\in\\{0,1,\ldots,k\\}$”, just needs to keep $\wedge_{j}$ the same sign.

In [22], Xiu and Zhang extended Eq. (1.2) to the EVLCP
($\mathbf{A},\mathbf{q}$) for $k=1$ under the row $\mathcal{W}$-property:

$\|x-x^{\ast}\|_{\infty}\leq\frac{\|A_{0}+A_{1}\|_{\infty}}{\alpha\\{A_{0},A_{1}\\}}\|r(x)\|_{\infty},\
\mbox{for any }\ x\in\mathbb{R}^{n},$ (3.9)

where

$\alpha\\{A_{0},A_{1}\\}:=\min_{\|x\|_{\infty}=1}\big{\\{}\max_{1\leq i\leq
n}(A_{0}x)_{i}(A_{1}x)_{i}\big{\\}}.$

In the sequel, what we’re interested in is whether we will extend (3.9) to the
EVLCP ($\mathbf{A},\mathbf{q}$) under the row $\mathcal{W}$-property,
meanwhile, and avoid the row rearrangement technique as well. To this end, we
require the following lemma, see Lemma 3.5.

######  Lemma 3.5.

If matrix $\mathbf{\bar{A}}=(A_{0},D_{1}A_{1}+D_{2}A_{2}+\ldots+D_{k}A_{k})$
has the row $\mathcal{W}$-property, where matrices $D_{j}=\mbox{diag}(d_{j})$
with $d_{j}\in[0,1]^{n}$ $(j=1,\ldots,k)$ are arbitrary nonnegative diagonal
matrices and $\sum_{j=1}^{k}D_{j}=I$, if and only if matrix
$\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ has the row $\mathcal{W}$-property.

Proof. ($\Rightarrow$) If
$\mathbf{\bar{A}}=(A_{0},D_{1}A_{1}+D_{2}A_{2}+\ldots+D_{k}A_{k})$ has the row
$\mathcal{W}$-property, matrices $D_{j}=\mbox{diag}(d_{j})$ with
$d_{j}\in[0,1]^{n}$ $(j=1,\ldots,k)$ are arbitrary nonnegative diagonal
matrices and $\sum_{j=1}^{k}D_{j}=I$, then, by using Lemma 3.4, for arbitrary
nonnegative diagonal matrix $\hat{D}=\mbox{diag}(\hat{d})$ with
$\hat{d}\in[0,1]^{n}$, matrix

$(I-\hat{D})A_{0}+\hat{D}(D_{1}A_{1}+D_{2}A_{2}+\ldots+D_{k}A_{k})$

is nonsingular. For arbitrary nonnegative diagonal matrix
$\bar{D}_{j}=\mbox{diag}(\bar{d}_{j})$ with $\bar{d}_{j}\in[0,1]^{n}$
$(j=0,1,\ldots,k)$ and $\sum_{j=0}^{k}\bar{D}_{j}=I$, we set
$\hat{D}=\mbox{diag}(\hat{d})=I-\bar{D}_{0}$, i.e.,
$\hat{d}_{i}=1-(\bar{d}_{0})_{i}$, $\hat{d}\in[0,1]^{n}$. Let
$D_{l}=\mbox{diag}(d_{l})$, $l=1,2,\ldots,k$, where
$(d_{l})_{i}=\frac{(\bar{d}_{l})_{i}}{\hat{d}_{i}}$ if $\hat{d}_{i}\neq 0$;
$(d_{l})_{i}=\frac{1}{k}$ if $\hat{d}_{i}=0$. Further,

$\bar{D}_{0}A_{0}+\bar{D}_{1}A_{1}+\ldots+\bar{D}_{k}A_{k}=(I-\hat{D})A_{0}+\hat{D}(D_{1}A_{1}+D_{2}A_{2}+\ldots+D_{k}A_{k}),$

and $\sum_{j=1}^{k}D_{j}=I$. So matrix

$\bar{D}_{0}A_{0}+\bar{D}_{1}A_{1}+\ldots+\bar{D}_{k}A_{k}$

is nonsingular. By using Lemma 3.4, $\mathbf{A}$ has the row
$\mathcal{W}$-property.

($\Leftarrow$) If $\mathbf{A}$ has the row $\mathcal{W}$-property, then for
arbitrary nonnegative diagonal matrix $\bar{D}_{j}=\mbox{diag}(\bar{d}_{j})$
with $\bar{d}_{j}\in[0,1]^{n}$ $(j=0,1,\ldots,k)$ and
$\sum_{j=0}^{k}\bar{D}_{j}=I$, matrix

$\bar{D}_{0}A_{0}+\bar{D}_{1}A_{1}+\ldots+\bar{D}_{k}A_{k}$

is nonsingular. In the following, we prove that
$\mathbf{\bar{A}}=(A_{0},D_{1}A_{1}+D_{2}A_{2}+\ldots+D_{k}A_{k})$ has the row
$\mathcal{W}$-property, where matrices $D_{j}=\mbox{diag}(d_{j})$ with
$d_{j}\in[0,1]^{n}$ $(j=1,\ldots,k)$ are arbitrary nonnegative diagonal
matrices and $\sum_{j=1}^{k}D_{j}=I$. In fact, for arbitrary nonnegative
diagonal matrix $\hat{D}=\mbox{diag}(\hat{d})$ with $\hat{d}\in[0,1]^{n}$,

$(I-\hat{D})A_{0}+\hat{D}(D_{1}A_{1}+D_{2}A_{2}+\ldots+D_{k}A_{k})=(I-\hat{D})A_{0}+\hat{D}D_{1}A_{1}+\hat{D}D_{2}A_{2}+\ldots+\hat{D}D_{k}A_{k}.$

Let $\bar{D}_{0}=I-\hat{D},\bar{D}_{1}=\hat{D}D_{1},\ldots,$
$\bar{D}_{k}=\hat{D}D_{k}$. Then $\bar{d}_{j}\in[0,1]^{n}$ $(j=0,1,\ldots,k)$,
$\sum_{j=0}^{k}\bar{D}_{j}=I$, and matrix

$(I-\hat{D})A_{0}+\hat{D}(D_{1}A_{1}+D_{2}A_{2}+\ldots+D_{k}A_{k})=\bar{D}_{0}A_{0}+\bar{D}_{1}A_{1}+\ldots+\bar{D}_{k}A_{k}$

is nonsingular. By using Lemma 3.4 again, $\mathbf{\bar{A}}$ has the row
$\mathcal{W}$-property. $\hfill{}\Box$

From Lemma 3.5, together with Lemma 3.1, for the (3.9) type error bound, we
give a general result for the EVLCP ($\mathbf{A},\mathbf{q}$) under the row
$\mathcal{W}$-property.

######  Theorem 3.4.

Let matrix $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ have the row
$\mathcal{W}$-property. Then for any $x\in\mathbb{R}^{n}$,

$\|x-x^{\ast}\|_{\infty}\leq\min_{D_{1},\ldots,D_{k}}\bigg{\\{}\frac{\|A_{0}+\sum_{j=1}^{k}D_{j}A_{j}\|_{\infty}}{\alpha\\{A_{0},A_{1},\ldots,A_{k}\\}}\bigg{\\}}\|r(x)\|_{\infty},$
(3.10)

where matrices $D_{j}=\mbox{diag}(d_{j})$ with $d_{j}\in[0,1]^{n}$
$(j=1,\ldots,k)$ are arbitrary nonnegative diagonal matrices and
$\sum_{j=1}^{k}D_{j}=I$, and

$\alpha\\{A_{0},A_{1},\ldots,A_{k}\\}=\min_{\|x\|_{\infty}=1}\Bigg{\\{}\max_{1\leq
i\leq
n}(A_{0}x)_{i}\bigg{(}\bigg{(}\sum_{j=1}^{k}D_{j}A_{j}\bigg{)}x\bigg{)}_{i}\Bigg{\\}}.$

Proof. Since matrix $\mathbf{A}=(A_{0},A_{1},\ldots,A_{k})$ has the row
$\mathcal{W}$-property, from Lemma 3.5, matrix
$\mathbf{\bar{A}}=(A_{0},D_{1}A_{1}+D_{2}A_{2}+\ldots+D_{k}A_{k})$ has the row
$\mathcal{W}$-property, where matrices $D_{j}=\mbox{diag}(d_{j})$ with
$d_{j}\in[0,1]^{n}$ $(j=1,\ldots,k)$ are arbitrary nonnegative diagonal
matrices and $\sum_{j=1}^{k}D_{j}=I$. As done in [22], we introduce a quantity
$\alpha\\{A_{0},A_{1},\ldots,A_{k}\\}$ below

$\alpha\\{A_{0},A_{1},\ldots,A_{k}\\}=\min_{\|x\|_{\infty}=1}\Bigg{\\{}\max_{1\leq
i\leq
n}(A_{0}x)_{i}\bigg{(}\bigg{(}\sum_{j=1}^{k}D_{j}A_{j}\bigg{)}x\bigg{)}_{i}\Bigg{\\}}.$

Further, we take

$s(x)=(A_{0}x+q_{0})-r(x)\ \mbox{and}\ t(x)=\min_{1\leq j\leq
k}\\{A_{j}x+q_{j}\\}-r(x).$

Then

$s(x)\geq 0,t(x)\geq 0,s(x)^{T}t(x)=0.$

Thus, for any $x\in\mathbb{R}^{n}$ and each $i=1,2,..,n$, together with Lemma
3.1, we obtain

$\displaystyle 0\geq$
$\displaystyle(s_{i}(x)-s_{i}(x^{\ast}))(t_{i}(x)-t_{i}(x^{\ast}))$
$\displaystyle=$
$\displaystyle((A(x-x^{*}))_{i}-r_{i}(x)+r_{i}(x^{\ast}))(((\sum_{j=1}^{k}D_{j}A_{j})(x-x^{\ast}))_{i}-r_{i}(x)+r_{i}(x^{\ast}))$
$\displaystyle\geq$
$\displaystyle((A(x-x^{\ast}))_{i}((\sum_{j=1}^{k}D_{j}A_{j})(x-x^{\ast}))_{i}$
$\displaystyle-(r_{i}(x)-r_{i}(x^{\ast}))((A+\sum_{j=1}^{k}D_{j}A_{j})(x-x^{\ast}))_{i}$
$\displaystyle\geq$
$\displaystyle((A(x-x^{\ast}))_{i}((\sum_{j=1}^{k}D_{j}A_{j})(x-x^{\ast}))_{i}$
$\displaystyle-\|r(x)\|_{\infty}\cdot\|A_{0}+\sum_{j=1}^{k}D_{j}A_{j}\|_{\infty}\cdot\|x-x^{\ast}\|_{\infty},$

from which it follows that

$\displaystyle\|A_{0}+\sum_{j=1}^{k}D_{j}A_{j}\|_{\infty}\cdot\|r(x)\|_{\infty}\cdot\|x-x^{\ast}\|_{\infty}\geq$
$\displaystyle\max_{1\leq i\leq
n}((A(x-x^{\ast}))_{i}((\sum_{j=1}^{k}D_{j}A_{j})(x-x^{\ast}))_{i}$
$\displaystyle\geq$
$\displaystyle\alpha\\{A_{0},A_{1},\ldots,A_{k}\\}\|x-x^{\ast}\|_{\infty}.$

This yields the desired inequality (3.10). $\hfill{}\Box$

##  4 Numerical examples

Since the advantages and disadvantages of Theorem 3.2 and Theorem 3.3 for
numerical examples have been presented in [19], we here compare Theorem 3.1
with Eq. (1.5), also see Theorem 4.1 in [19]. Numerical examples used by us
are from two aspects: (1) on the one hand, we still adopt a example in [19];
on the other hand, we list some new examples.

Example 4.1 [19] Let $\mathbf{A}=(A_{0},A_{1},A_{2})$, where

$A_{0}=\left[\begin{array}[]{cc}1&1\\\ -1&1\end{array}\right],\
A_{1}=\left[\begin{array}[]{cc}2&1\\\ -1&1\end{array}\right],\
A_{2}=\left[\begin{array}[]{cc}1&3\\\ -1&1\end{array}\right].$

Since $(A_{0})_{2\cdot}=(A_{1})_{2\cdot}=(A_{2})_{2\cdot}$, this particularity
avoids the row rearrangement. Even so, by Eq. (1.5), their calculations take 3
times:

$\displaystyle\mu_{1}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)A_{0}+DA_{1})^{-1}\|_{\infty}=1,$
$\displaystyle\mu_{2}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)A_{0}+DA_{2})^{-1}\|_{\infty}=1,$
$\displaystyle\mu_{3}=$
$\displaystyle\max_{d\in[0,1]^{n}}\|((I-D)A_{1}+DA_{2})^{-1}\|_{\infty}=1.$

Based on this,

$\alpha_{\infty}(\textbf{A})=\max\\{\mu_{1},\mu_{2},\mu_{3}\\}=1.$

Directly using Theorem 3.1, for $d_{1}+d_{2}\leq 1$ with
$d_{1},d_{2}\in[0,1]$, we calculate

$\max_{d\in[0,1]^{n}}\|(D_{0}A_{0}+D_{1}A_{1}+D_{2}A_{2})^{-1}\|_{\infty}=\max\\{\frac{2+2d_{2}}{2+2d_{2}+d_{1}},\frac{2+d_{1}}{2+d_{1}+2d_{2}}\\}=1=\alpha_{\infty}(\textbf{A}).$

Whereas, once the system matrix occurs the row rearrangement, the calculation
times by Eq. (1.5) increases dramatically, see Example 2.1 and Example 4.3.

Example 4.2 Here, Example 2.1 is used as Example 4.2 to investigate Theorem
3.1. Because the main diagonal elements of $A_{0},A_{1},A_{2}$ are positive,
the second element in the first row is non-negative, and the first element in
the second row is non-positive, so, for matrix
$D_{0}A_{0}+D_{1}A_{1}+D_{2}A_{2}$, its main diagonal elements are positive,
the second element in the first row is non-negative, and the first element in
the second row is non-positive for any $D_{j}=\mbox{diag}(d_{j})$ with
$d_{j}\in[0,1]^{n}$ $(j=0,1,2)$ being nonnegative diagonal matrices and
$\sum_{j=0}^{2}D_{j}=I$. According to the definition of the determinant, we
can get $\det(D_{0}A_{0}+D_{1}A_{1}+D_{2}A_{2})>0$, i.e.,
$D_{0}A_{0}+D_{1}A_{1}+D_{2}A_{2}$ is nonsingular. From Lemma 3.4, the block
matrix $\mathbf{A}=(A_{0},A_{1},A_{2})$ has row $\mathcal{W}$-property.

Directly using Theorem 3.1, for $D_{j}=\mbox{diag}(d_{j})$ with
$d_{j}\in[0,1]^{n}$ $(j=0,1,2)$ are nonnegative diagonal matrices and
$\sum_{j=0}^{2}D_{j}=I$, we calculate

$\max_{d\in[0,1]^{n}}\|(D_{0}A_{0}+D_{1}A_{1}+D_{2}A_{2})^{-1}\|_{\infty}=\max\left\\{w_{1},w_{2}\right\\}=3,$

where

$w_{1}=\frac{2-(d_{1})_{11}}{1+(d_{2})_{11}+(1-(d_{1})_{11})(1+(d_{1})_{22}-(d_{2})_{22})},$

and

$w_{2}=\frac{2+(d_{1})_{22}}{1+(d_{2})_{11}+(1-(d_{1})_{11})(1+(d_{1})_{22}-(d_{2})_{22})}.$

When using Eq. (1.5), although $\alpha_{\infty}(\textbf{A})=3$, see Example
2.1, which is equal to our this result, its calculation times are 9.

It is easy to check that $\mathbf{A}_{0}$, $\mathbf{A}_{1}$ in Example 4.2 are
not strictly diagonal dominant, and that

$\rho(\max\\{\Lambda_{0}^{-1}|C_{0}|,\Lambda_{1}^{-1}|C_{1}|,\Lambda_{2}^{-1}|C_{2}|\\})=1.4142>1.$

Hence the error bounds in Theorem 3.2, Theorem 3.3, Theorem 4.3 and Theorem
4.4 of [19] cannot work for this case.

Example 4.3 Let $\mathbf{A}=(A_{0},A_{1},A_{2},A_{3})$, where

$A_{0}=\left[\begin{array}[]{cc}1&1\\\ -1&1\end{array}\right],\
A_{1}=\left[\begin{array}[]{cc}1&0\\\ -2&1\end{array}\right],\
A_{2}=\left[\begin{array}[]{cc}4&1\\\ 0&1\end{array}\right],\
A_{3}=\left[\begin{array}[]{cc}2&0\\\ -3&1\end{array}\right]$

With the same analysis method as in Example 4.3, it is easy to get that
$\mathbf{A}=(A_{0},A_{1},A_{2},A_{3})$ has row $\mathcal{W}$-property.
Directly using Theorem 3.1, for $D_{0},D_{1},D_{2},D_{3}$ are nonnegative
diagonal matrices with $d_{j}\in[0,1]^{n}$ $(j=0,1,2,3)$ and
$\sum_{j=0}^{3}D_{j}=I$, we calculate

$\max_{d\in[0,1]^{n}}\|(D_{0}A_{0}+D_{1}A_{1}+D_{2}A_{2}+D_{3}A_{3})^{-1}\|_{\infty}=\max\left\\{\bar{w}_{1},\bar{w}_{2}\right\\}=4=\alpha_{\infty}(\textbf{A}).$

where

$\bar{w}_{1}=\frac{2-(d_{1})_{22}-(d_{2})_{22}+2(d_{3})_{22}}{1+3(d_{2})_{11}+(d_{3})_{11}+(1+(d_{1})_{22}-(d_{2})_{22}+2(d_{3})_{22})(1-(d_{1})_{11}-(d_{3})_{11})}$

and

$\bar{w}_{2}=\frac{2-(d_{1})_{11}+3(d_{2})_{11}}{1+3(d_{2})_{11}+(d_{3})_{11}+(1+(d_{1})_{22}-(d_{2})_{22}+2(d_{3})_{22})(1-(d_{1})_{11}-(d_{3})_{11})}.$

Of course, we can adopt Eq. (1.5) to obtain $\alpha_{\infty}(\textbf{A})=4$,
whereas, its calculation times are 36. Here, the computational process of Eq.
(1.5) is omitted.

It is easy to check that $\mathbf{A}_{0}$, $\mathbf{A}_{1}$, $\mathbf{A}_{3}$
are not strictly diagonal dominant, and that

$\rho(\max\\{\Lambda_{0}^{-1}|C_{0}|,\Lambda_{1}^{-1}|C_{1}|,\Lambda_{2}^{-1}|C_{2}|,\Lambda_{3}^{-1}|C_{3}|\\})=2.3028>1.$

For Example 4.3, the error bounds in Theorem 3.2, Theorem 3.3, Theorem 4.3 and
Theorem 4.4 of [19] cannot work for this case as well.

From Example 2.1 and Example 4.3, it is easy to find that although the latter
has one more matrix than the former, the calculation times of the latter are
more 27 times than the former.

##  5 Conclusion

In this paper, by introducing a general equivalent form of the minimum
function, some new error bounds for the EVLCP($\mathbf{A},\mathbf{q}$) under
the row $\mathcal{W}$-property have been presented. These new error bounds not
only cover some existing results, but also keep away from the row
rearrangement of the system matrix. In addition, with respect to the row
$\mathcal{W}$-property, we also obtain two new sufficient and necessary
conditions. Finally, by numerical examples, we show that the new error bound
is feasible. Compared with the error bound by the row rearrangement technique,
our new error bound greatly reduces the computation workload in a way.

In addition, by a lot of numerical experiments, we find that the error bound
(3.5) is equal to the error bound (1.5). There exists an interested problem
how to prove the equality of both, which is necessary for us to study it in
the future.

## References

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# The NIFFTE project

J. Ruza Corresponding author. D.M Asnerb R.G Bakerc J. Bundgaardd E. Burgette
M. Cunninghama J. Deavene D.L. Duked U. Greifed S. Grimesf M. Heffnera T.
Hillg D. Isenhowerh J.L. Klayc V. Kleinrathe N. Kornilovf A.B. Laptevi W.
Lovelandj T.N. Masseyf R. Meharchandi H. Quh S. Sangiorgioa B. Seilhana L.
Snydera S. Staveb G. Tatishvilib R.T. Thorntonh F. Tovessoni D. Towellh R.S.
Towellh S. Watsonh B. Wendte and L. Woodb
aLawrence Livermore National Laboratory  Livermore  CA 94550  USA
bPacific Northwest National Laboratory  Richland  WA 99354  USA
cCalifornia Polytechnic State University  San Luis Obispo  CA 93407  USA
dColorado School of Mines  Golden  CO 80401  USA
eIdaho State University  Pocatello  ID 83209  USA
fOhio University  Athens  OH 45701  USA
gIdaho National Laboratory  Idaho Falls  ID 83415  USA
hAbilene Christian University  Abilene  TX 79699  USA
iLos Alamos National Laboratory  Los Alamos  NM 87545  USA
jOregon State University  Corvallis  OR 97331  USA
E-mail<EMAIL_ADDRESS>
###### Abstract

The Neutron Induced Fission Fragment Tracking Experiment (NIFFTE) is a double-
sided Time Projection Chamber (TPC) with micromegas readout designed to
measure the energy-dependent neutron-induced fission cross sections of the
major and minor actinides with unprecedented accuracy. The NIFFTE project
addresses the challenge of minimizing major sources of systematic
uncertainties from previous fission chamber measurements such as: target and
beam non-uniformities, misidentification of alpha and light charged particles
as fission fragments, and uncertainties inherent to the reference standards
used. In-beam tests of the NIFFTE TPC at the Los Alamos Neutron Science Center
(LANSCE) started in 2010 and have continued in 2011, 2012 and 2013. An
overview of the NIFFTE TPC status and performance at LANSCE will be presented.

###### keywords:

Fission; TPC; Cross Section

##  1 Introduction

Sensitivity studies indicate that high-accuracy measurements of fission cross
sections are needed and the Neutron Induced Fission Fragment Tracking
Experiment (NIFFTE) collaboration is developing a Time Projection Chamber
(TPC) to meet this need. Neutron-induced fission cross sections have been
measured with fission chambers (ICs) for decades, typically by placing a thin
target of the fissile material of interest into a neutron beam and detecting
the ionization left in a gas by the resulting fission products. To date,
fission chambers have been limited to measuring the energy deposited in a gas
by charged particles, and were therefore unable to distinguish between
different particles of the same energy. Fragments are subject to energy loss
and straggling within the target111A non negligible fraction of fission
fragments might become indistinguishable from lighter charged particles within
the fission chamber due to energy loss and straggling. resulting in a
systematic error for the measured cross section.

Based on well-established technology originally developed for high-energy
physics experiments, the NIFFTE TPC is uniquely suited to achieve
unprecedented accuracy. The detection principle of a TPC is rather simple:
particles ionizing the gas traverse a uniform electric field. The electrons
produced by the ionization are drifted by the electric field and collected at
a segmented readout plane. A two-dimensional projection of the particle’s
track is obtained by interpreting the amount of charge deposited in each
segment. Combining this two dimensional projection with the drift time of the
electrons provides a full, three-dimensional representation of the particle’s
path through the detector.

A schematic of a neutron-induced fission in the TPC is shown in figure 1. The
fissionable sample to be studied is placed in the center of the detector at
the midpoint of the cathode. The field cage covers the $\rm{54\,mm}$ drift
distance at each side of the sample and it is placed inside a pressure vessel
of $\rm{15\,cm}$ diameter that is filled up with a suitable gas mixture e.g.
Argon($90\,\%$)-Methane($10\,\%$) at a nominal $\rm{550\,torr}$. At each end
of the chamber charge is amplified and read out on a MICROMEGAS “pad” plane.
The signals are digitized and processed by custom-made EtherDAQ card
assemblies [1, 2] sampling at $50\,\rm{MHz}$ together with a software
framework developed for the data acquisition, analysis, and simulation tasks
[3]. An article describing the TPC hardware is in preparation.

Figure 1: Schematic of a neutron-induced fission in the target of the NIFFTE
TPC (left) and picture of the TPC installed in the LANSCE-WNR facility during
the 2012 run cycle.

##  2 Beam experiments

Building a TPC suitable for fission research is an ambitious task, involving
complex systems and state-of-the-art technology. As new hardware and software
capabilities are implemented, frequent evaluation is required. In-beam testing
takes place at the 4FP90L flight path of Los Alamos Neutron Science Center
(LANSCE) Weapons Neutron Research (WNR) facility [4] that makes use of an
unmoderate tungsten spallation target to produce a high-energy neutron
spectrum with energies ranging from $100\,\rm{keV}$ to $600\,\rm{MeV}$.

The first TPC experiments at LANSCE-WNR took place in 2010. Data were
collected first with 64 then 192 individual channels ($\rm{1\,\%}$ and
$\rm{3\,\%}$ of the full TPC), with blank carbon samples and on carbon
backings with $\rm{{}^{238}U}$ deposited onto them [5]. These early
measurements validated the TPC technique, as particle tracks could be
reconstructed for both fission fragments and alpha particles. During the 2011
LANSCE run cycle, one complete sextant (496 channels, $\rm{8\,\%}$ of the full
TPC) was instrumented, and data were collected with samples containing
$\rm{{}^{238}U}$, $\rm{{}^{235}U}$, and $\rm{{}^{239}Pu}$. During the most
recent run cycle (2012), the TPC was instrumented with one complete pad plane
(2976 channels, 1/2 of the full detector), and notably, for the first time in
the TPC project, information about the neutron time-of-flight was available.

##  3 Performance

Preliminary analysis of the data taken during the 2010 and 2011 LANSCE run
cycles have been reported [6, 7] and the 2012 run cycle data is currently
being analyzed. Data were collected on three samples: a split sample
(semicircular deposits) containing $\rm{{}^{238}U}$ and $\rm{{}^{235}U}$; a
split sample with $\rm{{}^{235}U}$ and $\rm{{}^{239}Pu}$; and a sample with
$\rm{{}^{239}Pu}$ electro-plated onto metallized polyproplylene. In adition,
off-beam data was collected with the use of $\rm{{}^{252}Cf}$ and
$\rm{{}^{244}Cm}$ bottom sources to fine tune the acquisition software and the
analysis codes. All results included in this paper correspond to the 2012 data
taking period and should be considered preliminary.

###  3.1 Neutron Time of Flight

Neutron-induced fission depends on the energy of the incident neutron. The
neutron Time-Of-Flight (nTOF) between the spallation target at WNR and the
fissionable target placed inside the TPC can be folded together with the
overall WNR beam spacing structure -nominal $1.8\,\rm{\mu s}$ pulse spacing
between neutron bunches -to reproduce the energy spectrum of the neutrons
inducing fission events in the TPC. Figure 2 shows a reconstruction of the
nTOF for the NIFFTE TPC during the 2012 run cycle. The peak around
$28\,\rm{nsec}$ corresponds to photo-induced fission in the TPC target and,
can be used to obtain an absolute calibration for both, the distance of the
TPC relative to the spallation target and the energy of the incoming neutrons.
The width of the photo-fission peak serves as a tool to determine the
systematic error on the energy of the incident neutron. During 2012, the
NIFFTE TPC was able to achieve $2.5-3.5\,\rm{nsec}$ FWHM for the photo-fission
peak, which is competitive with the resolution obtained by previous fission
chamber experiments.

Figure 2: Neutron time of flight reconstruction for neutron-induced fission
events in the TPC and zoom of the observed photo-fission peak. A split sample
(semicircle deposits) containing $\rm{{}^{238}U}$ and $\rm{{}^{235}U}$ was
used during this measurement.

###  3.2 Tracking and Particle Identification

One of the major advantages of using a TPC to study neutron-induced fission
cross sections is the ability to reconstruct and study the features of the
ionizing tracks that the different particles generate on their way through the
active volume of NIFFTE.

One example of how particle identification (distinguishing alpha and light
charged particles from fission fragments) might be done using the tracking
information of the TPC can be shown by comparing the length of the particle’s
track to its total energy deposited in the TPC (see figure 3). Fission
fragments, have relatively short tracks and deposit large amount of energy in
the fill gas, while light particles (alpha particles) have longer tracks and
deposit less energy.

Figure 3: Plot of length versus energy of track reconstructions in the TPC for
a $\rm{{}^{252}Cf}$ buttom source. On the plot there are three regions that
can be easily spotted, an almost vertical line at low energies with lengths
from $\rm{1\,cm}$ up to $\rm{6\,cm}$ containing $\alpha$-particles and
protons, a big elliptical zone centered around $\rm{80\,MeV}$ with tracks from
$\rm{1.5\,cm}$ up to $\rm{3\,cm}$ gathering most of the fission fragment
population and a small shadow to the elipse with track lengths around
$\rm{1\,cm}$ and energies between $\rm{5}$ and $\rm{40\,MeV}$. The shadow
region has been succesfully identified as fission fragments that come out of
the target at high polar angles, suffering from straggling, and hit on the
concentric copper ring of $\rm{2}$ by $\rm{4\,cm}$ diameter that holds the
polypropylene backing foil where the $\rm{2\,mm}$ diameter $\rm{{}^{252}Cf}$
spot was deposited.

Using precise tracking information, e.g. the differential ionization pattern
of the particle within the gas, allows the search of the specific Bragg peaks
created by the particles and the comparison of their ionization pattern with
simulations and theoretical models to provide an ultimate tool for particle
identification. Fission fragments deposit most of the energy at the begining
of their short track through the TPC, while $\alpha$ particles and protons
penetrate the gas further and have their Bragg peak towards the end of their
path through the TPC (see figures 4 and 5).

Figure 4: Ionization pattern of a fission fragment in the NIFFTE TPC. The
Bragg peak is located at the begining of the track for these particles.

Finally, tracking information also enables tracing of each individual particle
back to a well-defined origin in the sample, which ultimately allows for a
minimization on the systematics errors such as beam spot size, and sample
uniformity. This makes it possible to study two or more actinides on the same
backing, as described in [5], and to check sample uniformity using the decay
activity of the actinide deposits.

Figure 5: Ionization pattern of an observed $\alpha$-track in the NIFFTE TPC
(black) and a theoretical prediction that includes the ionization nature of an
$\alpha$-like particle together with the specific diffusion expected in the
TPC for this particular track (blue). In red the residual of observed and
modeled ionization pattern is shown. Figure 6: Track vertex reconstructions of
$\alpha$-like events emerging from a $\rm{2\,mm}$ spot source of
$\rm{{}^{252}Cf}$ placed in the center of the NIFFTE TPC. In the figure a
picture of the sample used can be seen.

Figure 6 shows the vertex reconstructions for spontaneous $\alpha$-like events
from a $\rm{{}^{252}Cf}$ bottom source. The tracks seem to originate from the
center of the cathode plane, coming out of a circle of approximately
$\rm{2\,mm}$ diameter, which is consistent with the actual dimension of the
$\rm{{}^{252}Cf}$ deposit. Note that the two different backing materials can
easily be distinguished: one external ring of copper with inner and outer
diameters of $\rm{2\,cm}$ and $\rm{4\,cm}$ respectively, and a $\rm{2\,cm}$
diameter polypropylene foil on top of which the $\rm{{}^{252}Cf}$ was
deposited. This figure clearly shows the radiographic capabilities of NIFFTE
to map sample uniformities.

##  4 Conclusion

The NIFFTE TPC is being tested off-beam and in-beam at the neutron facility
LANSCE-WNR. Preliminary results (with half detector instrumented) indicate
that the TPC performance is meeting expectations. As the NIFFTE project moves
from development into production phase, the collaboration is optmistic that
the project is on-track to deliver neutron-induced fission cross sections with
unprecedented accuracy.

###### Acknowledgements.

This work was performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under contract No. DE-AC52-07NA27344.

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