1 Introduction
Matrix theory is one of the most fundamental tools of mathematics exploration and scientific research [2, 12]. As a higher-order generalization of a matrix, tensors and their properties are widely used in a great variety of fields, such as gravitational theory and quantum mechanics in physics [32, 42], large-scale date analysis [18], hypergraph spectral theory [33, 43], social network data analytics [16, 48], automatical control [27], the best rank-one approximations in statistical data analysis [17, 49], complementarity problems [1, 7, 9, 10, 15, 24‐26, 37, 38, 40, 41], etc. As a significant knowledge point of tensor theory, tensor eigenvalues is one of the most popular research topics in recent years, and gradually appears in many research and application fields.
In 2005, Qi [28] introduced the concept of eigenvalues for symmetric tensors. At the same time, this concept was simultaneously introduced by Lim [23], but he only considered the case when the eigenpairs are real. Since then, the tensor eigenvalue theory has attracted great attention and developed rapidly over the last decades. However, in order to find an eigenvalue or eigenvector of a higher-order tensor, it is necessary to solve a system of higher-degree polynomial equations with multiple variables [29, 31]. This means that it will be extremely difficult to solve the tensor eigenvalue problem when the order of such a tensor is very high. Therefore, many mathematical researchers pay attention to how to find more accurate range and numerical methods of eigenvalues and eigenvectors of higher-order tensors. For example, there is a lot of literature on bounds and calculation methods of the spectral radius (H-eigenvalue) of nonnegative tensors [3, 5, 8, 10, 19, 21, 31, 34‐36, 39, 43‐46].
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Equally important, the Z-eigenpair for nonnegative tensors plays a fundamental role in many applications such as high order Markov chains [13, 22], geometric measure of quantum entanglement [14], best rank-one approximation [6, 30, 47], and so on. Recently, due to the joint efforts of mathematicians, there are a series of theoretical conclusions and numerical methods to bound the Z-spectral radius for nonnegative tensors, these results are beneficial to further research and applications of the field.
In this paper, we mainly consider the bounds of Z-eigenpair of an irreducible nonnegative tensor. By estimating the ratio of the smallest and largest components of a Perron vector, we present some bounds for the eigenvector and Z-spectral radius of an irreducible and weakly symmetric nonnegative tensor. These proposed bounds extend and complement some existing ones. Furthermore, two examples are given to illustrate the proposed bounds.
This paper is organized as follows. In Sect. 2, we will give some basic facts and symbols. The concept of Z-eigenvalue and a Peron–Frobenius-type theorem is given [4]. In Sect. 3, we calculate the ratio of the smallest and largest components of a Perron vector. Moreover, a sharper bound of Z-spectral radius is shown for an irreducible and weakly symmetric nonnegative tensor. Two examples are given and the corresponding comparison is made intuitively and in detail. Some concluding remarks are presented in the final section.
2 Preliminaries and basic facts
For a positive integer n, \(I_{n}\) denotes the set \(I_{n}=\{1, 2, \dots , n\}\). Let \(\mathbb{R}\) and \(\mathbb{C}\) be the real and complex field, respectively. We call \(\mathcal{A}=(a_{i_{1} i_{2} \cdots i _{m}})\) a real (complex) tensor of mth order and dimension n if \(a_{i_{1} i_{2} \cdots i_{m}}\in \mathbb{R}\ (\mathbb{C})\), \(i_{1},i _{2},\dots ,i_{m}\in I_{n} \). Clearly, an mth order n-dimensional tensor consists of \(n^{m}\) entries from the real field \(\mathbb{R}\). The set of all mth order n-dimensional real tensors is denoted by \(T_{m,n}\). For any tensor \(\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in T _{m,n}\), if their entries \(a_{i_{1}\cdots i_{m}}\) are invariant under any permutation of their indices, then \(\mathcal{A}\) is called a symmetric tensor. We denote the set of all mth order n-dimensional real symmetric tensors as \(S_{m,n}\). Let \(\pi (1, 2,\dots , n)\) be set of all permutations of \(\{1, 2,\dots , n\}\). Let \(\mathcal{A} = (a _{i_{1}\cdots i_{m}})\in T_{m,n}\) and consider a vector \(x=(x_{1},x _{2},\dots ,x_{n})^{\top }\in \mathbb{R}^{n}\) or \(\mathbb{C}^{n}\). Then \(\mathcal{A}x^{m-1}\) is a vector with its ith component defined by
and \(\mathcal{A}x^{m}\) is a homogeneous polynomial of degree m,
where \(x^{\top }\) is the transposition of x.
$$ \bigl(\mathcal{A}x^{m-1}\bigr)_{i}:=\sum _{i_{2},\dots ,i_{m}=1}^{n}a_{ii_{2} \cdots i_{m}}x_{i_{2}} \cdots x_{i_{m}}, \quad \forall i\in I_{n}, $$
$$ \mathcal{A}x^{m}:=x^{\top }\bigl(\mathcal{A}x^{m-1} \bigr)= \sum_{i_{1},i_{2},\dots ,i_{m}=1}^{n}a_{i_{1}i_{2}\cdots i_{m}}x_{i _{1}}x_{i_{2}} \cdots x_{i_{m}}, $$
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Definition 2.1
Let \(\mathcal{A}=(a_{i_{1}i_{2}\cdots i_{m}})\in T_{m,n}\). We call a number \(\lambda \in \mathbb{C}\) an E-eigenvalue of \(\mathcal{A}\) if there is a nonzero vector \(x\in \mathbb{C}^{n}\) which solves the following system of polynomial equations:
and call the solution x an E-eigenvector of \(\mathcal{A}\) associated with the eigenvalue λ. Any such pair \((\lambda ,x)\) is called an E-eigenpair of \(\mathcal{A}\). We call \((\lambda ,x)\) a Z-eigenpair if they are both real.
$$ \begin{aligned} &\mathcal{A}x^{m-1}=\lambda x, \\ &x^{T}x=1, \end{aligned} $$
Definition 2.2
The set of all Z-eigenvalues of \(\mathcal{A}\) is called the Z-spectrum of \(\mathcal{A}\), denoted as \(\sigma _{z}(\mathcal{A})\). The largest modulus of the elements in the Z-spectrum of \(\mathcal{A}\) is called the Z-spectral radius of \(\mathcal{A}\), denoted as \(\rho _{z}(\mathcal{A})\).
Definition 2.3
For any given tensor \(\mathcal{A}=(a_{i_{1}i_{2}\cdots i_{m}})\in T _{m,n}\), we say that \(\mathcal{A}\) is reducible if there exists a nonempty proper index subset \(J\subset I_{n}\) such that
\(\mathcal{A}\) is called irreducible if it is not reducible.
$$ a_{i_{1}\cdots i_{m}}=0,\quad \forall i_{1}\in J,\forall i_{2},\dots ,i_{m}\notin J; $$
Definition 2.4
A real tensor \(\mathcal{A}\) is called weakly symmetric if the associated homogeneous polynomial
satisfies \(\nabla f_{\mathcal{A}}(x)=m\mathcal{A}x^{m-1}\).
$$ f_{\mathcal{A}}(x):=\mathcal{A}x^{m}=\sum _{i_{2},\dots ,i_{m}=1} ^{n} a_{i_{1}i_{2}\cdots i_{m}} x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}} $$
An mth order n-dimensional tensor \(\mathcal{A}\) is called nonnegative (or, respectively, positive) if \(a_{i_{1}\cdots i_{m}} \geq 0\) (or, respectively, \(a_{i_{1}\cdots i_{m}}>0\)) for all \(i_{1},\dots , i_{m}\in I_{n}\). We denote the set of all nonnegative (or, respectively, positive) tensors of mth order and dimension n by \(\mathbb{R}_{+}^{[m,n]}\) (or, respectively, \(\mathbb{R}_{++} ^{[m,n]}\)).
Theorem 2.1
([4])
Let
\(\mathcal{A}\)
be an
mth order
n-dimensional nonnegative tensor. Then
(i)
There exists a
Z-eigenvalue
\(\lambda _{0}\geq 0\)
of
\(\mathcal{A}\)
with a nonnegative
Z-eigenvector
\(x_{0}\neq 0\), i.e.,
$$ \mathcal{A}x_{0}^{m-1}=\lambda _{0}x_{0}, \quad x_{0}^{\top }x_{0}=1; $$
(ii)
The above
Z-eigenvalue
\(\lambda _{0}\)
and its
Z-eigenvector
\(x_{0}\)
are positive if
\(\mathcal{A}\)
is irreducible;
(iii)
The
Z-spectral radius
\(\rho _{z}(A)\)
is a positive
Z-eigenvalue with a positive
Z-eigenvector if
\(\mathcal{A}\)
is weakly symmetric and irreducible.
Recently, there appeared a series of theoretical conclusions and numerical methods to bound the Z-spectral radius for nonnegative tensors. For instance, Chang, Pearson and Zhang [4] studied some variation principles of Z-eigenvalues of nonnegative tensors. As a corollary of the main results, they presented the lower bound of Z-spectral radius for irreducible weakly symmetric nonnegative tensors (see Corollary 4.10 of [4]) as follows:
where \(c_{1}=\max_{i} a_{i \cdots i}\) and \(c_{2}=(\frac{1}{ \sqrt{n}})^{m-2}\min_{i}\sum_{i_{2},\dots ,i_{m}=1}^{n}a _{ii_{2}\ldots i_{m}}\). For a nonnegative tensor, they also gave an upper bound for the Z-spectral radius (see Proposition 3.3 of [4]):
$$ \max \{c_{1},c_{2}\}\leq \rho _{z}(\mathcal{A}), $$
(2.1)
$$ \rho _{z}(\mathcal{A})\leq \sqrt{n}\max _{i} \sum_{i_{2},\dots ,i_{m}=1}^{n}a_{ii_{2}\ldots i_{m}}. $$
(2.2)
Song and Qi [34] proved a sharper upper bound for the Z-spectral radius of any mth order n-dimensional tensor (see Corollary 4.5 of [34]):
$$ \rho _{z}(\mathcal{A})\leq \max _{i}\sum _{i_{2},\dots ,i_{m}=1}^{n} \vert a_{ii_{2}\ldots i_{m}} \vert . $$
(2.3)
He and Huang [11] obtained an upper bound of the Z-spectral radius for a weakly symmetric positive tensor (see Theorem 2.7 of [11]):
where \(r_{i}=\sum_{i_{2},\dots ,i_{m}=1}^{n}a_{ii_{2}\ldots i_{m}}\), \(R=\max_{i}r_{i}\), \(r=\min_{i}r_{i}\), \(l=\min_{i_{1},\dots , i_{m}}a_{i_{1}\cdots i_{m}}\), and \(\theta =( \frac{r}{R})^{\frac{1}{m}}\).
$$ \rho _{z}(\mathcal{A})\leq R-l(1-\theta ), $$
(2.4)
Li, Liu and Vong [20] gave an upper bound of the Z-spectral radius for any tensor:
Moreover, they also presented two-sided bounds of the Z-spectral radius for an irreducible weakly symmetric nonnegative tensor:
where \(\delta =\frac{\min_{i,j}a_{i j \cdots j}}{r-\min_{i,j}a_{i j \cdots j}}(\gamma ^{\frac{m-1}{m}} - \gamma ^{\frac{1}{m}})+\gamma \), \(\gamma =\frac{R-\min_{i,j}a _{ij\cdots j}}{r-\min_{i,j}a_{ij\cdots j}}\), \(r_{i}=\sum_{i_{2},\dots ,i_{m}=1}^{n}a_{ii_{2}\ldots i_{m}}\), \(R=\max_{i}r_{i}\), \(r=\min_{i}r_{i}\), and
$$ \rho _{z}(\mathcal{A})\leq \min _{k\in [m]}\max _{i_{k}} \sum_{i_{t}=1,t\in [m]\backslash \{k\}}^{n} \vert a_{i_{1}\cdots i _{k}\cdots i_{m}} \vert . $$
(2.5)
$$ d_{m,n}\leq \rho _{z}(\mathcal{A})\leq \max _{i,j}\bigl\{ r_{i}+a_{ij \cdots j}\bigl(\delta ^{-\frac{m-1}{m}}-1\bigr)\bigr\} , $$
(2.6)
$$ \begin{aligned} &d_{m,n}=\max _{k\in [m]\backslash \{1\}}\min _{i_{1}} \Biggl[\bigl(\delta ^{\frac{1}{m}}-1\bigr)\min _{i_{t},t\in [m]\backslash \{1\}}a_{i_{1}\cdots i_{k}\cdots i_{m}}+ \min _{i_{t},t\in [m]\backslash \{1,k\}}\sum_{i_{k}=1}^{n}a _{i_{1}\cdots i_{k}\cdots i_{m}} \Biggr] .\end{aligned} $$
Recently, Li, Liu and Vong [21] obtained an upper bound of the Z-spectral radius for an irreducible weakly symmetric nonnegative tensor by the following equation: for a Perron vector \(x=(x_{1}, \dots ,x_{n})^{\top }\),
and
where \(x_{\min }=\min_{1\leq i\leq n}x_{i}\), \(x_{\max }= \max_{1\leq i\leq n}x_{i}\),
\(\gamma =\frac{\max_{i\in I_{n}}r_{i}-\min_{i,j}a_{ij \cdots j}}{\min_{i\in I_{n}} r_{i}-\min_{i,j}a_{ij \cdots j}}\), \(r_{i}=\sum_{i_{2},\dots ,i_{m}=1}^{n}a_{ii_{2}\ldots i _{m}}\), and \(t=[\frac{m}{2}]\). From (2.8), they have the following conclusion:
where \(\mathcal{S}^{\prime }=\frac{1}{m!}\mathcal{S}\), \(\eta \equiv \eta ( \mathcal{S}^{\prime })\), \(\mathcal{S}=(s_{i_{1}\cdots i_{m}})\in R^{[m,n]}\), and \(s_{i_{1}\cdots i_{m}=\sum _{(j_{1},\dots , j_{m})\in \pi (i_{1},\dots , i_{m})}a_{j_{1} \cdots j_{m}}}\). However, there is a small negligence here since they use \(t\geq m-t\) in their proof, but the fact that \(t=[\frac{m}{2}]\) may not imply \(t\geq m-t\) (for example, for \(m=3\), \(t=[\frac{m}{2}]=1\) and \(m-t=2\)). In this paper, we will modify this negligence by taking \(t=m-[\frac{m}{2}]\).
$$ \frac{x_{\max }}{x_{\min }}\geq \eta (\mathcal{A})^{\frac{1}{m}} $$
(2.7)
$$ \rho _{z}(\mathcal{A})\leq \max _{i,j\in I_{n}} \Biggl( \sum_{k=0}^{m-1}\mathcal{A}_{i,\alpha (k,j)} \eta ^{-\frac{k}{m}} \Biggr), $$
(2.8)
$$\begin{aligned}& \begin{aligned} \eta (\mathcal{A})=\frac{\sum_{k=t}^{m-1}\min _{i,j \in I_{n}}\mathcal{A}_{i,\alpha (k,j)} [\gamma ^{\frac{k}{m}}- \gamma ^{\frac{m-k}{m}}]+\max _{i\in I_{n} }r_{i}-\sum_{k=t}^{m-1}\min _{i,j\in I_{n}}\mathcal{A}_{i,\alpha (k,j)}}{ \min _{i\in I_{n} }r_{i}-\sum_{k=1}^{t-1}\min _{i,j\in I_{n}}\mathcal{A}_{i,\alpha (k,j)}(1-\gamma ^{- \frac{k}{m}})- \sum_{k=t}^{m-1}\min _{i,j\in I_{n}} \mathcal{A}_{i,\alpha (k,j)}}, \end{aligned} \\& \mathcal{A}_{i,\alpha (k,j)}=\sum_{\substack{s_{1}< \cdots < s_{k} \\ s_{k+1}< \cdots < s_{m-1} \\ \{s_{1},\dots ,s_{k},\dots ,s_{m-1}\}\in \pi (2,\dots ,m)}}\sum _{\substack{i_{s_{1}}=\cdots =i_{s_{k}}=j \\ i_{s_{k+1}}=\cdots =i_{s_{m-1}}\neq j}}a_{i_{1}i_{2}\cdots i_{m}},\quad 0 \leq k\leq m-1, \end{aligned}$$
$$ \rho _{z}(\mathcal{A})\leq \max _{i,j\in I_{n}} \Biggl( \sum_{k=0}^{m-1}\mathcal{S}_{i,\alpha (k,j)}^{\prime } \eta ^{-\frac{k}{m}} \Biggr), $$
(2.9)
Obviously, the bound (2.5) is sharper than those in (2.2) and (2.3) for any tensor. Since \(\delta \geq 1\), it’s easy to see that the upper bound in (2.6) is sharper than that in (2.4) when the tensor is assumed to be weakly symmetric positive. Since \(\eta (\mathcal{A})\geq \delta \geq \gamma \geq 1\), hence the upper bound in (2.8) is always better than that in (2.6). When the tensor is irreducible symmetric nonnegative, the bound in (2.9) becomes that in (2.8).
3 Bounds for the Z-spectral radius of nonnegative tensors
Theorem 3.1
Let
\(\mathcal{A}=(a_{i_{1}i_{2}\cdots i_{m}})\in \mathbb{R}^{[m,n]}\)
be a nonnegative tensor having a positive
Z-eigenpair. Then for any
Z-eigenpair
\((\lambda ,x)\)
of
\(\mathcal{A}\)
with a positive
Z-eigenvector
x, we have
where
\(x_{s}=x_{\min }=\min_{i\in I_{n}} x_{i}\), \(x_{l}=x_{ \max }=\max_{i\in I_{n}} x_{i}\),
$$ \begin{aligned} \frac{x_{\max }}{x_{\min }}\geq \varphi ( \mathcal{A})^{\frac{1}{m}}, \end{aligned} $$
Proof
According to Theorem 2.1, there exists an \(x=(x_{1},x_{2}, \dots ,x_{n})^{T}>0\) such that \(\mathcal{A}x^{m-1}=\lambda x\). For \(x_{s}=x_{\min }=\min_{i\in I_{n}} x_{i}\), it follows that
Taking \(r_{i}=r_{q}=r\), since \(x_{s}>0\), we have
For \(x_{l}=x_{\max }=\max_{i\in I_{n}} x_{i}\), we similarly have
Taking \(r_{i}=r_{p}\), since \(x_{l}>0\), we have that
Combining (3.1) and (3.2) together gives
Multiplying by \(\frac{x_{l}}{x_{s}^{m-1}}\) on both sides gives
and so we have
Since \((\frac{x_{l}}{x_{s}})^{m}\geq (\frac{x_{l}}{x_{s}})^{m-1} \geq \cdots \geq \frac{x_{l}}{x_{s}}\geq 1 \), by (3.3), we get
i.e.,
Hence
Let \(\gamma =\frac{R-\min_{i,j\in I_{n}}a_{ij\cdots j}}{r- \min_{i,j\in I_{n}}a_{ij\cdots j}}\). Then \(\frac{x_{l}}{x_{s}} \geq \gamma ^{\frac{1}{m}}\geq 1\). Let \(t=m-[\frac{m}{2}]\). Then \(t\geq m-t\), so by (3.3) again, we have
Since \(\frac{x_{l}}{x_{s}}\geq \gamma ^{\frac{1}{m}}\geq 1\) and \(t\geq m-t\), we have
and hence, \(\frac{x_{l}}{x_{s}}\geq (\varphi (\mathcal{A}))^{ \frac{1}{m}}\), where
$$\begin{aligned} \lambda x_{s}\leq \lambda x_{i} =&\sum_{i_{2},\dots ,i_{m}\in I_{n}} a _{ii_{2}\cdots i_{m}} x_{i_{2}}\cdots x_{i_{m}} \\ \leq& a_{is\cdots s}x _{s}^{m-1} + \sum _{(i_{2},\dots ,i_{m})\in \Delta (s,m-2)} a_{ii_{2}\cdots i_{m}} x_{s}^{m-2}x_{l} + \cdots \\ &{} + \sum_{(i_{2},\dots ,i_{m})\in \Delta (s,k)} a_{ii_{2}\cdots i_{m}} x_{s}^{k}x_{l}^{m - k - 1} + \cdots \\ &{} + \sum_{(i_{2},\dots ,i_{m})\in \Delta (s,1)} a_{ii_{2}\cdots i_{m}} x_{s}x_{l}^{m-2} + \sum_{(i_{2},\ldots ,i_{m})\in \Delta (s,0)} a_{ii_{2}\cdots i_{m}}x_{l}^{m-1} \\ =&a_{is\cdots s}\bigl(x_{s}^{m-1} - x _{l}^{m-1}\bigr)+ \sum _{(i_{2},\dots ,i_{m})\in \Delta (s,m-2)}a_{ii_{2}\cdots i _{m}} \bigl(x_{s}^{m-2}x_{l} - x_{l}^{m-1}\bigr) + \cdots \\ &{} + \sum_{(i_{2},\dots ,i_{m})\in \Delta (s,k)} a_{ii_{2}\cdots i_{m}} \bigl(x_{s}^{k}x_{l}^{m - k - 1} - x_{l}^{m-1}\bigr)+ \cdots \\ &{} + \sum_{(i_{2},\ldots ,i_{m})\in \Delta (s,1)} a_{ii_{2}\cdots i_{m}} \bigl(x_{s}x_{l}^{m-2} - x_{l}^{m-1}\bigr) + r_{i}(A)x _{l}^{m-1} \\ \leq& \min _{i,j\in I_{n}}a_{ij\cdots j}\bigl(x_{s}^{m-1} - x_{l}^{m-1}\bigr) + \begin{pmatrix} m-1 \\ 1 \end{pmatrix} (n-1)\beta _{1}(\mathcal{A}) \bigl(x_{s}^{m-2}x_{l} - x_{l}^{m-1} \bigr) + \cdots \\ &{} + \begin{pmatrix} m-1 \\ m-2 \end{pmatrix} (n-1)^{m-2} \beta _{m-2}(\mathcal{A}) \bigl(x_{s}x_{l}^{m-2} - x_{l}^{m-1}\bigr) + r_{i}( \mathcal{A})x_{l}^{m-1} \\ =&\sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}( \mathcal{A}) \bigl(x_{s}^{m-k-1}x_{l}^{k} - x_{l}^{m-1}\bigr) + r_{i}( \mathcal{A})x_{l}^{m-1}. \end{aligned}$$
$$\begin{aligned} \lambda \leq& \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})x_{s}^{m-k-2}x_{l}^{k} \\ &{} + \left (r-\sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A}) \right )\frac{x_{l}^{m-1}}{x _{s}} . \end{aligned}$$
(3.1)
$$\begin{aligned} \lambda x_{l}\geq \lambda x_{i} =&\sum_{i_{2},\dots ,i_{m}\in I_{n}} a _{ii_{2}\cdots i_{m}} x_{i_{2}}\cdots x_{i_{m}} \\ \geq& a_{il\cdots l}x _{l}^{m-1} + \sum _{(i_{2},\dots ,i_{m})\in \Delta (l,m-2)} a_{ii_{2}\cdots i_{m}} x_{l}^{m-2}x_{s} + \cdots \\ &{} + \sum_{(i_{2},\dots ,i_{m})\in \Delta (l,k)} a_{ii_{2}\cdots i_{m}}x_{l}^{k}x_{s}^{m - k - 1} + \cdots \\ &{} + \sum_{(i_{2},\dots ,i_{m})\in \Delta (l,1)} a_{ii_{2}\cdots i_{m}} x_{l}x_{s}^{m-2} + \sum_{(i_{2},\dots ,i_{m})\in \Delta (l,0)} a_{ii_{2}\cdots i_{m}}x_{s}^{m-1} \\ =&a_{il\cdots l}\bigl(x_{l}^{m-1} - x _{s}^{m-1}\bigr)+ \sum _{(i_{2},\dots ,i_{m})\in \Delta (l,m-2)}a_{ii_{2}\cdots i _{m}} \bigl(x_{l}^{m-2}x_{s} - x_{s}^{m-1}\bigr) + \cdots \\ &{} + \sum_{(i_{2},\dots ,i_{m})\in \Delta (l,k)} a_{ii_{2}\cdots i_{m}} \bigl(x_{l}^{k}x_{s}^{m - k - 1} - x_{s}^{m-1}\bigr)+ \cdots \\ &{} + \sum_{(i_{2},\dots ,i_{m})\in \Delta (l,1)} a_{ii_{2}\cdots i_{m}} \bigl(x_{l}x_{s}^{m-2} - x_{s}^{m-1}\bigr) + r_{i}( \mathcal{A})x_{s}^{m-1} \\ \geq &\min _{i,j\in I_{n}}a_{ij\cdots j}\bigl(x_{l}^{m-1} - x_{s}^{m-1}\bigr) + \begin{pmatrix} m-1 \\ 1 \end{pmatrix} (n-1)\beta _{1}(\mathcal{A}) \bigl(x_{l}^{m-2}x_{s} - x_{s}^{m-1} \bigr) + \cdots \\ &{} + \begin{pmatrix} m-1 \\ m-2 \end{pmatrix} (n-1)^{m-2} \beta _{m-2}(A) \bigl(x_{l}x_{s}^{m-2} - x_{s}^{m-1}\bigr) + r_{i}( \mathcal{A})x_{s}^{m-1} \\ =&\sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}( \mathcal{A}) \bigl(x_{l}^{m-k-1 }x_{s}^{k} - x_{s} ^{m-1}\bigr) + r_{i}( \mathcal{A})x_{s}^{m-1}. \end{aligned}$$
$$\begin{aligned} \lambda \geq& \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})x_{l}^{m-k-2}x_{s}^{k} \\ &{}+ \left (R- \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})\right )\frac{x_{s}^{m-1}}{x_{l}} . \end{aligned}$$
(3.2)
$$ \begin{aligned} & \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})x_{l}^{m-k-2}x_{s}^{k}+ \left (R- \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})\right )\frac{x_{s}^{m-1}}{x_{l}} \\ &\quad \leq \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}( \mathcal{A})x_{s}^{m-k-2}x_{l}^{k}\\ &\qquad {}+ \left (r- \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})\right )\frac{x_{l}^{m-1}}{x_{s}}. \end{aligned} $$
$$ \begin{aligned} & \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})\frac{x_{l}^{m-k-2}x_{s}^{k}x_{l}}{x _{s}^{m-1}}+\left (R-\sum _{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})\right ) \\ &\quad\leq \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}( \mathcal{A})\frac{x_{s}^{m-k-2}x_{l}^{k}x_{l}}{x _{s}^{m-1}}\\ &\qquad {}+\left (r-\sum _{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})\right ) \biggl(\frac{x_{l}}{x_{s}}\biggr)^{m}, \end{aligned} $$
$$ \begin{aligned}[b] & \sum _{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A}) \biggl( \frac{x_{l}}{x_{s}}\biggr)^{m-k-1}+\left (R- \sum _{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})\right ) \\ &\quad\leq \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}( \mathcal{A}) \biggl(\frac{x_{l}}{x_{s}}\biggr)^{k+1}\\ &\qquad {}+\left (r- \sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}( \mathcal{A})\right ) \biggl(\frac{x_{l}}{x_{s}}\biggr)^{m}. \end{aligned} $$
(3.3)
$$ \begin{aligned} & \min _{i,j\in I_{n}}a_{ij\cdots j}\biggl( \frac{x_{l}}{x_{s}}\biggr)^{m-1}+ \sum_{k=1}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(A)+ \left (R-\sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})\right ) \\ &\quad\leq \min _{i,j \in I_{n}}a_{ij\cdots j}\frac{x_{l}}{x_{s}}+\sum _{k=1}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A}) \biggl( \frac{x_{l}}{x_{s}}\biggr)^{m}\\ &\qquad {}+ \left (r- \sum _{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(A)\right ) \biggl(\frac{x_{l}}{x_{s}}\biggr)^{m}, \end{aligned} $$
$$ \begin{aligned} & \min _{i,j\in I_{n}}a_{ij\cdots j}\biggl( \frac{x_{l}}{x_{s}}\biggr)^{m-1} + \Bigl(R-\min _{i,j\in I_{n}}a_{ij\cdots j} \Bigr)\leq \min _{i,j\in I_{n}}a_{ij\cdots j}\frac{x_{l}}{x_{s}} + \Bigl(r- \min _{i,j\in I_{n}}a_{ij\cdots j}\Bigr) \biggl(\frac{x_{l}}{x_{s}} \biggr)^{m}. \end{aligned} $$
$$ \biggl(\frac{x_{l}}{x_{s}}\biggr)^{m}\geq \frac{R-\min _{i,j\in I_{n}}a_{ij \cdots j}}{r-\min _{i,j\in I_{n}}a_{ij\cdots j}}. $$
$$\begin{aligned}& \begin{aligned} & \min _{i,j\in I_{n}}a_{ij\cdots j}\biggl( \frac{x_{l}}{x_{s}}\biggr)^{m-1}+ \begin{pmatrix} m-1 \\ 1 \end{pmatrix} (n-1)\beta _{1}(\mathcal{A}) \biggl(\frac{x_{l}}{x_{s}} \biggr)^{m-2}+\cdots \\ & \qquad {}+ \begin{pmatrix} m-1 \\ m-t-1 \end{pmatrix} (n-1)^{m-t-1} \beta _{m-t-1}(\mathcal{A}) \biggl(\frac{x_{l}}{x_{s}} \biggr)^{t} \\ & \qquad{} +\left (R- \sum_{k=t}^{m-1} \begin{pmatrix} m-1 \\ m-k-1 \end{pmatrix} (n-1)^{m-k-1}\beta _{m-k-1}(\mathcal{A})\right ) \\ &\quad \leq \min _{i,j\in I_{n}}a_{ij\cdots j}\frac{x_{l}}{x_{s}}+ \begin{pmatrix} m-1 \\ 1 \end{pmatrix} (n-1)\beta _{1}(A) \biggl( \frac{x_{l}}{x_{s}}\biggr)^{2}+\cdots \\ &\qquad{} + \begin{pmatrix} m-1 \\ m-t-1 \end{pmatrix} (n-1)^{m-t-1} \beta _{m-t-1}(\mathcal{A}) \biggl(\frac{x_{l}}{x_{s}} \biggr)^{m-t} \\ & \qquad {}+ \begin{pmatrix} m-1 \\ m-t \end{pmatrix} (n-1)^{m-t} \beta _{m-t}(\mathcal{A}) \biggl(\frac{x_{l}}{x_{s}} \biggr)^{m} \gamma ^{-\frac{t-1}{m}}+\cdots \\ &\qquad {}+ \begin{pmatrix} m-1 \\ m-2 \end{pmatrix} (n-1)^{m-2} \beta _{m-2}(\mathcal{A}) \biggl(\frac{x_{l}}{x_{s}} \biggr)^{m} \gamma ^{-\frac{1}{m}} \\ &\qquad{} +\left (r-\sum_{k=0}^{m-2} \begin{pmatrix} m-1 \\ k \end{pmatrix} (n-1)^{k}\beta _{k}(\mathcal{A})\right ) \biggl(\frac{x_{l}}{x_{s}} \biggr)^{m} \end{aligned} \\& \begin{aligned} &\quad =\min _{i,j\in I_{n}}a_{ij\cdots j}\biggl( \frac{x_{l}}{x_{s}}\biggr)+ \begin{pmatrix} m-1 \\ 1 \end{pmatrix} (n-1) \beta _{1}(\mathcal{A}) \biggl(\frac{x_{l}}{x_{s}} \biggr)^{2}+\cdots \\ & \qquad {}+ \begin{pmatrix} m-1 \\ m-t-1 \end{pmatrix} (n-1)^{m-t-1} \beta _{m-t-1}(\mathcal{A}) \biggl(\frac{x_{l}}{x_{s}} \biggr)^{m-t} \\ & \qquad {}+ \left[r-\sum_{k=1}^{t-1} \begin{pmatrix} m-1 \\ m-k-1 \end{pmatrix} (n-1)^{m-k-1}\beta _{m-k-1}(\mathcal{A}) \bigl(1-\gamma ^{-\frac{k}{m}}\bigr)\right. \\ &\left. \qquad{} -\sum_{k=t}^{m-1} \begin{pmatrix} m-1 \\ m-k-1 \end{pmatrix} (n-1)^{m-k-1}\beta _{m-k-1}( \mathcal{A}) \right]\biggl(\frac{x_{l}}{x_{s}}\biggr)^{m}. \end{aligned} \end{aligned}$$
$$ \begin{aligned} & \left[r-\sum_{k=1}^{t-1} \begin{pmatrix} m-1 \\ m-k-1 \end{pmatrix} (n-1)^{m-k-1}\beta _{m-k-1}(\mathcal{A}) \bigl(1-\gamma ^{-\frac{k}{m}}\bigr)\right. \\ &\left. \qquad{} -\sum_{k=t}^{m-1} \begin{pmatrix} m-1 \\ m-k-1 \end{pmatrix} (n-1)^{m-k-1}\beta _{m-k-1}( \mathcal{A}) \right]\biggl(\frac{x_{l}}{x_{s}}\biggr)^{m} \\ &\quad \geq \min _{i,j\in I_{n}}a_{ij\cdots j} \biggl[\biggl(\frac{x_{l}}{x _{s}} \biggr)^{m-1}-\frac{x_{l}}{x_{s}} \biggr] + \begin{pmatrix} m-1 \\ 1 \end{pmatrix} (n-1)\beta _{1}(\mathcal{A}) \biggl[\biggl(\frac{x_{l}}{x_{s}}\biggr)^{m-2}-\biggl( \frac{x _{l}}{x_{s}}\biggr)^{2} \biggr] +\cdots \\ &\qquad{} + \begin{pmatrix} m-1 \\ m-t-1 \end{pmatrix} (n-1)^{m-t-1} \beta _{m-t-1}(\mathcal{A}) \biggl[\biggl(\frac{x_{l}}{x_{s}} \biggr)^{t}-\biggl(\frac{x _{l}}{x_{s}}\biggr)^{m-t} \biggr] \\ &\qquad{} +\left (R-\sum_{k=t}^{m-1} \begin{pmatrix} m-1 \\ m-k-1 \end{pmatrix} (n-1)^{m-k-1}\beta _{m-k-1}(\mathcal{A})\right ) \\ &\quad \geq \min _{i,j\in I_{n}}a_{ij\cdots j} \bigl[\gamma ^{\frac{m-1}{m}}- \gamma ^{\frac{1}{m}} \bigr] + \begin{pmatrix} m-1 \\ 1 \end{pmatrix} (n-1)\beta _{1}(A) \bigl[\gamma ^{\frac{m-2}{m}}-\gamma ^{\frac{2}{m}} \bigr]+\cdots \\ &\qquad{} + \begin{pmatrix} m-1 \\ m-t-1 \end{pmatrix} (n-1)^{m-t-1}\beta _{m-t-1}(\mathcal{A}) \bigl[\gamma ^{\frac{t}{m}}- \gamma ^{\frac{m-t}{m}} \bigr] \\ &\qquad{} +\left (R-\sum_{k=t}^{m-1} \begin{pmatrix} m-1 \\ m-k-1 \end{pmatrix} (n-1)^{m-k-1}\beta _{m-k-1}(\mathcal{A})\right ), \end{aligned} $$
□
From Theorem 3.1, we have the following upper bound.
Theorem 3.2
Let
\(A\in \mathbb{R}^{[m,n]}\)
be an irreducible and weakly symmetric nonnegative tensor. Then we have
where
\(\varphi \equiv \varphi (A)\)
is given in Theorem
3.1.
$$ \rho _{z}(A)\leq \max _{i,j\in I_{n}} \Biggl(\sum _{k=0} ^{m-1}\sum_{(i_{2},\dots ,i_{m})\in \Delta (j,k)}a_{ii_{2} \cdots i_{m}} \varphi ^{-\frac{k}{m}} \Biggr), $$
Proof
Since \(\mathcal{A}\) is a weakly irreducible nonnegative symmetric tensor, we know that the Z-spectral radius \(\rho \equiv \rho _{z}(A)\) is a positive Z-eigenvalue with a positive Z-eigenvector x. Since \(x^{T}x=1\) and \(x>0\), then \(x_{i}^{m-1}\leq x_{i}\) for any i. Let \(x_{s}=x_{\min }=\min_{i\in I_{n}} x_{i}\), \(x_{l}=x_{\max }= \min_{i\in I_{n}} x_{i}\). Then we have
Taking \(i=l\) and multiplying by \(x_{l}^{1-m}\) on both sides of the above inequality, from Theorem 3.1, we get
□
$$ \begin{aligned} \rho (\mathcal{A})x_{i}^{m-1} \leq{}& \rho (\mathcal{A})x_{i} \\ \leq{}& a_{is \cdots s}x_{s}^{m-1}+ \sum_{(i_{2},\dots ,i_{m})\in \Delta (s,m-2)}a _{ii_{2}\cdots i_{m}} x_{s}^{m-2}x_{l}+ \cdots \\ &{} +\sum_{(i_{2},\dots ,i_{m})\in \Delta (s,1)}a_{ii_{2}\cdots i_{m}} x _{s}x_{l}^{m-2}+\sum _{(i_{2},\dots ,i_{m})\in \Delta (s,0)}a _{ii_{2}\cdots i_{m}}x_{l}^{m-1}. \end{aligned} $$
$$\begin{aligned} \rho (\mathcal{A}) \leq& a_{is\cdots s}\biggl( \frac{x_{s}}{x_{l}}\biggr)^{m-1}+ \sum_{(i_{2},\dots ,i_{m})\in \Delta (s,m-2)}a_{ii_{2}\cdots i _{m}} \biggl(\frac{x_{s}}{x_{l}}\biggr)^{m-2} + \cdots \\ &{} +\sum_{(i_{2},\dots ,i_{m})\in \Delta (s,1)}a_{ii_{2}\cdots i_{m}} \frac{x _{s}}{x_{l}}+\sum_{(i_{2},\dots ,i_{m})\in \Delta (s,0)}a_{ii _{2}\cdots i_{m}} \\ \leq& \max _{i,j\in I_{n}}\biggl\{ a_{ij\cdots j} \varphi ^{-\frac{m-1}{m}}+ \sum_{(i_{2},\dots ,i_{m})\in \Delta (j,m-2)}a_{ii_{2}\cdots i_{m}} \varphi ^{-\frac{m-2}{m}}+\cdots \\ &{} +\sum_{(i_{2},\dots ,i_{m})\in \Delta (j,1)}a_{ii_{2}\cdots i_{m}} \varphi ^{-\frac{1}{m}}+\sum_{(i_{2},\dots ,i_{m})\in \Delta (j,0)}a _{ii_{2}\cdots i_{m}} \biggr\} \\ =&\max _{i,j\in I_{n}} \Biggl(\sum_{k=0}^{m-1} \sum_{(i_{2},\dots ,i_{m})\in \Delta (j,k)}a _{ii_{2}\cdots i_{m}}\varphi ^{-\frac{k}{m}} \Biggr). \end{aligned}$$
Remark 3.1
For the matrix case \(A=(a_{ij})\in \mathbb{R}^{[n\times n]}\), \(i,j\in I_{n}\), i.e., when \(m=2\), it is easy to see that the bound in Theorem 3.2 reduces to the following one:
where \(\xi =(\frac{R-\min_{i,j}a_{ij}}{r-\min_{i,j}a _{ij}})^{-\frac{1}{2}}\), \(R=\max_{i}r_{i}\), \(r=\min_{i}r _{i}\), and \(r_{i}=\sum_{j=1}^{n}a_{ij}\), which is the same as the bound in (2.6) for the matrix case.
$$ \rho _{z}(A)\leq \max _{i,j}\bigl\{ r_{i}(A)+a_{ij}(\xi -1)\bigr\} , $$
(3.4)
Remark 3.2
Let \(\delta =\frac{a}{c}\) and \(\varphi =\frac{a+b}{c-d}\), where δ is given by (2.6), φ is given by Theorem 3.1 and
Since \(\delta \geq \gamma \geq 1\) and \(a,c,d,a+b,c-d\geq 0\), it’s easy to conclude that
when \(d>0\), and , when \(d=0\).
Then we have \(\frac{a+b}{c-d}\geq \frac{a}{c}\), i.e., \(\varphi \geq \delta \geq \gamma \geq 1\).
The authors presented the following bound in [20]:
where
$$ \frac{x_{\max }}{x_{\min }}\geq \delta ^{\frac{1}{m}}, $$
$$ \delta =\frac{\min _{i,j\in I_{n}}a_{i j \cdots j}}{r-\min _{i,j\in I_{n}}a_{i j \cdots j}}\bigl(\gamma ^{\frac{m-1}{m}}- \gamma ^{\frac{1}{m}}\bigr)+\gamma ,\qquad \gamma =\frac{R-\min _{i,j\in I_{n}}a_{i j \cdots j}}{r-\min _{i,j\in I_{n}}a _{i j \cdots j}} . $$
(3.5)
Example 3.1
Let \(\mathcal{A}=(a_{i_{1}i_{2}i_{3}})\in \mathbb{R}_{+}^{[3,2]}\) with
Then \(\mathcal{A}\) is a positive tensor. A simple computation with (3.5) gives \(R=2k+3\), \(r=7\), \(\gamma =\frac{k+1}{3}\), and then \(\delta =\frac{1}{6}[(\frac{k+1}{3})^{\frac{2}{3}}-(\frac{k+1}{3})^{ \frac{1}{3}}]+\frac{k+1}{3}\). However, by Theorem 3.1, we have \(t=2\), \(\beta _{0}(\mathcal{A})=\min_{i,j}a_{ij\cdots j}=1\), \(\beta _{1}(\mathcal{A})=1\), thus
Now we take \(k=10\), and then in (3.5) we have
while in Theorem 3.1,
$$ a_{111}=a_{121}=k,\qquad a_{112}=a_{222}=1, \qquad a_{211}=a_{212}=a_{221}=a _{122}=2, \quad k>4. $$
$$ \varphi =\frac{[(\frac{k+1}{3})^{\frac{2}{3}}-(\frac{k+1}{3})^{ \frac{1}{3}}]+2k+3-1}{7-2[1-(\frac{k+1}{3})^{-\frac{1}{3}}]-1}=\frac{[( \frac{k+1}{3})^{\frac{2}{3}}-(\frac{k+1}{3})^{\frac{1}{3}}]+2k+2}{4+2( \frac{k+1}{3})^{-\frac{1}{3}}}. $$
$$ 1.5613\leq \frac{x_{\max }}{x_{\min }}, $$
$$ 1.6275\leq \frac{x_{\max }}{x_{\min }}. $$
Remark 3.3
Let \(\eta =\frac{e}{g}\) and \(\varphi =\frac{f}{h}\), where η is given by (2.7), φ is given by Theorem 3.1 and
Then \((1-\gamma ^{-\frac{k}{m}})\geq 0\), so it’s easy to conclude that \(h\leq g\). Since
when \(\gamma ^{\frac{k}{m}}-\gamma ^{\frac{m-k}{m}}\geq 1\), \(k=t,\dots ,m-1\), we have \(f\geq e\), i.e., \(\varphi \geq \eta \).
Obviously, we have
$$ \sum_{k=t}^{m-1} \begin{pmatrix} m-1 \\ m-k-1 \end{pmatrix} (n-1)^{m-k-1}\beta _{m-k-1}( \mathcal{A})\geq \sum_{k=t}^{m-1} \min _{i,j\in In}\mathcal{A}_{i,\alpha (k,j)}. $$
$$ f-e=\sum_{k=t}^{m-1}\left ( \begin{pmatrix} m-1 \\ m-k-1 \end{pmatrix} (n-1)^{m-k-1}\beta _{m-k-1}(\mathcal{A})-\min _{i,j\in In} \mathcal{A}_{i,\alpha (k,j)}\right ) \bigl(\gamma ^{\frac{k}{m}}- \gamma ^{\frac{m-k}{m}}-1\bigr), $$
Example 3.2
Let \(\mathcal{A}=(a_{i_{1}i_{2}i_{3}})\in \mathbb{R}_{+}^{[3,5]}\) with \(k>4\),
and other
Then \(\mathcal{A}\) is a positive tensor, and we have \(t=2\). A simple computation with (2.7) gives \(\min_{i,j\in 5} \mathcal{A}_{i,\alpha (2,j)}=1\), \(\min_{i,j\in 5}\mathcal{A} _{i,\alpha (1,j)}=4\), \(R=5k+40\), \(r=49\), so
However, in Theorem 3.1, \(\beta _{0}(\mathcal{A})=\min_{i,j}a_{ij\cdots j}=1\), \(\beta _{1}(\mathcal{A})=2\), thus
Similarly, we take \(k=10\),
So we have
Since \(\gamma \geq 1\), we get \((1-(\gamma )^{-\frac{1}{3}})\geq 0\), and so
$$ a_{111}=a_{1i1}=k, \qquad a_{222}=a_{333}=a_{444}=a_{555}=1, \qquad a _{ijj}=2, $$
$$ a_{i_{1}i_{2}i_{3}}=2. $$
$$ \eta =\frac{[(\gamma )^{\frac{2}{3}}-(\gamma )^{\frac{1}{3}}]+5k+40-1}{49-4(1-( \gamma )^{-\frac{1}{3}})-1}. $$
$$ \varphi =\frac{[(\gamma )^{\frac{2}{3}}-(\gamma )^{\frac{1}{3}}]+5k+40-1}{49-2 \times 4\times 2(1-(\gamma )^{-\frac{1}{3}})-1}=\frac{[(\gamma )^{ \frac{2}{3}}-(\gamma )^{\frac{1}{3}}]+5k+40-1}{49-16(1-(\gamma )^{- \frac{1}{3}})-1}. $$
$$\begin{aligned}& \eta =\frac{[(\gamma )^{\frac{2}{3}}-(\gamma )^{\frac{1}{3}}]+89}{44+4( \gamma )^{-\frac{1}{3}}}, \\& \varphi =\frac{[(\gamma )^{\frac{2}{3}}-(\gamma )^{\frac{1}{3}}]+89}{32+16( \gamma )^{-\frac{1}{3}}}. \end{aligned}$$
$$ 44+4(\gamma )^{-\frac{1}{3}}-32-16(\gamma )^{-\frac{1}{3}}=12\bigl(1-( \gamma )^{-\frac{1}{3}}\bigr). $$
$$ \varphi \geq \eta . $$
Acknowledgements
The authors would like to express their sincere thanks the anonymous referees and editors for their constructive comments and valuable suggestions.
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