A tighter S-type singular value inclusion set for rectangular tensors

TL;DR

This paper introduces a new, tighter S-type singular value inclusion set for rectangular tensors, leading to improved bounds for the largest singular value of nonnegative rectangular tensors, verified through numerical examples.

Contribution

It presents a novel, more precise singular value inclusion set for rectangular tensors and derives sharper bounds for their largest singular value, improving upon previous methods.

Findings

New tighter S-type singular value inclusion set for rectangular tensors

Improved bounds for the largest singular value of nonnegative rectangular tensors

Numerical example confirms theoretical improvements

Abstract

A new S-type singular value inclusion set for rectangular tensors is given and proved to be tighter than that in [Sang C.L., An S-type singular value inclusion set for rectangular tensors, J. Inequal. Appl. 2017: 141, 2017]. Based on this set, new bounds for the largest singular value of nonnegative rectangular tensors are obtained and proved to be better than some existing results. Compared with the results in the paper mentioned above, the advantage of the new results is that, under the same computations, we can obtain a tighter singular value inclusion set for rectangular tensors and sharper bounds for the largest singular value of nonnegative rectangular tensors. Finally, a numerical example is given to verify the theoretical results.

Equations

$$a_{i_1\cdots i_p{j}_1\cdots j_q}\in \mathbb{R},1\le i_1\cdots i_p\le m,1\le j_1\cdots j_q\le n.$$

$$\{\begin{array}{r}\mathcal{A}{x}^{p-1}{y}^q=\lambda x^{[l-1]},\\ \mathcal{A}{x}^p{y}^{q-1}=\lambda y^{[l-1]},\end{array}$$

$$(\mathcal{A}{x}^{p-1}{y}^q{)}_i=\sum _{i_2,\cdots ,i_p=1}^m\sum _{j_1,\cdots ,j_q=1}^n{a}_{i{i}_2\cdots i_p{j}_1\cdots j_q}{x}_{i_2}\cdots x_{i_p}{y}_{j_1}\cdots y_{j_q}$$

$$(x^{[l-1]}{)}_i=x_i^{l-1};$$

$$(\mathcal{A}{x}^p{y}^{q-1}{)}_j=\sum _{i_1,\cdots ,i_p=1}^m\sum _{j_2,\cdots ,j_q=1}^n{a}_{i_1\cdots i_{p}j{j}_2\cdots j_q}{x}_{i_1}\cdots x_{i_p}{y}_{j_2}\cdots y_{j_q}$$

$$(y^{[l-1]}{)}_j=y_j^{l-1}.$$

$${\lambda }_0=\max \{|\lambda |:\lambda isasingularvalueof\mathcal{A}\}$$

$$\sigma (\mathcal{A})\subseteq {{\rm Y}}^S(\mathcal{A})=\left(\bigcup _{i\in S,j\in \bar{S}}\left({\widehat{{\rm Y}}}_{i,j}(\mathcal{A})\bigcup {\widetilde{{\rm Y}}}_{i,j}(\mathcal{A})\right)\right)\bigcup \left(\bigcup _{i\in \bar{S},j\in S}\left({\widehat{{\rm Y}}}_{i,j}(\mathcal{A})\bigcup {\widetilde{{\rm Y}}}_{i,j}(\mathcal{A})\right)\right),$$

$${\widehat{{\rm Y}}}_{i,j}(\mathcal{A})=\left\{z\in \mathbb{C}:(|z|-r_i^j(\mathcal{A}))|z|\le |a_{ij\cdots jj\cdots j}|\max \{R_j(\mathcal{A}),C_j(\mathcal{A})\}\right\},$$

$${\widetilde{{\rm Y}}}_{i,j}(\mathcal{A})=\left\{z\in \mathbb{C}:(|z|-c_i^j(\mathcal{A}))|z|\le |a_{j\cdots jij\cdots j}|\max \{R_j(\mathcal{A}),C_j(\mathcal{A})\}\right\},$$

$$R_i(\mathcal{A})=\sum _{i_2,\cdots ,i_p,j_1,\cdots ,j_q\in N}|a_{i{i}_2\cdots i_p{j}_1\cdots j_q}|,$$

$$r_i^j(\mathcal{A})=\sum _{{\delta }_{j{i}_2\cdots i_p{j}_1\cdots j_q}=0}|a_{i{i}_2\cdots i_p{j}_1\cdots j_q}|=R_i(\mathcal{A})-|a_{ij\cdots jj\cdots j}|,$$

$$C_j(\mathcal{A})=\sum _{i_1,\cdots ,i_p,j_2,\cdots ,j_q\in N}|a_{i_1\cdots i_{p}j{j}_2\cdots j_q}|,$$

$$c_j^i(\mathcal{A})=\sum _{{\delta }_{i_1\cdots i_{p}i{j}_2\cdots j_q}=0}|a_{i_1\cdots i_{p}j{j}_2\cdots j_q}|=C_j(\mathcal{A})-|a_{i\cdots iji\cdots i}|,$$

$$\displaystyle\delta_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}=\left\{\begin{array}[]{rl}1&if~{}i_{1}=\cdots=i_{p}=j_{1}=\cdots=j_{q},\\ 0&otherwise.\end{array}\right.$$

$$L^S(\mathcal{A})\le {\lambda }_0\le U^S(\mathcal{A}),$$

$$L^S(\mathcal{A})=\min \{{\hat{L}}^S(\mathcal{A}),{\hat{L}}^{\bar{S}}(\mathcal{A}),{\tilde{L}}^S(\mathcal{A}),{\tilde{L}}^{\bar{S}}(\mathcal{A})\}$$

$$U^S(\mathcal{A})=\max \{{\hat{U}}^S(\mathcal{A}),{\hat{U}}^{\bar{S}}(\mathcal{A}),{\tilde{U}}^S(\mathcal{A}),{\tilde{U}}^{\bar{S}}(\mathcal{A})\},$$

$${\hat{L}}^S(\mathcal{A})=\underset{i\in S,j\in \bar{S}}{\min }\frac{1}{2}\left\{r_i^j(\mathcal{A})+[(r_i^j(\mathcal{A}){)}^2+4a_{ij\cdots jj\cdots j}\min \{R_j(\mathcal{A}),C_j(\mathcal{A})\}{]}^{\frac{1}{2}}\right\},$$

$${\tilde{L}}^S(\mathcal{A})=\underset{i\in S,j\in \bar{S}}{\min }\frac{1}{2}\left\{c_i^j(\mathcal{A})+[(c_i^j(\mathcal{A}){)}^2+4a_{j\cdots jij\cdots j}\min \{R_j(\mathcal{A}),C_j(\mathcal{A})\}{]}^{\frac{1}{2}}\right\},$$

$${\hat{U}}^S(\mathcal{A})=\underset{i\in S,j\in \bar{S}}{\max }\frac{1}{2}\left\{r_i^j(\mathcal{A})+[(r_i^j(\mathcal{A}){)}^2+4a_{ij\cdots jj\cdots j}\max \{R_j(\mathcal{A}),C_j(\mathcal{A})\}{]}^{\frac{1}{2}}\right\},$$

$${\tilde{U}}^S(\mathcal{A})=\underset{i\in S,j\in \bar{S}}{\max }\frac{1}{2}\left\{c_i^j(\mathcal{A})+[(c_i^j(\mathcal{A}){)}^2+4a_{j\cdots jij\cdots j}\max \{R_j(\mathcal{A}),C_j(\mathcal{A})\}{]}^{\frac{1}{2}}\right\}.$$

$${\Delta }^N:=\{(i_2,\cdots i_p,j_1,\cdots ,j_q):i_2,\cdots i_p,j_1,\cdots ,j_q\in N\},$$

$${\Delta }^S:=\{(i_2,\cdots i_p,j_1,\cdots ,j_q):i_2,\cdots i_p,j_1,\cdots ,j_q\in S\},$$

$${\Omega }^N:=\{(i_1,\cdots i_p,j_2,\cdots ,j_q):i_1,\cdots i_p,j_2,\cdots ,j_q\in N\},$$

$${\Omega }^S:=\{(i_1,\cdots i_p,j_2,\cdots ,j_q):i_1,\cdots i_p,j_2,\cdots ,j_q\in S\},$$

$$\overline{{\Delta }^S}={\Delta }^N\backslash {\Delta }^S,\overline{{\Omega }^S}={\Omega }^N\backslash {\Omega }^S.$$

$$R_i(\mathcal{A})=\sum _{i_2,\cdots ,i_p,j_1,\cdots ,j_q\in N}|a_{i{i}_2\cdots i_p{j}_1\cdots j_q}|=r_i^{{\Delta }^S}(\mathcal{A})+r_i^{\overline{{\Delta }^S}}(\mathcal{A}),$$

$$C_j(\mathcal{A})=\sum _{i_1,\cdots ,i_p,j_2,\cdots ,j_q\in N}|a_{i_1\cdots i_{p}j{j}_2\cdots j_q}|=c_j^{{\Omega }^S}(\mathcal{A})+c_j^{\overline{{\Omega }^S}}(\mathcal{A}),$$

$$r_i^{{\Delta }^S}(\mathcal{A})=\sum _{(i_2,\cdots ,i_p,j_1,\cdots ,j_q)\in {\Delta }^S}|a_{i{i}_2\cdots i_p{j}_1\cdots j_q}|,r_i^{\overline{{\Delta }^S}}(\mathcal{A})=\sum _{(i_2,\cdots ,i_p,j_1,\cdots ,j_q)\in \overline{{\Delta }^S}}|a_{i{i}_2\cdots i_p{j}_1\cdots j_q}|,$$

$$c_j^{{\Omega }^S}(\mathcal{A})=\sum _{(i_1,\cdots ,i_p,j_2,\dots ,j_q)\in {\Omega }^S}|a_{i_1\cdots i_{p}j{j}_2\cdots j_q}|,c_j^{\overline{{\Omega }^S}}(\mathcal{A})=\sum _{(i_1,\cdots ,i_p,j_2,\cdots ,j_q)\in \overline{{\Omega }^S}}|a_{i_1\cdots i_{p}j{j}_2\cdots j_q}|.$$

$$\sigma (\mathcal{A})\subseteq {\Psi }^S(\mathcal{A})=\left(\bigcup _{i\in S,j\in \bar{S}}\left({\hat{\Psi }}_{i,j}^S(\mathcal{A})\bigcup {\tilde{\Psi }}_{i,j}^S(\mathcal{A})\right)\right)\bigcup \left(\bigcup _{i\in \bar{S},j\in S}\left({\hat{\Psi }}_{i,j}^{\bar{S}}(\mathcal{A})\bigcup {\tilde{\Psi }}_{i,j}^{\bar{S}}(\mathcal{A})\right)\right),$$

$${\hat{\Psi }}_{i,j}^S(\mathcal{A})=\left\{z\in \mathbb{C}:(|z|-r_j^{\overline{{\Delta }^S}}(\mathcal{A}))|z|\le r_j^{{\Delta }^S}(\mathcal{A})\max \{R_i(\mathcal{A}),C_i(\mathcal{A})\}\right\},$$

$${\hat{\Psi }}_{i,j}^{\bar{S}}(\mathcal{A})=\left\{z\in \mathbb{C}:(|z|-r_j^{\overline{{\Delta }^{\bar{S}}}}(\mathcal{A}))|z|\le r_j^{{\Delta }^{\bar{S}}}(\mathcal{A})\max \{R_i(\mathcal{A}),C_i(\mathcal{A})\}\right\},$$

$${\tilde{\Psi }}_{i,j}^S(\mathcal{A})=\left\{z\in \mathbb{C}:(|z|-c_j^{\overline{{\Delta }^S}}(\mathcal{A}))|z|\le c_j^{{\Delta }^S}(\mathcal{A})\max \{R_i(\mathcal{A}),C_i(\mathcal{A})\}\right\},$$

$${\tilde{\Psi }}_{i,j}^{\bar{S}}(\mathcal{A})=\left\{z\in \mathbb{C}:(|z|-c_j^{\overline{{\Delta }^{\bar{S}}}}(\mathcal{A}))|z|\le c_j^{{\Delta }^{\bar{S}}}(\mathcal{A})\max \{R_i(\mathcal{A}),C_i(\mathcal{A})\}\right\}.$$

$$\mathcal{A}{x}^{p-1}{y}^q=\lambda x^{[l-1]},$$

$$\mathcal{A}{x}^p{y}^{q-1}=\lambda y^{[l-1]}.$$

$$|x_t|=\max \{|x_i|:i\in S\},|x_h|=\max \{|x_i|:i\in \bar{S}\};$$

Full text

A tighter $S$-type singular value inclusion set for rectangular tensors

Caili Sang††Corresponding author, E-mail: sangcl@126.com; sangcaili@gzmu.edu.cn (Caili Sang)

*College of Data Science and Information Engineering, Guizhou Minzu University,

Guiyang 550025, P.R.China

Abstract. A new $S$-type singular value inclusion set for rectangular tensors is given and proved to be tighter than that in [Sang C.L., An $S$-type singular value inclusion set for rectangular tensors, J. Inequal. Appl. 2017: 141, 2017]. Based on this set, new bounds for the largest singular value of nonnegative rectangular tensors are obtained and proved to be better than some existing results. Compared with the results in the paper mentioned above, the advantage of the new results is that, under the same computations, we can obtain a tighter singular value inclusion set for rectangular tensors and sharper bounds for the largest singular value of nonnegative rectangular tensors. Finally, a numerical example is given to verify the theoretical results.

Keywords: rectangular tensor; nonnegative tensors; singular value; inclusion set

AMS Subject Classification: 15A18; 15A42; 15A69.

1 Introduction

Let $\mathbb{R}(\mathbb{C})$ be the real (complex) field, $p,q,m,n$ be positive integers, $l=p+q$, $m,n\geq 2$ and $N=\{1,2,\cdots,n\}$. $\mathcal{A}=(a_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}})$ is called a real $(p,q)$th order $m\times n$ dimensional rectangular tensor, or simply a real rectangular tensor, denoted by $\mathcal{A}\in\mathbb{R}^{[p,q;m,n]}$, if

[TABLE]

When $p=q=1,\mathcal{A}$ is simply a real $m\times n$ rectangular matrix. $\mathcal{A}$ is called a nonnegative rectangular tensor, denoted by $\mathcal{A}\in\mathbb{R}^{[p,q;m,n]}_{+}$, if each of its entries $a_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}\geq 0$.

If there are a number $\lambda\in\mathbb{C},$ vectors $x=(x_{1},x_{2},\cdots,x_{m})^{\textmd{T}}\in\mathbb{C}^{m}\backslash\{0\}$ and $y=(y_{1},y_{2},\cdots,y_{n})^{\textmd{T}}\in\mathbb{C}^{n}\backslash\{0\}$ such that

[TABLE]

then $\lambda$ is called a singular value of $\mathcal{A}$, and $(x,y)$ is the left and right eigenvectors pair of $\mathcal{A}$, associated with $\lambda$, respectively, where $\mathcal{A}x^{p-1}y^{q}$ and $x^{[l-1]}$ are vectors in $\mathbb{R}^{m}$, whose $i$th component is

[TABLE]

and

[TABLE]

$\mathcal{A}x^{p}y^{q-1}$ and $x^{[l-1]}$ are vectors in $\mathbb{R}^{n}$ whose $j$th component is

[TABLE]

and

[TABLE]

Furthermore, if $\lambda\in\mathbb{R},x\in\mathbb{R}^{m},$ and $y\in\mathbb{R}^{n}$, then we say that $\lambda$ is an H-singular value of $\mathcal{A}$, and $(x,y)$ is the left and right H-eigenvectors pair associated with $\lambda$, respectively. If a singular value is not an H-singular value, we call it an N-singular value of $\mathcal{A}$. We call

[TABLE]

is the largest singular value; see [2, 3, 1] for details.

When $p=q,m=n$, such real rectangular tensors have a wide range of practical applications in the strong ellipticity condition problem in solid mechanics [4, 5] and the entanglement problem in quantum physics [6, 7]. For example, the elasticity tensor is a tensor with $p=q=2$ and $m=n=2$ or 3; for details, see [2].

Because it is not easy to compute all singular values (eigenvalues) of tensors when the order and dimension are large, one always tries to give a set including all singular values (eigenvalues) in the complex plane [9, 10, 11, 8, 12], or give upper and lower bounds for the largest singular value of nonnegative rectangular tensors[14, 13, 15]. Very recently, Sang [8] proposed the following $S$-type singular value inclusion set for rectangular tensors by breaking $N=\{1,2,\cdots,n\}$ into disjoint subsets $S$ and its complement $\bar{S}$.

Theorem 1**.**

[8, Theorem 1]* Let $\mathcal{A}\in\mathbb{R}^{[p,q;n,n]}$, $S$ be a nonempty proper subset of $N$, $\bar{S}$ be the complement of $S$ in N. Then*

[TABLE]

where

[TABLE]

and

[TABLE]

Based on the set in Theorem 1, Sang in [8] obtained the following upper and lower bounds for the largest singular value of nonnegative rectangular tensors.

Theorem 2**.**

[8, Theorem 2]* Let $\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb{R}^{[p,q;n,n]}_{+}$, $S$ be a nonempty proper subset of $N$, $\bar{S}$ be the complement of $S$ in $N$. Then*

[TABLE]

where

[TABLE]

and

[TABLE]

In this paper, by the technique in [9], we give a new $S$-type singular value inclusion set for a real rectangular tensor $\mathcal{A}$ and prove that the new set is tighter than $\Upsilon^{S}(\mathcal{A})$. As an application, we obtain new upper and lower bounds for the largest singular value of nonnegative rectangular tensors and prove that the new bounds are better than those in Theorem 2 and Theorem 4 of [3].

2 Main results

We begin with some notation. Given a nonempty proper subset $S$ of $N$, we denote

[TABLE]

and then

[TABLE]

This implies that for a rectangular tensor $\mathcal{A}=(a_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}),$ we have that for $i,j\in S$,

[TABLE]

where

[TABLE]

Theorem 3**.**

Let $\mathcal{A}\in\mathbb{R}^{[p,q;n,n]}$, $S$ be a nonempty proper subset of $N$, $\bar{S}$ be the complement of $S$ in $N$. Then

[TABLE]

where

[TABLE]

Proof. For any $\lambda\in\sigma(\mathcal{A})$, let $x=(x_{1},x_{2},\cdots,x_{m})^{\mathrm{T}}\in\mathbb{C}^{m}\backslash\{0\}$ and $y=(y_{1},y_{2},\cdots,y_{n})^{\mathrm{T}}\in\mathbb{C}^{n}\backslash\{0\}$ be the left and right associated eigenvectors, that is,

[TABLE]

Let

[TABLE]

Then, at least one of $|x_{t}|$ and $|x_{h}|$ is nonzero, and at least of $|y_{f}|$ and $|y_{g}|$ is nonzero. We divide the proof into four parts.

Case I: Suppose that $w_{S}=|x_{t}|,w_{\bar{S}}=|x_{h}|$, then $|x_{t}|\geq|y_{t}|,|x_{h}|\geq|y_{h}|$.

(i) If $|x_{h}|\geq|x_{t}|$, then $|x_{h}|=\max\{w_{i}:i\in N\}$. By the $h$th equality in (2), i.e.,

[TABLE]

we have

[TABLE]

i.e.,

[TABLE]

If $|x_{t}|=0$, then $|\lambda|-r_{h}^{\overline{\Delta^{S}}}(\mathcal{A})\leq 0$ as $|x_{h}|>0,$ and it is obvious that

[TABLE]

which implies that $\lambda\in\hat{\Psi}_{t,h}(\mathcal{A})$.

Otherwise, $|x_{t}|>0$. From the $t$th equality in (2), we have

[TABLE]

i.e.,

[TABLE]

Multiplying (4) by (5) and noting that $|x_{t}|^{l-1}|x_{h}|^{l-1}>0$, we have

[TABLE]

which also implies that $\lambda\in\hat{\Psi}_{t,h}^{S}(\mathcal{A})\subseteq\bigcup\limits_{i\in S,j\in\bar{S}}\hat{\Psi}_{i,j}^{S}(\mathcal{A})$.

(ii) If $|x_{t}|\geq|x_{h}|$, then $|x_{t}|=\max\{w_{i}:i\in N\}$. Similar to the proof of (i), we have

[TABLE]

which implies that $\lambda\in\hat{\Psi}_{h,t}^{\bar{S}}(\mathcal{A})\subseteq\bigcup\limits_{i\in\bar{S},j\in S}\hat{\Psi}_{i,j}^{\bar{S}}(\mathcal{A})$.

Case II: Suppose that $w_{S}=|y_{f}|,w_{\bar{S}}=|y_{g}|$, then $|y_{f}|\geq|x_{f}|,|y_{g}|\geq|x_{g}|$. If $|y_{g}|\geq|y_{f}|$, then $|y_{g}|=\max\{w_{i}:i\in N\}$. Similar to the proof of (i) in Case I, we have

[TABLE]

which implies that $\lambda\in\tilde{\Psi}_{f,g}^{S}(\mathcal{A})\subseteq\bigcup\limits_{i\in S,j\in\bar{S}}\tilde{\Psi}_{i,j}^{S}(\mathcal{A})$.

If $|y_{f}|\geq|y_{g}|$, then $|y_{f}|=\max\{w_{i}:i\in N\}$. Similar to the proof of (ii) in Case I, we have

[TABLE]

and $\lambda\in\tilde{\Psi}_{g,f}^{\bar{S}}(\mathcal{A})\subseteq\bigcup\limits_{i\in\bar{S},j\in S}\tilde{\Psi}_{i,j}^{\bar{S}}(\mathcal{A})$.

Case III: Suppose that $w_{S}=|x_{t}|,w_{\bar{S}}=|y_{g}|$, then $|x_{t}|\geq|y_{t}|,|y_{g}|\geq|x_{g}|$. If $|y_{g}|\geq|x_{t}|$, then $|y_{g}|=\max\{w_{i}:i\in N\}$. Similarly, we have

[TABLE]

which implies that $\lambda\in\tilde{\Psi}_{t,g}^{S}(\mathcal{A})\subseteq\bigcup\limits_{i\in S,j\in\bar{S}}\tilde{\Psi}_{i,j}^{S}(\mathcal{A})$.

If $|x_{t}|\geq|y_{g}|$, then $|x_{t}|=\max\{w_{i}:i\in N\}$. Similarly, we have

[TABLE]

which implies that $\lambda\in\hat{\Psi}_{g,t}^{\bar{S}}(\mathcal{A})\subseteq\bigcup\limits_{i\in\bar{S},j\in S}\hat{\Psi}_{i,j}^{\bar{S}}(\mathcal{A})$.

Case IV: Suppose that $w_{S}=|y_{f}|,w_{\bar{S}}=|x_{h}|$, then $|y_{f}|\geq|x_{f}|,|x_{h}|\geq|y_{h}|$. If $|x_{h}|\geq|y_{f}|$, then $|x_{h}|=\max\{w_{i}:i\in N\}$. Similarly, we have

[TABLE]

which also implies that $\lambda\in\hat{\Psi}_{t,h}^{S}(\mathcal{A})\subseteq\bigcup\limits_{i\in S,j\in\bar{S}}\hat{\Psi}_{i,j}^{S}(\mathcal{A})$.

If $|y_{f}|\geq|x_{h}|$, then $|y_{f}|=\max\{w_{i}:i\in N\}$. Similarly, we have

[TABLE]

and $\lambda\in\tilde{\Psi}_{h,f}^{\bar{S}}(\mathcal{A})\subseteq\bigcup\limits_{i\in\bar{S},j\in S}\tilde{\Psi}_{i,j}^{\bar{S}}(\mathcal{A})$.

The conclusion follows from Case I, II, III and IV. $\blacksquare$

Next, we give the comparison theorem for Theorem 1 and Theorem 3.

Theorem 4**.**

Let $\mathcal{A}\in\mathbb{R}^{[p,q;n,n]}$, $S$ be a nonempty proper subset of $N$, $\bar{S}$ be the complement of $S$ in N. Then

[TABLE]

Proof. Let $z\in\Psi^{S}(\mathcal{A})$. Then

[TABLE]

Without loss of generality, suppose that $z\in\bigcup\limits_{i\in S,j\in\bar{S}}\hat{\Psi}_{i,j}^{S}(\mathcal{A})$, i.e., there are $u\in S,v\in\bar{S}$, such that

[TABLE]

We divide into two cases to prove that $z\in\Upsilon^{S}(\mathcal{A})$ (for other cases, we can prove them similarly).

Case I. If $r_{v}^{\Delta^{S}}(\mathcal{A})\max\{R_{u}(\mathcal{A}),C_{u}(\mathcal{A})\}=0$, then $r_{v}^{\Delta^{S}}(\mathcal{A})=0$ or $\max\{R_{u}(\mathcal{A}),C_{u}(\mathcal{A})\}=0$. When $r_{v}^{\Delta^{S}}(\mathcal{A})=0$, we have $a_{vu\cdots uu\cdots u}=0$, $r_{v}^{\overline{\Delta^{S}}}(\mathcal{A})=r_{v}^{u}(\mathcal{A})$ and

[TABLE]

which implies that $z\in\hat{\Upsilon}_{v,u}(\mathcal{A})\subseteq\bigcup\limits_{i\in\bar{S},j\in S}\hat{\Upsilon}_{i,j}(\mathcal{A})\subseteq\Upsilon^{S}(\mathcal{A}).$

When $\max\{R_{u}(\mathcal{A}),C_{u}(\mathcal{A})\}=0$, we have

[TABLE]

which also leads to $z\in\hat{\Upsilon}_{v,u}(\mathcal{A})\subseteq\bigcup\limits_{i\in\bar{S},j\in S}\hat{\Upsilon}_{i,j}(\mathcal{A})\subseteq\Upsilon^{S}(\mathcal{A}).$

Case II. If $r_{v}^{\Delta^{S}}(\mathcal{A})\max\{R_{u}(\mathcal{A}),C_{u}(\mathcal{A})\}>0$, then by (6), we have

[TABLE]

From (7), we have

[TABLE]

or

[TABLE]

Let $a=|z|,b=r_{v}^{\overline{\Delta^{S}}}(\mathcal{A}),c=r_{v}^{\Delta^{S}}(\mathcal{A})-|a_{vu\cdots uu\cdots u}|$ and $d=|a_{vu\cdots uu\cdots u}|$. When (8) holds and $d=|a_{vu\cdots uu\cdots u}|>0,$ by Lemma 2.2 in [10] and (7), we have

[TABLE]

equivalently,

[TABLE]

which implies that $z\in\hat{\Upsilon}_{v,u}(\mathcal{A})\subseteq\bigcup\limits_{i\in\bar{S},j\in S}\hat{\Upsilon}_{i,j}(\mathcal{A})\subseteq\Upsilon^{S}(\mathcal{A}).$ When (8) holds and $d=|a_{vu\cdots uu\cdots u}|=0,$ we have

[TABLE]

Hence,

[TABLE]

which also implies that $z\in\hat{\Upsilon}_{v,u}(\mathcal{A})\subseteq\bigcup\limits_{i\in\bar{S},j\in S}\hat{\Upsilon}_{i,j}(\mathcal{A})\subseteq\Upsilon^{S}(\mathcal{A}).$

On the other hand, when inequality (9) holds, i.e., $|z|\leq\max\{R_{u}(\mathcal{A}),C_{u}(\mathcal{A})\}$, we only need to prove $z\in\Upsilon^{S}(\mathcal{A})$ under the case that

[TABLE]

When $|z|\leq\max\{R_{u}(\mathcal{A}),C_{u}(\mathcal{A})\}=R_{u}(\mathcal{A})$ and $|a_{uv\cdots vv\cdots v}|>0$, then from Lemma 2.2, Lemma 2.3 of [10] and (7), we have

[TABLE]

equivalently,

[TABLE]

which implies that $z\in\hat{\Upsilon}_{u,v}(\mathcal{A})\subseteq\bigcup\limits_{i\in S,j\in\bar{S}}\hat{\Upsilon}_{i,j}(\mathcal{A})\subseteq\Upsilon^{S}(\mathcal{A}).$ And when $|z|\leq\max\{R_{u}(\mathcal{A}),C_{u}(\mathcal{A})\}=R_{u}(\mathcal{A})$ and $|a_{uv\cdots vv\cdots v}|=0$, then

[TABLE]

and

[TABLE]

which also implies that $z\in\hat{\Upsilon}_{u,v}(\mathcal{A})\subseteq\bigcup\limits_{i\in S,j\in\bar{S}}\hat{\Upsilon}_{i,j}(\mathcal{A})\subseteq\Upsilon^{S}(\mathcal{A}).$

When $|z|\leq\max\{R_{u}(\mathcal{A}),C_{u}(\mathcal{A})\}=C_{u}(\mathcal{A})$, similarly, we can obtain that

[TABLE]

which implies that $z\in\tilde{\Upsilon}_{u,v}(\mathcal{A})\subseteq\bigcup\limits_{i\in S,j\in\bar{S}}\tilde{\Upsilon}_{i,j}(\mathcal{A})\subseteq\Upsilon^{S}(\mathcal{A}).$ The proof is completed. $\blacksquare$

Remark 1**.**

For a complex tensor $\mathcal{A}\in\mathbb{R}^{[p,q;n,n]}$, the set $\Upsilon^{S}(\mathcal{A})$ consists of $2|S|(n-|S|)$ sets $\hat{\Upsilon}_{i,j}(\mathcal{A})$ and $2|S|(n-|S|)$ sets $\tilde{\Upsilon}_{i,j}(\mathcal{A})$, and the set $\Psi^{S}(\mathcal{A})$ consists of $|S|(n-|S|)$ sets $\hat{\Psi}_{i,j}^{S}(\mathcal{A})$, $|S|(n-|S|)$ sets $\hat{\Psi}_{i,j}^{\bar{S}}(\mathcal{A})$, $|S|(n-|S|)$ sets $\tilde{\Psi}_{i,j}^{S}(\mathcal{A})$ and $|S|(n-|S|)$ sets $\tilde{\Psi}_{i,j}^{\bar{S}}(\mathcal{A})$, where $S$ is a nonempty proper subset of $N$. Hence, under the same computations, $\Psi^{S}(\mathcal{A})$ captures all singular values of $\mathcal{A}$ more precisely than $\Upsilon^{S}(\mathcal{A})$.***

Based on Theorem 3 and Theorem 4, and similar to the proof of Theorem 2 of [8] and Theorem 5 of [13], the following theorems for the largest singular value of nonnegative rectangular tensors can be obtained easily.

Theorem 5**.**

Let $\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb{R}^{[p,q;n,n]}_{+}$, $S$ be a nonempty proper subset of $N$, $\bar{S}$ be the complement of $S$ in N. Then

[TABLE]

where

[TABLE]

and

[TABLE]

Theorem 6**.**

Let $\mathcal{A}=(a_{i_{1}\cdots i_{m}})\in\mathbb{R}^{[p,q;n,n]}_{+}$, $S$ be a nonempty proper subset of $N$, $\bar{S}$ be the complement of $S$ in N. Then

[TABLE]

In the end, a numerical example is given to verify the theoretical results.

Example 1**.**

Let $\mathcal{A}\in\mathbb{R}^{[2,2;3,3]}_{+}$ with entries defined as follows:*

[TABLE]

[TABLE]

*other $a_{ijkl}=0.$ By computation, we obtain that all different H-singular values of $\mathcal{A}$ are $-9.9319$, $-9.1950,-9.1673,-7.5415,-2.9723,-2.1787,-1.9725,-1.0403,~{}0,~{}0.4172,~{}3.1373,$

$3.3683$, $5.7147,6.9150$ and $18.0755.$ Let $S=\{3\}$ and $\bar{S}=\{1,2\}$. Next, we consider the singular value inclusion sets and the bounds for the largest singular value of $\mathcal{A}$.***

(i) $S$-type singular value inclusion sets.**

By Theorem 1, we have

[TABLE]

By Theorem 3, we have

[TABLE]

The singular value inclusion sets $\Upsilon^{S}(\mathcal{A})$, $\Psi^{S}(\mathcal{A})$ and the exact H-singular values are drawn in Figure 1, where $\Upsilon^{S}(\mathcal{A})$, $\Psi^{S}(\mathcal{A})$ and the exact H-singular values are represented by black solid boundary, blue dashed boundary and red “$+$”, respectively. It is easy to see that $\Psi^{S}(\mathcal{A})$ is tighter than $\Upsilon^{S}(\mathcal{A})$ from Figure 1.**

(ii) Bounds for the largest singular value $\lambda_{0}$.**

By Theorem 4 of [3], we have

[TABLE]

By Theorem 2, we have

[TABLE]

By Theorem 5, we have

[TABLE]

In fact, $\lambda_{0}=18.0755.$ This example shows that the bounds in Theorem 5 are better than those in Theorem 2 and Theorem 4 of [3].**

3 Conclusion

In this paper, by breaking $N=\{1,2,\cdots,n\}$ into disjoint subsets $S$ and its complement $\bar{S}$, we propose a new $S$-type singular inclusion sets $\Psi^{S}(\mathcal{A})$ for a real rectangular tensor $\mathcal{A}$ and prove that $\Psi^{S}(\mathcal{A})$ is tighter than $\Upsilon^{S}(\mathcal{A})$ in [8]. Based on the set $\Psi^{S}(\mathcal{A})$, we obtain a new $S$-type upper bound $\Phi^{S}(\mathcal{A})$ and a new $S$-type lower bound $\Theta^{S}(\mathcal{A})$ for the largest singular value of nonnegative rectangular tensors, and show that $\Phi^{S}(\mathcal{A})$ and $\Theta^{S}(\mathcal{A})$ are sharper than $U^{S}(\mathcal{A})$ and $L^{S}(\mathcal{A})$ in [8], respectively.

Compared with the results in [8], the advantage of the new results is that, under the same computations, $\Psi^{S}(\mathcal{A})$ can capture all singular values of $\mathcal{A}$ more precisely than $\Upsilon^{S}(\mathcal{A})$, and $\Phi^{S}(\mathcal{A})$ and $\Theta^{S}(\mathcal{A})$ can obtain more accurate upper and lower bounds than $U^{S}(\mathcal{A})$ and $L^{S}(\mathcal{A})$, respectively.

Competing interests

The author declares that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by National Natural Science Foundations of China (Grant No.11501141), Foundation of Guizhou Science and Technology Department (Grant No.[2015]2073) and Natural Science Programs of Education Department of Guizhou Province (Grant No.[2016]066).