Open Access

New inclusion sets for singular values

Journal of Inequalities and Applications20172017:64

https://doi.org/10.1186/s13660-017-1337-8

Received: 31 January 2017

Accepted: 14 March 2017

Published: 21 March 2017

Abstract

In this paper, for a given matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), in terms of \(r_{i}\) and \(c_{i}\), where \(r_{i} = \sum _{j = 1,j \ne i}^{n} {\vert {a_{ij} } \vert }\), \(c_{i} = \sum _{j = 1,j \ne i}^{n} {\vert {a_{ji} } \vert }\), some new inclusion sets for singular values of the matrix are established. It is proved that the new inclusion sets are tighter than the Geršgorin-type sets (Qi in Linear Algebra Appl. 56:105-119, 1984) and the Brauer-type sets (Li in Comput. Math. Appl. 37:9-15, 1999). A numerical experiment shows the efficiency of our new results.

Keywords

singular value matrix inclusion sets

MSC

15A18 15A57 65F15

1 Introduction

Singular values and the singular value decomposition play an important role in numerical analysis and many other applied fields [38]. First, we will use the following notations and definitions. Let \(N :=\{1, 2, \ldots, n\}\), and assume \(n \geq2\) throughout. For a given matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), we define \(a_{i} = |a_{ii } |\), \(s_{i} = \max\{r_{i}, c_{i} \} \) for any \(i \in N\) and \(u_{+} = \max\{0, u\}\), u is a real number, and where
$$r_{i} := \sum _{j = 1,j \ne i}^{n} {\vert {a_{ij} } \vert }, \qquad c_{i} := \sum _{j = 1,j \ne i}^{n} {\vert {a_{ji} } \vert }. $$

In terms of \(s_{i}\), the Geršgorin-type, Brauer-type and Ky Fan-type inclusion sets of the matrix singular values are given in [1, 2, 9, 10], we list the results as follows.

Theorem 1

If a matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), then
  1. (i)
    (Geršgorin-type, see [1]) all singular values of A are contained in
    $$ C(A):=\bigcup _{i = 1}^{n} C_{i} \quad\textit{with } C_{i}=\bigl[(a_{i}-s_{i})_{+},(a_{i}+s_{i}) \bigr]\in R; $$
    (1)
     
  2. (ii)
    (Brauer-type, see [2]) all singular values of A are contained in
    $$ D(A):=\bigcup _{i = 1}^{n} \bigcup _{j = 1,j\neq i}^{n} \bigl\{ z\geq0: |z-a_{i}||z-a_{j}| \leq s_{i}s_{j}\bigr\} ; $$
    (2)
     
  3. (iii)
    (Ky Fan-type, see [2]) let \(B=(b_{ij}) \in\mathbb{R}^{n\times n}\) be a nonnegative matrix satisfying \(b_{ij} \geq\max\{|a_{ij}|, |a_{ji}|\}\) for any \(i \neq j\), then all singular values of A are contained in
    $$E(A):=\bigcup _{i = 1}^{n} \bigl\{ z\geq0: |z-a_{i}|\leq\rho(B)-b_{ii}\bigr\} . $$
     

We observe that all the results in Theorem 1 are based on the values of \(s_{i} = \max\{r_{i}, c_{i} \}\), if \(r_{i}\ll c_{i}\) or \(r_{i}\gg c_{i}\), all these singular value localization sets in Theorem 1 become very crude. In this paper, we give some new singular value localization sets which are based on the values of \(r_{i}\) and \(c_{i}\). The remainder of the paper is organized as follows. In Section 2, we give our main results. In Section 3, a numerical experiment is given to show the efficiency of our new results.

2 New inclusion sets for singular values

Based on the idea of Li in [2], we give our main results as follows.

Theorem 2

If a matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), then all singular values of A are contained in
$$\Gamma(A):=\Gamma_{1}(A) \cup\Gamma_{2}(A), $$
where
$$\Gamma_{1}(A):=\bigcup _{i = 1}^{n} { \bigl\{ {\sigma \ge0:\bigl\vert {\sigma^{2} - \vert {a_{ii} } \vert ^{2} } \bigr\vert \le \vert {a_{ii} } \vert r_{i} (A) + \sigma c_{i} (A)} \bigr\} } $$
and
$$\Gamma_{2}(A):=\bigcup _{i = 1}^{n} { \bigl\{ {\sigma \ge0:\bigl\vert {\sigma^{2} - \vert {a_{ii} } \vert ^{2} } \bigr\vert \le \vert {a_{ii} } \vert c_{i} (A) + \sigma r_{i} (A)} \bigr\} }. $$

Proof

Let σ be an arbitrary singular value of A. Then there exist two nonzero vectors \(x = (x_{1}, x_{2}, \ldots, x_{n})^{T}\) and \(y = (y_{1}, y_{2},\ldots, y_{n})^{T}\) such that
$$ \sigma x = A^{*}y \quad\text{and} \quad\sigma y = Ax. $$
(3)
Denote
$$|x_{p}| = \max\bigl\{ |x_{i}|, 1\leq i \leq n \bigr\} ,\qquad |y_{q}| = \max\bigl\{ |y_{i}|, 1\leq i \leq n\bigr\} . $$
Now, we assume that \(|x_{p}| \leq|y_{q}|\), the qth equations in (3) imply
$$\begin{aligned}& \sigma x_{q}-\overline{a}_{qq}y_{q}= \sum _{j = 1,j \ne q}^{n} {\overline {a}_{jq} } y_{j}, \end{aligned}$$
(4)
$$\begin{aligned}& \sigma y_{q}-a_{qq}x_{q}=\sum _{j = 1,j \ne q}^{n} {a_{qj} } x_{j}. \end{aligned}$$
(5)
Solving for \(y_{q}\) we can get
$$ \bigl( \sigma^{2}-a_{qq}\overline{a}_{qq} \bigr)y_{q}=a_{qq}\sum _{j = 1,j \ne q}^{n} {\overline{a}_{jq} } y_{j} +\sigma\sum _{j = 1,j \ne q}^{n} {a_{qj} } x_{j}. $$
(6)
Taking the absolute value on both sides of the equation and using the triangle inequality yield
$$ \bigl\vert \sigma^{2}-|a_{qq}|^{2} \bigr\vert |y_{q}| \leq|a_{qq}|\sum _{j = 1,j \ne q}^{n} {|\overline{a}_{jq}| } |y_{j} |+\sigma\sum _{j = 1,j \ne q}^{n} {|a_{qj}| } |x_{j}|. $$
(7)
Then we can get
$$\bigl\vert \sigma^{2}-|a_{qq}|^{2} \bigr\vert \leq|a_{qq}|c_{q} (A) + \sigma r_{q} (A). $$
Similarly, if \(|y_{q}|\leq|x_{p}|\), we can get
$$\bigl\vert \sigma^{2}-|a_{pp}|^{2} \bigr\vert \leq \vert {a_{pp} } \vert r_{p} (A) + \sigma c_{p} (A). $$
Thus, we complete the proof. □

Remark 1

Since
$$\vert {a_{ii} } \vert r_{i} (A) + \sigma c_{i} (A) \leq \bigl(\vert {a_{ii} } \vert + \sigma \bigr)s_{i} $$
and
$$\vert {a_{ii} } \vert c_{i} (A) + \sigma r_{i} (A) \leq \bigl(\vert {a_{ii} } \vert + \sigma \bigr)s_{i}, $$
the results in Theorem 2 are always better than the results in Theorem 1(i).

Theorem 3

If a matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), then all singular values of A are contained in
$$\Omega(A):=\Omega_{1}(A) \cup\Omega_{2}(A) \cup\Omega_{3}(A), $$
where
$$\begin{aligned}& \begin{aligned} \Omega_{1}(A):={}&\bigcup _{i \neq j} \bigl\{ \sigma \ge0:\\ &\bigl\vert {\sigma^{2} - \vert {a_{ii} } \vert ^{2} }\bigr\vert \bigl\vert {\sigma^{2} - \vert {a_{jj} } \vert ^{2} }\bigr\vert \le \bigl(\vert {a_{ii} } \vert r_{i} (A) + \sigma c_{i} (A) \bigr) \bigl(\vert {a_{jj} } \vert r_{j} (A) + \sigma c_{j} (A) \bigr) \bigr\} , \end{aligned}\\& \begin{aligned} \Omega_{2}(A):={}&\bigcup _{i \neq j} \bigl\{ \sigma \ge0:\\ &\bigl\vert {\sigma^{2} - \vert {a_{ii} } \vert ^{2} }\bigr\vert \bigl\vert {\sigma^{2} - \vert {a_{jj} } \vert ^{2} }\bigr\vert \le \bigl( \vert {a_{ii} } \vert c_{i} (A) + \sigma r_{i} (A) \bigr) \bigl( \vert {a_{jj} } \vert c_{j} (A) + \sigma r_{j} (A) \bigr) \bigr\} , \end{aligned}\\& \begin{aligned} \Omega_{3}(A):={}&\bigcup _{i \neq j} \bigl\{ \sigma \ge0:\\ &\bigl\vert {\sigma^{2} - \vert {a_{ii} } \vert ^{2} }\bigr\vert \bigl\vert {\sigma^{2} - \vert {a_{jj} } \vert ^{2} }\bigr\vert \le \bigl(\vert {a_{ii} } \vert c_{i} (A) + \sigma r_{i} (A) \bigr) \bigl( \vert {a_{jj} \vert c_{j} (A) + \sigma r_{j} (A) \bigr) } \bigr\} \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \Omega_{4}(A):={}&\bigcup _{i \neq j} \bigl\{ \sigma \ge0:\\ &\bigl\vert {\sigma^{2} - \vert {a_{ii} } \vert ^{2} }\bigr\vert \bigl\vert {\sigma^{2} - \vert {a_{jj} } \vert ^{2} }\bigr\vert \le \bigl(\vert {a_{ii} } \vert r_{i} (A) + \sigma c_{i} (A) \bigr) \bigl( \vert {a_{jj} } \vert c_{j} (A) + \sigma r_{j} (A) \bigr) \bigr\} . \end{aligned}$$

Proof

Let σ be an arbitrary singular value of A. Then there exist two nonzero vectors \(x = (x_{1}, x_{2}, \ldots, x_{n})^{T}\) and \(y = (y_{1}, y_{2},\ldots, y_{n})^{T}\) such that
$$ \sigma x = A^{*}y \quad\text{and} \quad\sigma y = Ax. $$
(8)
Denote \(\omega_{i} = \max\{|x_{i}|, |y_{i}|\}\). Let q be an index such that \(\omega_{q}= \max\{|\omega_{i}|, i \in N\}\). Obviously, \(\omega_{q}\neq0\). Let p be an index such that \(\omega_{p} = \max\{|\omega_{i}|, i \in N, i\neq q\}\).
Case I: We suppose \(\omega_{q}=|x_{q}|\), \(\omega_{p}=|x_{p}|\), similar to the proof of Theorem 2, the qth equations in (8) imply
$$\begin{aligned} \bigl\vert \sigma^{2}-|a_{qq}|^{2} \bigr\vert \omega_{q} &\leq|a_{qq}|\sum _{j = 1,j \ne q}^{n} {|a_{qj}| } |y_{j} |+\sigma \sum _{j = 1,j \ne q}^{n} {|a_{jq}| } |x_{j}| \\ &\leq \Biggl( |a_{qq}|\sum _{j = 1,j \ne q}^{n} {|a_{qj}| } +\sigma \sum _{j = 1,j \ne q}^{n} {|a_{jq}| } \Biggr) \omega_{p}. \end{aligned}$$
(9)
Similarly, the pth equations in (8) imply
$$ \bigl\vert \sigma^{2}-|a_{pp}|^{2} \bigr\vert \omega_{p} \leq \Biggl( |a_{pp}|\sum _{j = 1,j \ne p}^{n} {|a_{pj}| } +\sigma\sum _{j = 1,j \ne p}^{n} {|a_{jp}| } \Biggr) \omega_{q}. $$
(10)
Multiplying inequalities (9) with (10), we have
$$\bigl\vert {\sigma^{2} - \vert {a_{pp} } \vert ^{2} }\bigr\vert \bigl\vert {\sigma ^{2} - \vert {a_{qq} } \vert ^{2} }\bigr\vert \le \bigl(\vert {a_{pp} } \vert r_{p} (A) + \sigma c_{p} (A) \bigr) \bigl( \vert {a_{qq} } \vert r_{q} (A) + \sigma c_{q} (A) \bigr). $$
Case II: We suppose \(\omega_{q}=|y_{q}|\), \(\omega_{p}=|y_{p}|\), similar to the proof of Theorem 2, the qth equations in (8) imply
$$\begin{aligned} \bigl\vert \sigma^{2}-|a_{qq}|^{2} \bigr\vert \omega_{q} &\leq|a_{qq}|\sum _{j = 1,j \ne q}^{n} {|a_{jq}| } |y_{j} |+\sigma \sum _{j = 1,j \ne q}^{n} {|a_{qj}| } |x_{j}| \\ &\leq \Biggl( |a_{qq}|\sum _{j = 1,j \ne q}^{n} {|a_{jq}| } +\sigma \sum _{j = 1,j \ne q}^{n} {|a_{qj}| } \Biggr) \omega_{p}. \end{aligned}$$
(11)
Similarly, the pth equations in (8) imply
$$ \bigl\vert \sigma^{2}-|a_{pp}|^{2} \bigr\vert \omega_{p} \leq \Biggl( |a_{pp}|\sum _{j = 1,j \ne p}^{n} {|a_{jp}| } +\sigma\sum _{j = 1,j \ne p}^{n} {|a_{pj}| } \Biggr) \omega_{q}. $$
(12)
Multiplying inequalities (11) with (12), we have
$$\bigl\vert {\sigma^{2} - \vert {a_{pp} } \vert ^{2} }\bigr\vert \bigl\vert {\sigma ^{2} - \vert {a_{qq} } \vert ^{2} }\bigr\vert \le \bigl( \vert {a_{pp} } \vert c_{p} (A) + \sigma r_{p} (A) \bigr) \bigl( \vert {a_{qq} } \vert c_{q} (A) + \sigma r_{q} (A) \bigr). $$
Case III: We suppose \(\omega_{q}=|y_{q}|\), \(\omega_{p}=|x_{p}|\), similar to the proof of Theorem 2, the qth equations in (8) imply
$$\begin{aligned} \bigl\vert \sigma^{2}-|a_{qq}|^{2} \bigr\vert \omega_{q} &\leq|a_{qq}|\sum _{j = 1,j \ne q}^{n} {|a_{jq}| } |y_{j} |+\sigma \sum _{j = 1,j \ne q}^{n} {|a_{qj}| } |x_{j}| \\ &\leq \Biggl( |a_{qq}|\sum _{j = 1,j \ne q}^{n} {|a_{jq}| } +\sigma \sum _{j = 1,j \ne q}^{n} {|a_{qj}| } \Biggr) \omega_{p}. \end{aligned}$$
(13)
Similarly, the pth equations in (8) imply
$$ \bigl\vert \sigma^{2}-|a_{pp}|^{2} \bigr\vert \omega_{p} \leq \Biggl( |a_{pp}|\sum _{j = 1,j \ne p}^{n} {|a_{pj}| } +\sigma\sum _{j = 1,j \ne p}^{n} {|a_{jp}| } \Biggr) \omega_{q}. $$
(14)
Multiplying inequalities (13) with (14), we have
$$\bigl\vert {\sigma^{2} - \vert {a_{pp} } \vert ^{2} }\bigr\vert \bigl\vert {\sigma ^{2} - \vert {a_{qq} } \vert ^{2} }\bigr\vert \le \bigl(\vert {a_{pp} } \vert r_{p} (A) + \sigma c_{p} (A) \bigr) \bigl( \vert {a_{qq} } \vert c_{q} (A) + \sigma r_{q} (A) \bigr). $$
Case IV: We suppose \(\omega_{q}=|x_{q}|\), \(\omega_{p}=|y_{p}|\), similar to the proof of Cases I, II, III, we can get
$$\bigl\vert {\sigma^{2} - \vert {a_{pp} } \vert ^{2} }\bigr\vert \bigl\vert {\sigma ^{2} - \vert {a_{qq} } \vert ^{2} }\bigr\vert \le \bigl(\vert {a_{pp} } \vert c_{p} (A) + \sigma r_{p} (A) \bigr) \bigl( \vert {a_{qq} } \vert c_{q} (A) + \sigma r_{q} (A) \bigr). $$

Thus, we complete the proof. □

Remark 2

Since
$$\begin{aligned}& \bigl(\vert {a_{ii} } \vert r_{i} (A) + \sigma c_{i} (A) \bigr) \bigl(\vert {a_{jj} } \vert r_{j} (A) + \sigma c_{j} (A) \bigr) \leq \bigl(\vert {a_{ii} } \vert + \sigma \bigr) \bigl(\vert {a_{jj} } \vert + \sigma \bigr)s_{i}s_{j}, \\& \bigl( \vert {a_{ii} } \vert c_{i} (A) + \sigma r_{i} (A) \bigr) \bigl( \vert {a_{jj} } \vert c_{j} (A) + \sigma r_{j} (A) \bigr) \leq \bigl(\vert {a_{ii} } \vert + \sigma \bigr) \bigl(\vert {a_{jj} } \vert + \sigma \bigr)s_{i}s_{j}, \\& \bigl(\vert {a_{ii} } \vert r_{i} (A) + \sigma c_{i} (A) \bigr) \bigl( \vert {a_{jj} } \vert c_{j} (A) + \sigma r_{j} (A) \bigr) \leq \bigl(\vert {a_{ii} } \vert + \sigma \bigr) \bigl(\vert {a_{jj} } \vert + \sigma \bigr)s_{i}s_{j} \end{aligned}$$
and
$$\bigl(\vert {a_{ii} } \vert r_{i} (A) + \sigma c_{i} (A) \bigr) \bigl( \vert {a_{jj} } \vert c_{j} (A) + \sigma r_{j} (A) \bigr) \leq \bigl(\vert {a_{ii} } \vert + \sigma \bigr) \bigl(\vert {a_{jj} } \vert + \sigma \bigr)s_{i}s_{j}, $$
the results in Theorem 3 are always better than the results in Theorem 1(ii).

We now establish comparison results between \(\Gamma(A)\) and \(\Omega(A)\).

Theorem 4

If a matrix \(A=(a_{ij}) \in\mathbb{C}^{n\times n}\), then
$$\sigma(A) \in \Omega(A) \subseteq\Gamma(A). $$

Proof

Let z be any point of \(\Omega_{3} (A)\). Then there are \(i , j \in N\), \(i \neq j\), such that \(z\in\Omega_{3} (A)\), i.e.,
$$ \bigl\vert {z ^{2} - \vert {a_{ii} } \vert ^{2} }\bigr\vert \bigl\vert {z ^{2} - \vert {a_{jj} } \vert ^{2} }\bigr\vert \le \bigl(\vert {a_{ii} } \vert r_{i} (A) + z c_{i} (A) \bigr) \bigl( \vert {a_{jj} } \vert c_{j} (A) + z r_{j} (A) \bigr). $$
(15)
If \((\vert {a_{ii} } \vert r_{i} (A) + z c_{i} (A) ) ( \vert {a_{jj} } \vert c_{j} (A) + z r_{j} (A) )=0\), then
$$\bigl\vert {z ^{2} - \vert {a_{ii} } \vert ^{2} }\bigr\vert =0 $$
or
$$\bigl\vert {z ^{2} - \vert {a_{jj} } \vert ^{2} }\bigr\vert =0. $$
Therefore, \(z \in\Gamma_{1} (A)\cup\Gamma_{2} (A)\). Moreover, if \((\vert {a_{ii} } \vert r_{i} (A) + z c_{i} (A) ) ( \vert {a_{jj} } \vert c_{j} (A) + z r_{j} (A) )> 0\), then from inequality (15), we have
$$ \frac{{\vert {z^{2} - \vert {a_{ii}^{2} } \vert } \vert }}{{\vert {a_{ii} } \vert r_{i} (A) + z c_{i} (A)}}\frac{{\vert {z^{2} - \vert {a_{jj}^{2} } \vert } \vert }}{{\vert {a_{jj} } \vert c_{j} (A) + z r_{j} (A)}} \le1. $$
(16)
Hence, from inequality (16), we have that
$$\frac{{\vert {z^{2} - \vert {a_{ii}^{2} } \vert } \vert }}{{\vert {a_{ii} } \vert r_{i} (A) + z c_{i} (A)}} \le1 $$
or
$$\frac{{\vert {z^{2} - \vert {a_{jj}^{2} } \vert } \vert }}{{\vert {a_{jj} } \vert c_{j} (A) + z r_{j} (A)}} \le1. $$
That is, \(z \in\Gamma_{1} (A)\) or \(z \in\Gamma_{2} (A)\), i.e., \(z \in \Gamma(A)\).
Similarly, if z is any point of \(\Omega_{1} (A)\) or \(\Omega_{2} (A)\), we can get
$$\sigma(A) \in \Omega_{1}(A) \subseteq\Gamma(A) $$
and
$$\sigma(A) \in \Omega_{2}(A) \subseteq\Gamma(A). $$

Thus, we complete the proof. □

3 Numerical example

Example 1

Let
$$A=\left [ { \textstyle\begin{array}{c@{\quad}c} 1 & 4 \\ {0.1} & {0.5} \end{array}\displaystyle } \right ]. $$
The singular values of A are \(\sigma_{1} = 4.1544\) and \(\sigma_{2} = 0.0241\). From Figure 1, it is easy to see that Theorem 2 is better than Theorem 1 for certain examples. In Figure 2, we can see that the results in Theorem 3 are tighter than the results in Theorem 2, which is analyzed in Theorem 4.
Figure 1

Comparisons of Theorem 1 (i), Theorem 1 (ii) and Theorem 2 for Example 1 .

Figure 2

Comparisons of Theorem 2 and Theorem 3 ( \(\pmb{\Omega_{3}}\) ) for Example 1 .

4 Conclusion

In this paper, some new inclusion sets for singular values are given. Theoretical analysis and numerical example show that these estimates are more efficient than recent corresponding results in some cases.

Declarations

Acknowledgements

He is supported by the Science and Technology Foundation of Guizhou province (Qian ke he Ji Chu [2016]1161); Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2016]255); the Doctoral Scientific Research Foundation of Zunyi Normal College (BS[2015]09). Liu is supported by the National Natural Science Foundation of China (71461027); Science and Technology Talent Training Object of Guizhou Province Outstanding Youth (Qian ke he ren zi [2015]06); Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2014]295); 2013, 2014 and 2015 Zunyi 15851 Talents Elite Project Funding; Zhunyi Innovative Talent Team (Zunyi KH(2015)38). Tian is supported by Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2015]451); the Science and Technology Foundation of Guizhou province (Qian ke he J zi [2015]2147). Ren is supported by the Science and Technology Foundation of Guizhou province (Qian ke he LH zi [2015]7006).

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Authors’ Affiliations

(1)
School of Mathematics, Zunyi Normal College

References

  1. Qi, L: Some simple estimates of singular values of a matrix. Linear Algebra Appl. 56, 105-119 (1984) MathSciNetView ArticleMATHGoogle Scholar
  2. Li, LL: Estimation for matrix singular values. Comput. Math. Appl. 37, 9-15 (1999) MathSciNetView ArticleMATHGoogle Scholar
  3. Brualdi, RA: Matrices, eigenvalues, and directed graphs. Linear Multilinear Algebra 11, 143-165 (1982) MathSciNetView ArticleMATHGoogle Scholar
  4. Golub, G, Kahan, W: Calculating the singular values and pseudo-inverse of a matrix. SIAM J. Numer. Anal. 2, 205-224 (1965) MathSciNetMATHGoogle Scholar
  5. Horn, RA, Johnson, CR: Matrix Analysis. Cambridge University Press, Cambridge (1985) View ArticleMATHGoogle Scholar
  6. Horn, RA, Johnson, CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991) View ArticleMATHGoogle Scholar
  7. Johnson, CR: A Geršgorin-type lower bound for the smallest singular value. Linear Algebra Appl. 112, 1-7 (1989) MathSciNetView ArticleMATHGoogle Scholar
  8. Johnson, CR, Szulc, T: Further lower bounds for the smallest singular value. Linear Algebra Appl. 272, 169-179 (1998) MathSciNetView ArticleMATHGoogle Scholar
  9. Li, W, Chang, Q: Inclusion intervals of singular values and applications. Comput. Math. Appl. 45, 1637-1646 (2003) MathSciNetView ArticleMATHGoogle Scholar
  10. Li, H-B, Huang, T-Z, Li, H: Inclusion sets for singular values. Linear Algebra Appl. 428, 2220-2235 (2008) MathSciNetView ArticleMATHGoogle Scholar

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