New inclusion sets for singular values

In this paper, for a given matrix A=(aij)∈Cn×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A=(a_{ij}) \in\mathbb{C}^{n\times n}$\end{document}, in terms of ri\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{i}$\end{document} and ci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{i}$\end{document}, where ri=∑j=1,j≠in|aij|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{i} = \sum _{j = 1,j \ne i}^{n} {\vert {a_{ij} } \vert }$\end{document}, ci=∑j=1,j≠in|aji|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{i} = \sum _{j = 1,j \ne i}^{n} {\vert {a_{ji} } \vert }$\end{document}, 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.


Introduction
Singular values and the singular value decomposition play an important role in numerical analysis and many other applied fields [-]. First, we will use the following notations and definitions. Let N := {, , . . . , n}, and assume n ≥  throughout. For a given matrix A = (a ij ) ∈ C n×n , we define a i = |a ii |, s i = max{r i , c i } for any i ∈ N and u + = max{, u}, u is a real number, and where In terms of s i , the Geršgorin-type, Brauer-type and Ky Fan-type inclusion sets of the matrix singular values are given in [, , , ], we list the results as follows.

Theorem  If a matrix
We observe that all the results in Theorem  are based on the values of s i = max{r i , c i }, if r i c i or r i c i , all these singular value localization sets in Theorem  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 , we give our main results. In Section , a numerical experiment is given to show the efficiency of our new results.

New inclusion sets for singular values
Proof Let σ be an arbitrary singular value of A. Then there exist two nonzero vectors x = (x  , x  , . . . , x n ) T and y = (y  , y  , . . . , y n ) T such that Now, we assume that |x p | ≤ |y q |, the qth equations in () imply Solving for y q we can get σ a qq a qq y q = a qq n j=,j =q Taking the absolute value on both sides of the equation and using the triangle inequality yield Then we can get Similarly, if |y q | ≤ |x p |, we can get Thus, we complete the proof.

Remark  Since
the results in Theorem  are always better than the results in Theorem (i).
Theorem  If a matrix A = (a ij ) ∈ C n×n , then all singular values of A are contained in Proof Let σ be an arbitrary singular value of A. Then there exist two nonzero vectors x = (x  , x  , . . . , x n ) T and y = (y  , y  , . . . , y n ) T such that σ x = A * y and σ y = Ax.
Denote ω i = max{|x i |, |y i |}. Let q be an index such that ω q = max{|ω i |, i ∈ N}. Obviously, ω q = . Let p be an index such that ω p = max{|ω i |, i ∈ N, i = q}. Case I: We suppose ω q = |x q |, ω p = |x p |, similar to the proof of Theorem , the qth equations in () imply Similarly, the pth equations in () imply Multiplying inequalities () with (), we have Case II: We suppose ω q = |y q |, ω p = |y p |, similar to the proof of Theorem , the qth equations in () imply Similarly, the pth equations in () imply Multiplying inequalities () with (), we have Case III: We suppose ω q = |y q |, ω p = |x p |, similar to the proof of Theorem , the qth equations in () imply Similarly, the pth equations in () imply Multiplying inequalities () with (), we have Case IV: We suppose ω q = |x q |, ω p = |y p |, similar to the proof of Cases I, II, III, we can get Thus, we complete the proof.

Remark  Since
Hence, from inequality (), we have that Thus, we complete the proof.

Numerical example
The singular values of A are σ  = . and σ  = .. From Figure , it is easy to see that Theorem  is better than Theorem  for certain examples. In Figure , we can see that the results in Theorem  are tighter than the results in Theorem , which is analyzed in Theorem .

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.