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New upper bounds for \(\|A^{-1}\|_{\infty}\) of strictly diagonally dominant M-matrices
Journal of Inequalities and Applications volume 2015, Article number: 172 (2015)
Abstract
A new upper bound for the infinity norm of inverse matrix of a strictly diagonally dominant M-matrix is given, and the lower bound for the minimum eigenvalue of the matrix is obtained. Furthermore, an upper bound for the infinity norm of inverse matrix of a strictly α-diagonally dominant M-matrix is presented. Finally, we give numerical examples to illustrate our results.
1 Introduction
Let \(R^{n\times n}\) denote the set of all \(n\times n\) real matrices, \(N=\{1, 2, \ldots, n\}\) and \(A=(a_{ij})\in R^{n\times n}\) (\(n\geq2\)). A matrix A is called a nonsingular M-matrix if there exist a nonnegative matrix B and some real number s such that
where I is the identity matrix, \(\rho(B)\) is the spectral radius of B. \(\tau(A)\) denotes the minimum of all real eigenvalues of the nonsingular M-matrix A.
Very often in numerical analysis, one needs a bound for the condition number of a square \(n\times n\) matrix A, \(\operatorname{Cond}(A)=\|A\|_{\infty}\cdot\|A^{-1}\|_{\infty}\). Bounding \(\|A\|_{\infty}\) is not usually difficult, but a bound of \(\|A^{-1}\|_{\infty}\) is not usually available unless \(A^{-1}\) is known explicitly.
However, if \(A=(a_{ij})\in R^{n\times n}\) is a strictly diagonally dominant matrix, Varah [1] bound \(\|A^{-1}\|_{\infty}\) quite easily by the following result:
Remark 1
[2]
If the diagonal dominance of A is weak, i.e., \(\min_{i\in N} \{|a_{ii}|-\sum_{j\neq i}|a_{ij}| \}\) is small, then using (1) in estimating \(\|A^{-1}\|_{\infty}\), the bound may yield a large value.
In 2007, Cheng and Huang [2] presented the following results.
If \(A=(a_{ij})\) is a strictly diagonally dominant M-matrix, then
If \(A=(a_{ij})\) is a strictly diagonally dominant M-matrix, then the bound in (2) is sharper than that in Theorem 3.3 in [3], i.e.,
In 2009, Wang [4] obtained the better result: Let \(A=(a_{ij})\) be a strictly diagonally dominant M-matrix. Then
In this paper, we present new upper bounds for \(\|A^{-1}\|_{\infty}\) of a strictly (α-)diagonally dominant M-matrix A, which improved the above results. As an application, a lower bound of \(\tau(A)\) is obtained.
For convenience, for \(i,j,k \in N\), \(j\neq i\), denote
We will denote by \(A^{(n_{1},n_{2})}\) the principal submatrix of A formed from all rows and all columns with indices between \(n_{1}\) and \(n_{2}\) inclusively; e.g., \(A^{(2,n)}\) is the submatrix of A obtained by deleting the first row and the first column of A.
Definition 1
[3]
\(A=(a_{ij})\in R^{n\times n}\) is a weakly chained diagonally dominant if for all \(i\in N\), \(d_{i}\leq1\) and \(J(A)\neq\phi\), and for all \(i\in N\), \(i\notin J(A)\), there exist indices \(i_{1}, i_{2},\ldots,i_{k}\) in N with \(a_{i_{r},i_{r+1}}\neq0\), \(0\leq r\leq k-1\), where \(i_{0}=i\) and \(i_{k}\in J(A)\).
Definition 2
[5]
\(A=(a_{ij})\in R^{n\times n}\) is called a strictly α-diagonally dominant matrix if there exists \(\alpha\in[0,1]\) such that
2 Upper bounds for \(\|A^{-1}\|_{\infty}\) of a strictly diagonally dominant M-matrix
In this section, we give several bounds of \(\|A^{-1}\|_{\infty}\) and \(\tau(A)\) for a strictly diagonally dominant M-matrix A.
Lemma 1
[2]
Let \(A=(a_{ij})\) be a weakly chained diagonally dominant M-matrix, \(B=A^{(2,n)}\), \(A^{-1}=(\alpha_{ij})\), and \(B^{-1}=(\beta_{ij})\). Then, for \(i,j=2,\ldots,n\),
Furthermore, if \(J(A)=N\), then
Lemma 2
[2]
If \(A=(a_{ij})\) is a strictly diagonally dominant M-matrix, then
Lemma 3
Let \(A=(a_{ij})\) be a strictly diagonally dominant M-matrix. Then, for \(A^{-1}=(\alpha_{ij})\),
Proof
This proof is similar to the one of Lemma 2 in [6]. □
Lemma 4
Let \(A=(a_{ij})\) be a strictly diagonally dominant M-matrix. Then, for \(A^{-1}=(\alpha_{ij})\),
Proof
This proof is similar to the one of Lemma 2.3 in [7]. □
Lemma 5
[3]
Let \(A=(a_{ij})\) be a weakly chained diagonally dominant M-matrix, \(A^{-1}=(\alpha_{ij})\), and \(\tau=\tau(A)\). Then
where
Theorem 1
Let \(A=(a_{ij})\) be a strictly diagonally dominant M-matrix, \(B=A^{(2,n)}\), \(A^{-1}=(\alpha_{ij})\), and \(B^{-1}=(\beta_{ij})\). Then
Proof
Let
Then
By Lemma 1, Lemma 2, and Lemma 4,
Let \(2\leq i\leq n\). Then, by Lemma 1 and Lemma 3,
Therefore, for \(2\leq i\leq n\), we have
Furthermore, from (4) and (5), we obtain
The result follows. □
Theorem 2
Let \(A=(a_{ij})\) be a strictly diagonally dominant M-matrix. Then
Proof
The result follows by applying the principle of mathematical induction with respect to k on \(A^{(k,n)}\) in (6). □
By Lemma 5 and Theorem 1, we can obtain a new bound of \(\tau(A)\).
Corollary 1
If \(A=(a_{ij})\) is a strictly diagonally dominant M-matrix, then
Theorem 3
Let \(A=(a_{ij})\) be a strictly diagonally dominant M-matrix. Then the bound in (7) is better than that in (3), i.e.,
Proof
Since A is a strictly diagonally dominant matrix, so \(0\leq u_{j}\), \(l_{j}<1\) for all j. By the definition of \(u_{i}\), \(l_{i}\), \(\omega_{ki}\), we have \(\omega_{ki}\leq l_{i}\) and \(a_{ii}u_{i}=\sum_{k=i+1}^{n}|a_{ik}|\) for all i. Obviously, the result follows. □
3 Upper bounds for \(\|A^{-1}\|_{\infty}\) of a strictly α-diagonally dominant M-matrix
In this section, we present an upper bound of \(\|A^{-1}\|_{\infty}\) for a strictly α-diagonally dominant M-matrix A.
Lemma 6
[8]
Let \(A, B\in R^{n\times n}\). If A and \(A-B\) are nonsingular, then
Lemma 7
Let \(A=(a_{ij})\in R^{n\times n}\) be a strictly diagonally dominant M-matrix, and \(B=(b_{ij}) \in R^{n\times n}\). If \(\varphi_{0} \cdot \|B\|_{\infty}<1\), then \(\|A^{-1}B\|_{\infty}<1\), where
Proof
By Theorem 2, we get
The result follows. □
Lemma 8
[8]
If \(\|A^{-1}\|_{\infty}<1\), then \(I-A\) is nonsingular and
Theorem 4
Let \(A=(a_{ij})\in R^{n\times n}\) be a strictly α-diagonally dominant matrix, \(\alpha\in(0, 1]\) and A be an M-matrix. If \(\{ i\in N| R_{i}(A)>C_{i}(A) \}\neq\emptyset\), and
then
where
Proof
Let \(A=B-C\), where \(B=(b_{ij})\), \(C=(c_{ij})\), and
For any \(i\in\{ i\in N| R_{i}(A)>C_{i}(A)\}\), we get
For any \(i\in\{ i\in N| R_{i}(A)\leq C_{i}(A)\}\), we have
Thus, B is a strictly diagonal dominant M-matrix. By Lemma 7, we get \(\|B^{-1}C\|_{\infty}<1\). By Lemma 6, Lemma 8, and Theorem 2, we have
Therefore
Furthermore, we have
The result follows. □
4 Numerical examples
In this section, we present numerical examples to illustrate the advantages of our derived results.
Example 1
Let
It is easy to see that A is a strictly diagonally dominant M-matrix. By calculations with Matlab 7.1, we have
respectively. It is obvious that the bound in (7) is the best result.
Example 2
Let
It is easy to see that A is a strictly α-diagonally dominant M-matrix by taking \(\alpha=0.5\), and A is not a strictly diagonally dominant matrix. Thus the bound of \(\|A^{-1}\|_{\infty}\) cannot be estimated by (1), (2), and (3), but it can be estimated by (8). By (8), we get
References
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (11361074, 71161020) and IRTSTYN, Applied Basic Research Programs of Science and Technology Department of Yunnan Province (2013FD002).
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Wang, F., Sun, Ds. & Zhao, Jx. New upper bounds for \(\|A^{-1}\|_{\infty}\) of strictly diagonally dominant M-matrices. J Inequal Appl 2015, 172 (2015). https://doi.org/10.1186/s13660-015-0696-2
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DOI: https://doi.org/10.1186/s13660-015-0696-2