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A simple estimation for the spectral radius of (block) H-matrices
Journal of Computational and Applied Mathematics 177 (2005) 455 – 459 www.elsevier.com/locate/cam
Letter to the Editor
A simple estimation for the spectral radius of (block) H-matrices夡 Ting-Zhu Huang∗ , Rui-Sheng Ran School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China Received 13 January 2004; received in revised form 29 September 2004; accepted 29 September 2004
Abstract Simple bounds for the spectral radius of an H-matrix and a block H-matrix are presented, respectively, i.e.,
−1 (A) < 2maxi |aii | for an H-matrix A = (aij ) and (A) < maxi (Aii + A−1 ii ) for a block H-matrix A = (Aij ), where (A) denote the spectral radius.
© 2004 Elsevier B.V. All rights reserved. MSC: 65F10; 65F15 Keywords: Spectral radius; H-matrix; Block H-matrix
1. Introduction In many such fields as numerical analysis and mathematical physics, a number of problems come down to the study of H-matrices. However, there are few bounds for spectral radius of H-matrices. In this note, an upper bound is presented. The estimate is practical and simple. Let Cn,n (Rn,n ) be n × n complex (real) matrix set. Let A ∈ Cn,n , |A| = (|aij |), and Z n,n = {(aij ) ∈ Rn,n : aij 0, i = j }, 夡
Cn,n = {A = (aij ) ∈ Cn,n : aii = 0}.
Supported by NSFC 60372012.
∗ Corresponding author. Tel.: +86 28 83201175; fax: +86 28 83200131.
E-mail address: [email protected] (T.-Z. Huang). 0377-0427/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2004.09.059
456
Letter to the Editor / Journal of Computational and Applied Mathematics 177 (2005) 455 – 459
We denote by Ri (A) and R(A) the sum of the ith row and the maximal row sum of A, respectively, i.e., Ri (A) =
n
|aij |,
R(A) = max Ri (A). i
i=1
By (A) we denote the spectral radius of A. Let A ∈ Z n,n , if A−1 0, then A is called an M-matrix. Let A=D−C, where D=diag(a11 , a22 , . . . , ann ); if the comparison matrix (A) = |D| − |C| is an M-matrix, then A is said to be an H-matrix and denoted ˜ by A ∈ D. If there exists a positive diagonal matrix X= diag(x1 , x2 , . . . , xn ) (x1 , x2 , . . . , xn > 0) such that AX is a strictly diagonally dominant matrix, then A is said to be a generalized strictly diagonally dominant matrix. It is well known that an H-matrix is a generalized strictly diagonally dominant matrix (e.g. see [1]). Let A = (aij ) ∈ Cn,n is partitioned as
A12 A22 ... Ak2
A11 A21 A= ... Ak1
. . . A1k . . . A2k , ... ... . . . Akk
where Aii are square matrices of order ni , −1 A−1 > ii
Aij
(1) k
i=1 ni
= n. If
for all i = 1, 2, . . . , k,
j =i
then A is said to be a block strictly diagonally dominant matrix, denoted by A ∈ BSSD [3]; if there exist positive numbers x1 , x2 , . . . , xk such that −1 xi A−1 > xj Aij for all i = 1, 2, . . . , k, ii j =i
then A is said to be a block H-matrix, denoted by A ∈ B D˜ [5].
2. An estimate for the spectral radius of (block) H-matrices A classical estimate for spectral radius of matrix is the following famous Frobenius inequality: Lemma 1 (Horn and Johnson [4]). Let A = (aij ) ∈ Cn,n , then (A) R(A). ˜ then Theorem 1. Let A = (aij ) ∈ Cn,n , if A ∈ D, (A) < 2 max |aii |. i
Letter to the Editor / Journal of Computational and Applied Mathematics 177 (2005) 455 – 459
457
Proof. If a matrix A belongs to the D˜ class, then there exists a positive diagonal matrix X=diag(x1 , x2 , . . ., xn ) such that AX is strictly diagonally dominant (SDD). Then X −1 AX is an SDD matrix, too, i.e., |aii | >
|aij |xj xi
j =i
for all i = 1, 2, . . . , n.
Since X−1 AX and A are similar, it follows that (A) = (X −1 AX) R(X −1 AX)
= max i
|aij |xj j
xi
|aij |xj = max |aii | + < 2 max |aii |. i i xi j =i
By the theorem, the following corollary can be obtained easily. ˜ then (A) min(R(A), 2maxi |aii |). Corollary. (1) If A ∈ D, (2) If A is an M-matrix, then (A) min(R(A), 2maxi aii ). Example 1. Let 0.9 0.9 A= 0.1 0.3
0.1 0.9 0.1 0.1
0.2 0.7 0.9 0.2
0.1 0.8 , 0.1 0.766
we can prove A is an H-matrix. Then, by Theorem 1: (A) < 1.8 (the accurate value is (A) = 1.5499); by Lemma 1: (A) 3.3. Example 2. Let 2.5 0 A= −0.1i 2.4
1 −2.8i 0.2 −2.6i
i 0 1 i , 1.9 0 2.5 2.8i
we can prove A is an H-matrix. Then, by Theorem 1: (A) < 2maxi |aii | = 5.6 (the accurate value is (A) = 2.6003); by Lemma 1: (A) 10.3. In the following, we will discuss the block H-matrix case. In [2], an estimation of the spectral radius of block matrices has been presented.
458
Letter to the Editor / Journal of Computational and Applied Mathematics 177 (2005) 455 – 459
Lemma 2 (Chen and Li [2]). Let A = (Aij ) be a block m × m matrix where Aij are nonnegative ni × nj matrices. If B = (Aij ), where · is a consistent matrix norm, then (A) (B).
˜ then Theorem 2. Let A = (aij ) ∈ Cn,n is partitioned as (1), if A ∈ B D, −1 (A) < max {Aii + A−1 ii }. i
˜ there exist positive numbers x1 , x2 , . . . , xk such that Proof. If A ∈ B D, −1 > xj Aij for all i = 1, 2, . . . , k. xi A−1 ii j =i
Denote X = diag(x1 In1 , x2 In2 , . . . , xk Ink ), i.e., X is a block positive diagonal matrix. By the above inequality, AX ∈ BSSD. Let B = X−1 AX, then xn x2 A11 x1 A12 · · · x1 A1k x1 A21 A22 · · · xxn2 A2k x B = . 2 . .. . . .. .. . . . x1 x2 xn Ak1 xn Ak2 · · · Akk It is clear that B = (Bij ) ∈ BSSD. Let C = (Bij )k×k , where · is a consistent norm. Then by Lemmas 2 and 1, (A) = (B) (C) R(C)
xj Aij = max Aii + i x j =i i −1 < max Aii + A−1 . ii i
Example 3. Let 1 0 A = · · · 1 1
0 1 ··· −4 −5
.. . .. . ··· .. . .. .
4.5 1 ··· 1
0.5 0.5 ··· . 0
0
1
−1 We can prove A is a block H-matrix [5]. Let · be · ∞ . It is clear that maxi {Aii + A−1 ii } = 2. Then
by Theorem 2: (A) < 2 (the accurate value is (A) = 1.8779), by Lemma 2: (A) 6.4772.
Letter to the Editor / Journal of Computational and Applied Mathematics 177 (2005) 455 – 459
459
Acknowledgements We would like to thank the referee very much for giving us the simpler proof of Theorem 1 of this note and thank the referee and editor Prof. Lothar Reichel very much for their helpful suggestions for rewriting this note. References [1] A. Berman, R.J. Plemmons, Nonnegative Matrices in Mathematical Sciences, SIAM Press, Philadelphia, 1994. [2] M.Q. Chen, X.Z. Li, An estimation of the spectral radius of a product of block matrices, Linear Algebra Appl. 379 (2004) 267–275. [3] D.G. Feingold, R.S. Varga, Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem, Pacific J. Math. 12 (1962) 1241–1249. [4] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. [5] B. Polman, Incomplete blockwise factorizations of (block) H-matrices, Linear Algebra Appl. 90 (1987) 119–132.
Letter to the Editor
A simple estimation for the spectral radius of (block) H-matrices夡 Ting-Zhu Huang∗ , Rui-Sheng Ran School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China Received 13 January 2004; received in revised form 29 September 2004; accepted 29 September 2004
Abstract Simple bounds for the spectral radius of an H-matrix and a block H-matrix are presented, respectively, i.e.,
−1 (A) < 2maxi |aii | for an H-matrix A = (aij ) and (A) < maxi (Aii + A−1 ii ) for a block H-matrix A = (Aij ), where (A) denote the spectral radius.
© 2004 Elsevier B.V. All rights reserved. MSC: 65F10; 65F15 Keywords: Spectral radius; H-matrix; Block H-matrix
1. Introduction In many such fields as numerical analysis and mathematical physics, a number of problems come down to the study of H-matrices. However, there are few bounds for spectral radius of H-matrices. In this note, an upper bound is presented. The estimate is practical and simple. Let Cn,n (Rn,n ) be n × n complex (real) matrix set. Let A ∈ Cn,n , |A| = (|aij |), and Z n,n = {(aij ) ∈ Rn,n : aij 0, i = j }, 夡
Cn,n = {A = (aij ) ∈ Cn,n : aii = 0}.
Supported by NSFC 60372012.
∗ Corresponding author. Tel.: +86 28 83201175; fax: +86 28 83200131.
E-mail address: [email protected] (T.-Z. Huang). 0377-0427/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2004.09.059
456
Letter to the Editor / Journal of Computational and Applied Mathematics 177 (2005) 455 – 459
We denote by Ri (A) and R(A) the sum of the ith row and the maximal row sum of A, respectively, i.e., Ri (A) =
n
|aij |,
R(A) = max Ri (A). i
i=1
By (A) we denote the spectral radius of A. Let A ∈ Z n,n , if A−1 0, then A is called an M-matrix. Let A=D−C, where D=diag(a11 , a22 , . . . , ann ); if the comparison matrix (A) = |D| − |C| is an M-matrix, then A is said to be an H-matrix and denoted ˜ by A ∈ D. If there exists a positive diagonal matrix X= diag(x1 , x2 , . . . , xn ) (x1 , x2 , . . . , xn > 0) such that AX is a strictly diagonally dominant matrix, then A is said to be a generalized strictly diagonally dominant matrix. It is well known that an H-matrix is a generalized strictly diagonally dominant matrix (e.g. see [1]). Let A = (aij ) ∈ Cn,n is partitioned as
A12 A22 ... Ak2
A11 A21 A= ... Ak1
. . . A1k . . . A2k , ... ... . . . Akk
where Aii are square matrices of order ni , −1 A−1 > ii
Aij
(1) k
i=1 ni
= n. If
for all i = 1, 2, . . . , k,
j =i
then A is said to be a block strictly diagonally dominant matrix, denoted by A ∈ BSSD [3]; if there exist positive numbers x1 , x2 , . . . , xk such that −1 xi A−1 > xj Aij for all i = 1, 2, . . . , k, ii j =i
then A is said to be a block H-matrix, denoted by A ∈ B D˜ [5].
2. An estimate for the spectral radius of (block) H-matrices A classical estimate for spectral radius of matrix is the following famous Frobenius inequality: Lemma 1 (Horn and Johnson [4]). Let A = (aij ) ∈ Cn,n , then (A) R(A). ˜ then Theorem 1. Let A = (aij ) ∈ Cn,n , if A ∈ D, (A) < 2 max |aii |. i
Letter to the Editor / Journal of Computational and Applied Mathematics 177 (2005) 455 – 459
457
Proof. If a matrix A belongs to the D˜ class, then there exists a positive diagonal matrix X=diag(x1 , x2 , . . ., xn ) such that AX is strictly diagonally dominant (SDD). Then X −1 AX is an SDD matrix, too, i.e., |aii | >
|aij |xj xi
j =i
for all i = 1, 2, . . . , n.
Since X−1 AX and A are similar, it follows that (A) = (X −1 AX) R(X −1 AX)
= max i
|aij |xj j
xi
|aij |xj = max |aii | + < 2 max |aii |. i i xi j =i
By the theorem, the following corollary can be obtained easily. ˜ then (A) min(R(A), 2maxi |aii |). Corollary. (1) If A ∈ D, (2) If A is an M-matrix, then (A) min(R(A), 2maxi aii ). Example 1. Let 0.9 0.9 A= 0.1 0.3
0.1 0.9 0.1 0.1
0.2 0.7 0.9 0.2
0.1 0.8 , 0.1 0.766
we can prove A is an H-matrix. Then, by Theorem 1: (A) < 1.8 (the accurate value is (A) = 1.5499); by Lemma 1: (A) 3.3. Example 2. Let 2.5 0 A= −0.1i 2.4
1 −2.8i 0.2 −2.6i
i 0 1 i , 1.9 0 2.5 2.8i
we can prove A is an H-matrix. Then, by Theorem 1: (A) < 2maxi |aii | = 5.6 (the accurate value is (A) = 2.6003); by Lemma 1: (A) 10.3. In the following, we will discuss the block H-matrix case. In [2], an estimation of the spectral radius of block matrices has been presented.
458
Letter to the Editor / Journal of Computational and Applied Mathematics 177 (2005) 455 – 459
Lemma 2 (Chen and Li [2]). Let A = (Aij ) be a block m × m matrix where Aij are nonnegative ni × nj matrices. If B = (Aij ), where · is a consistent matrix norm, then (A) (B).
˜ then Theorem 2. Let A = (aij ) ∈ Cn,n is partitioned as (1), if A ∈ B D, −1 (A) < max {Aii + A−1 ii }. i
˜ there exist positive numbers x1 , x2 , . . . , xk such that Proof. If A ∈ B D, −1 > xj Aij for all i = 1, 2, . . . , k. xi A−1 ii j =i
Denote X = diag(x1 In1 , x2 In2 , . . . , xk Ink ), i.e., X is a block positive diagonal matrix. By the above inequality, AX ∈ BSSD. Let B = X−1 AX, then xn x2 A11 x1 A12 · · · x1 A1k x1 A21 A22 · · · xxn2 A2k x B = . 2 . .. . . .. .. . . . x1 x2 xn Ak1 xn Ak2 · · · Akk It is clear that B = (Bij ) ∈ BSSD. Let C = (Bij )k×k , where · is a consistent norm. Then by Lemmas 2 and 1, (A) = (B) (C) R(C)
xj Aij = max Aii + i x j =i i −1 < max Aii + A−1 . ii i
Example 3. Let 1 0 A = · · · 1 1
0 1 ··· −4 −5
.. . .. . ··· .. . .. .
4.5 1 ··· 1
0.5 0.5 ··· . 0
0
1
−1 We can prove A is a block H-matrix [5]. Let · be · ∞ . It is clear that maxi {Aii + A−1 ii } = 2. Then
by Theorem 2: (A) < 2 (the accurate value is (A) = 1.8779), by Lemma 2: (A) 6.4772.
Letter to the Editor / Journal of Computational and Applied Mathematics 177 (2005) 455 – 459
459
Acknowledgements We would like to thank the referee very much for giving us the simpler proof of Theorem 1 of this note and thank the referee and editor Prof. Lothar Reichel very much for their helpful suggestions for rewriting this note. References [1] A. Berman, R.J. Plemmons, Nonnegative Matrices in Mathematical Sciences, SIAM Press, Philadelphia, 1994. [2] M.Q. Chen, X.Z. Li, An estimation of the spectral radius of a product of block matrices, Linear Algebra Appl. 379 (2004) 267–275. [3] D.G. Feingold, R.S. Varga, Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem, Pacific J. Math. 12 (1962) 1241–1249. [4] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. [5] B. Polman, Incomplete blockwise factorizations of (block) H-matrices, Linear Algebra Appl. 90 (1987) 119–132.