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On conditioning of saddle-point matrices with Lagrangian augmentation
Applied Mathematics and Computation 248 (2014) 4–7
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On conditioning of saddle-point matrices with Lagrangian augmentation q Fang Chen School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
a r t i c l e
i n f o
a b s t r a c t For saddle-point matrix and its Lagrangian augmentation, we derive bounds on eigenvalues and on Euclidean condition numbers, and discuss the asymptotic behavior of the Euclidean condition numbers with respect to the parameter in the augmentation. Ó 2014 Published by Elsevier Inc.
Keywords: Linear system Matrix splitting Series expansion Truncated inverse Preconditioning
1. Introduction Let
A
B
E
E
0
ð1:1Þ
be a saddle-point matrix, with B 2 Cpp being nonsingular, Hermitian and indefinite, and E 2 Cpq being of full column-rank. Note that the matrix A is Hermitian and indefinite [2,5]. Let
Ac
Bc
E
0
E
ð1:2Þ
be the corresponding saddle-point matrix with Lagrangian augmentation, where
Bc ¼ B þ cE W 1 E;
ð1:3Þ qq
with c being a positive constant and W 2 R being a Hermitian positive definite matrix [3]. Here and in the sequel, we use ðÞ to denote the conjugate transpose of either a vector or a matrix, and represent by n ¼ p þ q. In this paper, for saddle-point matrix A and the augmented matrix Ac , we derive bounds on eigenvalues and bound on Euclidean condition number of the augmented matrix Ac in terms of those of the matrix A, and discuss the asymptotic behavior of the Euclidean condition number jðAc Þ with respect to the parameter c. Alternatively, we also derive bounds on eigenvalues and bound on Euclidean condition number of the matrix A in terms of those of the augmented matrix Ac . These estimates further improve those given in [4]. 2. Conditioning analysis of Ac In [4], based on the matrix identity
q
Supported by the National Natural Science Foundation (Nos. 11401032; 11371075; 61473325), P.R. China. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.amc.2014.09.087 0096-3003/Ó 2014 Published by Elsevier Inc.
F. Chen / Applied Mathematics and Computation 248 (2014) 4–7 1 1 A1 c ¼ A Diagð0; cW Þ
5
ð2:1Þ
the authors presented the following upper bound on the condition number of the matrix Ac :
jðAc Þ 6 jðAÞ þ cðkEk2 þ 1ÞkW 1 kkA1 k þ c2 kW 1 k2 kEk2 : As a result, when W ¼ I, the identity matrix, it holds that
jðAc Þ !kEk2 c2 as c ! 1. Also, based on (2.1), we can demonstrate the following bounding the eigenvalues of the augmented matrix Ac , with the notations that kmin ðMÞ and kmax ðMÞ represent the smallest and the largest eigenvalues of a Hermitian matrix M, and spðMÞ indicates the set of all of its eigenvalues. Theorem 2.1. Let spðAÞ # ½H; h [ ½d; D, with h; H and d; D being positive reals. Then spðAc Þ # ½Hc ; hc [ ½dc ; Dc , where Hc ¼ H; dc ¼ d, and
hc ¼
h 1 þ chkW 1 k
;
Dc ¼ D þ ckW 1=2 Ek2 :
Proof. From
Ac ¼ A þ DiagðcE W 1 E; 0Þ we can straightforwardly obtain the estimates
kmax ðAc Þ 6 D þ ckW 1=2 Ek2
and kmin ðAc Þ P H;
which imply that
Dc ¼ D þ ckW 1=2 Ek2
and Hc ¼ H:
And from (2.1) and spðA1 Þ # 1h ; 1d we can derive the bounds 1 kmax ðA1 c Þ 6 kmax ðA Þ 6
1 d
and
1 1 1 1 kmin ðA1 c Þ P kmin ðA Þ ckW k P ckW k: h It then follows from Lemma 2.2 in [1] that
1 1 ¼ dc d
or dc ¼ d;
and
1 1 h ¼ ckW 1 k or hc ¼ : hc h 1 þ chkW 1 k
Theorem 2.1 immediately results in an estimate about the Euclidean condition number jðAc Þ of the augmented matrix Ac . Theorem 2.2. Let spðAÞ # ½H; h [ ½d; D, with h; H and d; D being positive reals. Then
jðAc Þ 6
maxfD; Hg þ ckW 1=2 Ek2
¼ jðAÞ þ
1 þ chkW 1 k
minfd; hg 1=2 kW Ek2 þ h maxfD; HgkW 1 k c þ hkW 1 kkW 1=2 Ek2 c2 minfd; hg
:
6
F. Chen / Applied Mathematics and Computation 248 (2014) 4–7
Proof. From Theorem 2.1 we have
n o 1=2 Ek2 ; H maxfDc ; Hc g max D þ ckW maxfD; Hg þ ckW 1=2 Ek2 n o 6 jðAc Þ 6 ¼ 1 h minfdc ; hc g minfd; hg 1 min d; 1þchkW
1þchkW 1 k
k
maxfD; Hg þ ckW 1=2 Ek2 1 þ chkW 1 k ¼
:
minfd; hg
Based on Theorem 2.2, we have the following asymptotic property of
jðAc Þ with respect to c.
Corollary 2.1. Let spðAÞ # ½H; h [ ½d; D, with h; H and d; D being positive reals. Then
jðAc Þ hkW 1=2 Ek2 kW 1 k ! minfd; hg c2 as c ! 1. 3. Conditioning analysis of A Alternatively, as it holds that
A ¼ Ac DiagðcE W 1 E; 0Þ
ð3:1Þ
1 A1 ¼ A1 c þ Diagð0; cW Þ;
ð3:2Þ
and
we can derive bounds on the eigenvalues of the matrix A in terms of those of the augmented matrix Ac as follows. Theorem 3.1. Let spðAc Þ # ½Hc ; hc [ ½dc ; Dc , with hc , Hc and dc ; Dc being positive reals. Then spðAÞ # ½H; h [ ½d; D, where h ¼ hc ; D ¼ Dc , and
H ¼ Hc þ ckW 1=2 Ek2 ;
d¼
dc 1 þ cdc kW 1 k
:
Proof. From (3.1) we can straightforwardly obtain the estimates
kmax ðAÞ 6 Dc
and kmin ðAÞ P Hc ckW 1=2 Ek2 ;
which imply that
D ¼ Dc
and H ¼ Hc þ ckW 1=2 Ek2 :
h i 1 1 And from (3.2) and spðA1 c Þ # hc ; dc we can derive the bounds 1 kmax ðA1 Þ 6 kmax ðA1 c Þ þ ckW k 6
1 þ ckW 1 k dc
and
kmin ðA1 Þ P kmin ðA1 c ÞP
1 : hc
Again, it then follows from Lemma 2.2 in [1] that
dc 1 1 ¼ þ ckW 1 k or d ¼ ; d dc 1 þ cdc kW 1 k and
1 1 ¼ h hc
or h ¼ hc :
Theorem 3.1 immediately results in an estimate about the Euclidean condition number
jðAÞ.
F. Chen / Applied Mathematics and Computation 248 (2014) 4–7
7
Theorem 3.2. Let spðAc Þ # ½Hc ; hc [ ½dc ; Dc , with hc , Hc and dc ; Dc being positive reals. Then
jðAÞ 6
maxfDc ; Hc g þ ckW 1=2 Ek2
1 þ cdc kW 1 k
minfdc ; hc g
¼ jðAc Þ þ
ðkW
1=2
Ek2 þ dc maxfDc ; Hc gkW 1 kÞc þ dc kW 1 kkW 1=2 Ek2 c2 : minfdc ; hc g
Proof. From Theorem 3.1 we have
jðAÞ 6 ¼
maxfD; Hg minfd; hg
maxfHc þ ckW 1=2 Ek2 ; Dc g minfhc ; 1þcd
dc c kW
1
k
g
maxfHc ; Dc g þ ckW Ek2 1 minfhc ; dc g 1þcd kW 1 k c maxfDc ; Hc g þ ckW 1=2 Ek2 1 þ cdc kW 1 k 1=2
6
¼
minfdc ; hc g
:
4. Concluding remarks Theoretical analyzes have shown that if the (1,1) block B is Hermitian positive definite, in particular, if it is well-conditioned, the augmentation makes the augmented matrices Ac and Bc be worse conditioning than the original matrices A and B, respectively. In this case, a choice of a value of c that could be significantly better than other choices may be less obvious or may not exist. As long as c is not large, the condition numbers of the (1,1) block and the Schur complement of the augmented matrix Ac may change slowly as functions of c. Hence, the Lagrangian augmentation is theoretically and practically useful and effective when the (1,1) block B is Hermitian indefinite or singular. References [1] [2] [3] [4] [5]
Z.-Z. Bai, Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks, J. Comput. Appl. Math. 237 (2013) 295–306. Z.-Z. Bai, M.K. Ng, Z.-Q. Wang, Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl. 31 (2009) 410–433. F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York and London, 1991. G.H. Golub, C. Greif, On solving block-structured indefinite linear systems, SIAM J. Sci. Comput. 24 (2003) 2076–2092. G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd Edition., The Johns Hopkins University Press, Baltimore and London, 1996.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On conditioning of saddle-point matrices with Lagrangian augmentation q Fang Chen School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
a r t i c l e
i n f o
a b s t r a c t For saddle-point matrix and its Lagrangian augmentation, we derive bounds on eigenvalues and on Euclidean condition numbers, and discuss the asymptotic behavior of the Euclidean condition numbers with respect to the parameter in the augmentation. Ó 2014 Published by Elsevier Inc.
Keywords: Linear system Matrix splitting Series expansion Truncated inverse Preconditioning
1. Introduction Let
A
B
E
E
0
ð1:1Þ
be a saddle-point matrix, with B 2 Cpp being nonsingular, Hermitian and indefinite, and E 2 Cpq being of full column-rank. Note that the matrix A is Hermitian and indefinite [2,5]. Let
Ac
Bc
E
0
E
ð1:2Þ
be the corresponding saddle-point matrix with Lagrangian augmentation, where
Bc ¼ B þ cE W 1 E;
ð1:3Þ qq
with c being a positive constant and W 2 R being a Hermitian positive definite matrix [3]. Here and in the sequel, we use ðÞ to denote the conjugate transpose of either a vector or a matrix, and represent by n ¼ p þ q. In this paper, for saddle-point matrix A and the augmented matrix Ac , we derive bounds on eigenvalues and bound on Euclidean condition number of the augmented matrix Ac in terms of those of the matrix A, and discuss the asymptotic behavior of the Euclidean condition number jðAc Þ with respect to the parameter c. Alternatively, we also derive bounds on eigenvalues and bound on Euclidean condition number of the matrix A in terms of those of the augmented matrix Ac . These estimates further improve those given in [4]. 2. Conditioning analysis of Ac In [4], based on the matrix identity
q
Supported by the National Natural Science Foundation (Nos. 11401032; 11371075; 61473325), P.R. China. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.amc.2014.09.087 0096-3003/Ó 2014 Published by Elsevier Inc.
F. Chen / Applied Mathematics and Computation 248 (2014) 4–7 1 1 A1 c ¼ A Diagð0; cW Þ
5
ð2:1Þ
the authors presented the following upper bound on the condition number of the matrix Ac :
jðAc Þ 6 jðAÞ þ cðkEk2 þ 1ÞkW 1 kkA1 k þ c2 kW 1 k2 kEk2 : As a result, when W ¼ I, the identity matrix, it holds that
jðAc Þ !kEk2 c2 as c ! 1. Also, based on (2.1), we can demonstrate the following bounding the eigenvalues of the augmented matrix Ac , with the notations that kmin ðMÞ and kmax ðMÞ represent the smallest and the largest eigenvalues of a Hermitian matrix M, and spðMÞ indicates the set of all of its eigenvalues. Theorem 2.1. Let spðAÞ # ½H; h [ ½d; D, with h; H and d; D being positive reals. Then spðAc Þ # ½Hc ; hc [ ½dc ; Dc , where Hc ¼ H; dc ¼ d, and
hc ¼
h 1 þ chkW 1 k
;
Dc ¼ D þ ckW 1=2 Ek2 :
Proof. From
Ac ¼ A þ DiagðcE W 1 E; 0Þ we can straightforwardly obtain the estimates
kmax ðAc Þ 6 D þ ckW 1=2 Ek2
and kmin ðAc Þ P H;
which imply that
Dc ¼ D þ ckW 1=2 Ek2
and Hc ¼ H:
And from (2.1) and spðA1 Þ # 1h ; 1d we can derive the bounds 1 kmax ðA1 c Þ 6 kmax ðA Þ 6
1 d
and
1 1 1 1 kmin ðA1 c Þ P kmin ðA Þ ckW k P ckW k: h It then follows from Lemma 2.2 in [1] that
1 1 ¼ dc d
or dc ¼ d;
and
1 1 h ¼ ckW 1 k or hc ¼ : hc h 1 þ chkW 1 k
Theorem 2.1 immediately results in an estimate about the Euclidean condition number jðAc Þ of the augmented matrix Ac . Theorem 2.2. Let spðAÞ # ½H; h [ ½d; D, with h; H and d; D being positive reals. Then
jðAc Þ 6
maxfD; Hg þ ckW 1=2 Ek2
¼ jðAÞ þ
1 þ chkW 1 k
minfd; hg 1=2 kW Ek2 þ h maxfD; HgkW 1 k c þ hkW 1 kkW 1=2 Ek2 c2 minfd; hg
:
6
F. Chen / Applied Mathematics and Computation 248 (2014) 4–7
Proof. From Theorem 2.1 we have
n o 1=2 Ek2 ; H maxfDc ; Hc g max D þ ckW maxfD; Hg þ ckW 1=2 Ek2 n o 6 jðAc Þ 6 ¼ 1 h minfdc ; hc g minfd; hg 1 min d; 1þchkW
1þchkW 1 k
k
maxfD; Hg þ ckW 1=2 Ek2 1 þ chkW 1 k ¼
:
minfd; hg
Based on Theorem 2.2, we have the following asymptotic property of
jðAc Þ with respect to c.
Corollary 2.1. Let spðAÞ # ½H; h [ ½d; D, with h; H and d; D being positive reals. Then
jðAc Þ hkW 1=2 Ek2 kW 1 k ! minfd; hg c2 as c ! 1. 3. Conditioning analysis of A Alternatively, as it holds that
A ¼ Ac DiagðcE W 1 E; 0Þ
ð3:1Þ
1 A1 ¼ A1 c þ Diagð0; cW Þ;
ð3:2Þ
and
we can derive bounds on the eigenvalues of the matrix A in terms of those of the augmented matrix Ac as follows. Theorem 3.1. Let spðAc Þ # ½Hc ; hc [ ½dc ; Dc , with hc , Hc and dc ; Dc being positive reals. Then spðAÞ # ½H; h [ ½d; D, where h ¼ hc ; D ¼ Dc , and
H ¼ Hc þ ckW 1=2 Ek2 ;
d¼
dc 1 þ cdc kW 1 k
:
Proof. From (3.1) we can straightforwardly obtain the estimates
kmax ðAÞ 6 Dc
and kmin ðAÞ P Hc ckW 1=2 Ek2 ;
which imply that
D ¼ Dc
and H ¼ Hc þ ckW 1=2 Ek2 :
h i 1 1 And from (3.2) and spðA1 c Þ # hc ; dc we can derive the bounds 1 kmax ðA1 Þ 6 kmax ðA1 c Þ þ ckW k 6
1 þ ckW 1 k dc
and
kmin ðA1 Þ P kmin ðA1 c ÞP
1 : hc
Again, it then follows from Lemma 2.2 in [1] that
dc 1 1 ¼ þ ckW 1 k or d ¼ ; d dc 1 þ cdc kW 1 k and
1 1 ¼ h hc
or h ¼ hc :
Theorem 3.1 immediately results in an estimate about the Euclidean condition number
jðAÞ.
F. Chen / Applied Mathematics and Computation 248 (2014) 4–7
7
Theorem 3.2. Let spðAc Þ # ½Hc ; hc [ ½dc ; Dc , with hc , Hc and dc ; Dc being positive reals. Then
jðAÞ 6
maxfDc ; Hc g þ ckW 1=2 Ek2
1 þ cdc kW 1 k
minfdc ; hc g
¼ jðAc Þ þ
ðkW
1=2
Ek2 þ dc maxfDc ; Hc gkW 1 kÞc þ dc kW 1 kkW 1=2 Ek2 c2 : minfdc ; hc g
Proof. From Theorem 3.1 we have
jðAÞ 6 ¼
maxfD; Hg minfd; hg
maxfHc þ ckW 1=2 Ek2 ; Dc g minfhc ; 1þcd
dc c kW
1
k
g
maxfHc ; Dc g þ ckW Ek2 1 minfhc ; dc g 1þcd kW 1 k c maxfDc ; Hc g þ ckW 1=2 Ek2 1 þ cdc kW 1 k 1=2
6
¼
minfdc ; hc g
:
4. Concluding remarks Theoretical analyzes have shown that if the (1,1) block B is Hermitian positive definite, in particular, if it is well-conditioned, the augmentation makes the augmented matrices Ac and Bc be worse conditioning than the original matrices A and B, respectively. In this case, a choice of a value of c that could be significantly better than other choices may be less obvious or may not exist. As long as c is not large, the condition numbers of the (1,1) block and the Schur complement of the augmented matrix Ac may change slowly as functions of c. Hence, the Lagrangian augmentation is theoretically and practically useful and effective when the (1,1) block B is Hermitian indefinite or singular. References [1] [2] [3] [4] [5]
Z.-Z. Bai, Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks, J. Comput. Appl. Math. 237 (2013) 295–306. Z.-Z. Bai, M.K. Ng, Z.-Q. Wang, Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl. 31 (2009) 410–433. F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York and London, 1991. G.H. Golub, C. Greif, On solving block-structured indefinite linear systems, SIAM J. Sci. Comput. 24 (2003) 2076–2092. G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd Edition., The Johns Hopkins University Press, Baltimore and London, 1996.