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A note on the generalization of parameterized inexact Uzawa method for singular saddle point problems
Applied Mathematics and Computation 235 (2014) 318–322
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A note on the generalization of parameterized inexact Uzawa method for singular saddle point problems Yuan Chen, Naimin Zhang ⇑,1 School of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t Recently, Zhang and Wang studied the generalized parameterized inexact Uzawa methods (GPIU) for solving singular saddle point problems (Zhang and Wang, 2013 [22]). In this note, we continue to discuss the semi-convergence of GPIU for solving singular saddle point problems, and weaken some semi-convergent conditions. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Singular linear systems Saddle point problems Parameterized inexact Uzawa method Semi-convergence
1. Introduction We consider the iterative solution of a linear system with 2 2 block structure:
AX
A
B
BT
0
x y
¼
f g
ð1Þ
where A 2 Rmm is a symmetric positive definite matrix, B 2 Rmn a rank-deficient matrix, and f 2 Rm and g 2 Rn are given vectors, with m P n. We use BT to denote the transpose of the matrix B. Linear system (1) is often called a saddle point problem, which arises in many application areas, such as computational fluid dynamics, mixed finite element approximation of elliptic partial differential equations, optimization, optimal control, constrained least-squares problems, electronic networks, computer graphics and others; see, e.g., [1,3,10,11,14,15,18,21] and references therein. When the coefficient matrix of linear system (1) is nonsingular, which requires B to be of full rank, a number of iterative methods have been proposed in the literature. For example, Uzawa-type methods which include parameterized Uzawa (PU) method and parameterized inexact Uzawa (PIU) method [6,7,13,16], Hermitian and skew-Hermitian splitting (HSS) methods [3,4], and a lot of preconditioned Krylov subspace iterative methods. In this note, since the matrix B in (1) is rank-deficient, the coefficient matrix of (1) is singular, and (1) is called a singular saddle point problem. Recently, many techniques have been proposed for solving singular saddle point problems, including preconditioned minimum residual (PMINRES) method [17], preconditioned conjugate gradient (PCG) method [21]. For conjugate gradient type methods, here we also cite Restrictively preconditioned conjugate gradient (RPCG) methods [5,8]. Since Bai et al. proposed the PU and PIU methods [6,7], some authors developed these methods and used them to solve singular saddle point problems. Zheng et al. [24] applied the PU method to solve singular saddle point problems. Chen and Jiang [13] extended these methods and proposed a class of generalized inexact parameterized Uzawa methods. Ma and Zhang [20] studied block-diagonally preconditioned parameterized inexact Uzawa methods for singular saddle point problem. ⇑ Corresponding author. E-mail address: [email protected] (N. Zhang). This author is supported by Zhejiang Provincial Natural Science Foundation of China under grant No. Y1110451 and National Natural Science Foundation of China under grant No. 61002039. 1
http://dx.doi.org/10.1016/j.amc.2014.02.089 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
Y. Chen, N. Zhang / Applied Mathematics and Computation 235 (2014) 318–322
319
Recently, Zhang and Wang [22] further studied the generalized parameterized inexact Uzawa (GPIU) methods for solving singular saddle point problems, and gave the corresponding semi-convergence analysis. In this note, we continue to discuss the GPIU methods for solving singular saddle point problems, and weaken some of the semi-convergence conditions in [22]. The rest of this note is organized as follow. In Section 2, we present the GPIU method for solving the singular saddle point problem (1), and give the corresponding semi-convergence analysis. 2. The semi-convergence of the GPIU method For solving the singular saddle point problem (1), we make the following matrix splitting
A¼
A
B
BT
0
¼MN
where
M¼
P
0
B þ Q 1
Q2
T
;
N ¼
P A B Q1
Q2
P 2 Rmm and Q 2 2 Rnn are prescribed symmetric positive definite matrices, and Q 1 2 Rnm is an arbitrary matrix. Then we present the following generalized parameterized inexact Uzawa (GPIU) iteration method [22] for solving the singular saddle point problem (1):
P
0
B þ Q 1
Q2
T
xkþ1
!
ykþ1
¼
P A B Q1
Q2
xk
!
yk
þ
f
ð2Þ
g
or in block form,
(
xkþ1 ¼ xk þ P1 ðf Axk Byk Þ
ð3Þ
T kþ1 kþ1 ykþ1 ¼ yk þ Q 1 þ gÞ Q 1 xk Þ 2 ðB x 2 Q 1 ðx
and the iteration matrix T is:
T ¼
P B þ Q 1 T
0 Q2
1
P A B Q2 Q1
¼ I M1 A
ð4Þ
As A is singular, then T has eigenvalue 1, and the spectral radius of the iteration matrix T , i.e., qðT Þ cannot be small than 1. For the iteration matrix T , we introduce its pseudo-spectral radius tðT Þ,
tðT Þ ¼ maxfjkj : k 2 rðT Þ; k – 1g where rðT Þ is the set of eigenvalues of T . For a matrix B 2 Rnn , the smallest nonnegative integer k such that rankðBk Þ ¼ rankðBkþ1 Þ is called the index of B, and we denote it by k ¼ indexðBÞ. In fact, indexðBÞ is the size of the largest Jordan block corresponding to the zero eigenvalue of B. We now discuss the conditions of semi-convergence for solving singular linear systems, which have been studied by several authors (cf. [2,12,23]). Lemma 2.1 [9]. The iterative method (2) is semi-convergent, if and only if indexðI T Þ ¼ 1 and tðT Þ < 1. Lemma 2.2 [22]. Let A; P and Q 2 be symmetric positive definite, and B be of column rank-deficient, Q 1 is an arbitrary matrix. T Suppose that k is an eigenvalue of the iteration matrix T and ðuT ; v T Þ 2 C mþn is the corresponding eigenvector. Then k ¼ 1 if and only if u ¼ 0. Lemma 2.3 [22]. Let A; P and Q 2 be symmetric positive definite, B be of rank-deficient and Q 1 is an arbitrary matrix. Suppose that T k – 1 is an eigenvalue of the iteration matrix T and ðuT ; v T Þ 2 C mþn is the corresponding eigenvector. Then k satisfies the following quadratic equation
k2 þ
b þ c 2a s
a
kþ
aþsb ¼0 a
ð5Þ
where
a¼
u Pu > 0; u u
b¼
u Au > 0; u u
c¼
T u BQ 1 2 B u P 0; uu
s¼
u BQ 1 2 Q 1u u u
320
Y. Chen, N. Zhang / Applied Mathematics and Computation 235 (2014) 318–322
Lemma 2.4 [19]. Both roots of the quadratic equation x2 px þ q ¼ 0 are less than one in modulus if and only if jp pqj < 1 jqj2 , where p is the conjugate number of p. Theorem 2.5. Assume that A; P and Q 2 be symmetric positive definite, B be of rank-deficient, and Q 1 is an arbitrary matrix. Then
tðT Þ < 1 if and only if the following conditions hold: (1) For c ¼ 0 : b < 2a; (2) For c > 0 : s > 0
2 2sðbcÞ c < ðcbÞ , where s ¼ b2 c2 d þ 2ab þ 2bc 2ac, and s ¼ c þ id, or, c and d are the real 2 þd2
and
part and imaginary part of s, respectively, and i ¼
pffiffiffiffiffiffiffi 1.
Proof. Suppose that k – 1 is an eigenvalue of T , (1) When c ¼ 0, i.e., BT u ¼ 0, then it is easy to see
k2 þ
b 2a
a
kþ
s ¼ 0. So (5) becomes
ab ¼0 a
and the two roots of the quadratic equation are k1 ¼ 1 and k2 ¼ ab a . Therefore,
a b < 1 () b < 2a jkj < 1 ()
a
(2) When c > 0, by Lemma 2.3 and Lemma 2.4, it is easy to see that jkj < 1 if and only if
8 < aþsb2 < 1 a : bþc2as þ bþc2as aþsb < 1 aþsb2 a a a a
ð6Þ
From the first equation of (6), we have 2
ja b þ c þ idj2 < a2 () ða b þ cÞ2 þ d < a2 and by some simple algebra, we obtain 2
s ¼ b2 c2 d þ 2ab þ 2bc 2ac > 0 From the second equation of (6), we have
j sa þ bs þ cs as ss b2 cb þ 2ab þ sbj < a2 ja þ s bj2 () j ðc idÞa þ bðc þ idÞ þ cðc þ idÞ aðc þ idÞ ðc idÞðc þ idÞ b2 cb þ 2ab þ bðc idÞj < a2 ja b þ c þ idj2 and by some algebra, it holds 2
2
js þ cc cb þ icdj < s () ðs þ cc cbÞ2 þ ðcdÞ < s2 () c2 ðc2 2cb þ b2 þ d Þ < 2scðb cÞ () c < which finishes the proof.
2sðb cÞ 2
ðc bÞ2 þ d
h
Remark 2.6. In the above Theorem, if BQ 1 2 Q 1 is symmetric, that is, clusion as that of Theorem 3.4 in [22].
s is a real number, then we can obtain the same con-
Lemma 2.7 [23]. IndexðI T Þ ¼ 1 if and only if, for any 0 – Y 2 RðAÞ, Y R N ðAM1 Þ. Theorem 2.8. Assume that A; P and Q 2 be symmetric positive definite, and B be of rank-deficient, Q 1 is an arbitrary matrix. Then indexðI T Þ ¼ 1. Proof. Let
0 – Y ¼ AX ¼
A
B T
B
0
z1 z2
¼
Az1 þ Bz2 T
B z1
ð7Þ
Y. Chen, N. Zhang / Applied Mathematics and Computation 235 (2014) 318–322
321
By Lemma 2.7, to prove indexðI T Þ ¼ 1, it suffices to prove AM1 Y – 0, we give the proof for AM1 Y – 0 by contradiction. Suppose AM1 Y ¼ 0. Notice
0 ; N ðAÞ ¼ span
where Bu ¼ 0:
u
Then there exists a vector u0 such that
M1 Y ¼
0
u0
;
Bu0 ¼ 0:
So
0 Y ¼M ¼
u0
P
0
BT þ Q 1
Q2
0
u0
¼
0
Q 2 u0
ð8Þ
1 T T Making use of Eqs. (7) and (8), we have Bu0 ¼ BQ 1 2 B z1 ¼ 0. Noticing Q 2 is symmetric positive definite, then B z1 ¼ 0, T which means u0 ¼ Q 1 B z ¼ 0. Hence, 1 2
0 ¼0 Y ¼M
u0
which contradicts with Y – 0.
h
Together with Lemma 2.1, Theorem 2.5 and Theorem 2.8, it holds the following result. Theorem 2.9. Assume that A; P and Q 2 be symmetric positive definite, B be of rank-deficient, and Q 1 is an arbitrary matrix, then the GPIU method (2) is semi-convergent to a solution of the singular saddle point problem (1) if and only if the following conditions hold: (1) For c ¼ 0 : b < 2a; (2) For c > 0 : s > 0 and
2 2sðbcÞ c < ðcbÞ , where s ¼ b2 c2 d þ 2ab þ 2bc 2ac, and s ¼ c þ id. 2 þd2
Remark 2.10. To solve the singular saddle point problem (1) by GPIU method in [22], it needs the two conditions T 1 T 1 ‘‘N ðBÞ RðQ 1 2 B A BÞ ¼ f0g and BQ 2 Q 1 is symmetric’’ for semi-convergence, see Theorem 3.4 and Theorem 3.5 in [22]. However, from Theorem 2.9 we see that the GPIU method is still semi-convergent without the two conditions. Acknowledgment The authors would like to thank the anonymous referees for their valuable comments and suggestions on the revision of this paper. References [1] Z.-Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput. 75 (2006) 791–815. [2] Z.-Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing 89 (2010) 171–197. [3] Z.-Z. Bai, G.H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal. 27 (2007) 1–23. [4] Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix. Anal. Appl. 24 (2003) 603–626. [5] Z.-Z. Bai, G.-Q. Li, Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal. 23 (2003) 561–580. [6] Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38. [7] Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932. [8] Z.-Z. Bai, Z.-Q. Wang, Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems, J. Comput. Appl. Math. 187 (2006) 202–226. [9] A. Berman, R. Plemmons, Nonnegative Matrices in Mathematical Science, Academic Press, New York, 1979. SIAM, 1994. [10] J.T. Betts, Practical Methods For Optimal Control Using Nonlinear Programming, SIAM, Philadelphia, PA, 2001. [11] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York and London, 1991. [12] Z.-H. Cao, On the convergence of iterative methods for solving singular linear systems, J. Comput. Appl. Math. 145 (2002) 1–9. [13] F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008) 765–771. [14] H.C. Elman, A. Ramage, D.J. Silvester, Algorithm 866: IFISS, a MatLab toolbox for modelling incompressible flow, ACM Trans. Math. Softw. 33 (2007) 1– 18. [15] H.C. Elman, D.J. Silvester, A.J. Wathen, Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math. 90 (2002) 665–688. [16] H.C. Elman, G.H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31 (1994) 1645–1661. [17] B. Fischer, R. Ramage, D.J. Silvester, A.J. Wathen, Minimum residual methods for augmented systems, BIT Numer. Math. 38 (1998) 527–543. [18] C.-J. Li, B.-J. Li, D.J. Evans, A generalized successive overrelaxation method for least squares problems, BIT 38 (1998) 347–356. [19] J.J.H. Miller, On the location of zeros of certain classes of polynomials with applications to numerical analysis, J. Inst. Math. Appl. 8 (1971) 397–406.
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[20] H.-F. Ma, N.-M. Zhang, A note on block-diagonally preconditioned PIU methods for singular saddle point problems, Int. J. Comput. Math. 88 (2011) 808. 17. [21] X. Wu, B.P.B. Silva, J.-Y. Yuan, Conjugate gradient method for rank deficient saddle point problems, Numer. Algor. 35 (2004) 139–154. [22] G.-F. Zhang, S.-S. Wang, A generalization of parameterized inexact Uzawa method for singular saddle point problems, Appl. Math. Comput. 219 (2013) 4225–4231. [23] N.-M. Zhang, Y.-M. Wei, On the convergence of general stationary iterative methods for Range-Hermitian singular linear systems, Numer. Linear Algebra Appl. 17 (2010) 139–154. [24] B. Zheng, Z.-Z. Bai, X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl. 431 (2009) 808–817.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A note on the generalization of parameterized inexact Uzawa method for singular saddle point problems Yuan Chen, Naimin Zhang ⇑,1 School of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t Recently, Zhang and Wang studied the generalized parameterized inexact Uzawa methods (GPIU) for solving singular saddle point problems (Zhang and Wang, 2013 [22]). In this note, we continue to discuss the semi-convergence of GPIU for solving singular saddle point problems, and weaken some semi-convergent conditions. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Singular linear systems Saddle point problems Parameterized inexact Uzawa method Semi-convergence
1. Introduction We consider the iterative solution of a linear system with 2 2 block structure:
AX
A
B
BT
0
x y
¼
f g
ð1Þ
where A 2 Rmm is a symmetric positive definite matrix, B 2 Rmn a rank-deficient matrix, and f 2 Rm and g 2 Rn are given vectors, with m P n. We use BT to denote the transpose of the matrix B. Linear system (1) is often called a saddle point problem, which arises in many application areas, such as computational fluid dynamics, mixed finite element approximation of elliptic partial differential equations, optimization, optimal control, constrained least-squares problems, electronic networks, computer graphics and others; see, e.g., [1,3,10,11,14,15,18,21] and references therein. When the coefficient matrix of linear system (1) is nonsingular, which requires B to be of full rank, a number of iterative methods have been proposed in the literature. For example, Uzawa-type methods which include parameterized Uzawa (PU) method and parameterized inexact Uzawa (PIU) method [6,7,13,16], Hermitian and skew-Hermitian splitting (HSS) methods [3,4], and a lot of preconditioned Krylov subspace iterative methods. In this note, since the matrix B in (1) is rank-deficient, the coefficient matrix of (1) is singular, and (1) is called a singular saddle point problem. Recently, many techniques have been proposed for solving singular saddle point problems, including preconditioned minimum residual (PMINRES) method [17], preconditioned conjugate gradient (PCG) method [21]. For conjugate gradient type methods, here we also cite Restrictively preconditioned conjugate gradient (RPCG) methods [5,8]. Since Bai et al. proposed the PU and PIU methods [6,7], some authors developed these methods and used them to solve singular saddle point problems. Zheng et al. [24] applied the PU method to solve singular saddle point problems. Chen and Jiang [13] extended these methods and proposed a class of generalized inexact parameterized Uzawa methods. Ma and Zhang [20] studied block-diagonally preconditioned parameterized inexact Uzawa methods for singular saddle point problem. ⇑ Corresponding author. E-mail address: [email protected] (N. Zhang). This author is supported by Zhejiang Provincial Natural Science Foundation of China under grant No. Y1110451 and National Natural Science Foundation of China under grant No. 61002039. 1
http://dx.doi.org/10.1016/j.amc.2014.02.089 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
Y. Chen, N. Zhang / Applied Mathematics and Computation 235 (2014) 318–322
319
Recently, Zhang and Wang [22] further studied the generalized parameterized inexact Uzawa (GPIU) methods for solving singular saddle point problems, and gave the corresponding semi-convergence analysis. In this note, we continue to discuss the GPIU methods for solving singular saddle point problems, and weaken some of the semi-convergence conditions in [22]. The rest of this note is organized as follow. In Section 2, we present the GPIU method for solving the singular saddle point problem (1), and give the corresponding semi-convergence analysis. 2. The semi-convergence of the GPIU method For solving the singular saddle point problem (1), we make the following matrix splitting
A¼
A
B
BT
0
¼MN
where
M¼
P
0
B þ Q 1
Q2
T
;
N ¼
P A B Q1
Q2
P 2 Rmm and Q 2 2 Rnn are prescribed symmetric positive definite matrices, and Q 1 2 Rnm is an arbitrary matrix. Then we present the following generalized parameterized inexact Uzawa (GPIU) iteration method [22] for solving the singular saddle point problem (1):
P
0
B þ Q 1
Q2
T
xkþ1
!
ykþ1
¼
P A B Q1
Q2
xk
!
yk
þ
f
ð2Þ
g
or in block form,
(
xkþ1 ¼ xk þ P1 ðf Axk Byk Þ
ð3Þ
T kþ1 kþ1 ykþ1 ¼ yk þ Q 1 þ gÞ Q 1 xk Þ 2 ðB x 2 Q 1 ðx
and the iteration matrix T is:
T ¼
P B þ Q 1 T
0 Q2
1
P A B Q2 Q1
¼ I M1 A
ð4Þ
As A is singular, then T has eigenvalue 1, and the spectral radius of the iteration matrix T , i.e., qðT Þ cannot be small than 1. For the iteration matrix T , we introduce its pseudo-spectral radius tðT Þ,
tðT Þ ¼ maxfjkj : k 2 rðT Þ; k – 1g where rðT Þ is the set of eigenvalues of T . For a matrix B 2 Rnn , the smallest nonnegative integer k such that rankðBk Þ ¼ rankðBkþ1 Þ is called the index of B, and we denote it by k ¼ indexðBÞ. In fact, indexðBÞ is the size of the largest Jordan block corresponding to the zero eigenvalue of B. We now discuss the conditions of semi-convergence for solving singular linear systems, which have been studied by several authors (cf. [2,12,23]). Lemma 2.1 [9]. The iterative method (2) is semi-convergent, if and only if indexðI T Þ ¼ 1 and tðT Þ < 1. Lemma 2.2 [22]. Let A; P and Q 2 be symmetric positive definite, and B be of column rank-deficient, Q 1 is an arbitrary matrix. T Suppose that k is an eigenvalue of the iteration matrix T and ðuT ; v T Þ 2 C mþn is the corresponding eigenvector. Then k ¼ 1 if and only if u ¼ 0. Lemma 2.3 [22]. Let A; P and Q 2 be symmetric positive definite, B be of rank-deficient and Q 1 is an arbitrary matrix. Suppose that T k – 1 is an eigenvalue of the iteration matrix T and ðuT ; v T Þ 2 C mþn is the corresponding eigenvector. Then k satisfies the following quadratic equation
k2 þ
b þ c 2a s
a
kþ
aþsb ¼0 a
ð5Þ
where
a¼
u Pu > 0; u u
b¼
u Au > 0; u u
c¼
T u BQ 1 2 B u P 0; uu
s¼
u BQ 1 2 Q 1u u u
320
Y. Chen, N. Zhang / Applied Mathematics and Computation 235 (2014) 318–322
Lemma 2.4 [19]. Both roots of the quadratic equation x2 px þ q ¼ 0 are less than one in modulus if and only if jp pqj < 1 jqj2 , where p is the conjugate number of p. Theorem 2.5. Assume that A; P and Q 2 be symmetric positive definite, B be of rank-deficient, and Q 1 is an arbitrary matrix. Then
tðT Þ < 1 if and only if the following conditions hold: (1) For c ¼ 0 : b < 2a; (2) For c > 0 : s > 0
2 2sðbcÞ c < ðcbÞ , where s ¼ b2 c2 d þ 2ab þ 2bc 2ac, and s ¼ c þ id, or, c and d are the real 2 þd2
and
part and imaginary part of s, respectively, and i ¼
pffiffiffiffiffiffiffi 1.
Proof. Suppose that k – 1 is an eigenvalue of T , (1) When c ¼ 0, i.e., BT u ¼ 0, then it is easy to see
k2 þ
b 2a
a
kþ
s ¼ 0. So (5) becomes
ab ¼0 a
and the two roots of the quadratic equation are k1 ¼ 1 and k2 ¼ ab a . Therefore,
a b < 1 () b < 2a jkj < 1 ()
a
(2) When c > 0, by Lemma 2.3 and Lemma 2.4, it is easy to see that jkj < 1 if and only if
8 < aþsb2 < 1 a : bþc2as þ bþc2as aþsb < 1 aþsb2 a a a a
ð6Þ
From the first equation of (6), we have 2
ja b þ c þ idj2 < a2 () ða b þ cÞ2 þ d < a2 and by some simple algebra, we obtain 2
s ¼ b2 c2 d þ 2ab þ 2bc 2ac > 0 From the second equation of (6), we have
j sa þ bs þ cs as ss b2 cb þ 2ab þ sbj < a2 ja þ s bj2 () j ðc idÞa þ bðc þ idÞ þ cðc þ idÞ aðc þ idÞ ðc idÞðc þ idÞ b2 cb þ 2ab þ bðc idÞj < a2 ja b þ c þ idj2 and by some algebra, it holds 2
2
js þ cc cb þ icdj < s () ðs þ cc cbÞ2 þ ðcdÞ < s2 () c2 ðc2 2cb þ b2 þ d Þ < 2scðb cÞ () c < which finishes the proof.
2sðb cÞ 2
ðc bÞ2 þ d
h
Remark 2.6. In the above Theorem, if BQ 1 2 Q 1 is symmetric, that is, clusion as that of Theorem 3.4 in [22].
s is a real number, then we can obtain the same con-
Lemma 2.7 [23]. IndexðI T Þ ¼ 1 if and only if, for any 0 – Y 2 RðAÞ, Y R N ðAM1 Þ. Theorem 2.8. Assume that A; P and Q 2 be symmetric positive definite, and B be of rank-deficient, Q 1 is an arbitrary matrix. Then indexðI T Þ ¼ 1. Proof. Let
0 – Y ¼ AX ¼
A
B T
B
0
z1 z2
¼
Az1 þ Bz2 T
B z1
ð7Þ
Y. Chen, N. Zhang / Applied Mathematics and Computation 235 (2014) 318–322
321
By Lemma 2.7, to prove indexðI T Þ ¼ 1, it suffices to prove AM1 Y – 0, we give the proof for AM1 Y – 0 by contradiction. Suppose AM1 Y ¼ 0. Notice
0 ; N ðAÞ ¼ span
where Bu ¼ 0:
u
Then there exists a vector u0 such that
M1 Y ¼
0
u0
;
Bu0 ¼ 0:
So
0 Y ¼M ¼
u0
P
0
BT þ Q 1
Q2
0
u0
¼
0
Q 2 u0
ð8Þ
1 T T Making use of Eqs. (7) and (8), we have Bu0 ¼ BQ 1 2 B z1 ¼ 0. Noticing Q 2 is symmetric positive definite, then B z1 ¼ 0, T which means u0 ¼ Q 1 B z ¼ 0. Hence, 1 2
0 ¼0 Y ¼M
u0
which contradicts with Y – 0.
h
Together with Lemma 2.1, Theorem 2.5 and Theorem 2.8, it holds the following result. Theorem 2.9. Assume that A; P and Q 2 be symmetric positive definite, B be of rank-deficient, and Q 1 is an arbitrary matrix, then the GPIU method (2) is semi-convergent to a solution of the singular saddle point problem (1) if and only if the following conditions hold: (1) For c ¼ 0 : b < 2a; (2) For c > 0 : s > 0 and
2 2sðbcÞ c < ðcbÞ , where s ¼ b2 c2 d þ 2ab þ 2bc 2ac, and s ¼ c þ id. 2 þd2
Remark 2.10. To solve the singular saddle point problem (1) by GPIU method in [22], it needs the two conditions T 1 T 1 ‘‘N ðBÞ RðQ 1 2 B A BÞ ¼ f0g and BQ 2 Q 1 is symmetric’’ for semi-convergence, see Theorem 3.4 and Theorem 3.5 in [22]. However, from Theorem 2.9 we see that the GPIU method is still semi-convergent without the two conditions. Acknowledgment The authors would like to thank the anonymous referees for their valuable comments and suggestions on the revision of this paper. References [1] Z.-Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput. 75 (2006) 791–815. [2] Z.-Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing 89 (2010) 171–197. [3] Z.-Z. Bai, G.H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal. 27 (2007) 1–23. [4] Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix. Anal. Appl. 24 (2003) 603–626. [5] Z.-Z. Bai, G.-Q. Li, Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal. 23 (2003) 561–580. [6] Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38. [7] Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932. [8] Z.-Z. Bai, Z.-Q. Wang, Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems, J. Comput. Appl. Math. 187 (2006) 202–226. [9] A. Berman, R. Plemmons, Nonnegative Matrices in Mathematical Science, Academic Press, New York, 1979. SIAM, 1994. [10] J.T. Betts, Practical Methods For Optimal Control Using Nonlinear Programming, SIAM, Philadelphia, PA, 2001. [11] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York and London, 1991. [12] Z.-H. Cao, On the convergence of iterative methods for solving singular linear systems, J. Comput. Appl. Math. 145 (2002) 1–9. [13] F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008) 765–771. [14] H.C. Elman, A. Ramage, D.J. Silvester, Algorithm 866: IFISS, a MatLab toolbox for modelling incompressible flow, ACM Trans. Math. Softw. 33 (2007) 1– 18. [15] H.C. Elman, D.J. Silvester, A.J. Wathen, Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math. 90 (2002) 665–688. [16] H.C. Elman, G.H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31 (1994) 1645–1661. [17] B. Fischer, R. Ramage, D.J. Silvester, A.J. Wathen, Minimum residual methods for augmented systems, BIT Numer. Math. 38 (1998) 527–543. [18] C.-J. Li, B.-J. Li, D.J. Evans, A generalized successive overrelaxation method for least squares problems, BIT 38 (1998) 347–356. [19] J.J.H. Miller, On the location of zeros of certain classes of polynomials with applications to numerical analysis, J. Inst. Math. Appl. 8 (1971) 397–406.
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