Towards an optimal condition number of certain augmented Lagrangian-type saddle-point matrices

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1 NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Nuer. Linear Algebra Appl. 06; 3: Published online 3 April 06 in Wiley Online Library (wileyonlinelibrary.co). DOI: 0.00/nla.050 Towards an optial condition nuber of certain augented Lagrangian-type saddle-point atrices R. Estrin and C. Greif*, The University of British Colubia, Departent of Coputer Science, 366 Main Mall, Vancouver, BC V6T Z4, Canada SUMMARY We present an analysis for iniizing the condition nuber of nonsingular paraeter-dependent block-structured saddle-point atrices with a axially rank-deficient (,) block. The atrices arise fro an augented Lagrangian approach. Using quasidirect sus, we show that a decoposition akin to siultaneous diagonalization leads to an optiization based on the extreal nonzero eigenvalues and singular values of the associated block atrices. Bounds on the condition nuber of the paraeter-dependent atrix are obtained, and we deonstrate their tightness on soe nuerical exaples. Copyright 06 John Wiley & Sons, Ltd. Received 5 Septeber 05; Revised 8 February 06; Accepted March 06 KEY WORDS: saddle-point atrices; condition nuber; singular values; eigenvalues. INTRODUCTION Consider the saddle-point syste AB T u f D ; () B 0 p g where A R nn is assued to be syetric-positive seidefinite with rank n, B R n with <n,andu; f R n ;p;g R. Let us denote the coefficient atrix of () by AB T K D : () B 0 We will assue throughout that B has full row rank and that K is nonsingular. The requireent rank.a/ D n is rather significant, as it liits us to consider a very specific class of probles. We say that A is axially rank-deficient because it has the inial rank that can still allow a nonsingular K. We have recently shown in [] that there are several applications that lead to saddlepoint atrices of this type, and these atrices have unique properties. For exaple, their inverse has a special nonzero structure, and specialized preconditioners can be designed for solving the corresponding linear systes. Suppose W R is a nonsingular (typically syetric positive-definite) weight atrix. Then () can be reforulated as follows: A C B T W BB T B 0 u f C B D T W g p g : (3) *Correspondence to: C. Greif, The University of British Colubia, Departent of Coputer Science, 366 Main Mall, Vancouver, BC V6T Z4, Canada E-ail: greif@cs.ubc.ca Copyright 06 John Wiley & Sons, Ltd.

2 694 R. ESTRIN AND C. GREIF We will denote the atrix of (3) as follows: A C B K.W / D T W BB T B 0 Notice that K.0/ is identical to the atrix K defined in () and associated with (). While (3) is atheatically equivalent to (), fro a nuerical point of view, K.W / and K ay be very different in ters of conditioning, spectral structure, and other aspects []. Under the assuptions we ake in this paper, A is singular, whereas A C B T W B is nonsingular, because a necessary condition for the nonsingularity of K is that the null spaces of A and B do not intersect except at the zero vector [3, Section 3]. This, in turn, potentially enriches the faily of solution ethods that ay be used for solving (3) in coparison with solution ethods for solving (); specifically, the Schur copleent B.A C B T W B/ B T is defined (which has a siple structure as shown in []), whereas the analogous Schur copleent of K associated with A is undefined owing to the singularity of A; for solution ethods and eigenvalue estiates based on Schur copleents see, for exaple, [3 7] and the references therein. A desirable goal is to find a nuerically good choice for W. In certain applications, such a choice ay often be based on the underlying application (e.g., [8]), where a scalar Laplacian is used. In the absence of specific characteristics of the underlying atrices, a possible consideration ay be to seek to iprove the conditioning of the linear syste using a siple choice of W. To that end, we will assue that W is a scaled identity atrix, W D I ; and study when we can expect the condition nuber of the leading block and the saddle-point atrix to iprove (desirably siultaneously) as a function of. A reduction in the condition nuber ay lead to ore accurate nuerical solutions. In the case of iterative solvers, it ay also result in faster convergence, although factors other than the condition nuber (such as clustering of eigenvalues) are just as iportant. With a slight abuse of notation, let us denote the associated atrix K.W / D K.I/ as A C B K./ D T BB T B 0 and the leading block as A./ D A C B T B: Notice that A.0/ A. The approach based on (3) or its siplified for with K./ has been extensively explored in the literature. It is related to the technique of augented Lagrangian in constrained optiization [9 ], and has been studied in [] and other places. Related ethods have been successfully applied in the solution of saddle-point systes arising fro nuerical solution of partial differential equations with constraints, notably in fluid flow [ 4] and tie-haronic Maxwell equations [8] (see [, 3] for additional references). The need to deal with linear systes involving A./ ay arise either as a subproble within the augented saddle-point proble or independently. For exaple, in the nuerical solution of partial differential equations arising in electroagnetics, the discrete operator A ay represent a curl-curl operator, and it is possible to reove the singularity associated with this operator by adopting the strategy discussed here [8, 5]. In the constrained optiization front, the need to solve systes associated with A./ has arisen in various articles as part of the revised interest in the alternating direction ethod of ultipliers [6]. However, to the best of our knowledge, the specific setting where A has rank n has not been studied. In Section we derive a decopositional relation that allows us to tie the choice of to the extreal nonzero eigenvalues of A and the extreal singular values of B. We show that our choice optiizes the condition nuber of A./. In Section 3, we optiize the condition nuber of K./. Our observations are accopanied by nuerical experients in Section 4 that validate our analysis. Finally, in Section 5, we draw soe conclusions. Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla :

3 TOWARDS AN OPTIMAL CONDITION NUMBER OF CERTAIN SADDLE-POINT MATRICES 695 Notation. Throughout the paper, we use for atrices the nor notation k:k to ean the induced two-nor, k:k.. OPTIMIZING THE CONDITION NUMBER OF A./ A sall condition nuber of A./ and K./ ay be beneficial in the derivation of nuerically stable solution ethods. We start our quest of optiizing the condition nuber by constructing a decoposition that is related to a siultaneous diagonalization of the atrices involved. The following result uses the quasidirect su of atrices [7]. Proposition. Let M; N R nn and let rank.m / D r, rank.n / D n r, such that M C N is nonsingular. Then there exist nonsingular atrices P;Q R nn and nonsingular S R rr, T R.n r/.n r/ such that M D P S0 Q 00 T I N D P 0 0 Q 0T T : Proof The proof is straightforward, using SVD, and the decoposition ay not be unique. We siply consider the reduced (econoy size) SVD of M and N : M D USV T I N D WTZ T ; where U; V R nr, W;Z R n.n r/, all with orthonoral coluns, and S R rr, T R.n r/.n r/ are diagonal. We can then construct the desired decoposition with P D ŒU W, Q D ŒV Z. Because M C N is nonsingular, it follows that P;Q are nonsingular. Proposition. does not require syetry, and thus, it applies to settings that go beyond the assuptions we ake in this paper. The proposition leads to a couple of interesting observations that are unique to atrices with the rank structure we are interested in: Corollary. For atrices M; N R nn, with rank.m / D r, rank.n / D n r and M C N nonsingular, we have that M.M C N/ N D 0; (4) and.m C N/ M has eigenvalues D 0; with ultiplicities n r and r, respectively. Proof We decopose M, N according to Proposition. and observe that M.M C N / S0 N D P 00 S0 D P 00 D 0: Q T S 0 P 0T S 0 0T 0 0 0T Q T 0 0 P 0T Q T Q T The ultiplicity of the eigenvalues follows because rank..m C N/ M/ D r, and so it is possible to show that this atrix has a projection property:.m C N/ M D.M C N/.M C N N /.M C N/ M D.M C N/ M.M C N/ N.M C N/ M D.M C N/ M; Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla

4 696 R. ESTRIN AND C. GREIF where we used (4) to transition fro the second to the third equation. Because null.m / D n r, we obtain the desired eigenvalue ultiplicities. The two-nors of the atrices P;Q, which are concatenations of rectangular atrices with orthonoral coluns, are bounded by a constant. Indeed, let P D ŒU U ::: U k, where each U i R nr i,rank.u i / D r i, Ui T U i D I and P k id r i D n. By the triangular inequality for atrix nors kx PP T D U i Ui T id kx (5) 6 Ui Ui T id D k; fro which it follows that kp k 6 p k. However, the nor of the inverse, naely kp k, cannot be siilarly bounded. We note here that we are particularly interested in the case k D. Let us turn our attention to the conditioning of A./. Let > > > n >0 and > > > >0 be the eigenvalues of A and singular values of B, respectively. Define μ kak D in kbk ; kb k ² D in ; ³ n I (6) ka k μ kak ˇ D ax kbk ; kb k D ax ka k ² ; ³ n ; (7) where the superscript denotes pseudo-inverse. It was experientally observed in [] that the optial, in ters of iniizing the condition nuber of A./, typically lies in a neighborhood kak of. We now show that if < < ˇ, then the condition nuber of A./ is reduced to kbk near-optiality. We decopose A and B T B as in Proposition.. Let A D USU T ; B D WTZ T ; (8) be the econoy-size SVD of A and B, respectively. Note that for A, this is not quite the spectral decoposition, because zero eigenvalues are not included; the atrix S is saller in diensions than A. It follows that the coluns of Z for the eigenvectors of B T B. Define Then we have P D ŒU Z : (9) A./ D P P T ; where S 0 0T : (0) Because is the only atrix that depends on, this siplifies our analysis for ka./k. We can write Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla

5 TOWARDS AN OPTIMAL CONDITION NUMBER OF CERTAIN SADDLE-POINT MATRICES 697 A./ D P T S 0 0 T P ; and can then study the effect of on A./. Although P is not orthogonal, we can still use. / to bound.a.// to within a constant fro above and below, as we now show. Theore. Given A, B, P, be as defined earlier, we have.p /. / 6.A.// 6.P /. / ; so the trend of the growth of.a.// is deterined by. /. Proof This result follows iediately fro the fact that P P T > P k k ; and by the triangle inequality on nors for atrix ultiplication. The aforeentioned bound is tight in the sense that it holds as an equality when P is orthogonal. If P is not near-orthogonal, that is, A and B T B span subspaces of R nn that greatly overlap, then this bound ay be weaker, because the condition nuber of P ight be large. Noting that affects the conditioning of, not of P, we now study the effect of on for various cases. As we shall see, different regions of values of relative to and ˇ give us a different characterization of the growth of ka./k and ka./k. We partition R C into three regions, for in intervals.0; /,.ˇ; /, andœ ; ˇ, and proceed to analyze each of these cases. Case. 0<<. In this case, we have that > and n <, and thus k k D kak I D B T B : We can see that for sall, ka./k will stay relatively constant, while the nor of the inverse would grow asyptotically like,as goes to infinity. Then.A.//. /, which shrinks as increases. Case. >ˇ. In this case, we have that < and n >, and thus k k D B T B I A D : We can see that for large, A./ will stay relatively constant, while ka./k would grow like.then.a.//./, which grows as increases. Case 3. <<ˇ. Note that we ay have either > n or the reverse, depending on the proble. In the following, we cover both scenarios. First, suppose that n <<. In this case, we have that > and n >, and thus k k D kak I A D : In this case, we see that.a.// '. / D.S/ is constant. Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla

6 698 R. ESTRIN AND C. GREIF Next, we consider, and thus k k << n. In this case, we have that < and n < D B T B I D B T B : Again, in this case, we see that.a.// '. / D T. It reains to show that. / is iniized when Œ ; ˇ. Because. / is continuous in, and because. / is decreasing in for < and increasing in for > ˇ, necessarily it ust be iniized within Œ ; ˇ. With. / approxiating.acb T B/, we would expect the true condition nuber to be iniized within that interval. Because the condition nuber of P does not depend on, the aforeentioned observations are valid for a sufficiently large. However,whenP is very ill-conditioned, it ay indeed take a large to identify the asyptotic behavior that we have characterized earlier. In suary, we see that our analysis of.a.// boils down to dealing erely with the algebraic relationships aong extreal nonzero eigenvalues and singular values, siplifying previous attepts to analyze this proble. Theore. can deterine the regions of that are non-optial, although the bounds require P to be orthogonal, in order for the to be tight. As we will see in Section 4, our bounds are rearkably accurate when.p/ is odest. When P is ill-conditioned, we cannot expect anyore the bounds to be very tight, but the trend is still fully captured. The ranges of, ˇ provided in (6), (7) provide useful bounds on practical choices for. 3. OPTIMIZING THE CONDITION NUMBER OF K./ We now perfor a siilar analysis to iniize the condition nuber of K./. Recalling the eigendecoposition of A and the SVD of B fro (8) and P defined in (9), we can decopose K./ as follows: K./ D P 0 R P 0 T () where P 0 D P 0 0 W and.t R D 0 / T T 0 ; with fro (0) and T 0 D Œ0T R n. Siilarly to our analysis of A.) in Section, does not affect the conditioning of P 0, and thus, we are concerned with iniizing the condition nuber of the iddle atrix in (), R. That said, an ill-conditioned P 0 ay require a larger to enter the asyptotic behavior that we are set out to characterize. We now study R by seeking a result analogous to Theore.. Theore 3. Let P, K./, andr be defined in (9) and (). Then.P /.R/ 6.K.// 6.P /.R/ ; so the trend of the growth of.k.// is deterined by.r/. The proof of this theore is exactly the sae as that of Theore., with the additional observation that because P 0 is block-diagonal and W is orthogonal, then.p 0 / D.P /. As before, this bound can be siplified using (5). To analyze R, we first apply a syetric perutation as was carried out in [, Lea.6]. We define a perutation vector in MATLAB notation as Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla

7 TOWARDS AN OPTIMAL CONDITION NUMBER OF CERTAIN SADDLE-POINT MATRICES 699 Sparsity Pattern of R Sparsity Pattern of R nz = nz = 70 Figure. Sparsity patterns of R and R 0 : the original atrix is on the left, and the peruted one is on the right. Np D Œ W n ; n C ; n C ; n C ; n C ; n C 3; n C 3;:::;n;nC ; and apply the syetric perutation to R. Let the peruted atrix be S 0 R 0 R. Np; Np/ D 0T d ; where S D diag. i / is the atrix of eigenvalues for A and T d is a block-diagonal atrix with blocks such that T d D diag i i : () i 0 The sparsity patterns of R and R 0 can be found in Figure. Having peruted R into a atrix with a ore favorable sparsity pattern, we can begin a siilar analysis of krk D kr 0 k and R D.R 0 /. Note that the eigenvalues of T d in () are i./ D i while the eigenvalues of the inverse are q i 4 i./ D s C 4 i C 4 i ; (3)! : (4) Reark 3. Fro here henceforth, unless otherwise noted, define./ D ˇˇ./ ˇˇ and./ D./ C. It should be noted that for fixed, i./ C decreases onotonically and i./ increases in agnitude (that is, becoes ore negative) as i increases, 0 6 i 6. Forafixed i, i./ C and i./ grow onotonically with. Thus, we have that krk D ax ; C./ and R D in./, because R 0 is block-diagonal and each block is syetric. Define n ; kak D in kbk kak ; A B μ A ² D in ; ³ n I n Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla (5)

8 700 R. ESTRIN AND C. GREIF kak! D ax kbk kak ; A B μ A ² D ax ; ³ n : n (6) Again, we partition R into three intervals,.0; /, Œ ;!, and.!;/, and exaine.r/ for each interval. Note that it is possible for ; n n to be negative, and we restrict >0. Case. 0<<.For<, it can be verified that >./ and that n >./. Thus, we have krk D kak I R A D : Thus, for sall,.r/ D.A/ and thus reains constant. As will be shown in the following cases, it is for sall (in fact, for D 0), that the condition nuber of R is iniized. Although using sall would be tepting, it was seen in Section that A C B T B is poorly conditioned. Note that assuing >0iposes a constraint on the eigenvalues of A and the singular values of B. Case. >!. In this case, it can be verified that <./ and that./.then krk D k./k I R D./ : n < It can be seen that.r/. / for large, agreeing with the results fro []. Case 3. <<!. We again split this case based on whether < n n reverse. Firstly, let << n n.then <./, while leading to In the second case, >>./, leading to krk D k./k I R A D : n n krk D kak I R D./ : n > or the./,.then >./, while n < Thus in both subcases, we find that.r/./. After analyzing the condition nubers of both and R, which asyptotically correlate (for sufficiently large) with the condition nubers of A C B T B and K./, respectively, we see that we are interested in.0; \ Œ ; ˇ. It is entirely possible that the intersection is epty, and so one will want to choose that falls near the endpoints of one of the desired intervals. Let us establish conditions under which.0; \ Œ ; ˇ ;. The following proposition captures the conditions under which we ay obtain a non-epty intersection. Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla

9 TOWARDS AN OPTIMAL CONDITION NUMBER OF CERTAIN SADDLE-POINT MATRICES 70 Proposition 3. Let A, B, be as given in previous sections. Then if. D, D n n and C n < n. D n, D and n >, > 3. D n, D n n and p n < then.0; \ Œ ; ˇ ;. Thus we have conditions where we can potentially find optial condition nubers for both A./ and K./ siultaneously.ifa or B are scaled individually, it is possible to shift, ˇ,,! to ore favorable positions, although the effectiveness of this is proble-dependent. We note that while our bounds depend only on two pairs of extreal (nonzero) eigenvalues and singular values, two of these four quantities ay be tough to copute. Indeed, while the largest eigenvalue of A,, and the largest singular value of B are expected to be relatively easy to copute at least when they are well separated fro their second largest counterparts (in which case, we can effectively use, for exaple, a power ethod-type ethod), it is expected that n and would be significantly harder to copute. 4. NUMERICAL EXPERIMENTS We gauge the accuracy of our estiation of the condition nuber for varying and the desired value of the optial by presenting two nuerical exaples. The first one deals with a wellconditioned atrix, and our bounds are shown to be rearkably tight. The second exaple is one of an ill-conditioned atrix, where we see that while the bounds are not as tight, they still capture the trend. Exaple 4. The atrices A and B arise fro the finite eleent ethod being applied to solving the tieharonic Maxwell s equation [8]; A is and represents a discrete curl-curl operator, and B represents a discrete divergence operator and is of diensions In the results presented in the following, we have scaled A and B by odest ultiplicative factors to better illustrate the erits of our analysis. We have.p/ D :67; so we see that A and B T B have nearly orthogonal colun spaces, allowing P to be well conditioned. For our augented atrix, D 3:88; ˇ D 734: We then plot A C B T B against our estiation of the condition nuber as derived in Section. The lower bound is taken to be.p /. /, while the upper bound is taken to be.p /. /. We can see in Figure the estiation of A C B T B atches the actual condition nuber well within an order of agnitude for both the upper and lower bounds. In theory, we cannot expect the variation of A C B T B to be constant over the interval <<ˇ, because our bound is affected by the spectru of P. At the sae tie, the condition nuber of is definitely constant over this interval, because we have n <i <. Pleasingly, we are seeing in Figure that the variation in the condition nuber is inor. We report a siilar situation for our estiation of.k.//. Wehave D 6:5;! D 734: As can be seen in Figure 3, the eigenvalues of A and singular values of B allow for three distinct regions where.r/ is flat, growing linearly, and growing quadratically. Therefore, we would be Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla

10 70 R. ESTRIN AND C. GREIF 0 0 κ (A + γ B T B) κ (Σ) κ (P) κ (Σ) κ (P) κ (Σ) 0 0 κ (A + γ B T B) λ n / σ λ / σ γ Figure. Condition nuber of augented leading block as a function of, for a proble arising fro the discretized tie-haronic Maxwell equations κ (K ) γ κ (R) κ (P ) κ (R) κ (P ) κ (R) 0 κ (K γ ) /λ n λ n / σ λ / σ /λ γ Figure 3. Condition nuber of augented saddle-point atrix as a function of, for a proble arising fro the discretized tie-haronic Maxwell equations. interested in keeping <. We note here that ight be zero or negative in certain instances. The practical eaning of such cases is that should be kept as sall as possible for the conditioning of the saddle-point atrix to be iniized. The choice D ay work well in this case. In both Figures and 3, it can be seen that while Theores. and 3. give true bounds on the condition nubers,. / and.r/ provide excellent estiates. Because Œ ; ˇ \ Œ0; ;, one would ideally choose in the range Œ ;, which would result in iniizing the condition nuber of both A C B T B and the saddle-point atrix itself. Exaple 4. Next, we consider a case where P is (relatively) ill conditioned and investigate its effect on our estiates. We note here that we cannot take P to have a condition nuber that is overly large, Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla

11 TOWARDS AN OPTIMAL CONDITION NUMBER OF CERTAIN SADDLE-POINT MATRICES κ( A+γ B T B ) λ / σ λ n /σ κ (A + γ B T B) κ (Σ) κ (P) κ (Σ) γ Figure 4. Condition nuber of ill-conditioned A C B T B for rando A and B over varying. because this quantity appears in squared for in the bounds for the condition nubers of A./ and K./ in Theores. and 3., respectively. Therefore, to avoid considering nuerically singular A./ and K./, we need to settle for P whose condition nuber is saller than the square root of the roundoff unit. We take a rando atrix for A and a rando atrix for B, suchthat.p/ ' Wehavethat A C B T B > 0 4. For these A and B, wehave D 0:4; ˇ D 9:7; and we plot.ac B T B/ in Figure 4. (Note that and ˇ are independent of the condition nuber of P.) We can ake soe observations about the quality of the estiation for ill-conditioned P in Figure 4 as copared with a well-conditioned P in Figure. As expected, the large condition nuber of P results in the true condition nuber better estiated by the upper bound of. /.P/ than just. /. The asyptotics for sall and large still grow like and, respectively; however, the transition regions for the condition nubers are not as well defined as they are in Figure. With 6 6 ˇ, we expect A C B T B to be fairly constant, but instead, optiality occurs at the slight dip where Cˇ. Next, we plot.k.// in Figure 5. For this case, we have D 0;! D 4:9: These values are sall and do not depend on or on.p/, and because D 0, no constraint is iposed in relation to the eigenvalues of A and the singular values of B. Siilarly to the situation for A./, we see that while.r/ no longer accurately captures.k.//, our bounds very well trap the condition nuber. Interestingly, the slope of the coputed condition nuber no longer follows that of the bounds. Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla

12 704 R. ESTRIN AND C. GREIF λ /σ /λ κ ( K(γ) ) γ κ (K γ ) κ (R) κ (P ) κ (R) Figure 5. Condition nuber of ill-conditioned.k.// for rando A and B over varying. 5. CONCLUDING REMARKS We have developed conditions for iniizing the condition nubers of A./ and K./, which depend on the extreal nonzero eigenvalues of A and singular values of B. We have also established conditions for a joint doain where both condition nubers are iniized. Our approach applies to the case of axial nullity of the leading block. While this is a restriction, it does represent a nuber of interesting applications [], and in our experients, the region of iniized condition nubers is predicted in a tight and precise fashion. Our analysis straightforwardly extends to the nonsyetric case, under the sae rank assuptions on the atrices A and B. If A were to be nonsyetric, then its eigendecoposition in (8) would be replaced by the SVD A D USV T, and we would have A./ D P Q T,where P D ŒU Z and Q D ŒV Z. Theores. and 3. would then be adjusted accordingly in a sealess fashion. Another potentially interesting issue to explore ay be a situation where the rank of the leading block is slightly larger than n. Under that scenario, our analysis (and our reliance on quasidirect sus) can no longer be carried out, but using eigenvalue interlacing arguents ay still provide a way to find an effective approxiation to the optial. We leave this as an ite for future investigation. ACKNOWLEDGEMENTS This anuscript has greatly benefited fro a set of sharp and helpful coents ade by two anonyous referees. The work of the second author was supported in part by NSERC Discovery grant REFERENCES. Estrin R, Greif C. On nonsingular saddle-point systes with a axially rank deficient leading block. SIAM Journal on Matrix Analysis and Applications 05; 36(): Golub GH, Greif C. On solving block-structured indefinite linear systes. SIAM Journal on Scientific Coputing 003; 4(6): Benzi M, Golub GH, Liesen J. Nuerical solution of saddle point probles. Acta Nuerica 005; 4: 37. Copyright 06 John Wiley & Sons, Ltd. Nuer. Linear Algebra Appl. 06; 3: DOI: 0.00/nla

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