An implicit Jacobi-like method for computing generalized hyperbolic SVD

Transcription

1 Linear Agebra and its Appications 358 (2003) wwweseviercom/ocate/aa An impicit Jacobi-ike method for computing generaized hyperboic SVD Adam W Bojanczyk Schoo of Eectrica and Computer Engineering Corne University Ithaca NY USA Received 28 March 2001; accepted 1 Apri 2002 Submitted by JL Barow Abstract In this paper we introduce a joint hyperboic-orthogona decomposition of two matrices which we ca a generaized hyperboic singuar vaue decomposition or GHSVD This decomposition can be used for finding the eigenvaues and eigenvectors of a symmetric definite penci X T X λy T ΦY where Φ = diag(±1) We aso present an impicit Jacobi-ike method for computing this GHSVD 2002 Esevier Science Inc A rights reserved Keywords: Hyperboic singuar vaue decomposition; Hyperboic transformations; Symmetric definite penci; Jacobi method 1 Introduction Reca from [520] that a square matrix H is caed a hyperexchange matrix with respect to a pair of weighting matrices Φ = diag(±1) and ˆΦ = diag(±1)or(φ ˆΦ)- hyperexchange iff H T ΦH = ˆΦ (1) In [20] the foowing hyperboic singuar vaue decomposition of a matrix was introduced Effort sponsored by the Defence Advanced Research Project Agency (DARPA) and Rome Laboratory Air Force Materia Command USAF under agreement number F The US Government is authorized to reproduce and distribute reprints for Governmenta purposes notwithstanding any copyright annotation thereon E-mai address: adamb@eecorneedu (AW Bojanczyk) /02/$ - see front matter 2002 Esevier Science Inc A rights reserved PII: S (02)

2 294 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) Theorem 1 Let Φ = diag(±1) be an m m diagona matrix and Y R m n m n be such that rank(y T ΦY) = n Then there exist a (Φ ˆΦ)-hyperexchange matrix H an orthogona matrix Q and an m n diagona matrix Γ with positive diagona eements Γ i arranged in nonincreasing order such that Y = H ΓQ (2) A more genera case when rank(y T ΦY) < rank(y ) was considered in [51828] The HSVD of a matrix Y can be used to compute the eigendecomposition of an indefinite matrix A = Y T ΦY in a simiar way as the SVD is used for computing the eigendecomposition of a matrix B given in the product form B = Y T Y Indeed from Theorem 1 we see that A = Y T ΦY = Q T Γ T ˆΦΓQ Thus Q is the matrix of eigenvectors of A and γi 2 are the corresponding eigenvaues up to signs Other appication of the HSVD is in soving indefinite inear east squares probems (ILLSP) [310] simiary as the SVD may be used for soving inear east squares probems The purpose of this paper is to introduce another hyperboic decomposition generaized hyperboic singuar vaue decomposition or GHSVD which is an anaogue to the generaized singuar vaue decomposition [26] The GHSVD is a joint decomposition of two matrices X and Y and its main appication is in soving a symmetric definite generaized eigenvaue probem Ax = λbx wherea and B are given in product forms A = X T X and B = Y T ΦY respectivey It can aso be used in soving a generaized ILLSP [3] The paper is organized as foows After introducing the GHSVD in Section 2 in Section 3 we show how this decomposition can be appied to symmetric definite generaized eigenprobems In Section 4 we describe a Jacobi-ike agorithm computing the GHSVD The agorithm uses both hyperboic and orthogona pane rotations The key eement in the agorithm is a procedure for computing the GHSVD of two 2 2 upper trianguar matrices In Section 5 some numerica resuts are presented Concuding remarks and open questions are incuded in Section 6 2 GHSVD Let A denote the pseudoinverse of a matrix A We wi denote by 0 m n an m n zero matrix Simiary I m n wi denote an m n diagona matrix with a ones on the main diagona If m = n we wi write 0 n and I n If the dimensions of 0 m n and I m n can be determined from the context the subscripts m and n wi be omitted We wi make use of hyperexchange matrices satisfying the reationship (1) When the form of the pair of weighting matrices (Φ ˆΦ) is obvious from the context or when ony its existence but not the form is important the weighting matrices wi not be expicity stated Reca the foowing hyperboic trianguarization of a matrix [71222]

3 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) Lemma 1 Let Y R n k n k and Φ = diag(±1) be an n n weighting matrix If rank(y ) = rank(y T ΦY) = p then there exist a ( ˆΦ Φ)-hyperexchange matrix H and an orthogona P such that R 0 HYP = 0 0 (n p) (k p) where R is a p p nonsinguar upper trianguar matrix We are ready to introduce the GHSVD of two matrices Theorem 2 Let X R s k Y R n k s k n k and Φ = diag(φ i ) Φ i = ±1 be an n n weighting matrix Assume that rank(x) = p (3) rank(y ) = rank(y T ΦY) = q (4) rank ( X(Y T ΦY) X T) = min(p q) = p (5) nu(x) nu(y ) ={0} (6) Then there exist orthogona matrices Q and P and a (Φ ˆΦ)-hyperexchange matrix H such that E Q 0 X 0 P = 0 H Y F (7) 0 where E and F are upper trianguar k k matrices having the foowing forms: E E m E r E = 0 0 q p 0 F 0 F = (8) k q k q where E r is a p (k q) fu rank matrix Moreover the matrix (E fu rank diagona p q matrix E m ) F 1 (E E m )F 1 = (D 0 p (q p) ) where D = diag(d i ) p i=1 (9) Proof The reations (3) and (4) impy that there exist orthogona P 1 and Q 1 such that YP 1 = ( ) Ẽ Ỹ 0 n (k q) Q1 XP 1 = 0 (s p) k where Ỹ isafurankn q matrix and Ẽ isafurankp k matrix Let us partition Ẽ as foows: Ẽ = ( Ẽ Ẽ m Ẽ r ) (10) is a

4 296 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) where Ẽ is a p p and Ẽ r is a p (k q) matrix By (6) Ẽ r must be fu coumn rank From Lemma 1 there exist a hyperexchange matrix H 1 and an orthogona P 2 P ˆ P 2 = I k q such that ˆF 0 H 1 (Y P 1 )P 2 = (11) 0 0 (n q) (k q) where ˆF is a q q nonsinguar upper trianguar matrix Let Ê = ẼP 2 = ( Ê Ê m Ẽ r ) (12) Consider now the matrix (Ê Ê m ) ˆF 1 From (5) and Theorem 1 there exist an orthogona ˆQ 2 and a hyperexchange Hˆ 2 such that ˆQ 2 (Ê Ê m ) ˆF 1 Hˆ 2 = (D 0) (13) where D = diag(d i ) and d p d p 1 d 1 > 0 Let ˆF 1 Hˆ 2 = Pˆ 3 F 1 (14) be the QR decomposition of ˆF 1 Hˆ 2 wherepˆ 3 is an orthogona and F is an upper trianguar matrix respectivey Then from (13) we must have that ˆQ 2 (Ê Ê m ) Pˆ 3 = (E E m ) (15) is a fu rank upper trapezoida matrix Hence (E E m )F 1 = (D 0) (16) By setting H ˆ H 2 = 2 0 ˆQ Q 0 I 2 = 2 0 P ˆ P n q 0 I 3 = 3 0 s p 0 I k q we see that H = H 1 2 H 1 Q = Q 2 Q 1 P = P 1 P 2 P 3 are the desired matrices in (7) Finay the matrix E r is given as E r = ˆQ 2 Ê r which competes the proof We wi ca the nonzero eements d i in (9) the generaized hyperboic singuar vaues of the pair (X Y ) Remark 1 The assumption (6) can be reaxed Indeed if rank(nu(x) nu(y )) = r then there exists an orthogona matrix G such that XG = (X 1 0 s r ) YG = (Y 1 0 k r ) and nu(x 1 ) nu(y 1 ) = 0 Theorem 2 can now be appied to matrices X 1 and Y 1

5 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) Symmetric definite generaized eigenprobem Let X R s k and Y R n k be two matrices as in Theorem 2 and et the assumption of Theorem 2 be satisfied Consider a definite symmetric eigenvaue probem in the foowing factored form: (X T X λy T ΦY)x = 0 (17) From the proof of Theorem 2 we find that (17) is equivaent to D T 0 E T r ( D 0 E r ) λ ( Iq q ) Iq q 0 Φ k ( FP T x) = where F 0 F = and Φ 0 I k = diag(φ i ) k i=1 k q From the reation (18) it foows that there are (k q) infinite eigenvaues λ and (q p) + (k q) = k p zero eigenvaues λ 0 = 0 Let e (r) i denote the ith coumn of the r r identity matrix Then the eigenvectors corresponding to zero eigenvaues are (18) x = P F 1 e (k) i i = p + 1q and x = P F 1 z i i = q + 1k (19) where D 1 E r e (k q) i q z i = 0 e (k q) i q Nonzero eigenvaues are aso easy to find in the case when q = k that is when the matrix F in Theorem 2 is fu rank Then there is no matrix E r in (18) Hence the nonzero eigenvaues and the corresponding eigenvectors are λ i = Φ i di 2 x i = P F 1 e (k) i for i = 1p We wi show now how one can find nonzero eigenvaues and the corresponding nonzero eigenvectors when k>q Let v = FP T x be an eigenvector in (18) From (18) note that eigenvectors corresponding to nonzero eigenvaues must have the foowing structure: a v = 0 (20) b

6 298 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) where a R p and b R k q These eigenvectors must satisfy the reationships D T (Da + E r b) λφ p a = 0 (21) Er T (Da + E rb) = 0 (22) where Φ p = diag(φ i ) p i=1 is the eading p p submatrix of Φ Let E r = UΣV T be the SVD of E r Then (21) and (22) can be rewritten as ( ) = λu T Φ 0 I p D 2 U (U T Da) p (k q) (23) b = VΣ U T Da (24) Define a p (p (k q)) matrix W by the reation U T Φ p D 2 0 UW = I p (k q) that is W = U T Φ p D 2 0 U I p (k q) Then the eigenvectors in (23) must be in the form U T Da = Wα for some α R p (k q) From (23) the vector α must satisfy the foowing (p (k q)) (p (k q)) eigenvaue probem: 0 Ip (k q) U T Φ p D 2 0 U α λα = 0 (25) I p (k q) Let 0 U r = U I p (k q) and consider the HSVD of DU r DU r = SΘZ T where S = (s 1 s p ) is a Φ p -orthogona matrix Θ = (θ) ij is a p (p (k q))diagona matrix with nonzero diagona eements θ ii andz = (z 1 z p (k q) ) is a (p (k q)) (p (k q)) orthogona matrix Then the eigenvaues λ i and the eigenvectors α i in (25) are given by α i = z i and λ i = Φ i θii 2 Hence vectors a i satisfying (23) must have the foowing form: a i = θ ii Φ p s i

7 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) The corresponding vectors v i in (20) wi then have the form I v i = 0 Φ p s i θ ii V Σ U T D and the eigenvectors of x i of (17) wi be I x i = P F 1 0 Φ p s i θ ii V Σ U T D Thus it is seen that the eigenvaues and eigenvectors of (17) can be determined without forming the expicit matrix products X T X and Y T ΦY if the GHSVD of X and Y andthehsvdofdu r are known As the GHSVD operates on the data matrices directy it is pausibe that the GHSVD can determine eigenvaues and eigenvectors of the penci (17) with higher accuracy than it woud have been possibe if the products were formed expicity 4 GHSVD Jacobi-ike agorithm From the proof of Theorem 2 it is cear that after initia transformation of X and Y to the forms (12) and (11) one needs to diagonaize the product (Ê Ê m ) ˆF 1 Without oss of generaity we can assume that p = q = k and that we are concerned with diagonaization of EF 1 wheree = (e ij ) and F = (f ij ) are two fu rank upper trianguar matrices However the inverse matrix F 1 and the product EF 1 shoud not be formed expicity This can be accompished by appying a two-sided Jacobi-ike procedure in a simiar fashion as it is done in computing the GSVD of two matrices [ ] In the GHSVD Jacobi-ike method the matrices E and F are subjected to a sequence of transformations E Q 1 EQ T 2 F H 3FQ T 2 where Q 1 Q 2 are orthogona rotations and H 3 is a generaized hyperboic rotation [6] so that the upper trianguar forms of E and F are preserved By preserving upper trianguar forms we can avoid expicit inversion of F [1516] In addition transformations Q 1 Q 2 and H 3 are chosen in such a way that a seected offdiagona eement in EF 1 is zeroed The order in which off-diagona eements are zeroed can be chosen in a variety of ways In our impementation of the method we chosen the so-caed odd even ordering of eimination described among others in [ ] or [21] In the odd-even scheme ony the eements ying on the first super-diagona are zeroed First in the odd cyce the eements in positions (1 2) (3 4)(n 1n) are zeroed Next in the even cyce the eements in positions (2 3) (4 5)(n 2n 1) are zeroed In each cyce transformations Q 1 Q 2 and H 3 are determined from 2 2 submatrices E ii+1 and F ii+1

8 300 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) eii e E ii+1 = ii+1 fii f and F 0 e ii+1 = ii+1 i+1i+1 0 f i+1i+1 of E and F respectivey The zeroing is foowed by suitabe permutations which move new eements to the first super-diagona Odd even cyces and corresponding permutations are repeated unti a off-diagona eements are zeroed exacty once This set of cyces is referred to as a sweep Sweeps are iterated unti EF 1 becomes numericay diagona The basic step of zeroing of a super-diagona eement is a computation of the GHSVD of two 2 2 upper trianguar matrices E ii+1 and F ii+1 [12811] This is simiar to the computation of the GSVD of two 2 2 presented in [18] and is described next The GHSVD of E ii+1 and F ii+1 is obtained in two steps First the HSVD of the product A = E ii+1 adj(f ii+1 ) is computed where adj denotes the adjoint of F ii+1 fi+1i+1 f adj(f ii+1 ) = ii+1 0 f ii If Q and H r are the orthogona and hyperexchange matrices respectivey from the HSVD decomposition of A then obviousy Q and H r are aso the orthogona and hyperexchange matrices from the HSVD of E ii+1 F 1 ii+1 This expains how one can avoid forming the inverse of F ii+1 in 2 2 subprobems Once Q and H r are determined the upper trianguar forms of the transformed matrices Q E ii+1 and adj(f ii+1 )H r are restored These two steps are impemented as foows Let A and A r denote the matrices E ii+1 and adj(f ii+1 ) respectivey Let a b A = ar b A 0 d r = r a b and A = = A 0 d r 0 d A r Our objective is to find a hyperexchange matrix H r and two orthogona matrices Q and Q m such that A = Q AH T r = a 0 0 d and ( A = Q A Q T a m = b ) ( 0 d A r = Q ma r Hr T a = r b r ) 0 d r Theorem 1 guarantees that this is possibe as ong ( ) 1 0 rank A A T = rank(a) (26) 0 1 We can compute the HSVD of A by a simpe modification of the haf-recursive tangent method deveoped in [1] for the product SVD computation In the haf-recursive tangent method we seek Q and H r in the form of a permuted refection

9 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) s c Q = c s where c 2 + s 2 = 1 and a generaized hyperboic rotation (which is a permuted hyperboic rotation) sc 1 H = c sh c = h 1 s c h s h c c where c 2 + s 2 = 1 respectivey The reason behind using permuted transformations is that we actuay dea with a k k probem The permutations that are incorporated into Q and H move new off-diagona eements to the first super diagona as needed in the odd even eimination scheme Foowing the exposition in [4] we consider the resut of appying the eft and right transformations Q and H r to a 2 2 upper trianguar matrix A: ( A = Q AHr T a b = ) T s c e d = a b sh c h (27) c s 0 d c h s h From (27) we can derive the reations: e = c c h ( at h + dt b) (28) b = c c h (at + dt h + bt t h ) (29) where t = s /c and t h = s h /c h The postuates that both e and b be zeros define two conditions on t and t h so that (27) represents the HSVD of A If one of t or t h is known the other can be computed from the first Simiary if a A = Q A Q T m = b T s c e d = a b sm c m (30) c s 0 d c m s m then e = c c m ( at m + dt b) where t m = s m /c m The postuate that e be zero defines a condition reating t to t m in such a way that if one is known the other can be computed so A is upper trianguar The postuate that e = 0andb = 0 in (28) and (29) eads either to an equation in t : where b = c c h ( bd a σ = a2 + d 2 b 2 2bd ) (t 2 + 2t σ 1)

10 302 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) or to an equation in t h : where b = c c h ( ab d ) (t 2 h + 2t hσ h + 1) σ h = d2 + b 2 + a 2 2ab If we assume that (26) hods and that E and F are fu rank we have that ad /= 0 This condition impies that c c h /= 0 and hence we get a quadratic equation in t t 2 + 2σ t 1 = 0 or in t h th 2 + 2σ ht h + 1 = 0 Our numerica experiments suggests that if a > d thent can be computed with higher accuracy than t h otherwiset h wi be more accurate than t Given t and t h one must compute t m Againif a > d t m shoud be computed from t Otherwise it shoud be computed from t h Thus the agorithm proceeds as foows Suppose that t has been computed first Then t m and t h are generated by the forward substitutions: t m = d t b t h = dt b a a On the other hand if t h has been computed first then t m and t are generated by the backward substitutions: t m = a rt h + b r t = at h + b d r d We used this dynamic criterion for computing the GHSVD of 2 2 subprobems in a experiments presented in Section 5 5 Numerica experiments A numerica experiments were conducted using MATLAB 6 on a De Precision workstation under RedHat Linux 71 operating system The experiments iustrate the numerica precision reaized in 2 2 GHSVD subprobems residua errors in n n GHSVD computations and residua errors in the symmetric definite generaized eigenvaue probems The first exampe iustrates that the right choice between the eft and the right transformation as the reference transformation in (27) from which a other transformations are computed is important Let the entries of A and A r be 10000e e + 04 A = e 04

11 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) e e + 04 A r = e 03 Then 10000e e + 09 A = e 07 Ceary the top diagona eement of A has greater magnitude than the bottom eement and hence we shoud compute t first and then compute t r and t m from t If we foow this strategy the resuts of the computation show that the transformed matrices A and A r remain numericay upper trianguar and A = Q AHr T is numericay diagona Q A Q T 99504e e 06 m = 27711e e + 05 Q m A r Hr T 30302e e + 03 = 63327e e e e 24 Q AH r = 54133e e + 09 If we instead decide to compute t r first and next compute t and t m from t r we obtain the foowing resuts: Q A Q T 11104e e + 04 m = 99935e e e e + 03 Q m A r H r = 13793e e e e + 09 Q AH r = 78813e e + 09 We can see that in this case neither A is numericay upper trianguar nor A is numericay diagona The second set of experiments iustrates the behavior of the Jacobi-ike method for computing the GHSVD of two n n matrices X and Y The matrices X and Y were created as products of random orthogona and upper trianguar matrices The weighting matrix Φ was taken as Φ = diag(( 1) i ) n i=1 The eements of trianguar matrices were generated by caing the MATLAB function rand Seected diagona eements of the trianguar matrices were changed to 10 i i =±1 ±2 ±3 ±4 to infuence condition numbers of the matrices Next the Jacobi-ike method described in Section 4 was appied to such created X and Y Residua errors res E and res F in the trianguarization of X and Y paraeism par EF of rows of the computed matrices E and F as we as the number of sweeps I performed by the Jacobi-ike method were recorded These quantities were defined as foows:

12 304 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) Tabe 1 Residua error in the Jacobi-ike GHSVD n cond(e) cond(f ) d 1 /d n I par EF res E res F maxres 10 13e e e e 16 29e 16 61e 16 59e 16 63e e e e 16 46e 16 10e 15 15e 13 63e e e e 16 43e 16 57e 16 91e 16 33e e e e 16 34e 16 52e 16 21e e e e e 16 10e 15 17e 15 89e 15 15e e e e 16 69e 16 13e 15 65e 13 15e e e e 16 13e 15 14e 15 51e 14 22e e e e 16 13e 15 27e 15 32e e e e e 16 28e 15 44e 15 57e 13 73e e e e 16 14e 15 18e 15 30e 12 73e e e e 16 87e 16 24e 15 63e 15 12e e e e 16 21e 15 16e 15 25e 08 res E = QXP E E and res F = HYP F F par EF = max i σ 2 ([E(i:) F(i:)]) σ 1 ([E(i:) F(i:)]) where σ j ([E(i:) F(i:)]) denotes the jth singuar vaue of the matrix composed from the ith row of E and the ith row of F Some representative resuts from this experiment are shown in Tabe 1 The ast coumn in Tabe 1 shows the maximum maxres of res E and res F when the dynamic criterion for soving 2 2 subprobems described at the end of Section 4 is repaced by a static criterion whereby one aways computes the right transformation first Entries in Tabe 1 indicate that the Jacobi-ike agorithm in this set of experiments produces the GHSVD decomposition with sma residua errors if 2 2 probems are soved by the dynamic criterion from Section 4 In the ast set of experiments the eigenvaues and the eigenvectors of the penci X T X λy T ΦY were computed using the GHSVD decomposition These were compared to the eigenvaues and the eigenvectors computed by the MATLAB qz agorithm The qz agorithm was appied to the penci X T X λy T ΦY as we as to the penci µx T X Y T ΦY The data matrices were generated as in the second set of experiments Let (y i λ i ) and (z i µ i ) denote the ith eigenvector/eigenvaue pair of the penci X T X λy T ΦY and µx T X Y T ΦY respectivey Let (z i e i /f i ) denote the ith eigenvector/eigenvaue pair computed by the GHSVD agorithm Here e i and f i are squares of the diagona eements of the upper trianguar matrices E and F and e i /f i

13 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) Tabe 2 Normaized residua errors in the generaized eigenvaue probem n cond(e) cond(f ) η 0 (x e f ) η 1 (y λ) η 2 (z µ) 10 13e e e 16 86e 16 29e 16 63e e e 17 81e 16* 32e 16* 63e e e 16 32e 16 47e 16* 33e e e 16 55e 16* 57e e e e 16 75e 15 32e 16 15e e e 13 89e 15* 35e 15* 15e e e 16 50e 16 18e 16* 22e e e 16 82e 16* 68e e e e 15 11e 14 72e 15 73e e e 13 89e 15* 24e 10* 73e e e 16 14e 15 63e 07* 12e e e 14 11e 15* 89e 11 is the (generaized) eigenvaue of X T X λy T ΦY Foowing [17] we introduce the normaized residua errors: Let η 0 (x i e i f i ) = η 1 (y i λ i ) = η 2 (z i µ i ) = (f i X T X e i Y T ΦY)x i ( f i X T X + e i Y T ΦY ) x i (X T X λ i Y T ΦY)y i ( X T X + λ i Y T ΦY ) y i (µ i X T X Y T ΦY)z i ( µ i X T X + Y T ΦY ) z i η 0 (xef)= max η 0 (x i e i f i ) i η 1 (y λ) = max η 1 (y λ i ) i η 2 (z µ) = max η 1 (y µ i ) i denote the maxima normaized residua errors for a given pair X and Y The performance of the three methods is iustrated in Tabe 2 An asterisk next to an entry indicates that the qz method approximated one or more of finite eigenvaues by or when signs of eigenvaues with sma magnitudes were not correct In such cases the corresponding normaized residua errors were not cacuated In two cases the performance of the GHSVD method was noticeaby worse than the performance of the qz methods However in these cases the qz methods either decared some of the eigenvaues as or faied to compute the right sign of sma magnitude eigenvaues

14 306 AW Bojanczyk / Linear Agebra and its Appications 358 (2003) Remarks In this paper we have proposed a generaization of the hyperboic singuar decomposition of a singe matrix to a simutaneous decomposition of two matrices This generaized decomposition can be used for computing eigenvaues and eigenvectors of a symmetric definite matrix penci given in a product form (17) In Theorem 2 we have isted sufficient condition under which the GHSVD exists It is of interest to reax some of the assumptions of Theorem 2 In particuar assumptions (4) and (5) appear to be unnecessary and coud be removed However under reaxed assumptions the form of Theorem 2 may somewhat change There is a strong numerica evidence that the accuracy of the proposed GHSVD- Jacobi method depends on how the GHSVD of 2 2 subprobems are computed In particuar it is evident that the magnitudes of the diagona eements a and d in the 2 2 subprobem (27) infuence whether to compute the eft or the right transformation as the reference transformation However ony a rigorous round-off error anaysis ike the ones presented in [12] can determine whether the criterion based on the magnitude of diagona eements of 2 2 subprobems is sufficient for numerica stabiity of the method A reated question is the sensitivity of the GHSVD decomposition to changes in data Some resuts for a reated hyperboic eigenvaue probem were obtained in [2324] It might be possibe to extend these resuts to the GHSVD decomposition Another issue not expicity addressed in this note is the convergence of the Jacobi method when appied to the computation of the GHSVD Convergence resuts for a reated J-symmetric Jacobi method were presented in [1425] and shoud carry over to the case of the GHSVD Acknowedgement The author wants to thank referees for their usefu comments References [1] GE Adams AW Bojanczyk FT Luk Computing the PSVD of two 2 2 trianguar matrices SIAM J Matrix Ana App 15 (2) (1994) [2] Z Bai JW Demme Computing the generaized singuar vaue decomposition SIAM J Sci Comput 14 (6) (1993) [3] AW Bojanczyk Hyperboic agorithms for indefinite east squares probems CSL Technica Reports Schoo of Eectrica and Computer Engineering Corne University January 2001 [4] AW Bojanczyk LM Ewerbring FT Luk P Van Dooren An accurate product SVD agorithm Signa Process 25 (1991) [5] AW Bojanczyk R Onn AO Steinhardt Existence of the hyperboic singuar vaue decomposition Linear Agebra App 185 (1993) 21 30

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