C. C. Paige and M. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18(1981), 398-405. Home/Search Document Not in Database Summary Related Articles Check |
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The Generalized Eigenvalue Problem for Non-Square.. - Boutry, Elad, Golub.. (Correct)
....which serves two di#erent objectives: i) the iterative algorithm requires less computations, and (ii) a bound is obtained on the error of the squaring initialization method. The proposed algorithm is a combination of the QR transformation and the GSV D, due to Van Loan (cf. page 465 in [12] and [16]) A similar decomposition was given in [14, 15] 6.1 The Factorization We start our description with the assumption that m 2n (there are at least twice as many rows as there are columns in the pencil) The general case will be treated later. Theorem 7: Consider a rectangular matrix pencil, ....
C.C. Paige and M.A. Saunders, Toward a generalized singular value decomposition, SIAM Journal on Numerical Analysis, Vol. 18, pp. 398--405, 1981.
Extrapolation Techniques for Ill-Conditioned Linear.. - Brezinski.. (1998) (Correct)
....prove [13] that (3) 4) Now, if the matrix H is rectangular with fewer rows than columns (i.e. q p) we first transform it into a square matrix by adding rows of zeros. The generalized singular value decomposition (GSVD) can then be applied to the pair (A; H) It consists [13, 23, 24, 33] of finding two orthogonal matrices U and V and a nonsingular (but not unique) matrix Z such that AZ = DA and V HZ = DH where DA and DH are diagonal matrices with nonnegative entries. Setting fi = Z x and fl = U b, we have J( fi) kDA fi Gamma flk kDH fik and it follows that ....
C.C. Paige, M.A. Saunders, Towards a Generalized Singular Value Decomposition, SIAM J. Numer. Anal., 18 (1981) 398--405.
Updating a Generalized URV Decomposition - Stewart, Van Dooren (2000) (2 citations) (Correct)
....is required to be orthogonal, then the best that can be done is to make #A and #B triangular. An appropriate choice of orthogonal X, VA , and VB guarantees that # 1 A #B will be diagonal. More generally, when A and B are possibly rank deficient m n a and m n b matrices, the generalized SVD [10, 13] has been defined by X 1 AV 1 = # #A 0 # r m r , X 1 BV 2 = # #B 0 # r m r , 1.1) where #A = # # I A SA 0A # # , #B = # # 0B SB I B # # # Received by the editors April 23, 1997; accepted for publication (in revised form) by L. Elden September 11, 1998; published ....
....reveals the singular values of A B, where A denotes the pseudoinverse of A. If A is rank deficient, then the decomposition reveals singular values associated with a quotient formed from the B weighted pseudoinverse of A [4, 2] An early development of the generalized SVD was given in [10]. A general description suitable for adaptation to a URV decomposition is as follows: the m (n a n b ) matrix # A B # is decomposed as U T # A # B # V = U T # A # B # # V 1 0 0 V 2 # = # # R 11 0 0 0 0 0 # # # # # # # # # # # # S 13 R 14 0 R 23 0 0 0 0 0 # # , 1.2) ....
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C. C. Paige and M. A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18 (1981), pp. 398--405.
Optimal Perturbation Bounds for the Hermitian Eigenvalue Problem - Barlow, al. (1993) (2 citations) (Correct)
....orthonormal rows and orthonormal nontrivial columns. That is, columns of U(i) which correspond to i 2 S are orthonormal, and columns of V (i) for which j i (i) 1 are orthonormal. Note that the form (3. 7) describes the quotient singular value decomposition (QSVD) of the pair (C(i; S) G(i; S) [15, 16]. The G(i; S) weighted pseudoinverse of C(i; S) 3, 8] is given by C y G (i; S) j X(i ) Phi y (i; S)U (i) 3.8) Likewise, the C(i; S) weighted psuedoinverse of G(i; S) is G y C (i; S) j X(i)J y (i; S)V (i) Using this structure, we can establish bounds on all of the eigenvalues ....
C.C. Paige and M.A. Saunders, Towards a generalized singular value decomposition, SIAM J. Num. Anal., 18:398--405, 1981.
Computing the Generalized Singular Value Decomposition - Bai, Demmel (1991) (10 citations) (Correct)
....and demonstrate it using examples on which all previous algorithms fail. 1 Introduction The purpose of this paper is to describe a variation of Paige s algorithm [28] for computing the following generalized singular value decomposition (GSVD) introduced by Van Loan [33] and Paige and Saunders [25]. This is also called the quotient singular value decomposition (QSVD) in [8] Theorem 1.1 Let A 2 IR m Thetan and B 2 IR p Thetan have rank(A T ; B T ) n. 1 Then there are orthogonal matrices U , V and Q such that U T AQ = Sigma 1 R; V T BQ = Sigma 2 R; 1.1) where R is a n ....
C. C. Paige and M. A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal. 18:398--405(1981).
Computing the Singular Values of the Product of two.. - Mollar, Hernández (Correct)
....AB are the products ff i fi i , for i = 1; n. The PSVD was introduced by Fernando and Hammarling [5] as a new generalization of the SVD, based on the product AB T . It complements the generalized singular value decomposition (GSVD) introduced by Van Loan [16] and Paige and Sanders [14], now called quotient SVD (QSVD) as proposed in [3] The parallel computation of the GSVD has been treated in [1, 2] A complete study of the properties of the PSVD can be found in [4] In this paper we present a parallel algorithm to compute the product singular values of (A; B) and we start ....
Paige, C. and Sanders, M.: Towards a generalized singular value decomposition. SIAM J. Numer. Anal., 18 pp., 398-405, (1981).
A Completely Rank Revealing Quotient URV Decomposition - Stewart (1998) (Correct)
....cation problem. 1 Introduction Let A and B be m a n and m b n matrices. De ne C = A B # : If r a = rank(A) r b : rank(B) r c = rank(C) then the dimension of the intersection of the row subspaces of A and B is r i = r a r b r c : 1) The quotient (or generalized) SVD (QSVD) [10, 12], is a decomposition that reveals these ranks. It has been de ned as U T AQ = h AR 0 i r n r V T BQ = h BR 0 i r n r (2) where A = 2 6 4 I A SA 0 A 3 7 5 B = 2 6 4 0 B SB I B 3 7 5 with square, diagonal, positive de nite SA and SB typically chosen, by appropriate ....
C. C. Paige and M. A. Saunders. Toward a generalized singular value decomposition. SIAM Journal on Numerical Analysis, 18:398-405, 1981.
Geometric Methods in Stochastic Realization and System.. - Picci (1996) (Correct)
....space at time k. This subspace in turn can be computed as the intersection of the extended future Y k = Y k U k with the past space of u at time k. Below we give an algorithm to compute a well conditioned basis in the intersection Y k U Gamma k , based on the GSVD [48, 64]. Note that in order to get the right constant parameters in the realization, the basis at time k 1 should be chosen so as to correspond to the stationarily time shifted state E[x u (k 1)j U ] x u (k 1) oex u (k) otherwise time varying matrices A u ; B u ; C u ; D u are obtained. This means ....
.... 3 5 = L 11 0 L 21 L 22 Q 0 1 Q 0 2 where Q 0 2 is obtained by stacking the rows of Q 0 2 over those of Q 0 4 . We have deleted the subscript k. We shall assume that [ L 21 L 22 ] has full row rank. From the theory of Generalized Singular Valued Decomposition, [48]) there exist orthogonal matrices V; Z and a nonsingular X of appropriate dimensions, such that L 21 = X C V 0 = X diag(c 1 ; Delta Delta Delta ; c )V 0 ; 1 c 1 Delta Delta Delta c 0 L 22 = X S Z 0 = Xdiag(s 1 ; Delta Delta Delta ; s )Z 0 ; 1 s Delta Delta Delta s ....
C.C. Paige and M.A. Saunders (1981). Towards a generalized Singular Value Decomposition, SIAM Journal on Numerical Analysis, 18, pp. 398-- 405.
The CSD, GSVD, their Applications and Computations - Bai (1992) (Correct)
....oe 2 B T B) 0g: Van Loan shows that oe(A; B) can obtained by factorizing A and B into the products of an orthogonal matrix, a diagonal matrix and a nonsingular matrix, respectively. In particular, if B is the identity matrix, the decomposition gives the SVD of A. In 1981, Paige and Saunders [40] described a more general formulation of Van Loan s BSVD, which they called the generalized singular value decomposition (GSVD) Under their formulation, the CSD can be regarded as a special case of the GSVD. The assessment of the conditioning of the decompositions comes from the perturbation ....
....0 l S C 0 n Gamma 2l 0 0 I 1 C A where the diagonal matrices C and S satisfy (2.2) in Theorem 2.1. Armed with the CSD of a partitioned orthonormal matrix, we may now have the following GSVD for any two matrices A and B having the same number of columns: Theorem 2. 3 (GSVD Triangular Form [40]) If A 2 IR m Thetan and B 2 IR p Thetan , with rank( A T ; B T ) T = k, then there are orthogonal matrices U 2 IR m Thetam , V 2 IR p Thetap and Q 2 IR n Thetan such that U T 0 0 V T A B Q = Sigma 1 Sigma 2 ( R; 0 ) 2.3) where R is a k Theta k ....
C. C. Paige and M. A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal. 18, (1981), pp.398--405.
On the QR algorithm and updating the SVD and URV.. - Moonen, Van Dooren, al. (1993) (13 citations) (Correct)
....is given as follows A = QA 1 UA 1 (6AR) 1 Q T z RA B = QB 1 UB 1 (6BR) 1 Q T z RB reveals the SVD of AR 01 B in an implicit way. Here 6A and 6B are diagonal matrices, R is upper triangular, and U T A UA = U T B UB = Q T Q = I . For details, the reader is referred to [17]. Starting from the square triangular factors RA and RB , the QSVD may be computed with an iterative procedure, similar to the SVD procedure : R 1 ( RA R 2 ( RB UA ( I UB ( I Q ( I for k = 1; 1 for i = 1; m0 1 2 6 6 6 6 6 6 6 6 4 R 1 ( UA T [i;k] 1 R 1 1 ....
C.C. Paige and M. Saunders. `Towards a generalized singular value decomposition'. SIAM J. Numer. Anal., Vol. 18, No. 3, pp 398-405, 1981.
Optimal Perturbation Bounds For The Hermitian Eigenvalue Problem - Barlow, Slapnicar (1996) (2 citations) (Correct)
....of a matrix A 2 C m Thetan is given by A = U SigmaV ; Sigma = k n Gamma k k Psi 0 m Gamma k 0 0 ; k = rank(A) where U 2 C m Thetam and V 2 C n Thetan are unitary, and Psi = diag(oe 1 ; oe k ) 2 R m Thetan is nonnegative. From Van Loan [13] and Paige and Saunders [14], the generalized quotient SVD (QSVD) of the matrix pair (A; B) A 2 C m Thetan , B 2 C p Thetan is given by A = U Sigma AX Gamma1 ; Sigma A = 0 s 1 s 2 s 3 s 4 s 1 Psi A 0 0 0 s 2 0 I s2 0 0 t 1 0 0 0 0 1 A ; s 1 s 2 = rank(A) B = V Sigma BX Gamma1 ; Sigma B = 0 B B ....
C.C. Paige and M.A. Saunders. Towards a generalized singular value decomposition. SIAM J. Num. Anal., 18:398--405, 1981.
On a Variational Formulation of QSVD and RSVD - Chu, De Moor (1998) (Correct)
.... y ra # , then U 1 = h x 1 kx1k Delta Delta Delta xra kxra k i : Moreover, if n = m = r a , then V 1 = h y1 ky1 k Delta Delta Delta yra kyra k i : Recently, in Chu, Funderlic and Golub [1] Theorem 1 has been generalized to the Quotient Singular Value Decomposition (QSVD) [3, 5, 6, 7, 8, 9, 10, 11, 13, 14] of two matrices A 2 R n Thetam ; C 2 R p Thetam based on the relationship between QSVD of two matrix A; C and the eigendecomposition of the matrix pencil (A T A; C T C) The purposes of this paper are twofold. Firstly, we present an alternative derivation of the variational formulation ....
....c Gamma r abc ; k 3 = r ac r b Gamma r abc ; k 4 = r a r abc Gamma r ab Gamma r ac : 2 A Variational Formulation for QSVD Nowadays, several generalizations of the OSVD have been proposed and analysed. One that is well known is the generalized SVD as introduced by Paige and Saunders in [5], which was proposed by De Moor and Golub [11] to rename as the QSVD. Another one is the RSVD, introduced in its explicit form by Zha [16] and further developed and discussed by De Moor and Golub [8] In this section we will give an alternative proof for the variational formulation for the QSVD of ....
C.C. Paige and M.A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18:398--405, 1981.
A Continuous Jacobi-like Approach to the - Simultaneous Reduction Of (Correct)
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C. C. Paige and M. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18(1981), 398-405.
On A Variational Formulation Of The Generalized - Singular Value Decomposition (Correct)
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C. C. Paige and M. A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18(1981), 398-405.
On the Error Analysis and Implementation of Some Eigenvalue.. - Ren (1996) (Correct)
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C. Paige and M. Saunders. Towards a generalized singular value decomposition. SIAM J. Num. Anal., 15:241--256, 1981.
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C. Paige and M.A.Saunders. Towards a generalized singular value decomposition. SIAM Journal on Numerical Analysis, 18:398--405, 1981.
A New Optimization Criterion for Generalized.. - Ye, Janardan, Park, Park (2003) (Correct)
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C.C. Paige, and M.A.Saunders. Towards a generalized singular value decomposition, SIAM Journal on Numerical Analysis. 18, pp. 398--405, 1981.
An Optimization Criterion for Generalized Discriminant.. - Ye, Janardan, Park, Park (Correct)
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C.C. Paige, and M.A.Saunders. Towards a generalized singular value decomposition, SIAM J. Numerical Analysis. 18, pp. 398--405, 1981.
The Generalized Eigenvalue Problem for Non-Square.. - Boutry, Elad, Golub.. (2004) (Correct)
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C.C. Paige and M.A. Saunders, Toward a generalized singular value decomposition, SIAM Journal on Numerical Analysis, Vol. 18, pp. 398-- 405, 1981.
On the Computation of the Restricted Singular Value.. - Chu, De Lathauwer, De.. (2000) (Correct)
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C.C. Paige and M.A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18 (1981), pp. 398-405.
Signal Subspace Methods for Speech Enhancement - Hansen (1997) (10 citations) (Correct)
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C. C. Paige and M. A. Saunders. Toward a Generalized Singular Value Decomposition. SIAM Journal on Numerical Analysis, 18:398--405, 1981.
The Generalized Eigenvalue Problem for Non-Square.. - Boutry, Elad, Golub.. (Correct)
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C.C. Paige and M.A. Saunders, Toward a generalized singular value decomposition, SIAM Journal on Numerical Analysis, Vol. 18, pp. 398--405, 1981.
Subspace Algorithms for the Stochastic Identification Problem - Van Overschee, De Moor (20 citations) (Correct)
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Paige C.C., Saunders M.A. Towards a generalized singular value decomposition. SIAM J. Numer. Anal., 18, pp. 398-405, (1981).
Relative Perturbation Theory for Matrix Spectral Decompositions - Truhar (2000) (Correct)
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C. C. Paige amd M. A. Saunders, Towards a generalized singular value decomposition, SIAM J. Num. Anal., 18:398--405 (1981).
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