F08VAF (DGGSVD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08VAF (DGGSVD)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08VAF (DGGSVD) computes the generalized singular value decomposition (GSVD) of an m by n real matrix A and a p by n real matrix B.

2  Specification

SUBROUTINE F08VAF ( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK, INFO)
INTEGER  M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, IWORK(N), INFO
REAL (KIND=nag_wp)  A(LDA,*), B(LDB,*), ALPHA(N), BETA(N), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(max(3*N,M,P)+N)
CHARACTER(1)  JOBU, JOBV, JOBQ
The routine may be called by its LAPACK name dggsvd.

3  Description

The generalized singular value decomposition is given by
UT A Q = D1 0 R ,   VT B Q = D2 0 R ,
where U, V and Q are orthogonal matrices. Let k+l be the effective numerical rank of the matrix A B , then R is a k+l by k+l nonsingular upper triangular matrix, D1 and D2 are m by k+l and p by k+l ‘diagonal’ matrices structured as follows:
if m-k-l0,
D1= klk(I0) l 0 C m-k-l 0 0
D2= kll(0S) p-l 0 0
0R = n-k-lklk(0R11R12) l 0 0 R22
where
C = diagαk+1,,αk+l ,
S = diagβk+1,,βk+l ,
and
C2 + S2 = I .
R is stored as a submatrix of A with elements Rij stored as Ai,n-k-l+j on exit.
If m-k-l<0 ,
D1= km-kk+l-mk(I00) m-k 0 C 0
D2= km-kk+l-mm-k(0S0) k+l-m 0 0 I p-l 0 0 0
0R = n-k-lkm-kk+l-mk(0R11R12R13) m-k 0 0 R22 R23 k+l-m 0 0 0 R33
where
C = diagαk+1,,αm ,
S = diagβk+1,,βm ,
and
C2 + S2 = I .
R11 R12 R13 0 R22 R23  is stored as a submatrix of A with Rij stored as Ai,n-k-l+j, and R33  is stored as a submatrix of B with R33ij stored as Bm-k+i,n+m-k-l+j.
The routine computes C, S, R and, optionally, the orthogonal transformation matrices U, V and Q.
In particular, if B is an n by n nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of AB-1:
A B-1 = U D1 D2-1 VT .
If A B  has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:
AT Ax=λ BT Bx .
In some literature, the GSVD of A and B is presented in the form
UT A X = 0D1 ,   VT B X = 0D2 ,
where U and V are orthogonal and X is nonsingular, and D1 and D2 are ‘diagonal’. The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as
X = Q I 0 0 R-1 .

4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Parameters

1:     JOBU – CHARACTER(1)Input
On entry: if JOBU='U', the orthogonal matrix U is computed.
If JOBU='N', U is not computed.
Constraint: JOBU='U' or 'N'.
2:     JOBV – CHARACTER(1)Input
On entry: if JOBV='V', the orthogonal matrix V is computed.
If JOBV='N', V is not computed.
Constraint: JOBV='V' or 'N'.
3:     JOBQ – CHARACTER(1)Input
On entry: if JOBQ='Q', the orthogonal matrix Q is computed.
If JOBQ='N', Q is not computed.
Constraint: JOBQ='Q' or 'N'.
4:     M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint: M0.
5:     N – INTEGERInput
On entry: n, the number of columns of the matrices A and B.
Constraint: N0.
6:     P – INTEGERInput
On entry: p, the number of rows of the matrix B.
Constraint: P0.
7:     K – INTEGEROutput
8:     L – INTEGEROutput
On exit: K and L specify the dimension of the subblocks k and l as described in Section 3; k+l is the effective numerical rank of AB.
9:     A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: the m by n matrix A.
On exit: contains the triangular matrix R, or part of R. See Section 3 for details.
10:   LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08VAF (DGGSVD) is called.
Constraint: LDAmax1,M.
11:   B(LDB,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least max1,N.
On entry: the p by n matrix B.
On exit: contains the triangular matrix R if m-k-l<0. See Section 3 for details.
12:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08VAF (DGGSVD) is called.
Constraint: LDBmax1,P.
13:   ALPHA(N) – REAL (KIND=nag_wp) arrayOutput
On exit: see the description of BETA.
14:   BETA(N) – REAL (KIND=nag_wp) arrayOutput
On exit: ALPHA and BETA contain the generalized singular value pairs of A and B, αi  and βi ;
  • ALPHA1:K = 1 ,
  • BETA1:K = 0 ,
and if m-k-l0 ,
  • ALPHAK+1:K+L = C ,
  • BETAK+1:K+L = S ,
or if m-k-l<0 ,
  • ALPHAK+1:M = C ,
  • ALPHAM+1:K+L = 0 ,
  • BETAK+1:M = S ,
  • BETAM+1:K+L = 1 , and
  • ALPHAK+L+1:N = 0 ,
  • BETAK+L+1:N = 0 .
The notation ALPHAK:N above refers to consecutive elements ALPHAi, for i=K,,N.
15:   U(LDU,*) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array U must be at least max1,M if JOBU='U', and at least 1 otherwise.
On exit: if JOBU='U', U contains the m by m orthogonal matrix U.
If JOBU='N', U is not referenced.
16:   LDU – INTEGERInput
On entry: the first dimension of the array U as declared in the (sub)program from which F08VAF (DGGSVD) is called.
Constraints:
  • if JOBU='U', LDUmax1,M;
  • otherwise LDU1.
17:   V(LDV,*) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array V must be at least max1,P if JOBV='V', and at least 1 otherwise.
On exit: if JOBV='V', V contains the p by p orthogonal matrix V.
If JOBV='N', V is not referenced.
18:   LDV – INTEGERInput
On entry: the first dimension of the array V as declared in the (sub)program from which F08VAF (DGGSVD) is called.
Constraints:
  • if JOBV='V', LDVmax1,P;
  • otherwise LDV1.
19:   Q(LDQ,*) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Q must be at least max1,N if JOBQ='Q', and at least 1 otherwise.
On exit: if JOBQ='Q', Q contains the n by n orthogonal matrix Q.
If JOBQ='N', Q is not referenced.
20:   LDQ – INTEGERInput
On entry: the first dimension of the array Q as declared in the (sub)program from which F08VAF (DGGSVD) is called.
Constraints:
  • if JOBQ='Q', LDQmax1,N;
  • otherwise LDQ1.
21:   WORK(max3×N,M,P+N) – REAL (KIND=nag_wp) arrayWorkspace
22:   IWORK(N) – INTEGER arrayOutput
On exit: stores the sorting information. More precisely, the following loop will sort ALPHA
for I=K+1, min(M,K+L) 
swap ALPHA(I) and ALPHA(IWORK(I)) 
endfor 
such that ALPHA1ALPHA2ALPHAN.
23:   INFO – INTEGEROutput

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO=1
If INFO=1, the Jacobi-type procedure failed to converge.

7  Accuracy

The computed generalized singular value decomposition is nearly the exact generalized singular value decomposition for nearby matrices A+E  and B+F , where
E2 = Oε A2 ​ and ​ F2 = Oε B2 ,
and ε  is the machine precision. See Section 4.12 of
Anderson et al. (1999) for further details.

8  Further Comments

The complex analogue of this routine is F08VNF (ZGGSVD).

9  Example

This example finds the generalized singular value decomposition
A = U Σ1 0R QT ,   B = V Σ2 0R QT ,
where
A = 1 2 3 3 2 1 4 5 6 7 8 8   and   B = -2 -3 3 4 6 5 ,
together with estimates for the condition number of R and the error bound for the computed generalized singular values.
The example program assumes that mn, and would need slight modification if this is not the case.

9.1  Program Text

Program Text (f08vafe.f90)

9.2  Program Data

Program Data (f08vafe.d)

9.3  Program Results

Program Results (f08vafe.r)


F08VAF (DGGSVD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011