DGGSVD
Purpose
DGGSVD computes the generalized singular value decomposition (GSVD)
of an M-by-N real matrix A and P-by-N real matrix B:
U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R )
where U, V and Q are orthogonal matrices.
Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
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 and of the
following structures, respectively:
If M-K-L >= 0,
K L
D1 = K ( I 0 )
L ( 0 C )
M-K-L ( 0 0 )
K L
D2 = L ( 0 S )
P-L ( 0 0 )
N-K-L K L
( 0 R ) = K ( 0 R11 R12 )
L ( 0 0 R22 )
where
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1), ... , BETA(K+L) ),
C**2 + S**2 = I.
R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L < 0,
K M-K K+L-M
D1 = K ( I 0 0 )
M-K ( 0 C 0 )
K M-K K+L-M
D2 = M-K ( 0 S 0 )
K+L-M ( 0 0 I )
P-L ( 0 0 0 )
N-K-L K M-K K+L-M
( 0 R ) = K ( 0 R11 R12 R13 )
M-K ( 0 0 R22 R23 )
K+L-M ( 0 0 0 R33 )
where
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1), ... , BETA(M) ),
C**2 + S**2 = I.
(R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
( 0 R22 R23 )
in B(M-K+1:L,N+M-K-L+1:N) on exit.
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 A*inv(B):
A*inv(B) = U*(D1*inv(D2))*V**T.
If ( A**T,B**T)**T 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:
A**T*A x = lambda* B**T*B x.
In some literature, the GSVD of A and B is presented in the form
U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, 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 inv(R) ).
of an M-by-N real matrix A and P-by-N real matrix B:
U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R )
where U, V and Q are orthogonal matrices.
Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
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 and of the
following structures, respectively:
If M-K-L >= 0,
K L
D1 = K ( I 0 )
L ( 0 C )
M-K-L ( 0 0 )
K L
D2 = L ( 0 S )
P-L ( 0 0 )
N-K-L K L
( 0 R ) = K ( 0 R11 R12 )
L ( 0 0 R22 )
where
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1), ... , BETA(K+L) ),
C**2 + S**2 = I.
R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L < 0,
K M-K K+L-M
D1 = K ( I 0 0 )
M-K ( 0 C 0 )
K M-K K+L-M
D2 = M-K ( 0 S 0 )
K+L-M ( 0 0 I )
P-L ( 0 0 0 )
N-K-L K M-K K+L-M
( 0 R ) = K ( 0 R11 R12 R13 )
M-K ( 0 0 R22 R23 )
K+L-M ( 0 0 0 R33 )
where
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1), ... , BETA(M) ),
C**2 + S**2 = I.
(R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
( 0 R22 R23 )
in B(M-K+1:L,N+M-K-L+1:N) on exit.
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 A*inv(B):
A*inv(B) = U*(D1*inv(D2))*V**T.
If ( A**T,B**T)**T 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:
A**T*A x = lambda* B**T*B x.
In some literature, the GSVD of A and B is presented in the form
U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, 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 inv(R) ).
Arguments
| JOBU |
(input) CHARACTER*1
= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed. |
| JOBV |
(input) CHARACTER*1
= 'V': Orthogonal matrix V is computed;
= 'N': V is not computed. |
| JOBQ |
(input) CHARACTER*1
= 'Q': Orthogonal matrix Q is computed;
= 'N': Q is not computed. |
| M |
(input) INTEGER
The number of rows of the matrix A. M >= 0.
|
| N |
(input) INTEGER
The number of columns of the matrices A and B. N >= 0.
|
| P |
(input) INTEGER
The number of rows of the matrix B. P >= 0.
|
| K |
(output) INTEGER
|
| L |
(output) INTEGER
On exit, K and L specify the dimension of the subblocks
described in the Purpose section. K + L = effective numerical rank of (A**T,B**T)**T. |
| A |
(input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular matrix R, or part of R. See Purpose for details. |
| LDA |
(input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
|
| B |
(input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix R if M-K-L < 0. See Purpose for details. |
| LDB |
(input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
|
| ALPHA |
(output) DOUBLE PRECISION array, dimension (N)
|
| BETA |
(output) DOUBLE PRECISION array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B; ALPHA(1:K) = 1, BETA(1:K) = 0, and if M-K-L >= 0, ALPHA(K+1:K+L) = C, BETA(K+1:K+L) = S, or if M-K-L < 0, ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 BETA(K+1:M) =S, BETA(M+1:K+L) =1 and ALPHA(K+L+1:N) = 0 BETA(K+L+1:N) = 0 |
| U |
(output) DOUBLE PRECISION array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
If JOBU = 'N', U is not referenced. |
| LDU |
(input) INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise. |
| V |
(output) DOUBLE PRECISION array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
If JOBV = 'N', V is not referenced. |
| LDV |
(input) INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise. |
| Q |
(output) DOUBLE PRECISION array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced. |
| LDQ |
(input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise. |
| WORK |
(workspace) DOUBLE PRECISION array,
dimension (max(3*N,M,P)+N)
|
| IWORK |
(workspace/output) INTEGER array, dimension (N)
On exit, IWORK 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 ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). |
| INFO |
(output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, the Jacobi-type procedure failed to converge. For further details, see subroutine DTGSJA. |
Internal Parameters
TOLA DOUBLE PRECISION
TOLB DOUBLE PRECISION
TOLA and TOLB are the thresholds to determine the effective
rank of (A',B')**T. Generally, they are set to
TOLA = MAX(M,N)*norm(A)*MAZHEPS,
TOLB = MAX(P,N)*norm(B)*MAZHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.
TOLB DOUBLE PRECISION
TOLA and TOLB are the thresholds to determine the effective
rank of (A',B')**T. Generally, they are set to
TOLA = MAX(M,N)*norm(A)*MAZHEPS,
TOLB = MAX(P,N)*norm(B)*MAZHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.
Further Details
2-96 Based on modifications by
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA