SGGSVP
Purpose
SGGSVP computes orthogonal matrices U, V and Q such that
N-K-L K L
U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
V**T*B*Q = L ( 0 0 B13 )
P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
SGGSVD.
N-K-L K L
U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
V**T*B*Q = L ( 0 0 B13 )
P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
SGGSVD.
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.
|
| P |
(input) INTEGER
The number of rows of the matrix B. P >= 0.
|
| N |
(input) INTEGER
The number of columns of the matrices A and B. N >= 0.
|
| A |
(input/output) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular (or trapezoidal) matrix described in the Purpose section. |
| LDA |
(input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
|
| B |
(input/output) REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix described in the Purpose section. |
| LDB |
(input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
|
| TOLA |
(input) REAL
|
| TOLB |
(input) REAL
TOLA and TOLB are the thresholds to determine the effective
numerical rank of matrix B and a subblock of A. Generally, they are set to TOLA = MAX(M,N)*norm(A)*MACHEPS, TOLB = MAX(P,N)*norm(B)*MACHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decomposition. |
| K |
(output) INTEGER
|
| L |
(output) INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose section. K + L = effective numerical rank of (A**T,B**T)**T. |
| U |
(output) REAL array, dimension (LDU,M)
If JOBU = 'U', U contains the 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) REAL array, dimension (LDV,P)
If JOBV = 'V', V contains the 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) REAL array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the 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. |
| IWORK |
(workspace) INTEGER array, dimension (N)
|
| TAU |
(workspace) REAL array, dimension (N)
|
| WORK |
(workspace) REAL array, dimension (max(3*N,M,P))
|
| INFO |
(output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value. |
Further Details
The subroutine uses LAPACK subroutine SGEQPF for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix. It may be replaced by a better rank determination strategy.
with column pivoting to detect the effective numerical rank of the
a matrix. It may be replaced by a better rank determination strategy.