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NAME
sggsvd - compute the generalized singular value decomposi-
tion (GSVD) of an M-by-N real matrix A and P-by-N real
matrix B
SYNOPSIS
SUBROUTINE SGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA,
B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
IWORK, INFO )
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
INTEGER IWORK( * )
REAL A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q(
LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )
#include <sunperf.h>
void sggsvd(char jobu, char jobv, char jobq, int m, int n,
int p, int *k, int *l, float *sa, int lda, float
*sb, int ldb, float *salpha, float *sbeta, float
*su, int ldu, float *v, int ldv, float *q, int
ldq, int *info);
PURPOSE
SGGSVD computes the generalized singular value decomposition
(GSVD) of an M-by-N real matrix A and P-by-N real matrix B:
U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R )
where U, V and Q are orthogonal matrices, and Z' is the
transpose of Z. Let K+L = 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 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'.
If ( A',B')' has orthonormal columns, then the GSVD of A and
B is also equal to the CS decomposition of A and B. Further-
more, the GSVD can be used to derive the solution of the
eigenvalue problem:
A'*A x = lambda* B'*B x.
In some literature, the GSVD of A and B is presented in the
form
U'*A*X = ( 0 D1 ), V'*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',B')'.
A (input/output) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, A con-
tains 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) REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B. On exit, B con-
tains the triangular matrix R if M-K-L < 0. See
Purpose for details.
LDB (input) INTEGER
The leading dimension of the array B. LDA >=
max(1,P).
ALPHA (output) REAL array, dimension (N)
BETA (output) REAL 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) REAL 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) REAL 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) REAL 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) REAL array,
dimension (max(3*N,M,P)+N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output)INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal value.
> 0: if INFO = 1, the Jacobi-type procedure
failed to converge. For further details, see sub-
routine STGSJA.
PARAMETERS
TOLA REAL TOLB REAL TOLA and TOLB are the thresholds
to determine the effective rank of (A',B')'. Gen-
erally, 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.
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