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( * )
PURPOSE
SGGSVD computes the generalized singular value decomposi
tion (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 struc
tures, 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.
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 orthogo
nal 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.
Furthermore, 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 con
verted to the latter form by taking the nonsingular matrix
X as
X = Q*( I 0 )
( 0 inv(R) ).
ARGUMENTS
= '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 sin
gular 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
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/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 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')'. Gener
ally, 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
Further Details ===============
2-96 Based on modifications by Ming Gu and Huan
Ren, Computer Science Division, University of Cal
ifornia at Berkeley, USA
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