SGGSVD(1) LAPACK driver routine (version 3.2) SGGSVD(1)NAME
SGGSVD - computes the generalized singular value decomposition (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 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 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.
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 transforma‐
tion 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 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 contains the trian‐
gular 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 contains the trian‐
gular 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) 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/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 con‐
verge. For further details, see subroutine STGSJA.
PARAMETERS
TOLA REAL
TOLB REAL TOLA and TOLB are the thresholds to determine the
effective rank of (A',B')'. 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. Further Details =============== 2-96 Based
on modifications by Ming Gu and Huan Ren, Computer Science
Division, University of California at Berkeley, USA
LAPACK driver routine (version 3November 2008 SGGSVD(1)
