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NAME
SGGSVD - compute the generalized singular value decomposi-
tion (GSVD) of the M-by-N matrix A and P-by-N 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 the M-by-N matrix A and P-by-N matrix B:
U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R )
(1)
where U, V and Q are orthogonal matrices, and Z' is the
transpose of Z. Let K+L = the numerical effective rank of
the matrix (A',B')', then R is a K+L-by-K+L nonsingular
upper tridiagonal matrix, D1 and D2 are "diagonal" matrices,
and of the following structures, respectively:
If M-K-L >= 0,
U'*A*Q = D1*( 0 R )
= K ( I 0 ) * ( 0 R11 R12 ) K
L ( 0 C ) ( 0 0 R22 ) L
M-K-L ( 0 0 ) N-K-L K L
K L
V'*B*Q = D2*( 0 R )
= L ( 0 S ) * ( 0 R11 R12 ) K
P-L ( 0 0 ) ( 0 0 R22 ) L
K L N-K-L K L
where
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1), ... , BETA(K+L) ), C**2 + S**2 = I.
The nonsingular triangular matrix R = ( R11 R12 ) is
stored
( 0 R22 )
in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L < 0,
U'*A*Q = D1*( 0 R )
= K ( I 0 0 ) * ( 0 R11 R12 R13 ) K
M-K ( 0 C 0 ) ( 0 0 R22 R23 )
M-K
K M-K K+L-M ( 0 0 0 R33 )
K+L-M
N-K-L K M-K K+L-M
V'*B*Q = D2*( 0 R )
= M-K ( 0 S 0 ) * ( 0 R11 R12 R13 ) K
K+L-M ( 0 0 I ) ( 0 0 R22 R23 )
M-K
P-L ( 0 0 0 ) ( 0 0 0 R33 )
K+L-M
K M-K K+L-M N-K-L K M-K K+L-M
where
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1), ... , BETA(M) ), C**2 + S**2 = I.
R = ( R11 R12 R13 ) is a nonsingular upper triangular
matrix,
( 0 R22 R23 )
( 0 0 R33 )
(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 the matrix
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 )
(2) where U and V are orthogonal and X is nonsingular, D1
and D2 are ``diagonal''. It is easy to see that the GSVD
form (1) can be converted to the form (2) by taking the non-
singular 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 Pur-
pose 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 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 contains
the triangular matrix R if necessary. See Purpose
for details.
LDB (input) INTEGER
The leading dimension of the array B. LDA >=
MAX(1,P).
ALPHA (output) REAL arrays, dimension (N)
BETA (output) REAL array, dimension (N) On exit,
ALPHA and BETA contain the generalized singular
value pairs of A and B; if M-K-L >= 0, ALPHA(1:K) =
ONE, ALPHA(K+1:K+L) = C,
BETA(1:K) = ZERO, BETA(K+1:K+L) = S, or if M-K-L <
0, ALPHA(1:K)=ONE, ALPHA(K+1:M)=C,
ALPHA(M+1:K+L)=ZERO
BETA(1:K) =ZERO, BETA(K+1:M) =S, BETA(M+1:K+L) =ONE
and ALPHA(K+L+1:N) = ZERO
BETA(K+L+1:N) = ZERO
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).
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. LDA >=
MAX(1,P).
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).
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 illegal
value.
> 0: if INFO = 1, the Jacobi-type procedure failed
to converge. 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.