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NAME
CGGSVD - compute the generalized singular value decomposi-
tion (GSVD) of the M-by-N complex matrix A and P-by-N com-
plex matrix B
SYNOPSIS
SUBROUTINE CGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA,
B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q,
LDQ, WORK, RWORK, IWORK, INFO )
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M,
N, P
INTEGER IWORK( * )
REAL ALPHA( * ), BETA( * ), RWORK( * )
COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U(
LDU, * ), V( LDV, * ), WORK( * )
PURPOSE
CGGSVD computes the generalized singular value decomposition
(GSVD) of the M-by-N complex matrix A and P-by-N complex
matrix B:
U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R )
(1)
where U, V and Q are unitary matrices, R is an upper tri-
angular matrix, and Z' means the conjugate transpose of Z.
Let K+L = the numerical effective rank of the matrix
(A',B')', then D1 and D2 are M-by-(K+L) and P-by-(K+L)
"diagonal" matrices and of the following structures, respec-
tively:
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 unitary
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, and
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': Unitary matrix U is computed;
= 'N': U is not computed.
JOBV (input) CHARACTER*1
= 'V': Unitary matrix V is computed;
= 'N': V is not computed.
JOBQ (input) CHARACTER*1
= 'Q': Unitary 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 Purpose.
K + L = effective numerical rank of (A',B')'.
A (input/output) COMPLEX 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) COMPLEX array, dimension (LDB,N)
On entry, the P-by-N matrix B. On exit, B contains
part of the triangular 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; 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) COMPLEX array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M unitary 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) COMPLEX array, dimension (LDV,P)
If JOBU = 'V', V contains the P-by-P unitary matrix
V. If JOBV = 'N', V is not referenced.
LDV (input) INTEGER
The leading dimension of the array V. LDV >=
max(1,P).
Q (output) COMPLEX array, dimension (LDQ,N)
If JOBU = 'Q', Q contains the N-by-N unitary 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) COMPLEX array, dimension (MAX(3*N,M,P)+N)
RWORK (workspace) REAL array, dimension (2*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
CTGSJA.
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.