
ZGGSVD(l) ) ZGGSVD(l)
NAME
ZGGSVD  compute the generalized singular value decomposition (GSVD) of an MbyN complex
matrix A and PbyN complex matrix B
SYNOPSIS
SUBROUTINE ZGGSVD( 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( * )
DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK(
* )
PURPOSE
ZGGSVD computes the generalized singular value decomposition (GSVD) of an MbyN complex
matrix A and PbyN complex matrix B:
U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R )
where U, V and Q are unitary matrices, and Z' means the conjugate transpose of Z. Let K+L
= the effective numerical rank of the matrix (A',B')', then R is a (K+L)by(K+L) nonsin
gular upper triangular matrix, D1 and D2 are Mby(K+L) and Pby(K+L) "diagonal" matrices
and of the following structures, respectively:
If MKL >= 0,
K L
D1 = K ( I 0 )
L ( 0 C )
MKL ( 0 0 )
K L
D2 = L ( 0 S )
PL ( 0 0 )
NKL 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,NKL+1:N) on exit.
If MKL < 0,
K MK K+LM
D1 = K ( I 0 0 )
MK ( 0 C 0 )
K MK K+LM
D2 = MK ( 0 S 0 )
K+LM ( 0 0 I )
PL ( 0 0 0 )
NKL K MK K+LM
( 0 R ) = K ( 0 R11 R12 R13 )
MK ( 0 0 R22 R23 )
K+LM ( 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, NKL+1:N), and R33 is stored
( 0 R22 R23 )
in B(MK+1:L,N+MKL+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 NbyN 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 orthnormal 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, and 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': 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*16 array, dimension (LDA,N)
On entry, the MbyN 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*16 array, dimension (LDB,N)
On entry, the PbyN matrix B. On exit, B contains part of the triangular matrix
R if MKL < 0. See Purpose for details.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
ALPHA (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION 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 MKL >= 0, ALPHA(K+1:K+L) = C,
BETA(K+1:K+L) = S, or if MKL < 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) COMPLEX*16 array, dimension (LDU,M)
If JOBU = 'U', U contains the MbyM unitary 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 oth
erwise.
V (output) COMPLEX*16 array, dimension (LDV,P)
If JOBV = 'V', V contains the PbyP unitary 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 oth
erwise.
Q (output) COMPLEX*16 array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the NbyN unitary 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 oth
erwise.
WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N)
RWORK (workspace) DOUBLE PRECISION array, dimension (2*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 ith argument had an illegal value.
> 0: if INFO = 1, the Jacobitype procedure failed to converge. For further
details, see subroutine ZTGSJA.
PARAMETERS
TOLA DOUBLE PRECISION
TOLB DOUBLE PRECISION TOLA and TOLB are the thresholds to determine the effec
tive rank of (A',B')'. Generally, they are set to TOLA = MAX(M,N)*norm(A)*MAZHEPS,
TOLB = MAX(P,N)*norm(B)*MAZHEPS. The size of TOLA and TOLB may affect the size of
backward errors of the decomposition.
Further Details ===============
296 Based on modifications by Ming Gu and Huan Ren, Computer Science Division,
University of California at Berkeley, USA
LAPACK version 3.0 15 June 2000 ZGGSVD(l) 
