ZTGSJA(3F) ZTGSJA(3F)
ZTGSJA - compute the generalized singular value decomposition (GSVD) of
two complex upper triangular (or trapezoidal) matrices A and B
SUBROUTINE ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA,
TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
NCYCLE, INFO )
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, NCYCLE, P
DOUBLE PRECISION TOLA, TOLB
DOUBLE PRECISION ALPHA( * ), BETA( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V(
LDV, * ), WORK( * )
ZTGSJA computes the generalized singular value decomposition (GSVD) of
two complex upper triangular (or trapezoidal) matrices A and B.
On entry, it is assumed that matrices A and B have the following forms,
which may be obtained by the preprocessing subroutine ZGGSVP from a
general M-by-N matrix A and P-by-N matrix B:
N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
B = L ( 0 0 B13 )
P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular upper
triangular; A23 is L-by-L upper triangular if M-K-L >= 0, otherwise A23
is (M-K)-by-L upper trapezoidal.
On exit,
U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),
where U, V and Q are unitary matrices, Z' denotes the conjugate transpose
of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are
``diagonal'' matrices, which are of the following structures:
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ZTGSJA(3F) ZTGSJA(3F)
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 ) K
L ( 0 0 R22 ) L
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
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.
R = ( 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 computation of the unitary transformation matrices U, V or Q is
optional. These matrices may either be formed explicitly, or they may be
postmultiplied into input matrices U1, V1, or Q1.
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ZTGSJA(3F) ZTGSJA(3F)
JOBU (input) CHARACTER*1
= 'U': U must contain a unitary matrix U1 on entry, and the
product U1*U is returned; = 'I': U is initialized to the unit
matrix, and the unitary matrix U is returned; = 'N': U is not
computed.
JOBV (input) CHARACTER*1
= 'V': V must contain a unitary matrix V1 on entry, and the
product V1*V is returned; = 'I': V is initialized to the unit
matrix, and the unitary matrix V is returned; = 'N': V is not
computed.
JOBQ (input) CHARACTER*1
= 'Q': Q must contain a unitary matrix Q1 on entry, and the
product Q1*Q is returned; = 'I': Q is initialized to the unit
matrix, and the unitary matrix Q is returned; = 'N': Q is not
computed.
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
P (input) INTEGER
The number of rows of the matrix B. P >= 0.
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
K (input) INTEGER
L (input) INTEGER K and L specify the subblocks in the
input matrices A and B:
A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N) of A
and B, whose GSVD is going to be computed by ZTGSJA. See Further
details.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, A(N-K+1:N,1:MIN(K+L,M) )
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 P-by-N matrix B. On exit, if necessary, B(MK+1:L,N+M-K-L+1:N)
contains a part of R. See Purpose for
details.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
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ZTGSJA(3F) ZTGSJA(3F)
TOLA (input) DOUBLE PRECISION
TOLB (input) DOUBLE PRECISION TOLA and TOLB are the
convergence criteria for the Jacobi- Kogbetliantz iteration
procedure. Generally, they are the same as used in the
preprocessing step, say TOLA = MAX(M,N)*norm(A)*MAZHEPS, TOLB =
MAX(P,N)*norm(B)*MAZHEPS.
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 M-K-L >= 0, ALPHA(K+1:K+L) = diag(C),
BETA(K+1:K+L) = diag(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. Furthermore, if K+L < N,
ALPHA(K+L+1:N) = 0
BETA(K+L+1:N) = 0.
U (input/output) COMPLEX*16 array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually the
unitary matrix returned by ZGGSVP). On exit, if JOBU = 'I', U
contains the unitary matrix U; if JOBU = 'U', U contains the
product U1*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 (input/output) COMPLEX*16 array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually the
unitary matrix returned by ZGGSVP). On exit, if JOBV = 'I', V
contains the unitary matrix V; if JOBV = 'V', V contains the
product V1*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 (input/output) COMPLEX*16 array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the
unitary matrix returned by ZGGSVP). On exit, if JOBQ = 'I', Q
contains the unitary matrix Q; if JOBQ = 'Q', Q contains the
product Q1*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) COMPLEX*16 array, dimension (2*N)
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ZTGSJA(3F) ZTGSJA(3F)
NCYCLE (output) INTEGER
The number of cycles required for convergence.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1: the procedure does not converge after MAXIT cycles.
MAXIT INTEGER
MAXIT specifies the total loops that the iterative procedure may
take. If after MAXIT cycles, the routine fails to converge, we
return INFO = 1.
Further Details ===============
ZTGSJA essentially uses a variant of Kogbetliantz algorithm to
reduce min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and
L-by-L matrix B13 to the form:
U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
where U1, V1 and Q1 are unitary matrix, and Z' is the conjugate
transpose of Z. C1 and S1 are diagonal matrices satisfying
C1**2 + S1**2 = I,
and R1 is an L-by-L nonsingular upper triangular matrix.
ZTGSJA(3F) ZTGSJA(3F)
ZTGSJA - compute the generalized singular value decomposition (GSVD) of
two complex upper triangular (or trapezoidal) matrices A and B
SUBROUTINE ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA,
TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
NCYCLE, INFO )
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, NCYCLE, P
DOUBLE PRECISION TOLA, TOLB
DOUBLE PRECISION ALPHA( * ), BETA( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V(
LDV, * ), WORK( * )
ZTGSJA computes the generalized singular value decomposition (GSVD) of
two complex upper triangular (or trapezoidal) matrices A and B.
On entry, it is assumed that matrices A and B have the following forms,
which may be obtained by the preprocessing subroutine ZGGSVP from a
general M-by-N matrix A and P-by-N matrix B:
N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
B = L ( 0 0 B13 )
P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular upper
triangular; A23 is L-by-L upper triangular if M-K-L >= 0, otherwise A23
is (M-K)-by-L upper trapezoidal.
On exit,
U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),
where U, V and Q are unitary matrices, Z' denotes the conjugate transpose
of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are
``diagonal'' matrices, which are of the following structures:
Page 1
ZTGSJA(3F) ZTGSJA(3F)
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 ) K
L ( 0 0 R22 ) L
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
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.
R = ( 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 computation of the unitary transformation matrices U, V or Q is
optional. These matrices may either be formed explicitly, or they may be
postmultiplied into input matrices U1, V1, or Q1.
Page 2
ZTGSJA(3F) ZTGSJA(3F)
JOBU (input) CHARACTER*1
= 'U': U must contain a unitary matrix U1 on entry, and the
product U1*U is returned; = 'I': U is initialized to the unit
matrix, and the unitary matrix U is returned; = 'N': U is not
computed.
JOBV (input) CHARACTER*1
= 'V': V must contain a unitary matrix V1 on entry, and the
product V1*V is returned; = 'I': V is initialized to the unit
matrix, and the unitary matrix V is returned; = 'N': V is not
computed.
JOBQ (input) CHARACTER*1
= 'Q': Q must contain a unitary matrix Q1 on entry, and the
product Q1*Q is returned; = 'I': Q is initialized to the unit
matrix, and the unitary matrix Q is returned; = 'N': Q is not
computed.
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
P (input) INTEGER
The number of rows of the matrix B. P >= 0.
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
K (input) INTEGER
L (input) INTEGER K and L specify the subblocks in the
input matrices A and B:
A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N) of A
and B, whose GSVD is going to be computed by ZTGSJA. See Further
details.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, A(N-K+1:N,1:MIN(K+L,M) )
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 P-by-N matrix B. On exit, if necessary, B(MK+1:L,N+M-K-L+1:N)
contains a part of R. See Purpose for
details.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
Page 3
ZTGSJA(3F) ZTGSJA(3F)
TOLA (input) DOUBLE PRECISION
TOLB (input) DOUBLE PRECISION TOLA and TOLB are the
convergence criteria for the Jacobi- Kogbetliantz iteration
procedure. Generally, they are the same as used in the
preprocessing step, say TOLA = MAX(M,N)*norm(A)*MAZHEPS, TOLB =
MAX(P,N)*norm(B)*MAZHEPS.
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 M-K-L >= 0, ALPHA(K+1:K+L) = diag(C),
BETA(K+1:K+L) = diag(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. Furthermore, if K+L < N,
ALPHA(K+L+1:N) = 0
BETA(K+L+1:N) = 0.
U (input/output) COMPLEX*16 array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually the
unitary matrix returned by ZGGSVP). On exit, if JOBU = 'I', U
contains the unitary matrix U; if JOBU = 'U', U contains the
product U1*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 (input/output) COMPLEX*16 array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually the
unitary matrix returned by ZGGSVP). On exit, if JOBV = 'I', V
contains the unitary matrix V; if JOBV = 'V', V contains the
product V1*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 (input/output) COMPLEX*16 array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the
unitary matrix returned by ZGGSVP). On exit, if JOBQ = 'I', Q
contains the unitary matrix Q; if JOBQ = 'Q', Q contains the
product Q1*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) COMPLEX*16 array, dimension (2*N)
Page 4
ZTGSJA(3F) ZTGSJA(3F)
NCYCLE (output) INTEGER
The number of cycles required for convergence.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1: the procedure does not converge after MAXIT cycles.
MAXIT INTEGER
MAXIT specifies the total loops that the iterative procedure may
take. If after MAXIT cycles, the routine fails to converge, we
return INFO = 1.
Further Details ===============
ZTGSJA essentially uses a variant of Kogbetliantz algorithm to
reduce min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and
L-by-L matrix B13 to the form:
U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
where U1, V1 and Q1 are unitary matrix, and Z' is the conjugate
transpose of Z. C1 and S1 are diagonal matrices satisfying
C1**2 + S1**2 = I,
and R1 is an L-by-L nonsingular upper triangular matrix.
PPPPaaaaggggeeee 5555 [ Back ]
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