DTGSJA(3F)DTGSJA(3F)NAMEDTGSJA - compute the generalized singular value decomposition (GSVD) of
two real upper triangular (or trapezoidal) matrices A and B
SYNOPSIS
SUBROUTINE DTGSJA( 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 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA(
* ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )
PURPOSEDTGSJA computes the generalized singular value decomposition (GSVD) of
two real 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 DGGSVP 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 orthogonal matrices, Z' denotes the transpose of Z,
R is a nonsingular upper triangular matrix, and D1 and D2 are
``diagonal'' matrices, which are of the following structures:
If M-K-L >= 0,
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DTGSJA(3F)DTGSJA(3F)
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 orthogonal 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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DTGSJA(3F)DTGSJA(3F)ARGUMENTS
JOBU (input) CHARACTER*1
= 'U': U must contain an orthogonal matrix U1 on entry, and the
product U1*U is returned; = 'I': U is initialized to the unit
matrix, and the orthogonal matrix U is returned; = 'N': U is not
computed.
JOBV (input) CHARACTER*1
= 'V': V must contain an orthogonal matrix V1 on entry, and the
product V1*V is returned; = 'I': V is initialized to the unit
matrix, and the orthogonal matrix V is returned; = 'N': V is not
computed.
JOBQ (input) CHARACTER*1
= 'Q': Q must contain an orthogonal matrix Q1 on entry, and the
product Q1*Q is returned; = 'I': Q is initialized to the unit
matrix, and the orthogonal 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 DTGSJA. See Further
details.
A (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B. On exit, if necessary, B(M-
K+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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DTGSJA(3F)DTGSJA(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 and
BETA(K+L+1:N) = 0.
U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually the
orthogonal matrix returned by DGGSVP). On exit, if JOBU = 'I', U
contains the orthogonal 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) DOUBLE PRECISION array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually the
orthogonal matrix returned by DGGSVP). On exit, if JOBV = 'I', V
contains the orthogonal 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) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the
orthogonal matrix returned by DGGSVP). On exit, if JOBQ = 'I', Q
contains the orthogonal 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) DOUBLE PRECISION array, dimension (2*N)
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DTGSJA(3F)DTGSJA(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.
PARAMETERS
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 ===============
DTGSJA 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 orthogonal matrix, and Z' is the
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.
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