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NAME
DTGSJA - compute the generalized singular value decomposi-
tion (GSVD) of two real upper ``triangular (or tra-
pezoidal)'' 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( * )
PURPOSE
DTGSJA 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 fol-
lowing forms, which may be obtained by the preprocessing
subroutine DGGSVP for two general M-by-N matrix A and P-by-N
matrix B:
If M-K-L >= 0
A = ( 0 A12 A13 ) K , B = ( 0 0 B13 ) L
( 0 0 A23 ) L ( 0 0 0 ) P-L
( 0 0 0 ) M-K-L N-K-L K L
N-K-L K L
if M-K-L < 0
A = ( 0 A12 A13 ) K , B = ( 0 0 B13 ) L
( 0 0 A23 ) M-K ( 0 0 0 ) P-L
N-K-L K L N-K-L K L
where K-by-K matrix A12 and L-by-L matrix B13 are nonsingu-
lar upper triangular. A23 is L-by-L upper triangular if M-
K-L > 0, otherwise A23 is L-by-(M-K) 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,
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, the
( 0 R22 R23 )
( 0 0 R33 )
first M rows of R are stored in A(1:M, N-K-L+1:N) and R33 is
stored in B(M-K+1:L,N+M-K-L+1:N) on exit.
The computations of the orthogonal transformation matrices
U, V and Q are optional and may also be applied to the input
orthogonal matrices U, V and Q.
ARGUMENTS
JOBU (input) CHARACTER*1
= 'U': U is overwritten on the input orthogonal
matrix U;
= 'I': U is initialized to the identity matrix;
= 'N': U is not computed.
JOBV (input) CHARACTER*1
= 'V': V is overwritten on the input orthogonal
matrix V;
= 'I': V is initialized to the identity matrix;
= 'N': V is not computed.
JOBQ (input) CHARACTER*1
= 'Q': Q is overwritten on the input orthogonal
matrix Q;
= 'I': Q is initialized to the identity matrix;
= '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 sub-
blocks 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 com-
puted 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 neces-
sary, 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).
TOLA (input) DOUBLE PRECISION
TOLB (input) DOUBLE PRECISION TOLA and TOLB are
the convergence criteria for the Jacobi- Kogbetli-
antz 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; If M-K-L >= 0,
ALPHA(1:K) = ONE, ALPHA(K+1:K+L) = diag(C),
BETA(1:K) = ZERO, BETA(K+1:K+L) = diag(S), and 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. Furthermore, if K+L < N, ALPHA(K+L+1:N) = ZERO
BETA(K+L+1:N) = ZERO.
U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
On entry, if JOBU = 'U', U contains the orthogonal
matrix U, On exit, if JOBU = 'U', U is overwritten
on the input orthogonal matrix U. If JOBU = 'I', U
is first set to the identity matrix. If JOBU = 'N',
U is not referenced.
LDU (input) INTEGER
The leading dimension of the array U. LDU >=
max(1,M).
V (input/output) DOUBLE PRECISION array, dimension (LDV,P)
On entry, if JOBV = 'V', V contains the orthogonal
matrix V. On exit, if JOBV = 'V', V is overwritten
on the input orthogonal matrix V. If JOBV = 'I', U
is first set to the identity matrix. If JOBV = 'N',
V is not referenced.
LDV (input) INTEGER
The leading dimension of the array V. LDV >=
max(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q contains the orthogonal
matrix Q, On exit, if JOBQ = 'Q', Q is overwritten
on the input orthogonal matrix Q. If JOBQ = 'I', Q
is first set to the identity matrix. If JOBQ = 'N',
Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >=
MAX(1,N).
WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
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 rou-
tine 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.