DTGSJA - compute the generalized singular value decomposi
tion (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( * )
PURPOSE
DTGSJA computes the generalized singular value decomposi
tion (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 non
singular upper triangular; A23 is L-by-L upper triangular
if M-K-L >= 0, otherwise A23 is (M-K)-by-L upper trape
zoidal.
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
of the following structures:
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 orthogonal transformation matrices
U, V or Q is optional. These matrices may either be
input matrices U1, V1, or Q1.
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 orthog
onal matrix U is returned; = 'N': U is not com
puted.
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 orthog
onal matrix V is returned; = 'N': V is not com
puted.
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 orthog
onal matrix Q is returned; = 'N': Q is not com
puted.
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).
(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- Kog
betliantz 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 general
ized 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.
(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)
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 ille
gal 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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