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NAME
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(
* )
#include <sunperf.h>
void dtgsja(char jobu, char jobv, char jobq, int m, int p,
int n, int k, int l, double *da, int lda,
double *db, int ldb, double tola, double tolb,
double *dalpha, double *dbeta, double *du, int
ldu, double *v, int ldv, double *q, int ldq, int
*ncycle, int *info) ;
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 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 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,
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.
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
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 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 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.
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)
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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