Contents
stgsja - compute the generalized singular value decomposi-
tion (GSVD) of two real upper triangular (or trapezoidal)
matrices A and B
SUBROUTINE STGSJA(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 * 1 JOBU, JOBV, JOBQ
INTEGER M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
REAL TOLA, TOLB
REAL A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), U(LDU,*),
V(LDV,*), Q(LDQ,*), WORK(*)
SUBROUTINE STGSJA_64(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 * 1 JOBU, JOBV, JOBQ
INTEGER*8 M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE,
INFO
REAL TOLA, TOLB
REAL A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), U(LDU,*),
V(LDV,*), Q(LDQ,*), WORK(*)
F95 INTERFACE
SUBROUTINE TGSJA(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(LEN=1) :: JOBU, JOBV, JOBQ
INTEGER :: M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE,
INFO
REAL :: TOLA, TOLB
REAL, DIMENSION(:) :: ALPHA, BETA, WORK
REAL, DIMENSION(:,:) :: A, B, U, V, Q
SUBROUTINE TGSJA_64(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(LEN=1) :: JOBU, JOBV, JOBQ
INTEGER(8) :: M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCY-
CLE, INFO
REAL :: TOLA, TOLB
REAL, DIMENSION(:) :: ALPHA, BETA, WORK
REAL, DIMENSION(:,:) :: A, B, U, V, Q
C INTERFACE
#include <sunperf.h>
void stgsja(char jobu, char jobv, char jobq, int m, int p,
int n, int k, int l, float *a, int lda, float *b,
int ldb, float tola, float tolb, float *alpha,
float *beta, float *u, int ldu, float *v, int ldv,
float *q, int ldq, int *ncycle, int *info);
void stgsja_64(char jobu, char jobv, char jobq, long m, long
p, long n, long k, long l, float *a, long lda,
float *b, long ldb, float tola, float tolb, float
*alpha, float *beta, float *u, long ldu, float *v,
long ldv, float *q, long ldq, long *ncycle, long
*info);
stgsja 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 SGGSVP 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.
STGSJA 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.
JOBU (input)
= '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)
= '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)
= '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) The number of rows of the matrix A. M >= 0.
P (input) The number of rows of the matrix B. P >= 0.
N (input) The number of columns of the matrices A and B. N
>= 0.
K (input) 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 STGSJA. See Further details.
L (input) See the description of K.
A (input/output)
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)
The leading dimension of the array A. LDA >=
max(1,M).
B (input/output)
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)
The leading dimension of the array B. LDB >=
max(1,P).
TOLA (input)
TOLA and TOLB are the convergence criteria for the
Jacobi- Kogbetliantz iteration procedure. Gen-
erally, they are the same as used in the prepro-
cessing step, say TOLA = max(M,N)*norm(A)*MACHEPS,
TOLB = max(P,N)*norm(B)*MACHEPS.
TOLB (input)
See the description of TOLA.
ALPHA (output)
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.
BETA (output)
See the description of ALPHA.
U (input) On entry, if JOBU = 'U', U must contain a matrix
U1 (usually the orthogonal matrix returned by
SGGSVP). 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)
The leading dimension of the array U. LDU >=
max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
V (input) On entry, if JOBV = 'V', V must contain a matrix
V1 (usually the orthogonal matrix returned by
SGGSVP). 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)
The leading dimension of the array V. LDV >=
max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
Q (input) On entry, if JOBQ = 'Q', Q must contain a matrix
Q1 (usually the orthogonal matrix returned by
SGGSVP). 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)
The leading dimension of the array Q. LDQ >=
max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK (workspace)
dimension(2*N)
NCYCLE (output)
The number of cycles required for convergence.
INFO (output)
= 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.