ztgsja(3P) Sun Performance Library ztgsja(3P)NAMEztgsja - compute the generalized singular value decomposition (GSVD) of
two complex upper triangular (or trapezoidal) matrices A and B
SYNOPSIS
SUBROUTINE ZTGSJA(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
DOUBLE COMPLEX A(LDA,*), B(LDB,*), U(LDU,*), V(LDV,*), Q(LDQ,*),
WORK(*)
INTEGER M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
DOUBLE PRECISION TOLA, TOLB
DOUBLE PRECISION ALPHA(*), BETA(*)
SUBROUTINE ZTGSJA_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
DOUBLE COMPLEX A(LDA,*), B(LDB,*), U(LDU,*), V(LDV,*), Q(LDQ,*),
WORK(*)
INTEGER*8 M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
DOUBLE PRECISION TOLA, TOLB
DOUBLE PRECISION ALPHA(*), BETA(*)
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
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, U, V, Q
INTEGER :: M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
REAL(8) :: TOLA, TOLB
REAL(8), DIMENSION(:) :: ALPHA, BETA
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
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, U, V, Q
INTEGER(8) :: M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
REAL(8) :: TOLA, TOLB
REAL(8), DIMENSION(:) :: ALPHA, BETA
C INTERFACE
#include <sunperf.h>
void ztgsja(char jobu, char jobv, char jobq, int m, int p, int n, int
k, int l, doublecomplex *a, int lda, doublecomplex *b, int
ldb, double tola, double tolb, double *alpha, double *beta,
doublecomplex *u, int ldu, doublecomplex *v, int ldv, double‐
complex *q, int ldq, int *ncycle, int *info);
void ztgsja_64(char jobu, char jobv, char jobq, long m, long p, long n,
long k, long l, doublecomplex *a, long lda, doublecomplex *b,
long ldb, double tola, double tolb, double *alpha, double
*beta, doublecomplex *u, long ldu, doublecomplex *v, long
ldv, doublecomplex *q, long ldq, long *ncycle, long *info);
PURPOSEztgsja computes the generalized singular value decomposition (GSVD) of
two complex 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 ZGGSVP 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 unitary matrices, Z' denotes the conjugate trans‐
pose 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 unitary 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.
ZTGSJA 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 unitary matrix, and Z' is the conjugate trans‐
pose 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.
ARGUMENTS
JOBU (input)
= 'U': U must contain a unitary matrix U1 on entry, and the
product U1*U is returned; = 'I': U is initialized to the
unit matrix, and the unitary matrix U is returned; = 'N': U
is not computed.
JOBV (input)
= 'V': V must contain a unitary matrix V1 on entry, and the
product V1*V is returned; = 'I': V is initialized to the
unit matrix, and the unitary matrix V is returned; = 'N': V
is not computed.
JOBQ (input)
= 'Q': Q must contain a unitary matrix Q1 on entry, and the
product Q1*Q is returned; = 'I': Q is initialized to the
unit matrix, and the unitary matrix Q is returned; = 'N': Q
is not computed.
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 ZTGSJA. See
the Further Details section below.
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 necessary, 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. Generally, they are the
same as used in the preprocessing 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
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 unitary matrix returned by ZGGSVP). On exit, if JOBU =
'I', U contains the unitary matrix U; if JOBU = 'U', U con‐
tains 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 unitary matrix returned by ZGGSVP). On exit, if JOBV =
'I', V contains the unitary matrix V; if JOBV = 'V', V con‐
tains 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 unitary matrix returned by ZGGSVP). On exit, if JOBQ =
'I', Q contains the unitary matrix Q; if JOBQ = 'Q', Q con‐
tains 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 illegal value.
= 1: the procedure does not converge after MAXIT cycles.
6 Mar 2009 ztgsja(3P)
