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NAME
ctgsja - compute the generalized singular value decomposi-
tion (GSVD) of two complex upper triangular (or trapezoidal)
matrices A and B
SYNOPSIS
SUBROUTINE CTGSJA( 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
REAL TOLA, TOLB
REAL ALPHA( * ), BETA( * )
COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ),
V( LDV, * ), WORK( * )
#include <sunperf.h>
void ctgsja(char jobu, char jobv, char jobq, int m, int p,
int n, int k, int l, complex *ca, int lda, complex
*cb, int ldb, float tola, float tolb, float *sal-
pha, float *sbeta, complex *cu, int ldu, complex
*v, int ldv, complex *q, int ldq, int *ncycle, int
* info) ;
PURPOSE
CTGSJA 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 fol-
lowing forms, which may be obtained by the preprocessing
subroutine CGGSVP 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 unitary matrices, Z' denotes the conju-
gate 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 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.
ARGUMENTS
JOBU (input) CHARACTER*1
= '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) CHARACTER*1
= '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) CHARACTER*1
= '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) 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 computed by CTGSJA. See Further details.
A (input/output) COMPLEX 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) COMPLEX 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) REAL
TOLB (input) REAL TOLA and TOLB are the conver-
gence criteria for the Jacobi- Kogbetliantz itera-
tion 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.
ALPHA (output) REAL array, dimension (N)
BETA (output) REAL array, dimension (N) 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.
U (input/output) COMPLEX array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix
U1 (usually the unitary matrix returned by
CGGSVP). On exit, if JOBU = 'I', U contains the
unitary 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) COMPLEX array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix
V1 (usually the unitary matrix returned by
CGGSVP). On exit, if JOBV = 'I', V contains the
unitary 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) COMPLEX array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix
Q1 (usually the unitary matrix returned by
CGGSVP). On exit, if JOBQ = 'I', Q contains the
unitary 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) COMPLEX 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 ===============
CTGSJA 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 transpose of Z. C1 and S1 are diag-
onal matrices satisfying
C1**2 + S1**2 = I,
and R1 is an L-by-L nonsingular upper triangular
matrix.
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