CTGSJA - compute the generalized singular value decomposi
tion (GSVD) of two complex upper triangular (or trape
zoidal) 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( * )
PURPOSE
CTGSJA computes the generalized singular value decomposi
tion (GSVD) of two complex upper triangular (or trape
zoidal) 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 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 trape
zoidal.
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
gular 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,
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.
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 sin
gular 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.
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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