.TH DTGSJA 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
DTGSJA - the generalized singular value decomposition (GSVD) of two real upper triangular (or trapezoidal) matrices A and B
.SH SYNOPSIS
.TP 19
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 )
.TP 19
.ti +4
CHARACTER
JOBQ, JOBU, JOBV
.TP 19
.ti +4
INTEGER
INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
NCYCLE, P
.TP 19
.ti +4
DOUBLE
PRECISION TOLA, TOLB
.TP 19
.ti +4
DOUBLE
PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
BETA( * ), Q( LDQ, * ), U( LDU, * ),
V( LDV, * ), WORK( * )
.SH 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 following
forms, which may be obtained by the preprocessing subroutine DGGSVP
from a general M-by-N matrix A and P-by-N matrix B:
.br
N-K-L K L
.br
A = K ( 0 A12 A13 ) if M-K-L >= 0;
.br
L ( 0 0 A23 )
.br
M-K-L ( 0 0 0 )
.br
N-K-L K L
.br
A = K ( 0 A12 A13 ) if M-K-L < 0;
.br
M-K ( 0 0 A23 )
.br
N-K-L K L
.br
B = L ( 0 0 B13 )
.br
P-L ( 0 0 0 )
.br
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.
.br
On exit,
.br
U\(aq*A*Q = D1*( 0 R ), V\(aq*B*Q = D2*( 0 R ),
.br
where U, V and Q are orthogonal matrices, Z\(aq denotes the transpose
of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are
``diagonal\(aq\(aq matrices, which are of the following structures:
If M-K-L >= 0,
.br
K L
.br
D1 = K ( I 0 )
.br
L ( 0 C )
.br
M-K-L ( 0 0 )
.br
K L
.br
D2 = L ( 0 S )
.br
P-L ( 0 0 )
.br
N-K-L K L
.br
( 0 R ) = K ( 0 R11 R12 ) K
.br
L ( 0 0 R22 ) L
.br
where
.br
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
.br
S = diag( BETA(K+1), ... , BETA(K+L) ),
.br
C**2 + S**2 = I.
.br
R is stored in A(1:K+L,N-K-L+1:N) on exit.
.br
If M-K-L < 0,
.br
K M-K K+L-M
.br
D1 = K ( I 0 0 )
.br
M-K ( 0 C 0 )
.br
K M-K K+L-M
.br
D2 = M-K ( 0 S 0 )
.br
K+L-M ( 0 0 I )
.br
P-L ( 0 0 0 )
.br
N-K-L K M-K K+L-M
.br
M-K ( 0 0 R22 R23 )
.br
K+L-M ( 0 0 0 R33 )
.br
where
.br
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
.br
S = diag( BETA(K+1), ... , BETA(M) ),
.br
C**2 + S**2 = I.
.br
R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
( 0 R22 R23 )
.br
in B(M-K+1:L,N+M-K-L+1:N) on exit.
.br
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.
.br
.SH ARGUMENTS
.TP 8
JOBU (input) CHARACTER*1
= \(aqU\(aq: U must contain an orthogonal matrix U1 on entry, and
the product U1*U is returned;
= \(aqI\(aq: U is initialized to the unit matrix, and the
orthogonal matrix U is returned;
= \(aqN\(aq: U is not computed.
.TP 8
JOBV (input) CHARACTER*1
.br
= \(aqV\(aq: V must contain an orthogonal matrix V1 on entry, and
the product V1*V is returned;
= \(aqI\(aq: V is initialized to the unit matrix, and the
orthogonal matrix V is returned;
= \(aqN\(aq: V is not computed.
.TP 8
JOBQ (input) CHARACTER*1
.br
= \(aqQ\(aq: Q must contain an orthogonal matrix Q1 on entry, and
the product Q1*Q is returned;
= \(aqI\(aq: Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= \(aqN\(aq: Q is not computed.
.TP 8
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
.TP 8
P (input) INTEGER
The number of rows of the matrix B. P >= 0.
.TP 8
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
.TP 8
K (input) INTEGER
L (input) INTEGER
K and L specify the subblocks in the input matrices A and B:
.br
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.
.TP 8
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.
.TP 8
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
.TP 8
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
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.
.TP 8
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
.TP 8
TOLA (input) DOUBLE PRECISION
TOLB (input) DOUBLE PRECISION
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)*MAZHEPS,
TOLB = max(P,N)*norm(B)*MAZHEPS.
.TP 8
ALPHA (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
ALPHA(1:K) = 1,
.br
BETA(1:K) = 0,
and if M-K-L >= 0,
ALPHA(K+1:K+L) = diag(C),
.br
BETA(K+1:K+L) = diag(S),
or if M-K-L < 0,
ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
.br
BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
Furthermore, if K+L < N,
ALPHA(K+L+1:N) = 0 and
.br
BETA(K+L+1:N) = 0.
.TP 8
U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
On entry, if JOBU = \(aqU\(aq, U must contain a matrix U1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBU = \(aqI\(aq, U contains the orthogonal matrix U;
if JOBU = \(aqU\(aq, U contains the product U1*U.
If JOBU = \(aqN\(aq, U is not referenced.
.TP 8
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = \(aqU\(aq; LDU >= 1 otherwise.
.TP 8
V (input/output) DOUBLE PRECISION array, dimension (LDV,P)
On entry, if JOBV = \(aqV\(aq, V must contain a matrix V1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBV = \(aqI\(aq, V contains the orthogonal matrix V;
if JOBV = \(aqV\(aq, V contains the product V1*V.
If JOBV = \(aqN\(aq, V is not referenced.
.TP 8
LDV (input) INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = \(aqV\(aq; LDV >= 1 otherwise.
.TP 8
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBQ = \(aqQ\(aq, Q must contain a matrix Q1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBQ = \(aqI\(aq, Q contains the orthogonal matrix Q;
if JOBQ = \(aqQ\(aq, Q contains the product Q1*Q.
If JOBQ = \(aqN\(aq, Q is not referenced.
.TP 8
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = \(aqQ\(aq; LDQ >= 1 otherwise.
.TP 8
WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
.TP 8
NCYCLE (output) INTEGER
The number of cycles required for convergence.
.TP 8
INFO (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value.
.br
= 1: the procedure does not converge after MAXIT cycles.
.SH PARAMETERS
.TP 8
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\(aq*A13*Q1 = C1*R1; V1\(aq*B13*Q1 = S1*R1,
where U1, V1 and Q1 are orthogonal matrix, and Z\(aq 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.