
dtgsja.f(3) LAPACK dtgsja.f(3)
NAME
dtgsja.f 
SYNOPSIS
Functions/Subroutines
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)
DTGSJA
Function/Subroutine Documentation
subroutine dtgsja (characterJOBU, characterJOBV, characterJOBQ, integerM, integerP, integerN,
integerK, integerL, double precision, dimension( lda, * )A, integerLDA, double precision,
dimension( ldb, * )B, integerLDB, double precisionTOLA, double precisionTOLB, double
precision, dimension( * )ALPHA, double precision, dimension( * )BETA, double precision,
dimension( ldu, * )U, integerLDU, double precision, dimension( ldv, * )V, integerLDV,
double precision, dimension( ldq, * )Q, integerLDQ, double precision, dimension( * )WORK,
integerNCYCLE, integerINFO)
DTGSJA
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 MbyN matrix A and PbyN matrix B:
NKL K L
A = K ( 0 A12 A13 ) if MKL >= 0;
L ( 0 0 A23 )
MKL ( 0 0 0 )
NKL K L
A = K ( 0 A12 A13 ) if MKL < 0;
MK ( 0 0 A23 )
NKL K L
B = L ( 0 0 B13 )
PL ( 0 0 0 )
where the KbyK matrix A12 and LbyL matrix B13 are nonsingular
upper triangular; A23 is LbyL upper triangular if MKL >= 0,
otherwise A23 is (MK)byL upper trapezoidal.
On exit,
U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
where U, V and Q are orthogonal matrices.
R is a nonsingular upper triangular matrix, and D1 and D2 are
``diagonal'' matrices, which are of the following structures:
If MKL >= 0,
K L
D1 = K ( I 0 )
L ( 0 C )
MKL ( 0 0 )
K L
D2 = L ( 0 S )
PL ( 0 0 )
NKL 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,NKL+1:N) on exit.
If MKL < 0,
K MK K+LM
D1 = K ( I 0 0 )
MK ( 0 C 0 )
K MK K+LM
D2 = MK ( 0 S 0 )
K+LM ( 0 0 I )
PL ( 0 0 0 )
NKL K MK K+LM
( 0 R ) = K ( 0 R11 R12 R13 )
MK ( 0 0 R22 R23 )
K+LM ( 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, NKL+1:N) and R33 is stored
( 0 R22 R23 )
in B(MK+1:L,N+MKL+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.
Parameters:
JOBU
JOBU is CHARACTER*1
= '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
orthogonal matrix U is returned;
= 'N': U is not computed.
JOBV
JOBV is CHARACTER*1
= '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
orthogonal matrix V is returned;
= 'N': V is not computed.
JOBQ
JOBQ is CHARACTER*1
= '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
orthogonal matrix Q is returned;
= 'N': Q is not computed.
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
P
P is INTEGER
The number of rows of the matrix B. P >= 0.
N
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
K
K is INTEGER
L
L is INTEGER
K and L specify the subblocks in the input matrices A and B:
A23 = A(K+1:MIN(K+L,M),NL+1:N) and B13 = B(1:L,NL+1:N)
of A and B, whose GSVD is going to be computed by DTGSJA.
See Further Details.
A
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the MbyN matrix A.
On exit, A(NK+1:N,1:MIN(K+L,M) ) contains the triangular
matrix R or part of R. See Purpose for details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the PbyN matrix B.
On exit, if necessary, B(MK+1:L,N+MKL+1:N) contains
a part of R. See Purpose for details.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,P).
TOLA
TOLA is DOUBLE PRECISION
TOLB
TOLB is 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.
ALPHA
ALPHA is DOUBLE PRECISION array, dimension (N)
BETA
BETA is DOUBLE PRECISION 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 MKL >= 0,
ALPHA(K+1:K+L) = diag(C),
BETA(K+1:K+L) = diag(S),
or if MKL < 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.
U
U is DOUBLE PRECISION array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually
the orthogonal matrix returned by DGGSVP).
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
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.
V
V is DOUBLE PRECISION array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually
the orthogonal matrix returned by DGGSVP).
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
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.
Q
Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
the orthogonal matrix returned by DGGSVP).
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
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK
WORK is DOUBLE PRECISION array, dimension (2*N)
NCYCLE
NCYCLE is INTEGER
The number of cycles required for convergence.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value.
= 1: the procedure does not converge after MAXIT cycles.
Internal 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..fi
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Further Details:
DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
min(L,MK)byL triangular (or trapezoidal) matrix A23 and LbyL
matrix B13 to the form:
U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
of Z. C1 and S1 are diagonal matrices satisfying
C1**2 + S1**2 = I,
and R1 is an LbyL nonsingular upper triangular matrix.
Definition at line 377 of file dtgsja.f.
Author
Generated automatically by Doxygen for LAPACK from the source code.
Version 3.4.2 Tue Sep 25 2012 dtgsja.f(3) 
