
ctgsja.f(3) LAPACK ctgsja.f(3)
NAME
ctgsja.f 
SYNOPSIS
Functions/Subroutines
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)
CTGSJA
Function/Subroutine Documentation
subroutine ctgsja (characterJOBU, characterJOBV, characterJOBQ, integerM, integerP, integerN,
integerK, integerL, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, *
)B, integerLDB, realTOLA, realTOLB, real, dimension( * )ALPHA, real, dimension( * )BETA,
complex, dimension( ldu, * )U, integerLDU, complex, dimension( ldv, * )V, integerLDV,
complex, dimension( ldq, * )Q, integerLDQ, complex, dimension( * )WORK, integerNCYCLE,
integerINFO)
CTGSJA
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 following
forms, which may be obtained by the preprocessing subroutine CGGSVP
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**H *A*Q = D1*( 0 R ), V**H *B*Q = D2*( 0 R ),
where U, V and Q are unitary 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 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.
Parameters:
JOBU
JOBU is 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
JOBV is 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
JOBQ is 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
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 CTGSJA.
See Further Details.
A
A is COMPLEX 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 COMPLEX 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 REAL
TOLB
TOLB is REAL
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.
ALPHA
ALPHA is REAL array, dimension (N)
BETA
BETA is 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 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
BETA(K+L+1:N) = 0.
U
U is 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
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.
V
V is 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
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.
Q
Q is 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
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK
WORK is COMPLEX 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.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Further Details:
CTGSJA 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**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,
where U1, V1 and Q1 are unitary matrix.
C1 and S1 are diagonal matrices satisfying
C1**2 + S1**2 = I,
and R1 is an LbyL nonsingular upper triangular matrix.
Definition at line 378 of file ctgsja.f.
Author
Generated automatically by Doxygen for LAPACK from the source code.
Version 3.4.2 Tue Sep 25 2012 ctgsja.f(3) 
