.TH "tgsja" 3 "Tue Jan 28 2025 00:54:31" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
tgsja \- tgsja: generalized SVD of trapezoidal matrices, step in ggsvd3
.SH SYNOPSIS
.br
.PP
.SS "Functions"
.in +1c
.ti -1c
.RI "subroutine \fBctgsja\fP (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)"
.br
.RI "\fBCTGSJA\fP "
.ti -1c
.RI "subroutine \fBdtgsja\fP (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)"
.br
.RI "\fBDTGSJA\fP "
.ti -1c
.RI "subroutine \fBstgsja\fP (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)"
.br
.RI "\fBSTGSJA\fP "
.ti -1c
.RI "subroutine \fBztgsja\fP (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)"
.br
.RI "\fBZTGSJA\fP "
.in -1c
.SH "Detailed Description"
.PP
.SH "Function Documentation"
.PP
.SS "subroutine ctgsja (character jobu, character jobv, character jobq, integer m, integer p, integer n, integer k, integer l, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real tola, real tolb, real, dimension( * ) alpha, real, dimension( * ) beta, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( * ) work, integer ncycle, integer info)"
.PP
\fBCTGSJA\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
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 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 nonsingular
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**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 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
( 0 R ) = K ( 0 R11 R12 R13 )
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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIJOBV\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIJOBQ\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of rows of the matrix A\&. M >= 0\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is INTEGER
The number of rows of the matrix B\&. P >= 0\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
The number of columns of the matrices A and B\&. N >= 0\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
K is INTEGER
.fi
.PP
.br
\fIL\fP
.PP
.nf
L is INTEGER
K and L specify the subblocks 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\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,M)\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is COMPLEX 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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
LDB is INTEGER
The leading dimension of the array B\&. LDB >= max(1,P)\&.
.fi
.PP
.br
\fITOLA\fP
.PP
.nf
TOLA is REAL
.fi
.PP
.br
\fITOLB\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is REAL array, dimension (N)
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
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 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\&.
.fi
.PP
.br
\fIU\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDU\fP
.PP
.nf
LDU is INTEGER
The leading dimension of the array U\&. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise\&.
.fi
.PP
.br
\fIV\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDV\fP
.PP
.nf
LDV is INTEGER
The leading dimension of the array V\&. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise\&.
.fi
.PP
.br
\fIQ\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDQ\fP
.PP
.nf
LDQ is INTEGER
The leading dimension of the array Q\&. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is COMPLEX array, dimension (2*N)
.fi
.PP
.br
\fINCYCLE\fP
.PP
.nf
NCYCLE is INTEGER
The number of cycles required for convergence\&.
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value\&.
= 1: the procedure does not converge after MAXIT cycles\&.
.fi
.PP
.RE
.PP
\fBInternal Parameters:\fP
.RS 4
.PP
.nf
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
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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**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 L-by-L nonsingular upper triangular matrix\&.
.fi
.PP
.RE
.PP
.SS "subroutine dtgsja (character jobu, character jobv, character jobq, integer m, integer p, integer n, integer k, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision tola, double precision tolb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) work, integer ncycle, integer info)"
.PP
\fBDTGSJA\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
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:
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 nonsingular
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**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 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
( 0 R ) = K ( 0 R11 R12 R13 )
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 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIJOBV\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIJOBQ\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of rows of the matrix A\&. M >= 0\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is INTEGER
The number of rows of the matrix B\&. P >= 0\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
The number of columns of the matrices A and B\&. N >= 0\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
K is INTEGER
.fi
.PP
.br
\fIL\fP
.PP
.nf
L is INTEGER
K and L specify the subblocks 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 DTGSJA\&.
See Further Details\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,M)\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is 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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
LDB is INTEGER
The leading dimension of the array B\&. LDB >= max(1,P)\&.
.fi
.PP
.br
\fITOLA\fP
.PP
.nf
TOLA is DOUBLE PRECISION
.fi
.PP
.br
\fITOLB\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
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 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 and
BETA(K+L+1:N) = 0\&.
.fi
.PP
.br
\fIU\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDU\fP
.PP
.nf
LDU is INTEGER
The leading dimension of the array U\&. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise\&.
.fi
.PP
.br
\fIV\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDV\fP
.PP
.nf
LDV is INTEGER
The leading dimension of the array V\&. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise\&.
.fi
.PP
.br
\fIQ\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fILDQ\fP
.PP
.nf
LDQ is INTEGER
The leading dimension of the array Q\&. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is DOUBLE PRECISION array, dimension (2*N)
.fi
.PP
.br
\fINCYCLE\fP
.PP
.nf
NCYCLE is INTEGER
The number of cycles required for convergence\&.
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value\&.
= 1: the procedure does not converge after MAXIT cycles\&.
.fi
.PP
.RE
.PP
.PP
.nf
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
.PP
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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**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 L-by-L nonsingular upper triangular matrix\&.
.fi
.PP
.RE
.PP
.SS "subroutine stgsja (character jobu, character jobv, character jobq, integer m, integer p, integer n, integer k, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real tola, real tolb, real, dimension( * ) alpha, real, dimension( * ) beta, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldq, * ) q, integer ldq, real, dimension( * ) work, integer ncycle, integer info)"
.PP
\fBSTGSJA\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
STGSJA 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 SGGSVP
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 nonsingular
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**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 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
( 0 R ) = K ( 0 R11 R12 R13 )
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 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIJOBV\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIJOBQ\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of rows of the matrix A\&. M >= 0\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is INTEGER
The number of rows of the matrix B\&. P >= 0\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
The number of columns of the matrices A and B\&. N >= 0\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
K is INTEGER
.fi
.PP
.br
\fIL\fP
.PP
.nf
L is INTEGER
K and L specify the subblocks 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 STGSJA\&.
See Further Details\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is REAL 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,M)\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is REAL 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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
LDB is INTEGER
The leading dimension of the array B\&. LDB >= max(1,P)\&.
.fi
.PP
.br
\fITOLA\fP
.PP
.nf
TOLA is REAL
.fi
.PP
.br
\fITOLB\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is REAL array, dimension (N)
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
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 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 and
BETA(K+L+1:N) = 0\&.
.fi
.PP
.br
\fIU\fP
.PP
.nf
U is REAL array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually
the orthogonal matrix returned by SGGSVP)\&.
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\&.
.fi
.PP
.br
\fILDU\fP
.PP
.nf
LDU is INTEGER
The leading dimension of the array U\&. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise\&.
.fi
.PP
.br
\fIV\fP
.PP
.nf
V is REAL array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually
the orthogonal matrix returned by SGGSVP)\&.
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\&.
.fi
.PP
.br
\fILDV\fP
.PP
.nf
LDV is INTEGER
The leading dimension of the array V\&. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise\&.
.fi
.PP
.br
\fIQ\fP
.PP
.nf
Q is REAL array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
the orthogonal matrix returned by SGGSVP)\&.
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\&.
.fi
.PP
.br
\fILDQ\fP
.PP
.nf
LDQ is INTEGER
The leading dimension of the array Q\&. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is REAL array, dimension (2*N)
.fi
.PP
.br
\fINCYCLE\fP
.PP
.nf
NCYCLE is INTEGER
The number of cycles required for convergence\&.
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value\&.
= 1: the procedure does not converge after MAXIT cycles\&.
.fi
.PP
.RE
.PP
.PP
.nf
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
.PP
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
STGSJA 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**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 L-by-L nonsingular upper triangular matrix\&.
.fi
.PP
.RE
.PP
.SS "subroutine ztgsja (character jobu, character jobv, character jobq, integer m, integer p, integer n, integer k, integer l, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision tola, double precision tolb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( * ) work, integer ncycle, integer info)"
.PP
\fBZTGSJA\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZTGSJA 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 ZGGSVP
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 nonsingular
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**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 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
( 0 R ) = K ( 0 R11 R12 R13 )
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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIJOBV\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIJOBQ\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of rows of the matrix A\&. M >= 0\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is INTEGER
The number of rows of the matrix B\&. P >= 0\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
The number of columns of the matrices A and B\&. N >= 0\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
K is INTEGER
.fi
.PP
.br
\fIL\fP
.PP
.nf
L is INTEGER
K and L specify the subblocks 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 ZTGSJA\&.
See Further Details\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
LDA is INTEGER
The leading dimension of the array A\&. LDA >= max(1,M)\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
LDB is INTEGER
The leading dimension of the array B\&. LDB >= max(1,P)\&.
.fi
.PP
.br
\fITOLA\fP
.PP
.nf
TOLA is DOUBLE PRECISION
.fi
.PP
.br
\fITOLB\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
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 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 and
BETA(K+L+1:N) = 0\&.
.fi
.PP
.br
\fIU\fP
.PP
.nf
U is COMPLEX*16 array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually
the unitary matrix returned by ZGGSVP)\&.
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\&.
.fi
.PP
.br
\fILDU\fP
.PP
.nf
LDU is INTEGER
The leading dimension of the array U\&. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise\&.
.fi
.PP
.br
\fIV\fP
.PP
.nf
V is COMPLEX*16 array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually
the unitary matrix returned by ZGGSVP)\&.
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\&.
.fi
.PP
.br
\fILDV\fP
.PP
.nf
LDV is INTEGER
The leading dimension of the array V\&. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise\&.
.fi
.PP
.br
\fIQ\fP
.PP
.nf
Q is COMPLEX*16 array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
the unitary matrix returned by ZGGSVP)\&.
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\&.
.fi
.PP
.br
\fILDQ\fP
.PP
.nf
LDQ is INTEGER
The leading dimension of the array Q\&. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is COMPLEX*16 array, dimension (2*N)
.fi
.PP
.br
\fINCYCLE\fP
.PP
.nf
NCYCLE is INTEGER
The number of cycles required for convergence\&.
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value\&.
= 1: the procedure does not converge after MAXIT cycles\&.
.fi
.PP
.RE
.PP
\fBInternal Parameters:\fP
.RS 4
.PP
.nf
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
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
ZTGSJA 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**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 L-by-L nonsingular upper triangular matrix\&.
.fi
.PP
.RE
.PP
.SH "Author"
.PP
Generated automatically by Doxygen for LAPACK from the source code\&.