.TH "ggsvp3" 3 "Tue Jan 28 2025 00:54:31" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
ggsvp3 \- ggsvp3: step in ggsvd
.SH SYNOPSIS
.br
.PP
.SS "Functions"
.in +1c
.ti -1c
.RI "subroutine \fBcggsvp3\fP (jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola, tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork, tau, work, lwork, info)"
.br
.RI "\fBCGGSVP3\fP "
.ti -1c
.RI "subroutine \fBdggsvp3\fP (jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola, tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, tau, work, lwork, info)"
.br
.RI "\fBDGGSVP3\fP "
.ti -1c
.RI "subroutine \fBsggsvp3\fP (jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola, tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, tau, work, lwork, info)"
.br
.RI "\fBSGGSVP3\fP "
.ti -1c
.RI "subroutine \fBzggsvp3\fP (jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola, tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork, tau, work, lwork, info)"
.br
.RI "\fBZGGSVP3\fP "
.in -1c
.SH "Detailed Description"
.PP
.SH "Function Documentation"
.PP
.SS "subroutine cggsvp3 (character jobu, character jobv, character jobq, integer m, integer p, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real tola, real tolb, integer k, integer l, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, real, dimension( * ) rwork, complex, dimension( * ) tau, complex, dimension( * ) work, integer lwork, integer info)"
.PP
\fBCGGSVP3\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
CGGSVP3 computes unitary matrices U, V and Q such that
N-K-L K L
U**H*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
V**H*B*Q = 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\&. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H\&.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
CGGSVD3\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP
.PP
.nf
JOBU is CHARACTER*1
= 'U': Unitary matrix U is computed;
= 'N': U is not computed\&.
.fi
.PP
.br
\fIJOBV\fP
.PP
.nf
JOBV is CHARACTER*1
= 'V': Unitary matrix V is computed;
= 'N': V is not computed\&.
.fi
.PP
.br
\fIJOBQ\fP
.PP
.nf
JOBQ is CHARACTER*1
= 'Q': Unitary matrix Q is computed;
= '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
\fIA\fP
.PP
.nf
A is COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A\&.
On exit, A contains the triangular (or trapezoidal) matrix
described in the Purpose section\&.
.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, B contains the triangular matrix described in
the Purpose section\&.
.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 thresholds to determine the effective
numerical rank of matrix B and a subblock of A\&. Generally,
they are set to
TOLA = MAX(M,N)*norm(A)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS\&.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
K is INTEGER
.fi
.PP
.br
\fIL\fP
.PP
.nf
L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose section\&.
K + L = effective numerical rank of (A**H,B**H)**H\&.
.fi
.PP
.br
\fIU\fP
.PP
.nf
U is COMPLEX array, dimension (LDU,M)
If JOBU = 'U', U contains the unitary matrix 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)
If JOBV = 'V', V contains the unitary matrix 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)
If JOBQ = 'Q', Q contains the unitary matrix 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
\fIIWORK\fP
.PP
.nf
IWORK is INTEGER array, dimension (N)
.fi
.PP
.br
\fIRWORK\fP
.PP
.nf
RWORK is REAL array, dimension (2*N)
.fi
.PP
.br
\fITAU\fP
.PP
.nf
TAU is COMPLEX array, dimension (N)
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP
.PP
.nf
LWORK is INTEGER
The dimension of the array WORK\&. LWORK >= 1\&.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA\&.
.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\&.
.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
The subroutine uses LAPACK subroutine CGEQP3 for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix\&. It may be replaced by a better rank determination strategy\&.
CGGSVP3 replaces the deprecated subroutine CGGSVP\&.
.fi
.PP
.RE
.PP
.SS "subroutine dggsvp3 (character jobu, character jobv, character jobq, integer m, integer p, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision tola, double precision tolb, integer k, integer l, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)"
.PP
\fBDGGSVP3\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DGGSVP3 computes orthogonal matrices U, V and Q such that
N-K-L K L
U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
V**T*B*Q = 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\&. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T\&.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
DGGSVD3\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP
.PP
.nf
JOBU is CHARACTER*1
= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed\&.
.fi
.PP
.br
\fIJOBV\fP
.PP
.nf
JOBV is CHARACTER*1
= 'V': Orthogonal matrix V is computed;
= 'N': V is not computed\&.
.fi
.PP
.br
\fIJOBQ\fP
.PP
.nf
JOBQ is CHARACTER*1
= 'Q': Orthogonal matrix Q is computed;
= '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
\fIA\fP
.PP
.nf
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A\&.
On exit, A contains the triangular (or trapezoidal) matrix
described in the Purpose section\&.
.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, B contains the triangular matrix described in
the Purpose section\&.
.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 thresholds to determine the effective
numerical rank of matrix B and a subblock of A\&. Generally,
they are set to
TOLA = MAX(M,N)*norm(A)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS\&.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
K is INTEGER
.fi
.PP
.br
\fIL\fP
.PP
.nf
L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose section\&.
K + L = effective numerical rank of (A**T,B**T)**T\&.
.fi
.PP
.br
\fIU\fP
.PP
.nf
U is DOUBLE PRECISION array, dimension (LDU,M)
If JOBU = 'U', U contains the orthogonal matrix 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)
If JOBV = 'V', V contains the orthogonal matrix 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)
If JOBQ = 'Q', Q contains the orthogonal matrix 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
\fIIWORK\fP
.PP
.nf
IWORK is INTEGER array, dimension (N)
.fi
.PP
.br
\fITAU\fP
.PP
.nf
TAU is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP
.PP
.nf
LWORK is INTEGER
The dimension of the array WORK\&. LWORK >= 1\&.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA\&.
.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\&.
.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
The subroutine uses LAPACK subroutine DGEQP3 for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix\&. It may be replaced by a better rank determination strategy\&.
DGGSVP3 replaces the deprecated subroutine DGGSVP\&.
.fi
.PP
.RE
.PP
.SS "subroutine sggsvp3 (character jobu, character jobv, character jobq, integer m, integer p, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real tola, real tolb, integer k, integer l, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)"
.PP
\fBSGGSVP3\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
SGGSVP3 computes orthogonal matrices U, V and Q such that
N-K-L K L
U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
V**T*B*Q = 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\&. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T\&.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
SGGSVD3\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP
.PP
.nf
JOBU is CHARACTER*1
= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed\&.
.fi
.PP
.br
\fIJOBV\fP
.PP
.nf
JOBV is CHARACTER*1
= 'V': Orthogonal matrix V is computed;
= 'N': V is not computed\&.
.fi
.PP
.br
\fIJOBQ\fP
.PP
.nf
JOBQ is CHARACTER*1
= 'Q': Orthogonal matrix Q is computed;
= '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
\fIA\fP
.PP
.nf
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A\&.
On exit, A contains the triangular (or trapezoidal) matrix
described in the Purpose section\&.
.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, B contains the triangular matrix described in
the Purpose section\&.
.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 thresholds to determine the effective
numerical rank of matrix B and a subblock of A\&. Generally,
they are set to
TOLA = MAX(M,N)*norm(A)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS\&.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
K is INTEGER
.fi
.PP
.br
\fIL\fP
.PP
.nf
L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose section\&.
K + L = effective numerical rank of (A**T,B**T)**T\&.
.fi
.PP
.br
\fIU\fP
.PP
.nf
U is REAL array, dimension (LDU,M)
If JOBU = 'U', U contains the orthogonal matrix 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)
If JOBV = 'V', V contains the orthogonal matrix 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)
If JOBQ = 'Q', Q contains the orthogonal matrix 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
\fIIWORK\fP
.PP
.nf
IWORK is INTEGER array, dimension (N)
.fi
.PP
.br
\fITAU\fP
.PP
.nf
TAU is REAL array, dimension (N)
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP
.PP
.nf
LWORK is INTEGER
The dimension of the array WORK\&. LWORK >= 1\&.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA\&.
.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\&.
.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
The subroutine uses LAPACK subroutine SGEQP3 for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix\&. It may be replaced by a better rank determination strategy\&.
SGGSVP3 replaces the deprecated subroutine SGGSVP\&.
.fi
.PP
.RE
.PP
.SS "subroutine zggsvp3 (character jobu, character jobv, character jobq, integer m, integer p, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision tola, double precision tolb, integer k, integer l, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, double precision, dimension( * ) rwork, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)"
.PP
\fBZGGSVP3\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZGGSVP3 computes unitary matrices U, V and Q such that
N-K-L K L
U**H*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
V**H*B*Q = 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\&. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H\&.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
ZGGSVD3\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP
.PP
.nf
JOBU is CHARACTER*1
= 'U': Unitary matrix U is computed;
= 'N': U is not computed\&.
.fi
.PP
.br
\fIJOBV\fP
.PP
.nf
JOBV is CHARACTER*1
= 'V': Unitary matrix V is computed;
= 'N': V is not computed\&.
.fi
.PP
.br
\fIJOBQ\fP
.PP
.nf
JOBQ is CHARACTER*1
= 'Q': Unitary matrix Q is computed;
= '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
\fIA\fP
.PP
.nf
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A\&.
On exit, A contains the triangular (or trapezoidal) matrix
described in the Purpose section\&.
.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, B contains the triangular matrix described in
the Purpose section\&.
.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 thresholds to determine the effective
numerical rank of matrix B and a subblock of A\&. Generally,
they are set to
TOLA = MAX(M,N)*norm(A)*MAZHEPS,
TOLB = MAX(P,N)*norm(B)*MAZHEPS\&.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
K is INTEGER
.fi
.PP
.br
\fIL\fP
.PP
.nf
L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose section\&.
K + L = effective numerical rank of (A**H,B**H)**H\&.
.fi
.PP
.br
\fIU\fP
.PP
.nf
U is COMPLEX*16 array, dimension (LDU,M)
If JOBU = 'U', U contains the unitary matrix 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)
If JOBV = 'V', V contains the unitary matrix 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)
If JOBQ = 'Q', Q contains the unitary matrix 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
\fIIWORK\fP
.PP
.nf
IWORK is INTEGER array, dimension (N)
.fi
.PP
.br
\fIRWORK\fP
.PP
.nf
RWORK is DOUBLE PRECISION array, dimension (2*N)
.fi
.PP
.br
\fITAU\fP
.PP
.nf
TAU is COMPLEX*16 array, dimension (N)
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP
.PP
.nf
LWORK is INTEGER
The dimension of the array WORK\&. LWORK >= 1\&.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA\&.
.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\&.
.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
The subroutine uses LAPACK subroutine ZGEQP3 for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix\&. It may be replaced by a better rank determination strategy\&.
ZGGSVP3 replaces the deprecated subroutine ZGGSVP\&.
.fi
.PP
.RE
.PP
.SH "Author"
.PP
Generated automatically by Doxygen for LAPACK from the source code\&.