.TH "SRC/cggrqf.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
SRC/cggrqf.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBcggrqf\fP (m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info)"
.br
.RI "\fBCGGRQF\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine cggrqf (integer m, integer p, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) taua, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) taub, complex, dimension( * ) work, integer lwork, integer info)"

.PP
\fBCGGRQF\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> CGGRQF computes a generalized RQ factorization of an M-by-N matrix A
!> and a P-by-N matrix B:
!>
!>             A = R*Q,        B = Z*T*Q,
!>
!> where Q is an N-by-N unitary matrix, Z is a P-by-P unitary
!> matrix, and R and T assume one of the forms:
!>
!> if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
!>                  N-M  M                           ( R21 ) N
!>                                                      N
!>
!> where R12 or R21 is upper triangular, and
!>
!> if P >= N,  T = ( T11 ) N  ,   or if P < N,  T = ( T11  T12 ) P,
!>                 (  0  ) P-N                         P   N-P
!>                    N
!>
!> where T11 is upper triangular\&.
!>
!> In particular, if B is square and nonsingular, the GRQ factorization
!> of A and B implicitly gives the RQ factorization of A*inv(B):
!>
!>              A*inv(B) = (R*inv(T))*Z**H
!>
!> where inv(B) denotes the inverse of the matrix B, and Z**H denotes the
!> conjugate transpose of the matrix Z\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\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, if M <= N, the upper triangle of the subarray
!>          A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
!>          if M > N, the elements on and above the (M-N)-th subdiagonal
!>          contain the M-by-N upper trapezoidal matrix R; the remaining
!>          elements, with the array TAUA, represent the unitary
!>          matrix Q as a product of elementary reflectors (see Further
!>          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
\fITAUA\fP 
.PP
.nf
!>          TAUA is COMPLEX array, dimension (min(M,N))
!>          The scalar factors of the elementary reflectors which
!>          represent the unitary matrix Q (see Further Details)\&.
!> 
.fi
.PP
.br
\fIB\fP 
.PP
.nf
!>          B is COMPLEX array, dimension (LDB,N)
!>          On entry, the P-by-N matrix B\&.
!>          On exit, the elements on and above the diagonal of the array
!>          contain the min(P,N)-by-N upper trapezoidal matrix T (T is
!>          upper triangular if P >= N); the elements below the diagonal,
!>          with the array TAUB, represent the unitary matrix Z as a
!>          product of elementary reflectors (see Further 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
\fITAUB\fP 
.PP
.nf
!>          TAUB is COMPLEX array, dimension (min(P,N))
!>          The scalar factors of the elementary reflectors which
!>          represent the unitary matrix Z (see Further Details)\&.
!> 
.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 >= max(1,N,M,P)\&.
!>          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
!>          where NB1 is the optimal blocksize for the RQ factorization
!>          of an M-by-N matrix, NB2 is the optimal blocksize for the
!>          QR factorization of a P-by-N matrix, and NB3 is the optimal
!>          blocksize for a call of CUNMRQ\&.
!>
!>          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 matrix Q is represented as a product of elementary reflectors
!>
!>     Q = H(1) H(2) \&. \&. \&. H(k), where k = min(m,n)\&.
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - taua * v * v**H
!>
!>  where taua is a complex scalar, and v is a complex vector with
!>  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
!>  A(m-k+i,1:n-k+i-1), and taua in TAUA(i)\&.
!>  To form Q explicitly, use LAPACK subroutine CUNGRQ\&.
!>  To use Q to update another matrix, use LAPACK subroutine CUNMRQ\&.
!>
!>  The matrix Z is represented as a product of elementary reflectors
!>
!>     Z = H(1) H(2) \&. \&. \&. H(k), where k = min(p,n)\&.
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - taub * v * v**H
!>
!>  where taub is a complex scalar, and v is a complex vector with
!>  v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
!>  and taub in TAUB(i)\&.
!>  To form Z explicitly, use LAPACK subroutine CUNGQR\&.
!>  To use Z to update another matrix, use LAPACK subroutine CUNMQR\&.
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB212\fP of file \fBcggrqf\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.