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version 1.8, 2011/07/22 07:38:14 version 1.9, 2011/11/21 20:43:10
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   *> \brief \b ZGGRQF
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZGGRQF + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggrqf.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggrqf.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggrqf.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
   *                          LWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, LDA, LDB, LWORK, M, N, P
   *       ..
   *       .. Array Arguments ..
   *       COMPLEX*16         A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
   *      $                   WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGGRQF 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.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of rows of the matrix A.  M >= 0.
   *> \endverbatim
   *>
   *> \param[in] P
   *> \verbatim
   *>          P is INTEGER
   *>          The number of rows of the matrix B.  P >= 0.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of columns of the matrices A and B. N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is COMPLEX*16 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).
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A. LDA >= max(1,M).
   *> \endverbatim
   *>
   *> \param[out] TAUA
   *> \verbatim
   *>          TAUA is COMPLEX*16 array, dimension (min(M,N))
   *>          The scalar factors of the elementary reflectors which
   *>          represent the unitary matrix Q (see Further Details).
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is COMPLEX*16 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).
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B. LDB >= max(1,P).
   *> \endverbatim
   *>
   *> \param[out] TAUB
   *> \verbatim
   *>          TAUB is COMPLEX*16 array, dimension (min(P,N))
   *>          The scalar factors of the elementary reflectors which
   *>          represent the unitary matrix Z (see Further Details).
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
   *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          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 ZUNMRQ.
   *>
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit
   *>          < 0:  if INFO=-i, the i-th argument had an illegal value.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup complex16OTHERcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  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 ZUNGRQ.
   *>  To use Q to update another matrix, use LAPACK subroutine ZUNMRQ.
   *>
   *>  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 ZUNGQR.
   *>  To use Z to update another matrix, use LAPACK subroutine ZUNMQR.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE ZGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,        SUBROUTINE ZGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
      $                   LWORK, INFO )       $                   LWORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.3.1) --  *  -- LAPACK computational routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, LDA, LDB, LWORK, M, N, P        INTEGER            INFO, LDA, LDB, LWORK, M, N, P
Line 14 Line 227
      $                   WORK( * )       $                   WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGGRQF 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.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  M       (input) INTEGER  
 *          The number of rows of the matrix A.  M >= 0.  
 *  
 *  P       (input) INTEGER  
 *          The number of rows of the matrix B.  P >= 0.  
 *  
 *  N       (input) INTEGER  
 *          The number of columns of the matrices A and B. N >= 0.  
 *  
 *  A       (input/output) COMPLEX*16 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).  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A. LDA >= max(1,M).  
 *  
 *  TAUA    (output) COMPLEX*16 array, dimension (min(M,N))  
 *          The scalar factors of the elementary reflectors which  
 *          represent the unitary matrix Q (see Further Details).  
 *  
 *  B       (input/output) COMPLEX*16 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).  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B. LDB >= max(1,P).  
 *  
 *  TAUB    (output) COMPLEX*16 array, dimension (min(P,N))  
 *          The scalar factors of the elementary reflectors which  
 *          represent the unitary matrix Z (see Further Details).  
 *  
 *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))  
 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.  
 *  
 *  LWORK   (input) 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 ZUNMRQ.  
 *  
 *          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.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit  
 *          < 0:  if INFO=-i, the i-th argument had an illegal value.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  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 ZUNGRQ.  
 *  To use Q to update another matrix, use LAPACK subroutine ZUNMRQ.  
 *  
 *  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 ZUNGQR.  
 *  To use Z to update another matrix, use LAPACK subroutine ZUNMQR.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Local Scalars ..  *     .. Local Scalars ..
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