text: *> \brief \b DGGQRF
  *
  *  =========== DOCUMENTATION ===========
  *
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  *
  *  Definition:
  *  ===========
  *
  *       SUBROUTINE DGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
  *                          LWORK, INFO )
  *
  *       .. Scalar Arguments ..
  *       INTEGER            INFO, LDA, LDB, LWORK, M, N, P
  *       ..
  *       .. Array Arguments ..
  *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
  *      $                   WORK( * )
  *       ..
  *
  *
  *> \par Purpose:
  *  =============
  *>
  *> \verbatim
  *>
  *> DGGQRF computes a generalized QR factorization of an N-by-M matrix A
  *> and an N-by-P matrix B:
  *>
  *>             A = Q*R,        B = Q*T*Z,
  *>
  *> where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
  *> matrix, and R and T assume one of the forms:
  *>
  *> if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
  *>                 (  0  ) N-M                         N   M-N
  *>                    M
  *>
  *> where R11 is upper triangular, and
  *>
  *> if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
  *>                  P-N  N                           ( T21 ) P
  *>                                                      P
  *>
  *> where T12 or T21 is upper triangular.
  *>
  *> In particular, if B is square and nonsingular, the GQR factorization
  *> of A and B implicitly gives the QR factorization of inv(B)*A:
  *>
  *>              inv(B)*A = Z**T*(inv(T)*R)
  *>
  *> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
  *> transpose of the matrix Z.
  *> \endverbatim
  *
  *  Arguments:
  *  ==========
  *
  *> \param[in] N
  *> \verbatim
  *>          N is INTEGER
  *>          The number of rows of the matrices A and B. N >= 0.
  *> \endverbatim
  *>
  *> \param[in] M
  *> \verbatim
  *>          M is INTEGER
  *>          The number of columns of the matrix A.  M >= 0.
  *> \endverbatim
  *>
  *> \param[in] P
  *> \verbatim
  *>          P is INTEGER
  *>          The number of columns of the matrix B.  P >= 0.
  *> \endverbatim
  *>
  *> \param[in,out] A
  *> \verbatim
  *>          A is DOUBLE PRECISION array, dimension (LDA,M)
  *>          On entry, the N-by-M matrix A.
  *>          On exit, the elements on and above the diagonal of the array
  *>          contain the min(N,M)-by-M upper trapezoidal matrix R (R is
  *>          upper triangular if N >= M); the elements below the diagonal,
  *>          with the array TAUA, represent the orthogonal matrix Q as a
  *>          product of min(N,M) elementary reflectors (see Further
  *>          Details).
  *> \endverbatim
  *>
  *> \param[in] LDA
  *> \verbatim
  *>          LDA is INTEGER
  *>          The leading dimension of the array A. LDA >= max(1,N).
  *> \endverbatim
  *>
  *> \param[out] TAUA
  *> \verbatim
  *>          TAUA is DOUBLE PRECISION array, dimension (min(N,M))
  *>          The scalar factors of the elementary reflectors which
  *>          represent the orthogonal matrix Q (see Further Details).
  *> \endverbatim
  *>
  *> \param[in,out] B
  *> \verbatim
  *>          B is DOUBLE PRECISION array, dimension (LDB,P)
  *>          On entry, the N-by-P matrix B.
  *>          On exit, if N <= P, the upper triangle of the subarray
  *>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
  *>          if N > P, the elements on and above the (N-P)-th subdiagonal
  *>          contain the N-by-P upper trapezoidal matrix T; the remaining
  *>          elements, with the array TAUB, represent the orthogonal
  *>          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,N).
  *> \endverbatim
  *>
  *> \param[out] TAUB
  *> \verbatim
  *>          TAUB is DOUBLE PRECISION array, dimension (min(N,P))
  *>          The scalar factors of the elementary reflectors which
  *>          represent the orthogonal matrix Z (see Further Details).
  *> \endverbatim
  *>
  *> \param[out] WORK
  *> \verbatim
  *>          WORK is DOUBLE PRECISION 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 QR factorization
  *>          of an N-by-M matrix, NB2 is the optimal blocksize for the
  *>          RQ factorization of an N-by-P matrix, and NB3 is the optimal
  *>          blocksize for a call of DORMQR.
  *>
  *>          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 December 2016
  *
  *> \ingroup doubleOTHERcomputational
  *
  *> \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(n,m).
  *>
  *>  Each H(i) has the form
  *>
  *>     H(i) = I - taua * v * v**T
  *>
  *>  where taua is a real scalar, and v is a real vector with
  *>  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
  *>  and taua in TAUA(i).
  *>  To form Q explicitly, use LAPACK subroutine DORGQR.
  *>  To use Q to update another matrix, use LAPACK subroutine DORMQR.
  *>
  *>  The matrix Z is represented as a product of elementary reflectors
  *>
  *>     Z = H(1) H(2) . . . H(k), where k = min(n,p).
  *>
  *>  Each H(i) has the form
  *>
  *>     H(i) = I - taub * v * v**T
  *>
  *>  where taub is a real scalar, and v is a real vector with
  *>  v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
  *>  B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
  *>  To form Z explicitly, use LAPACK subroutine DORGRQ.
  *>  To use Z to update another matrix, use LAPACK subroutine DORMRQ.
  *> \endverbatim
  *>
  *  =====================================================================
        SUBROUTINE DGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
       $                   LWORK, INFO )
  *
  *  -- LAPACK computational routine (version 3.7.0) --
  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  *     December 2016
  *
  *     .. Scalar Arguments ..
        INTEGER            INFO, LDA, LDB, LWORK, M, N, P
  *     ..
  *     .. Array Arguments ..
        DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
       $                   WORK( * )
  *     ..
  *
  *  =====================================================================
  *
  *     .. Local Scalars ..
        LOGICAL            LQUERY
        INTEGER            LOPT, LWKOPT, NB, NB1, NB2, NB3
  *     ..
  *     .. External Subroutines ..
        EXTERNAL           DGEQRF, DGERQF, DORMQR, XERBLA
  *     ..
  *     .. External Functions ..
        INTEGER            ILAENV
        EXTERNAL           ILAENV
  *     ..
  *     .. Intrinsic Functions ..
        INTRINSIC          INT, MAX, MIN
  *     ..
  *     .. Executable Statements ..
  *
  *     Test the input parameters
  *
        INFO = 0
        NB1 = ILAENV( 1, 'DGEQRF', ' ', N, M, -1, -1 )
        NB2 = ILAENV( 1, 'DGERQF', ' ', N, P, -1, -1 )
        NB3 = ILAENV( 1, 'DORMQR', ' ', N, M, P, -1 )
        NB = MAX( NB1, NB2, NB3 )
        LWKOPT = MAX( N, M, P )*NB
        WORK( 1 ) = LWKOPT
        LQUERY = ( LWORK.EQ.-1 )
        IF( N.LT.0 ) THEN
           INFO = -1
        ELSE IF( M.LT.0 ) THEN
           INFO = -2
        ELSE IF( P.LT.0 ) THEN
           INFO = -3
        ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
           INFO = -5
        ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
           INFO = -8
        ELSE IF( LWORK.LT.MAX( 1, N, M, P ) .AND. .NOT.LQUERY ) THEN
           INFO = -11
        END IF
        IF( INFO.NE.0 ) THEN
           CALL XERBLA( 'DGGQRF', -INFO )
           RETURN
        ELSE IF( LQUERY ) THEN
           RETURN
        END IF
  *
  *     QR factorization of N-by-M matrix A: A = Q*R
  *
        CALL DGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO )
        LOPT = WORK( 1 )
  *
  *     Update B := Q**T*B.
  *
        CALL DORMQR( 'Left', 'Transpose', N, P, MIN( N, M ), A, LDA, TAUA,
       $             B, LDB, WORK, LWORK, INFO )
        LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
  *
  *     RQ factorization of N-by-P matrix B: B = T*Z.
  *
        CALL DGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO )
        WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
  *
        RETURN
  *
  *     End of DGGQRF
  *
        END
  