dorgrq.f

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*> \brief \b DORGRQ
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DORGRQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DORGRQ generates an M-by-N real matrix Q with orthonormal rows,
*> which is defined as the last M rows of a product of K elementary
*> reflectors of order N
*>
*>       Q  =  H(1) H(2) . . . H(k)
*>
*> as returned by DGERQF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q. N >= M.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the (m-k+i)-th row must contain the vector which
*>          defines the elementary reflector H(i), for i = 1,2,...,k, as
*>          returned by DGERQF in the last k rows of its array argument
*>          A.
*>          On exit, the M-by-N matrix Q.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The first dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is DOUBLE PRECISION array, dimension (K)
*>          TAU(i) must contain the scalar factor of the elementary
*>          reflector H(i), as returned by DGERQF.
*> \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,M).
*>          For optimum performance LWORK >= M*NB, where NB is the
*>          optimal blocksize.
*>
*>          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 has an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ungrq
*
*  =====================================================================
      SUBROUTINE dorgrq( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      parameter( zero = 0.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, II, IINFO, IWS, J, KK, L, LDWORK,
     $                   LWKOPT, NB, NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlarfb, dlarft, dorgr2, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ilaenv
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      lquery = ( lwork.EQ.-1 )
      IF( m.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.m ) THEN
         info = -2
      ELSE IF( k.LT.0 .OR. k.GT.m ) THEN
         info = -3
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -5
      END IF
*
      IF( info.EQ.0 ) THEN
         IF( m.LE.0 ) THEN
            lwkopt = 1
         ELSE
            nb = ilaenv( 1, 'DORGRQ', ' ', m, n, k, -1 )
            lwkopt = m*nb
         END IF
         work( 1 ) = lwkopt
*
         IF( lwork.LT.max( 1, m ) .AND. .NOT.lquery ) THEN
            info = -8
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DORGRQ', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( m.LE.0 ) THEN
         RETURN
      END IF
*
      nbmin = 2
      nx = 0
      iws = m
      IF( nb.GT.1 .AND. nb.LT.k ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         nx = max( 0, ilaenv( 3, 'DORGRQ', ' ', m, n, k, -1 ) )
         IF( nx.LT.k ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            ldwork = m
            iws = ldwork*nb
            IF( lwork.LT.iws ) THEN
*
*              Not enough workspace to use optimal NB:  reduce NB and
*              determine the minimum value of NB.
*
               nb = lwork / ldwork
               nbmin = max( 2, ilaenv( 2, 'DORGRQ', ' ', m, n, k, -1 ) )
            END IF
         END IF
      END IF
*
      IF( nb.GE.nbmin .AND. nb.LT.k .AND. nx.LT.k ) THEN
*
*        Use blocked code after the first block.
*        The last kk rows are handled by the block method.
*
         kk = min( k, ( ( k-nx+nb-1 ) / nb )*nb )
*
*        Set A(1:m-kk,n-kk+1:n) to zero.
*
         DO 20 j = n - kk + 1, n
            DO 10 i = 1, m - kk
               a( i, j ) = zero
   10       CONTINUE
   20    CONTINUE
      ELSE
         kk = 0
      END IF
*
*     Use unblocked code for the first or only block.
*
      CALL dorgr2( m-kk, n-kk, k-kk, a, lda, tau, work, iinfo )
*
      IF( kk.GT.0 ) THEN
*
*        Use blocked code
*
         DO 50 i = k - kk + 1, k, nb
            ib = min( nb, k-i+1 )
            ii = m - k + i
            IF( ii.GT.1 ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i+ib-1) . . . H(i+1) H(i)
*
               CALL dlarft( 'Backward', 'Rowwise', n-k+i+ib-1, ib,
     $                      a( ii, 1 ), lda, tau( i ), work, ldwork )
*
*              Apply H**T to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
*
               CALL dlarfb( 'Right', 'Transpose', 'Backward', 'Rowwise',
     $                      ii-1, n-k+i+ib-1, ib, a( ii, 1 ), lda, work,
     $                      ldwork, a, lda, work( ib+1 ), ldwork )
            END IF
*
*           Apply H**T to columns 1:n-k+i+ib-1 of current block
*
            CALL dorgr2( ib, n-k+i+ib-1, ib, a( ii, 1 ), lda, tau( i ),
     $                   work, iinfo )
*
*           Set columns n-k+i+ib:n of current block to zero
*
            DO 40 l = n - k + i + ib, n
               DO 30 j = ii, ii + ib - 1
                  a( j, l ) = zero
   30          CONTINUE
   40       CONTINUE
   50    CONTINUE
      END IF
*
      work( 1 ) = iws
      RETURN
*
*     End of DORGRQ
*
      END