sgerqf.f

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*> \brief \b SGERQF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SGERQF computes an RQ factorization of a real M-by-N matrix A:
*> A = R * Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is REAL 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 TAU, represent the
*>          orthogonal matrix Q as a product of min(m,n) 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] TAU
*> \verbatim
*>          TAU is REAL array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL 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 >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
*>          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 had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gerqf
*
*> \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 - tau * v * v**T
*>
*>  where tau is a real scalar, and v is a real 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 tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE sgerqf( M, N, 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, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
     $                   MU, NB, NBMIN, NU, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgerq2, slarfb, slarft, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      REAL               SROUNDUP_LWORK
      EXTERNAL           ilaenv, sroundup_lwork
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      lquery = ( lwork.EQ.-1 )
      IF( m.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -4
      END IF
*
      IF( info.EQ.0 ) THEN
         k = min( m, n )
         IF( k.EQ.0 ) THEN
            lwkopt = 1
         ELSE
            nb = ilaenv( 1, 'SGERQF', ' ', m, n, -1, -1 )
            lwkopt = m*nb
         END IF
         work( 1 ) = sroundup_lwork(lwkopt)
*
         IF ( .NOT.lquery ) THEN
            IF( lwork.LE.0 .OR. ( n.GT.0 .AND. lwork.LT.max( 1, m ) ) )
     $         info = -7
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SGERQF', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( k.EQ.0 ) THEN
         RETURN
      END IF
*
      nbmin = 2
      nx = 1
      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, 'SGERQF', ' ', m, n, -1, -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, 'SGERQF', ' ', m, n, -1,
     $                 -1 ) )
            END IF
         END IF
      END IF
*
      IF( nb.GE.nbmin .AND. nb.LT.k .AND. nx.LT.k ) THEN
*
*        Use blocked code initially.
*        The last kk rows are handled by the block method.
*
         ki = ( ( k-nx-1 ) / nb )*nb
         kk = min( k, ki+nb )
*
         DO 10 i = k - kk + ki + 1, k - kk + 1, -nb
            ib = min( k-i+1, nb )
*
*           Compute the RQ factorization of the current block
*           A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1)
*
            CALL sgerq2( ib, n-k+i+ib-1, a( m-k+i, 1 ), lda,
     $                   tau( i ),
     $                   work, iinfo )
            IF( m-k+i.GT.1 ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i+ib-1) . . . H(i+1) H(i)
*
               CALL slarft( 'Backward', 'Rowwise', n-k+i+ib-1, ib,
     $                      a( m-k+i, 1 ), lda, tau( i ), work, ldwork )
*
*              Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
*
               CALL slarfb( 'Right', 'No transpose', 'Backward',
     $                      'Rowwise', m-k+i-1, n-k+i+ib-1, ib,
     $                      a( m-k+i, 1 ), lda, work, ldwork, a, lda,
     $                      work( ib+1 ), ldwork )
            END IF
   10    CONTINUE
         mu = m - k + i + nb - 1
         nu = n - k + i + nb - 1
      ELSE
         mu = m
         nu = n
      END IF
*
*     Use unblocked code to factor the last or only block
*
      IF( mu.GT.0 .AND. nu.GT.0 )
     $   CALL sgerq2( mu, nu, a, lda, tau, work, iinfo )
*
      work( 1 ) = sroundup_lwork(iws)
      RETURN
*
*     End of SGERQF
*
      END