dgeqrf.f

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*> \brief \b DGEQRF
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQRF computes a QR factorization of a real M-by-N matrix A:
*>
*>    A = Q * ( R ),
*>            ( 0 )
*>
*> where:
*>
*>    Q is a M-by-M orthogonal matrix;
*>    R is an upper-triangular N-by-N matrix;
*>    0 is a (M-N)-by-N zero matrix, if M > N.
*>
*> \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 DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, the elements on and above the diagonal of the array
*>          contain the min(M,N)-by-N upper trapezoidal matrix R (R is
*>          upper triangular if m >= n); the elements below the diagonal,
*>          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 DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (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 >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
*>          For optimum performance LWORK >= N*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 geqrf
*
*> \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(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
*>  and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE dgeqrf( 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 ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
     $                   NBMIN, NX
*     ..
*     .. External Subroutines ..
      EXTERNAL           dgeqr2, dlarfb, dlarft, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ilaenv
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      k = min( m, n )
      info = 0
      nb = ilaenv( 1, 'DGEQRF', ' ', m, n, -1, -1 )
      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
      ELSE IF( .NOT.lquery ) THEN
         IF( lwork.LE.0 .OR. ( m.GT.0 .AND. lwork.LT.max( 1, n ) ) )
     $      info = -7
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DGEQRF', -info )
         RETURN
      ELSE IF( lquery ) THEN
         IF( k.EQ.0 ) THEN
            lwkopt = 1
         ELSE
            lwkopt = n*nb
         END IF
         work( 1 ) = lwkopt
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( k.EQ.0 ) THEN
         work( 1 ) = 1
         RETURN
      END IF
*
      nbmin = 2
      nx = 0
      iws = n
      IF( nb.GT.1 .AND. nb.LT.k ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
         nx = max( 0, ilaenv( 3, 'DGEQRF', ' ', m, n, -1, -1 ) )
         IF( nx.LT.k ) THEN
*
*           Determine if workspace is large enough for blocked code.
*
            ldwork = n
            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, 'DGEQRF', ' ', 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
*
         DO 10 i = 1, k - nx, nb
            ib = min( k-i+1, nb )
*
*           Compute the QR factorization of the current block
*           A(i:m,i:i+ib-1)
*
            CALL dgeqr2( m-i+1, ib, a( i, i ), lda, tau( i ), work,
     $                   iinfo )
            IF( i+ib.LE.n ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i) H(i+1) . . . H(i+ib-1)
*
               CALL dlarft( 'Forward', 'Columnwise', m-i+1, ib,
     $                      a( i, i ), lda, tau( i ), work, ldwork )
*
*              Apply H**T to A(i:m,i+ib:n) from the left
*
               CALL dlarfb( 'Left', 'Transpose', 'Forward',
     $                      'Columnwise', m-i+1, n-i-ib+1, ib,
     $                      a( i, i ), lda, work, ldwork, a( i, i+ib ),
     $                      lda, work( ib+1 ), ldwork )
            END IF
   10    CONTINUE
      ELSE
         i = 1
      END IF
*
*     Use unblocked code to factor the last or only block.
*
      IF( i.LE.k )
     $   CALL dgeqr2( m-i+1, n-i+1, a( i, i ), lda, tau( i ), work,
     $                iinfo )
*
      work( 1 ) = iws
      RETURN
*
*     End of DGEQRF
*
      END