dgelqf()

subroutine dgelqf	(	integer	m,
		integer	n,
		double precision, dimension( lda, * )	a,
		integer	lda,
		double precision, dimension( * )	tau,
		double precision, dimension( * )	work,
		integer	lwork,
		integer	info )

DGELQF

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Purpose:

!>
!> DGELQF computes an LQ factorization of a real M-by-N matrix A:
!>
!>    A = ( L 0 ) *  Q
!>
!> where:
!>
!>    Q is a N-by-N orthogonal matrix;
!>    L is a lower-triangular M-by-M matrix;
!>    0 is a M-by-(N-M) zero matrix, if M < N.
!>
!> 

Parameters

[in]	M	!> M is INTEGER !> The number of rows of the matrix A. M >= 0. !>
[in]	N	!> N is INTEGER !> The number of columns of the matrix A. N >= 0. !>
[in,out]	A	!> A is DOUBLE PRECISION array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, the elements on and below the diagonal of the array !> contain the m-by-min(m,n) lower trapezoidal matrix L (L is !> lower triangular if m <= n); the elements above the diagonal, !> with the array TAU, represent the orthogonal matrix Q as a !> product of elementary reflectors (see Further Details). !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !>
[out]	TAU	!> TAU is DOUBLE PRECISION array, dimension (min(M,N)) !> The scalar factors of the elementary reflectors (see Further !> Details). !>
[out]	WORK	!> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
[in]	LWORK	!> 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. !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

!>
!>  The matrix Q is represented as a product of elementary reflectors
!>
!>     Q = H(k) . . . H(2) H(1), 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:n) is stored on exit in A(i,i+1:n),
!>  and tau in TAU(i).
!> 

Definition at line 141 of file dgelqf.f.

*
*  -- 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           dgelq2, dlarfb, dlarft, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ilaenv
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      k = min( m, n )
      nb = ilaenv( 1, 'DGELQF', ' ', 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. ( n.GT.0 .AND. lwork.LT.max( 1, m ) ) )
     $      info = -7
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DGELQF', -info )
         RETURN
      ELSE IF( lquery ) THEN
         IF( k.EQ.0 ) THEN
            lwkopt = 1
         ELSE
            lwkopt = m*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 = 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, 'DGELQF', ' ', 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, 'DGELQF', ' ', 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 LQ factorization of the current block
*           A(i:i+ib-1,i:n)
*
            CALL dgelq2( ib, n-i+1, a( i, i ), lda, tau( i ), work,
     $                   iinfo )
            IF( i+ib.LE.m ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i) H(i+1) . . . H(i+ib-1)
*
               CALL dlarft( 'Forward', 'Rowwise', n-i+1, ib, a( i,
     $                      i ),
     $                      lda, tau( i ), work, ldwork )
*
*              Apply H to A(i+ib:m,i:n) from the right
*
               CALL dlarfb( 'Right', 'No transpose', 'Forward',
     $                      'Rowwise', m-i-ib+1, n-i+1, ib, a( i, i ),
     $                      lda, work, ldwork, a( i+ib, i ), 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 dgelq2( m-i+1, n-i+1, a( i, i ), lda, tau( i ), work,
     $                iinfo )
*
      work( 1 ) = iws
      RETURN
*
*     End of DGELQF
*

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