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      SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )

*

*  -- LAPACK routine (version 3.2) --

*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

*     November 2006

*

*     .. Scalar Arguments ..

      INTEGER            INFO, LDA, LWORK, M, N

*     ..

*     .. Array Arguments ..

      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )

*     ..

*

*  Purpose

*  =======

*

*  DGELQF computes an LQ factorization of a real M-by-N matrix A:

*  A = L * Q.

*

*  Arguments

*  =========

*

*  M       (input) INTEGER

*          The number of rows of the matrix A.  M >= 0.

*

*  N       (input) INTEGER

*          The number of columns of the matrix A.  N >= 0.

*

*  A       (input/output) 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).

*

*  LDA     (input) INTEGER

*          The leading dimension of the array A.  LDA >= max(1,M).

*

*  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))

*          The scalar factors of the elementary reflectors (see Further

*          Details).

*

*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))

*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*

*  LWORK   (input) 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.

*

*  INFO    (output) INTEGER

*          = 0:  successful exit

*          < 0:  if INFO = -i, the i-th argument had an illegal value

*

*  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'

*

*  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).

*

*  =====================================================================

*

*     .. 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

      NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )

      LWKOPT = M*NB

      WORK( 1 ) = LWKOPT

      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( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN

         INFO = -7

      END IF

      IF( INFO.NE.0 ) THEN

         CALL XERBLA( 'DGELQF', -INFO )

         RETURN

      ELSE IF( LQUERY ) THEN

         RETURN

      END IF

*

*     Quick return if possible

*

      K = MIN( M, N )

      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

*

      END