```      SUBROUTINE SGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
REAL               A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  SGEQRF computes a QR factorization of a real M-by-N matrix A:
*  A = Q * R.
*
*  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) REAL 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).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  TAU     (output) REAL array, dimension (min(M,N))
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  WORK    (workspace/output) REAL 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,N).
*          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.
*
*  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(1) H(2) . . . H(k), 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:m) is stored on exit in A(i+1:m,i),
*  and tau in TAU(i).
*
*  =====================================================================
*
*     .. Local Scalars ..
LOGICAL            LQUERY
INTEGER            I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
\$                   NBMIN, NX
*     ..
*     .. External Subroutines ..
EXTERNAL           SGEQR2, SLARFB, SLARFT, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX, MIN
*     ..
*     .. External Functions ..
INTEGER            ILAENV
EXTERNAL           ILAENV
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
INFO = 0
NB = ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 )
LWKOPT = N*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, N ) .AND. .NOT.LQUERY ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGEQRF', -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 = 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, 'SGEQRF', ' ', 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, 'SGEQRF', ' ', 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 SGEQR2( 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 SLARFT( 'Forward', 'Columnwise', M-I+1, IB,
\$                      A( I, I ), LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H' to A(i:m,i+ib:n) from the left
*
CALL SLARFB( '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 SGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
\$                IINFO )
*
WORK( 1 ) = IWS
RETURN
*
*     End of SGEQRF
*
END

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