subroutine sgelqf	(	integer	M,
		integer	N,
		real, dimension( lda, * )	A,
		integer	LDA,
		real, dimension( * )	TAU,
		real, dimension( * )	WORK,
		integer	LWORK,
		integer	INFO
	)

SGELQF

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Purpose:

 SGELQF computes an LQ factorization of a real M-by-N matrix A:
 A = L * Q.

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 REAL 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 REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details).
[out]	WORK	WORK is REAL 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 >= 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.
[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.

Date

November 2011

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 137 of file sgelqf.f.

*
*  -- LAPACK computational routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. 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, ldwork, lwkopt, nb,
     $                   nbmin, nx
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgelq2, 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, 'SGELQF', ' ', 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( 'SGELQF', -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, 'SGELQF', ' ', 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, 'SGELQF', ' ', 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 sgelq2( 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 slarft( '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 slarfb( '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 sgelq2( m-i+1, n-i+1, a( i, i ), lda, tau( i ), work,
     $                iinfo )
*
      work( 1 ) = iws
      RETURN
*
*     End of SGELQF
*

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