sgelq2.f

*> \brief \b SGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SGELQ2( M, N, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SGELQ2 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.
*>
*> \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 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).
*> \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 REAL array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (M)
*> \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 gelq2
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  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).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE sgelq2( M, N, A, LDA, TAU, WORK, 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, M, N
*     ..
*     .. Array Arguments ..
      REAL               A( lda, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE
      parameter( one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K
      REAL               AII
*     ..
*     .. External Subroutines ..
      EXTERNAL           slarf, slarfg, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      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
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SGELQ2', -info )
         RETURN
      END IF
*
      k = min( m, n )
*
      DO 10 i = 1, k
*
*        Generate elementary reflector H(i) to annihilate A(i,i+1:n)
*
         CALL slarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
     $                tau( i ) )
         IF( i.LT.m ) THEN
*
*           Apply H(i) to A(i+1:m,i:n) from the right
*
            aii = a( i, i )
            a( i, i ) = one
            CALL slarf( 'Right', m-i, n-i+1, a( i, i ), lda, tau( i ),
     $                  a( i+1, i ), lda, work )
            a( i, i ) = aii
         END IF
   10 CONTINUE
      RETURN
*
*     End of SGELQ2
*
      END