dgeql2.f

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*> \brief \b DGEQL2 computes the QL 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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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGEQL2( M, N, A, LDA, TAU, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGEQL2 computes a QL factorization of a real m by n matrix A:
*> A = Q * L.
*> \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 DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the m by n matrix A.
*>          On exit, if m >= n, the lower triangle of the subarray
*>          A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
*>          if m <= n, the elements on and below the (n-m)-th
*>          superdiagonal contain the m by n lower trapezoidal matrix L;
*>          the remaining elements, 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 DOUBLE PRECISION array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors (see Further
*>          Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \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 doubleGEcomputational
*
*> \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(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
*>  A(1:m-k+i-1,n-k+i), and tau in TAU(i).
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE dgeql2( 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 ..
      DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      parameter( one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K
      DOUBLE PRECISION   AII
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlarf, dlarfg, 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( 'DGEQL2', -info )
         RETURN
      END IF
*
      k = min( m, n )
*
      DO 10 i = k, 1, -1
*
*        Generate elementary reflector H(i) to annihilate
*        A(1:m-k+i-1,n-k+i)
*
         CALL dlarfg( m-k+i, a( m-k+i, n-k+i ), a( 1, n-k+i ), 1,
     $                tau( i ) )
*
*        Apply H(i) to A(1:m-k+i,1:n-k+i-1) from the left
*
         aii = a( m-k+i, n-k+i )
         a( m-k+i, n-k+i ) = one
         CALL dlarf( 'Left', m-k+i, n-k+i-1, a( 1, n-k+i ), 1, tau( i ),
     $               a, lda, work )
         a( m-k+i, n-k+i ) = aii
   10 CONTINUE
      RETURN
*
*     End of DGEQL2
*
      END