dlapll.f

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*> \brief \b DLAPLL measures the linear dependence of two vectors.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
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*> \endhtmlonly 
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
* 
*       .. Scalar Arguments ..
*       INTEGER            INCX, INCY, N
*       DOUBLE PRECISION   SSMIN
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   X( * ), Y( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Given two column vectors X and Y, let
*>
*>                      A = ( X Y ).
*>
*> The subroutine first computes the QR factorization of A = Q*R,
*> and then computes the SVD of the 2-by-2 upper triangular matrix R.
*> The smaller singular value of R is returned in SSMIN, which is used
*> as the measurement of the linear dependency of the vectors X and Y.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The length of the vectors X and Y.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCX)
*>          On entry, X contains the N-vector X.
*>          On exit, X is overwritten.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between successive elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCY)
*>          On entry, Y contains the N-vector Y.
*>          On exit, Y is overwritten.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*>          INCY is INTEGER
*>          The increment between successive elements of Y. INCY > 0.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*>          SSMIN is DOUBLE PRECISION
*>          The smallest singular value of the N-by-2 matrix A = ( X Y ).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*  =====================================================================
      SUBROUTINE dlapll( N, X, INCX, Y, INCY, SSMIN )
*
*  -- LAPACK auxiliary routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      INTEGER            incx, incy, n
      DOUBLE PRECISION   ssmin
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   x( * ), y( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   zero, one
      parameter( zero = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   a11, a12, a22, c, ssmax, tau
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   ddot
      EXTERNAL           ddot
*     ..
*     .. External Subroutines ..
      EXTERNAL           daxpy, dlarfg, dlas2
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( n.LE.1 ) THEN
         ssmin = zero
         return
      END IF
*
*     Compute the QR factorization of the N-by-2 matrix ( X Y )
*
      CALL dlarfg( n, x( 1 ), x( 1+incx ), incx, tau )
      a11 = x( 1 )
      x( 1 ) = one
*
      c = -tau*ddot( n, x, incx, y, incy )
      CALL daxpy( n, c, x, incx, y, incy )
*
      CALL dlarfg( n-1, y( 1+incy ), y( 1+2*incy ), incy, tau )
*
      a12 = y( 1 )
      a22 = y( 1+incy )
*
*     Compute the SVD of 2-by-2 Upper triangular matrix.
*
      CALL dlas2( a11, a12, a22, ssmin, ssmax )
*
      return
*
*     End of DLAPLL
*
      END