          SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INCX, INCY, N
          DOUBLE PRECISION   SSMIN
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   X( * ), Y( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  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.
    *
    *  Arguments
    *  =========
    *
    *  N       (input) INTEGER
    *          The length of the vectors X and Y.
    *
    *  X       (input/output) DOUBLE PRECISION array,
    *                         dimension (1+(N-1)*INCX)
    *          On entry, X contains the N-vector X.
    *          On exit, X is overwritten.
    *
    *  INCX    (input) INTEGER
    *          The increment between successive elements of X. INCX > 0.
    *
    *  Y       (input/output) DOUBLE PRECISION array,
    *                         dimension (1+(N-1)*INCY)
    *          On entry, Y contains the N-vector Y.
    *          On exit, Y is overwritten.
    *
    *  INCY    (input) INTEGER
    *          The increment between successive elements of Y. INCY > 0.
    *
    *  SSMIN   (output) DOUBLE PRECISION
    *          The smallest singular value of the N-by-2 matrix A = ( 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