```      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

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