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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/
*
*> \htmlonly
*> Download DLAPLL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapll.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapll.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapll.f">
*> [TXT]</a>
*> \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 December 2016
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. 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
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