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*> \brief \b SLAPLL
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLAPLL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slapll.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slapll.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slapll.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition
* ==========
*
* SUBROUTINE SLAPLL( N, X, INCX, Y, INCY, SSMIN )
*
* .. Scalar Arguments ..
* INTEGER INCX, INCY, N
* REAL SSMIN
* ..
* .. Array Arguments ..
* REAL X( * ), Y( * )
* ..
*
* Purpose
* =======
*
*>\details \b 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 REAL 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 REAL 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 REAL
*> 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 November 2011
*
*> \ingroup realOTHERauxiliary
*
* =====================================================================
SUBROUTINE SLAPLL( N, X, INCX, Y, INCY, SSMIN )
*
* -- LAPACK auxiliary routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INCX, INCY, N
REAL SSMIN
* ..
* .. Array Arguments ..
REAL X( * ), Y( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
REAL A11, A12, A22, C, SSMAX, TAU
* ..
* .. External Functions ..
REAL SDOT
EXTERNAL SDOT
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SLARFG, SLAS2
* ..
* .. 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 SLARFG( N, X( 1 ), X( 1+INCX ), INCX, TAU )
A11 = X( 1 )
X( 1 ) = ONE
*
C = -TAU*SDOT( N, X, INCX, Y, INCY )
CALL SAXPY( N, C, X, INCX, Y, INCY )
*
CALL SLARFG( 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 SLAS2( A11, A12, A22, SSMIN, SSMAX )
*
RETURN
*
* End of SLAPLL
*
END
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