*> \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 *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \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