*> \brief \b DLASD5 computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix. Used by sbdsdc. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DLASD5 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK ) * * .. Scalar Arguments .. * INTEGER I * DOUBLE PRECISION DSIGMA, RHO * .. * .. Array Arguments .. * DOUBLE PRECISION D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> This subroutine computes the square root of the I-th eigenvalue *> of a positive symmetric rank-one modification of a 2-by-2 diagonal *> matrix *> *> diag( D ) * diag( D ) + RHO * Z * transpose(Z) . *> *> The diagonal entries in the array D are assumed to satisfy *> *> 0 <= D(i) < D(j) for i < j . *> *> We also assume RHO > 0 and that the Euclidean norm of the vector *> Z is one. *> \endverbatim * * Arguments: * ========== * *> \param[in] I *> \verbatim *> I is INTEGER *> The index of the eigenvalue to be computed. I = 1 or I = 2. *> \endverbatim *> *> \param[in] D *> \verbatim *> D is DOUBLE PRECISION array, dimension ( 2 ) *> The original eigenvalues. We assume 0 <= D(1) < D(2). *> \endverbatim *> *> \param[in] Z *> \verbatim *> Z is DOUBLE PRECISION array, dimension ( 2 ) *> The components of the updating vector. *> \endverbatim *> *> \param[out] DELTA *> \verbatim *> DELTA is DOUBLE PRECISION array, dimension ( 2 ) *> Contains (D(j) - sigma_I) in its j-th component. *> The vector DELTA contains the information necessary *> to construct the eigenvectors. *> \endverbatim *> *> \param[in] RHO *> \verbatim *> RHO is DOUBLE PRECISION *> The scalar in the symmetric updating formula. *> \endverbatim *> *> \param[out] DSIGMA *> \verbatim *> DSIGMA is DOUBLE PRECISION *> The computed sigma_I, the I-th updated eigenvalue. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension ( 2 ) *> WORK contains (D(j) + sigma_I) in its j-th component. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup OTHERauxiliary * *> \par Contributors: * ================== *> *> Ren-Cang Li, Computer Science Division, University of California *> at Berkeley, USA *> * ===================================================================== SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK ) * * -- LAPACK auxiliary routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER I DOUBLE PRECISION DSIGMA, RHO * .. * .. Array Arguments .. DOUBLE PRECISION D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0, $ THREE = 3.0D+0, FOUR = 4.0D+0 ) * .. * .. Local Scalars .. DOUBLE PRECISION B, C, DEL, DELSQ, TAU, W * .. * .. Intrinsic Functions .. INTRINSIC ABS, SQRT * .. * .. Executable Statements .. * DEL = D( 2 ) - D( 1 ) DELSQ = DEL*( D( 2 )+D( 1 ) ) IF( I.EQ.1 ) THEN W = ONE + FOUR*RHO*( Z( 2 )*Z( 2 ) / ( D( 1 )+THREE*D( 2 ) )- $ Z( 1 )*Z( 1 ) / ( THREE*D( 1 )+D( 2 ) ) ) / DEL IF( W.GT.ZERO ) THEN B = DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 1 )*Z( 1 )*DELSQ * * B > ZERO, always * * The following TAU is DSIGMA * DSIGMA - D( 1 ) * D( 1 ) * TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) ) * * The following TAU is DSIGMA - D( 1 ) * TAU = TAU / ( D( 1 )+SQRT( D( 1 )*D( 1 )+TAU ) ) DSIGMA = D( 1 ) + TAU DELTA( 1 ) = -TAU DELTA( 2 ) = DEL - TAU WORK( 1 ) = TWO*D( 1 ) + TAU WORK( 2 ) = ( D( 1 )+TAU ) + D( 2 ) * DELTA( 1 ) = -Z( 1 ) / TAU * DELTA( 2 ) = Z( 2 ) / ( DEL-TAU ) ELSE B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 2 )*Z( 2 )*DELSQ * * The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 ) * IF( B.GT.ZERO ) THEN TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) ) ELSE TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO END IF * * The following TAU is DSIGMA - D( 2 ) * TAU = TAU / ( D( 2 )+SQRT( ABS( D( 2 )*D( 2 )+TAU ) ) ) DSIGMA = D( 2 ) + TAU DELTA( 1 ) = -( DEL+TAU ) DELTA( 2 ) = -TAU WORK( 1 ) = D( 1 ) + TAU + D( 2 ) WORK( 2 ) = TWO*D( 2 ) + TAU * DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU ) * DELTA( 2 ) = -Z( 2 ) / TAU END IF * TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) ) * DELTA( 1 ) = DELTA( 1 ) / TEMP * DELTA( 2 ) = DELTA( 2 ) / TEMP ELSE * * Now I=2 * B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 2 )*Z( 2 )*DELSQ * * The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 ) * IF( B.GT.ZERO ) THEN TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO ELSE TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) ) END IF * * The following TAU is DSIGMA - D( 2 ) * TAU = TAU / ( D( 2 )+SQRT( D( 2 )*D( 2 )+TAU ) ) DSIGMA = D( 2 ) + TAU DELTA( 1 ) = -( DEL+TAU ) DELTA( 2 ) = -TAU WORK( 1 ) = D( 1 ) + TAU + D( 2 ) WORK( 2 ) = TWO*D( 2 ) + TAU * DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU ) * DELTA( 2 ) = -Z( 2 ) / TAU * TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) ) * DELTA( 1 ) = DELTA( 1 ) / TEMP * DELTA( 2 ) = DELTA( 2 ) / TEMP END IF RETURN * * End of DLASD5 * END