forked from lijiext/lammps
232 lines
6.8 KiB
Fortran
232 lines
6.8 KiB
Fortran
*> \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.
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download DLASD5 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd5.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd5.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd5.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
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*
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* .. Scalar Arguments ..
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* INTEGER I
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* DOUBLE PRECISION DSIGMA, RHO
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> This subroutine computes the square root of the I-th eigenvalue
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*> of a positive symmetric rank-one modification of a 2-by-2 diagonal
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*> matrix
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*>
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*> diag( D ) * diag( D ) + RHO * Z * transpose(Z) .
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*>
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*> The diagonal entries in the array D are assumed to satisfy
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*>
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*> 0 <= D(i) < D(j) for i < j .
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*>
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*> We also assume RHO > 0 and that the Euclidean norm of the vector
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*> Z is one.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] I
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*> \verbatim
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*> I is INTEGER
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*> The index of the eigenvalue to be computed. I = 1 or I = 2.
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*> \endverbatim
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*>
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*> \param[in] D
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*> \verbatim
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*> D is DOUBLE PRECISION array, dimension ( 2 )
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*> The original eigenvalues. We assume 0 <= D(1) < D(2).
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*> \endverbatim
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*>
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*> \param[in] Z
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*> \verbatim
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*> Z is DOUBLE PRECISION array, dimension ( 2 )
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*> The components of the updating vector.
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*> \endverbatim
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*>
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*> \param[out] DELTA
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*> \verbatim
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*> DELTA is DOUBLE PRECISION array, dimension ( 2 )
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*> Contains (D(j) - sigma_I) in its j-th component.
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*> The vector DELTA contains the information necessary
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*> to construct the eigenvectors.
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*> \endverbatim
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*>
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*> \param[in] RHO
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*> \verbatim
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*> RHO is DOUBLE PRECISION
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*> The scalar in the symmetric updating formula.
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*> \endverbatim
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*>
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*> \param[out] DSIGMA
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*> \verbatim
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*> DSIGMA is DOUBLE PRECISION
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*> The computed sigma_I, the I-th updated eigenvalue.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is DOUBLE PRECISION array, dimension ( 2 )
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*> WORK contains (D(j) + sigma_I) in its j-th component.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date December 2016
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*
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*> \ingroup OTHERauxiliary
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*
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*> \par Contributors:
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* ==================
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*>
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*> Ren-Cang Li, Computer Science Division, University of California
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*> at Berkeley, USA
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*>
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* =====================================================================
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SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
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*
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* -- LAPACK auxiliary routine (version 3.7.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* December 2016
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*
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* .. Scalar Arguments ..
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INTEGER I
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DOUBLE PRECISION DSIGMA, RHO
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
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$ THREE = 3.0D+0, FOUR = 4.0D+0 )
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* ..
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* .. Local Scalars ..
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DOUBLE PRECISION B, C, DEL, DELSQ, TAU, W
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, SQRT
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* ..
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* .. Executable Statements ..
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*
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DEL = D( 2 ) - D( 1 )
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DELSQ = DEL*( D( 2 )+D( 1 ) )
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IF( I.EQ.1 ) THEN
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W = ONE + FOUR*RHO*( Z( 2 )*Z( 2 ) / ( D( 1 )+THREE*D( 2 ) )-
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$ Z( 1 )*Z( 1 ) / ( THREE*D( 1 )+D( 2 ) ) ) / DEL
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IF( W.GT.ZERO ) THEN
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B = DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
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C = RHO*Z( 1 )*Z( 1 )*DELSQ
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*
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* B > ZERO, always
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*
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* The following TAU is DSIGMA * DSIGMA - D( 1 ) * D( 1 )
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*
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TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
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*
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* The following TAU is DSIGMA - D( 1 )
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*
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TAU = TAU / ( D( 1 )+SQRT( D( 1 )*D( 1 )+TAU ) )
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DSIGMA = D( 1 ) + TAU
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DELTA( 1 ) = -TAU
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DELTA( 2 ) = DEL - TAU
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WORK( 1 ) = TWO*D( 1 ) + TAU
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WORK( 2 ) = ( D( 1 )+TAU ) + D( 2 )
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* DELTA( 1 ) = -Z( 1 ) / TAU
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* DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
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ELSE
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B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
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C = RHO*Z( 2 )*Z( 2 )*DELSQ
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*
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* The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
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*
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IF( B.GT.ZERO ) THEN
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TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
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ELSE
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TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
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END IF
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*
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* The following TAU is DSIGMA - D( 2 )
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*
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TAU = TAU / ( D( 2 )+SQRT( ABS( D( 2 )*D( 2 )+TAU ) ) )
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DSIGMA = D( 2 ) + TAU
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DELTA( 1 ) = -( DEL+TAU )
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DELTA( 2 ) = -TAU
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WORK( 1 ) = D( 1 ) + TAU + D( 2 )
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WORK( 2 ) = TWO*D( 2 ) + TAU
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* DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
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* DELTA( 2 ) = -Z( 2 ) / TAU
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END IF
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* TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
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* DELTA( 1 ) = DELTA( 1 ) / TEMP
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* DELTA( 2 ) = DELTA( 2 ) / TEMP
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ELSE
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*
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* Now I=2
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*
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B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
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C = RHO*Z( 2 )*Z( 2 )*DELSQ
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*
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* The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
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*
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IF( B.GT.ZERO ) THEN
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TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
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ELSE
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TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
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END IF
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*
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* The following TAU is DSIGMA - D( 2 )
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*
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TAU = TAU / ( D( 2 )+SQRT( D( 2 )*D( 2 )+TAU ) )
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DSIGMA = D( 2 ) + TAU
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DELTA( 1 ) = -( DEL+TAU )
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DELTA( 2 ) = -TAU
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WORK( 1 ) = D( 1 ) + TAU + D( 2 )
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WORK( 2 ) = TWO*D( 2 ) + TAU
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* DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
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* DELTA( 2 ) = -Z( 2 ) / TAU
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* TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
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* DELTA( 1 ) = DELTA( 1 ) / TEMP
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* DELTA( 2 ) = DELTA( 2 ) / TEMP
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END IF
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RETURN
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*
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* End of DLASD5
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*
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END
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