forked from lijiext/lammps
232 lines
6.8 KiB
FortranFixed
232 lines
6.8 KiB
FortranFixed
|
*> \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
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd5.f">
|
||
|
|
*> [TGZ]</a>
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd5.f">
|
||
|
|
*> [ZIP]</a>
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd5.f">
|
||
|
|
*> [TXT]</a>
|
||
|
|
*> \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.
|
||
|
|
*
|
||
|
|
*> \date December 2016
|
||
|
|
*
|
||
|
|
*> \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 (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 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
|