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*> \brief \b DLAED5 used by sstedc. Solves the 2-by-2 secular equation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAED5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed5.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
*
* .. Scalar Arguments ..
* INTEGER I
* DOUBLE PRECISION DLAM, RHO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( 2 ), DELTA( 2 ), Z( 2 )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine computes the I-th eigenvalue of a symmetric rank-one
*> modification of a 2-by-2 diagonal matrix
*>
*> diag( D ) + RHO * Z * transpose(Z) .
*>
*> The diagonal elements in the array D are assumed to satisfy
*>
*> 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 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)
*> 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] DLAM
*> \verbatim
*> DLAM is DOUBLE PRECISION
*> The computed lambda_I, the I-th updated eigenvalue.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM )
*
* -- LAPACK computational 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 DLAM, RHO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( 2 ), DELTA( 2 ), Z( 2 )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, FOUR
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ FOUR = 4.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION B, C, DEL, TAU, TEMP, W
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
DEL = D( 2 ) - D( 1 )
IF( I.EQ.1 ) THEN
W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL
IF( W.GT.ZERO ) THEN
B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 1 )*Z( 1 )*DEL
*
* B > ZERO, always
*
TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
DLAM = D( 1 ) + TAU
DELTA( 1 ) = -Z( 1 ) / TAU
DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
ELSE
B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 2 )*Z( 2 )*DEL
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
DLAM = 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 = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
C = RHO*Z( 2 )*Z( 2 )*DEL
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
DLAM = 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 DLAED5
*
END
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