190 lines
5.2 KiB
Fortran
190 lines
5.2 KiB
Fortran
*> \brief \b DLAED5 used by sstedc. Solves the 2-by-2 secular equation.
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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 DLAED5 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed5.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/dlaed5.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/dlaed5.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 DLAED5( I, D, Z, DELTA, RHO, DLAM )
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*
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* .. Scalar Arguments ..
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* INTEGER I
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* DOUBLE PRECISION DLAM, RHO
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION D( 2 ), DELTA( 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 I-th eigenvalue of a symmetric rank-one
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*> modification of a 2-by-2 diagonal matrix
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*>
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*> diag( D ) + RHO * Z * transpose(Z) .
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*>
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*> The diagonal elements in the array D are assumed to satisfy
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*>
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*> 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 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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*> 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] DLAM
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*> \verbatim
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*> DLAM is DOUBLE PRECISION
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*> The computed lambda_I, the I-th updated eigenvalue.
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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 auxOTHERcomputational
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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 DLAED5( I, D, Z, DELTA, RHO, DLAM )
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*
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* -- LAPACK computational 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 DLAM, RHO
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION D( 2 ), DELTA( 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, FOUR
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
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$ FOUR = 4.0D0 )
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* ..
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* .. Local Scalars ..
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DOUBLE PRECISION B, C, DEL, TAU, TEMP, 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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IF( I.EQ.1 ) THEN
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W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL
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IF( W.GT.ZERO ) THEN
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B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
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C = RHO*Z( 1 )*Z( 1 )*DEL
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*
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* B > ZERO, always
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*
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TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
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DLAM = D( 1 ) + TAU
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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 = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
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C = RHO*Z( 2 )*Z( 2 )*DEL
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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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DLAM = 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 = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
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C = RHO*Z( 2 )*Z( 2 )*DEL
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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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DLAM = 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 DLAED5
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*
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END
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