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      SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )

*

*  -- LAPACK auxiliary routine (version 3.2) --

*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

*     November 2006

*

*     .. Scalar Arguments ..

      INTEGER            I

      DOUBLE PRECISION   DSIGMA, RHO

*     ..

*     .. Array Arguments ..

      DOUBLE PRECISION   D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )

*     ..

*

*  Purpose

*  =======

*

*  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.

*

*  Arguments

*  =========

*

*  I      (input) INTEGER

*         The index of the eigenvalue to be computed.  I = 1 or I = 2.

*

*  D      (input) DOUBLE PRECISION array, dimension ( 2 )

*         The original eigenvalues.  We assume 0 <= D(1) < D(2).

*

*  Z      (input) DOUBLE PRECISION array, dimension ( 2 )

*         The components of the updating vector.

*

*  DELTA  (output) 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.

*

*  RHO    (input) DOUBLE PRECISION

*         The scalar in the symmetric updating formula.

*

*  DSIGMA (output) DOUBLE PRECISION

*         The computed sigma_I, the I-th updated eigenvalue.

*

*  WORK   (workspace) DOUBLE PRECISION array, dimension ( 2 )

*         WORK contains (D(j) + sigma_I) in its  j-th component.

*

*  Further Details

*  ===============

*

*  Based on contributions by

*     Ren-Cang Li, Computer Science Division, University of California

*     at Berkeley, USA

*

*  =====================================================================

*

*     .. 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