```      SUBROUTINE SLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
*
*  -- LAPACK auxiliary routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
INTEGER            I
REAL               DSIGMA, RHO
*     ..
*     .. Array Arguments ..
REAL               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) REAL array, dimension (2)
*         The original eigenvalues.  We assume 0 <= D(1) < D(2).
*
*  Z      (input) REAL array, dimension (2)
*         The components of the updating vector.
*
*  DELTA  (output) REAL 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) REAL
*         The scalar in the symmetric updating formula.
*
*  DSIGMA (output) REAL
*         The computed sigma_I, the I-th updated eigenvalue.
*
*  WORK   (workspace) REAL 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 ..
REAL               ZERO, ONE, TWO, THREE, FOUR
PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0,
\$                   THREE = 3.0E+0, FOUR = 4.0E+0 )
*     ..
*     .. Local Scalars ..
REAL               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 SLASD5
*
END

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