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    1:       SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK )
    2: *
    3: *  -- LAPACK auxiliary routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       INTEGER            I
   10:       DOUBLE PRECISION   DSIGMA, RHO
   11: *     ..
   12: *     .. Array Arguments ..
   13:       DOUBLE PRECISION   D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 )
   14: *     ..
   15: *
   16: *  Purpose
   17: *  =======
   18: *
   19: *  This subroutine computes the square root of the I-th eigenvalue
   20: *  of a positive symmetric rank-one modification of a 2-by-2 diagonal
   21: *  matrix
   22: *
   23: *             diag( D ) * diag( D ) +  RHO *  Z * transpose(Z) .
   24: *
   25: *  The diagonal entries in the array D are assumed to satisfy
   26: *
   27: *             0 <= D(i) < D(j)  for  i < j .
   28: *
   29: *  We also assume RHO > 0 and that the Euclidean norm of the vector
   30: *  Z is one.
   31: *
   32: *  Arguments
   33: *  =========
   34: *
   35: *  I      (input) INTEGER
   36: *         The index of the eigenvalue to be computed.  I = 1 or I = 2.
   37: *
   38: *  D      (input) DOUBLE PRECISION array, dimension ( 2 )
   39: *         The original eigenvalues.  We assume 0 <= D(1) < D(2).
   40: *
   41: *  Z      (input) DOUBLE PRECISION array, dimension ( 2 )
   42: *         The components of the updating vector.
   43: *
   44: *  DELTA  (output) DOUBLE PRECISION array, dimension ( 2 )
   45: *         Contains (D(j) - sigma_I) in its  j-th component.
   46: *         The vector DELTA contains the information necessary
   47: *         to construct the eigenvectors.
   48: *
   49: *  RHO    (input) DOUBLE PRECISION
   50: *         The scalar in the symmetric updating formula.
   51: *
   52: *  DSIGMA (output) DOUBLE PRECISION
   53: *         The computed sigma_I, the I-th updated eigenvalue.
   54: *
   55: *  WORK   (workspace) DOUBLE PRECISION array, dimension ( 2 )
   56: *         WORK contains (D(j) + sigma_I) in its  j-th component.
   57: *
   58: *  Further Details
   59: *  ===============
   60: *
   61: *  Based on contributions by
   62: *     Ren-Cang Li, Computer Science Division, University of California
   63: *     at Berkeley, USA
   64: *
   65: *  =====================================================================
   66: *
   67: *     .. Parameters ..
   68:       DOUBLE PRECISION   ZERO, ONE, TWO, THREE, FOUR
   69:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
   70:      $                   THREE = 3.0D+0, FOUR = 4.0D+0 )
   71: *     ..
   72: *     .. Local Scalars ..
   73:       DOUBLE PRECISION   B, C, DEL, DELSQ, TAU, W
   74: *     ..
   75: *     .. Intrinsic Functions ..
   76:       INTRINSIC          ABS, SQRT
   77: *     ..
   78: *     .. Executable Statements ..
   79: *
   80:       DEL = D( 2 ) - D( 1 )
   81:       DELSQ = DEL*( D( 2 )+D( 1 ) )
   82:       IF( I.EQ.1 ) THEN
   83:          W = ONE + FOUR*RHO*( Z( 2 )*Z( 2 ) / ( D( 1 )+THREE*D( 2 ) )-
   84:      $       Z( 1 )*Z( 1 ) / ( THREE*D( 1 )+D( 2 ) ) ) / DEL
   85:          IF( W.GT.ZERO ) THEN
   86:             B = DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
   87:             C = RHO*Z( 1 )*Z( 1 )*DELSQ
   88: *
   89: *           B > ZERO, always
   90: *
   91: *           The following TAU is DSIGMA * DSIGMA - D( 1 ) * D( 1 )
   92: *
   93:             TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) )
   94: *
   95: *           The following TAU is DSIGMA - D( 1 )
   96: *
   97:             TAU = TAU / ( D( 1 )+SQRT( D( 1 )*D( 1 )+TAU ) )
   98:             DSIGMA = D( 1 ) + TAU
   99:             DELTA( 1 ) = -TAU
  100:             DELTA( 2 ) = DEL - TAU
  101:             WORK( 1 ) = TWO*D( 1 ) + TAU
  102:             WORK( 2 ) = ( D( 1 )+TAU ) + D( 2 )
  103: *           DELTA( 1 ) = -Z( 1 ) / TAU
  104: *           DELTA( 2 ) = Z( 2 ) / ( DEL-TAU )
  105:          ELSE
  106:             B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
  107:             C = RHO*Z( 2 )*Z( 2 )*DELSQ
  108: *
  109: *           The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
  110: *
  111:             IF( B.GT.ZERO ) THEN
  112:                TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) )
  113:             ELSE
  114:                TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO
  115:             END IF
  116: *
  117: *           The following TAU is DSIGMA - D( 2 )
  118: *
  119:             TAU = TAU / ( D( 2 )+SQRT( ABS( D( 2 )*D( 2 )+TAU ) ) )
  120:             DSIGMA = D( 2 ) + TAU
  121:             DELTA( 1 ) = -( DEL+TAU )
  122:             DELTA( 2 ) = -TAU
  123:             WORK( 1 ) = D( 1 ) + TAU + D( 2 )
  124:             WORK( 2 ) = TWO*D( 2 ) + TAU
  125: *           DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
  126: *           DELTA( 2 ) = -Z( 2 ) / TAU
  127:          END IF
  128: *        TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
  129: *        DELTA( 1 ) = DELTA( 1 ) / TEMP
  130: *        DELTA( 2 ) = DELTA( 2 ) / TEMP
  131:       ELSE
  132: *
  133: *        Now I=2
  134: *
  135:          B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) )
  136:          C = RHO*Z( 2 )*Z( 2 )*DELSQ
  137: *
  138: *        The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 )
  139: *
  140:          IF( B.GT.ZERO ) THEN
  141:             TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO
  142:          ELSE
  143:             TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) )
  144:          END IF
  145: *
  146: *        The following TAU is DSIGMA - D( 2 )
  147: *
  148:          TAU = TAU / ( D( 2 )+SQRT( D( 2 )*D( 2 )+TAU ) )
  149:          DSIGMA = D( 2 ) + TAU
  150:          DELTA( 1 ) = -( DEL+TAU )
  151:          DELTA( 2 ) = -TAU
  152:          WORK( 1 ) = D( 1 ) + TAU + D( 2 )
  153:          WORK( 2 ) = TWO*D( 2 ) + TAU
  154: *        DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU )
  155: *        DELTA( 2 ) = -Z( 2 ) / TAU
  156: *        TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) )
  157: *        DELTA( 1 ) = DELTA( 1 ) / TEMP
  158: *        DELTA( 2 ) = DELTA( 2 ) / TEMP
  159:       END IF
  160:       RETURN
  161: *
  162: *     End of DLASD5
  163: *
  164:       END