dlasd5.f

          SUBROUTINE DLASD5( 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
          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