          SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
         $                   DSIGMA, WORK, INFO )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            ICOMPQ, INFO, K, LDDIFR
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   D( * ), DIFL( * ), DIFR( LDDIFR, * ),
         $                   DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
         $                   Z( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLASD8 finds the square roots of the roots of the secular equation,
    *  as defined by the values in DSIGMA and Z. It makes the appropriate
    *  calls to DLASD4, and stores, for each  element in D, the distance
    *  to its two nearest poles (elements in DSIGMA). It also updates
    *  the arrays VF and VL, the first and last components of all the
    *  right singular vectors of the original bidiagonal matrix.
    *
    *  DLASD8 is called from DLASD6.
    *
    *  Arguments
    *  =========
    *
    *  ICOMPQ  (input) INTEGER
    *          Specifies whether singular vectors are to be computed in
    *          factored form in the calling routine:
    *          = 0: Compute singular values only.
    *          = 1: Compute singular vectors in factored form as well.
    *
    *  K       (input) INTEGER
    *          The number of terms in the rational function to be solved
    *          by DLASD4.  K >= 1.
    *
    *  D       (output) DOUBLE PRECISION array, dimension ( K )
    *          On output, D contains the updated singular values.
    *
    *  Z       (input) DOUBLE PRECISION array, dimension ( K )
    *          The first K elements of this array contain the components
    *          of the deflation-adjusted updating row vector.
    *
    *  VF      (input/output) DOUBLE PRECISION array, dimension ( K )
    *          On entry, VF contains  information passed through DBEDE8.
    *          On exit, VF contains the first K components of the first
    *          components of all right singular vectors of the bidiagonal
    *          matrix.
    *
    *  VL      (input/output) DOUBLE PRECISION array, dimension ( K )
    *          On entry, VL contains  information passed through DBEDE8.
    *          On exit, VL contains the first K components of the last
    *          components of all right singular vectors of the bidiagonal
    *          matrix.
    *
    *  DIFL    (output) DOUBLE PRECISION array, dimension ( K )
    *          On exit, DIFL(I) = D(I) - DSIGMA(I).
    *
    *  DIFR    (output) DOUBLE PRECISION array,
    *                   dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
    *                   dimension ( K ) if ICOMPQ = 0.
    *          On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
    *          defined and will not be referenced.
    *
    *          If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
    *          normalizing factors for the right singular vector matrix.
    *
    *  LDDIFR  (input) INTEGER
    *          The leading dimension of DIFR, must be at least K.
    *
    *  DSIGMA  (input) DOUBLE PRECISION array, dimension ( K )
    *          The first K elements of this array contain the old roots
    *          of the deflated updating problem.  These are the poles
    *          of the secular equation.
    *
    *  WORK    (workspace) DOUBLE PRECISION array, dimension at least 3 * K
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit.
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *          > 0:  if INFO = 1, an singular value did not converge
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Ming Gu and Huan Ren, Computer Science Division, University of
    *     California at Berkeley, USA
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE
          PARAMETER          ( ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J
          DOUBLE PRECISION   DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, RHO, TEMP
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DCOPY, DLASCL, DLASD4, DLASET, XERBLA
    *     ..
    *     .. External Functions ..
          DOUBLE PRECISION   DDOT, DLAMC3, DNRM2
          EXTERNAL           DDOT, DLAMC3, DNRM2
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, SIGN, SQRT
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
    *
          IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
             INFO = -1
          ELSE IF( K.LT.1 ) THEN
             INFO = -2
          ELSE IF( LDDIFR.LT.K ) THEN
             INFO = -9
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DLASD8', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( K.EQ.1 ) THEN
             D( 1 ) = ABS( Z( 1 ) )
             DIFL( 1 ) = D( 1 )
             IF( ICOMPQ.EQ.1 ) THEN
                DIFL( 2 ) = ONE
                DIFR( 1, 2 ) = ONE
             END IF
             RETURN
          END IF
    *
    *     Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can
    *     be computed with high relative accuracy (barring over/underflow).
    *     This is a problem on machines without a guard digit in
    *     add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
    *     The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),
    *     which on any of these machines zeros out the bottommost
    *     bit of DSIGMA(I) if it is 1; this makes the subsequent
    *     subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation
    *     occurs. On binary machines with a guard digit (almost all
    *     machines) it does not change DSIGMA(I) at all. On hexadecimal
    *     and decimal machines with a guard digit, it slightly
    *     changes the bottommost bits of DSIGMA(I). It does not account
    *     for hexadecimal or decimal machines without guard digits
    *     (we know of none). We use a subroutine call to compute
    *     2*DSIGMA(I) to prevent optimizing compilers from eliminating
    *     this code.
    *
          DO 10 I = 1, K
             DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I )
       10 CONTINUE
    *
    *     Book keeping.
    *
          IWK1 = 1
          IWK2 = IWK1 + K
          IWK3 = IWK2 + K
          IWK2I = IWK2 - 1
          IWK3I = IWK3 - 1
    *
    *     Normalize Z.
    *
          RHO = DNRM2( K, Z, 1 )
          CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
          RHO = RHO*RHO
    *
    *     Initialize WORK(IWK3).
    *
          CALL DLASET( 'A', K, 1, ONE, ONE, WORK( IWK3 ), K )
    *
    *     Compute the updated singular values, the arrays DIFL, DIFR,
    *     and the updated Z.
    *
          DO 40 J = 1, K
             CALL DLASD4( K, J, DSIGMA, Z, WORK( IWK1 ), RHO, D( J ),
         $                WORK( IWK2 ), INFO )
    *
    *        If the root finder fails, the computation is terminated.
    *
             IF( INFO.NE.0 ) THEN
                RETURN
             END IF
             WORK( IWK3I+J ) = WORK( IWK3I+J )*WORK( J )*WORK( IWK2I+J )
             DIFL( J ) = -WORK( J )
             DIFR( J, 1 ) = -WORK( J+1 )
             DO 20 I = 1, J - 1
                WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
         $                        WORK( IWK2I+I ) / ( DSIGMA( I )-
         $                        DSIGMA( J ) ) / ( DSIGMA( I )+
         $                        DSIGMA( J ) )
       20    CONTINUE
             DO 30 I = J + 1, K
                WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
         $                        WORK( IWK2I+I ) / ( DSIGMA( I )-
         $                        DSIGMA( J ) ) / ( DSIGMA( I )+
         $                        DSIGMA( J ) )
       30    CONTINUE
       40 CONTINUE
    *
    *     Compute updated Z.
    *
          DO 50 I = 1, K
             Z( I ) = SIGN( SQRT( ABS( WORK( IWK3I+I ) ) ), Z( I ) )
       50 CONTINUE
    *
    *     Update VF and VL.
    *
          DO 80 J = 1, K
             DIFLJ = DIFL( J )
             DJ = D( J )
             DSIGJ = -DSIGMA( J )
             IF( J.LT.K ) THEN
                DIFRJ = -DIFR( J, 1 )
                DSIGJP = -DSIGMA( J+1 )
             END IF
             WORK( J ) = -Z( J ) / DIFLJ / ( DSIGMA( J )+DJ )
             DO 60 I = 1, J - 1
                WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJ )-DIFLJ )
         $                   / ( DSIGMA( I )+DJ )
       60    CONTINUE
             DO 70 I = J + 1, K
                WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJP )+DIFRJ )
         $                   / ( DSIGMA( I )+DJ )
       70    CONTINUE
             TEMP = DNRM2( K, WORK, 1 )
             WORK( IWK2I+J ) = DDOT( K, WORK, 1, VF, 1 ) / TEMP
             WORK( IWK3I+J ) = DDOT( K, WORK, 1, VL, 1 ) / TEMP
             IF( ICOMPQ.EQ.1 ) THEN
                DIFR( J, 2 ) = TEMP
             END IF
       80 CONTINUE
    *
          CALL DCOPY( K, WORK( IWK2 ), 1, VF, 1 )
          CALL DCOPY( K, WORK( IWK3 ), 1, VL, 1 )
    *
          RETURN
    *
    *     End of DLASD8
    *
          END