      SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
     $                   DSIGMA, WORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
*     Courant Institute, NAG Ltd., and Rice University
*     June 30, 1999
*
*     .. 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*DLAMBDA(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
