```      SUBROUTINE SLASD8( 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 ..
REAL               D( * ), DIFL( * ), DIFR( LDDIFR, * ),
\$                   DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
\$                   Z( * )
*     ..
*
*  Purpose
*  =======
*
*  SLASD8 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 SLASD4, 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.
*
*  SLASD8 is called from SLASD6.
*
*  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 SLASD4.  K >= 1.
*
*  D       (output) REAL array, dimension ( K )
*          On output, D contains the updated singular values.
*
*  Z       (input) REAL array, dimension ( K )
*          The first K elements of this array contain the components
*          of the deflation-adjusted updating row vector.
*
*  VF      (input/output) REAL 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) REAL 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) REAL array, dimension ( K )
*          On exit, DIFL(I) = D(I) - DSIGMA(I).
*
*  DIFR    (output) REAL 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) REAL 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) REAL 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 ..
REAL               ONE
PARAMETER          ( ONE = 1.0E+0 )
*     ..
*     .. Local Scalars ..
INTEGER            I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J
REAL               DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, RHO, TEMP
*     ..
*     .. External Subroutines ..
EXTERNAL           SCOPY, SLASCL, SLASD4, SLASET, XERBLA
*     ..
*     .. External Functions ..
REAL               SDOT, SLAMC3, SNRM2
EXTERNAL           SDOT, SLAMC3, SNRM2
*     ..
*     .. 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( 'SLASD8', -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 ) = SLAMC3( 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 = SNRM2( K, Z, 1 )
CALL SLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
RHO = RHO*RHO
*
*     Initialize WORK(IWK3).
*
CALL SLASET( '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 SLASD4( 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 ) / ( SLAMC3( DSIGMA( I ), DSIGJ )-DIFLJ )
\$                   / ( DSIGMA( I )+DJ )
60    CONTINUE
DO 70 I = J + 1, K
WORK( I ) = Z( I ) / ( SLAMC3( DSIGMA( I ), DSIGJP )+DIFRJ )
\$                   / ( DSIGMA( I )+DJ )
70    CONTINUE
TEMP = SNRM2( K, WORK, 1 )
WORK( IWK2I+J ) = SDOT( K, WORK, 1, VF, 1 ) / TEMP
WORK( IWK3I+J ) = SDOT( K, WORK, 1, VL, 1 ) / TEMP
IF( ICOMPQ.EQ.1 ) THEN
DIFR( J, 2 ) = TEMP
END IF
80 CONTINUE
*
CALL SCOPY( K, WORK( IWK2 ), 1, VF, 1 )
CALL SCOPY( K, WORK( IWK3 ), 1, VL, 1 )
*
RETURN
*
*     End of SLASD8
*
END

```