          SUBROUTINE DLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W,
         $                   S, LDS, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, K, KSTART, KSTOP, LDQ, LDS, N
          DOUBLE PRECISION   RHO
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
         $                   W( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLAED9 finds the roots of the secular equation, as defined by the
    *  values in D, Z, and RHO, between KSTART and KSTOP.  It makes the
    *  appropriate calls to DLAED4 and then stores the new matrix of
    *  eigenvectors for use in calculating the next level of Z vectors.
    *
    *  Arguments
    *  =========
    *
    *  K       (input) INTEGER
    *          The number of terms in the rational function to be solved by
    *          DLAED4.  K >= 0.
    *
    *  KSTART  (input) INTEGER
    *  KSTOP   (input) INTEGER
    *          The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
    *          are to be computed.  1 <= KSTART <= KSTOP <= K.
    *
    *  N       (input) INTEGER
    *          The number of rows and columns in the Q matrix.
    *          N >= K (delation may result in N > K).
    *
    *  D       (output) DOUBLE PRECISION array, dimension (N)
    *          D(I) contains the updated eigenvalues
    *          for KSTART <= I <= KSTOP.
    *
    *  Q       (workspace) DOUBLE PRECISION array, dimension (LDQ,N)
    *
    *  LDQ     (input) INTEGER
    *          The leading dimension of the array Q.  LDQ >= max( 1, N ).
    *
    *  RHO     (input) DOUBLE PRECISION
    *          The value of the parameter in the rank one update equation.
    *          RHO >= 0 required.
    *
    *  DLAMDA  (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.
    *
    *  W       (input) DOUBLE PRECISION array, dimension (K)
    *          The first K elements of this array contain the components
    *          of the deflation-adjusted updating vector.
    *
    *  S       (output) DOUBLE PRECISION array, dimension (LDS, K)
    *          Will contain the eigenvectors of the repaired matrix which
    *          will be stored for subsequent Z vector calculation and
    *          multiplied by the previously accumulated eigenvectors
    *          to update the system.
    *
    *  LDS     (input) INTEGER
    *          The leading dimension of S.  LDS >= max( 1, K ).
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit.
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *          > 0:  if INFO = 1, an eigenvalue did not converge
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Jeff Rutter, Computer Science Division, University of California
    *     at Berkeley, USA
    *
    *  =====================================================================
    *
    *     .. Local Scalars ..
          INTEGER            I, J
          DOUBLE PRECISION   TEMP
    *     ..
    *     .. External Functions ..
          DOUBLE PRECISION   DLAMC3, DNRM2
          EXTERNAL           DLAMC3, DNRM2
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DCOPY, DLAED4, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, SIGN, SQRT
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
    *
          IF( K.LT.0 ) THEN
             INFO = -1
          ELSE IF( KSTART.LT.1 .OR. KSTART.GT.MAX( 1, K ) ) THEN
             INFO = -2
          ELSE IF( MAX( 1, KSTOP ).LT.KSTART .OR. KSTOP.GT.MAX( 1, K ) )
         $          THEN
             INFO = -3
          ELSE IF( N.LT.K ) THEN
             INFO = -4
          ELSE IF( LDQ.LT.MAX( 1, K ) ) THEN
             INFO = -7
          ELSE IF( LDS.LT.MAX( 1, K ) ) THEN
             INFO = -12
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DLAED9', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( K.EQ.0 )
         $   RETURN
    *
    *     Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(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 DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
    *     which on any of these machines zeros out the bottommost
    *     bit of DLAMDA(I) if it is 1; this makes the subsequent
    *     subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
    *     occurs. On binary machines with a guard digit (almost all
    *     machines) it does not change DLAMDA(I) at all. On hexadecimal
    *     and decimal machines with a guard digit, it slightly
    *     changes the bottommost bits of DLAMDA(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, N
             DLAMDA( I ) = DLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I )
       10 CONTINUE
    *
          DO 20 J = KSTART, KSTOP
             CALL DLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO )
    *
    *        If the zero finder fails, the computation is terminated.
    *
             IF( INFO.NE.0 )
         $      GO TO 120
       20 CONTINUE
    *
          IF( K.EQ.1 .OR. K.EQ.2 ) THEN
             DO 40 I = 1, K
                DO 30 J = 1, K
                   S( J, I ) = Q( J, I )
       30       CONTINUE
       40    CONTINUE
             GO TO 120
          END IF
    *
    *     Compute updated W.
    *
          CALL DCOPY( K, W, 1, S, 1 )
    *
    *     Initialize W(I) = Q(I,I)
    *
          CALL DCOPY( K, Q, LDQ+1, W, 1 )
          DO 70 J = 1, K
             DO 50 I = 1, J - 1
                W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
       50    CONTINUE
             DO 60 I = J + 1, K
                W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
       60    CONTINUE
       70 CONTINUE
          DO 80 I = 1, K
             W( I ) = SIGN( SQRT( -W( I ) ), S( I, 1 ) )
       80 CONTINUE
    *
    *     Compute eigenvectors of the modified rank-1 modification.
    *
          DO 110 J = 1, K
             DO 90 I = 1, K
                Q( I, J ) = W( I ) / Q( I, J )
       90    CONTINUE
             TEMP = DNRM2( K, Q( 1, J ), 1 )
             DO 100 I = 1, K
                S( I, J ) = Q( I, J ) / TEMP
      100    CONTINUE
      110 CONTINUE
    *
      120 CONTINUE
          RETURN
    *
    *     End of DLAED9
    *
          END