          SUBROUTINE SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
         $                   CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
         $                   GIVCOL, GIVNUM, INDXP, INDX, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
         $                   QSIZ
          REAL               RHO
    *     ..
    *     .. Array Arguments ..
          INTEGER            GIVCOL( 2, * ), INDX( * ), INDXP( * ),
         $                   INDXQ( * ), PERM( * )
          REAL               D( * ), DLAMDA( * ), GIVNUM( 2, * ),
         $                   Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SLAED8 merges the two sets of eigenvalues together into a single
    *  sorted set.  Then it tries to deflate the size of the problem.
    *  There are two ways in which deflation can occur:  when two or more
    *  eigenvalues are close together or if there is a tiny element in the
    *  Z vector.  For each such occurrence the order of the related secular
    *  equation problem is reduced by one.
    *
    *  Arguments
    *  =========
    *
    *  ICOMPQ  (input) INTEGER
    *          = 0:  Compute eigenvalues only.
    *          = 1:  Compute eigenvectors of original dense symmetric matrix
    *                also.  On entry, Q contains the orthogonal matrix used
    *                to reduce the original matrix to tridiagonal form.
    *
    *  K      (output) INTEGER
    *         The number of non-deflated eigenvalues, and the order of the
    *         related secular equation.
    *
    *  N      (input) INTEGER
    *         The dimension of the symmetric tridiagonal matrix.  N >= 0.
    *
    *  QSIZ   (input) INTEGER
    *         The dimension of the orthogonal matrix used to reduce
    *         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
    *
    *  D      (input/output) REAL array, dimension (N)
    *         On entry, the eigenvalues of the two submatrices to be
    *         combined.  On exit, the trailing (N-K) updated eigenvalues
    *         (those which were deflated) sorted into increasing order.
    *
    *  Q      (input/output) REAL array, dimension (LDQ,N)
    *         If ICOMPQ = 0, Q is not referenced.  Otherwise,
    *         on entry, Q contains the eigenvectors of the partially solved
    *         system which has been previously updated in matrix
    *         multiplies with other partially solved eigensystems.
    *         On exit, Q contains the trailing (N-K) updated eigenvectors
    *         (those which were deflated) in its last N-K columns.
    *
    *  LDQ    (input) INTEGER
    *         The leading dimension of the array Q.  LDQ >= max(1,N).
    *
    *  INDXQ  (input) INTEGER array, dimension (N)
    *         The permutation which separately sorts the two sub-problems
    *         in D into ascending order.  Note that elements in the second
    *         half of this permutation must first have CUTPNT added to
    *         their values in order to be accurate.
    *
    *  RHO    (input/output) REAL
    *         On entry, the off-diagonal element associated with the rank-1
    *         cut which originally split the two submatrices which are now
    *         being recombined.
    *         On exit, RHO has been modified to the value required by
    *         SLAED3.
    *
    *  CUTPNT (input) INTEGER
    *         The location of the last eigenvalue in the leading
    *         sub-matrix.  min(1,N) <= CUTPNT <= N.
    *
    *  Z      (input) REAL array, dimension (N)
    *         On entry, Z contains the updating vector (the last row of
    *         the first sub-eigenvector matrix and the first row of the
    *         second sub-eigenvector matrix).
    *         On exit, the contents of Z are destroyed by the updating
    *         process.
    *
    *  DLAMDA (output) REAL array, dimension (N)
    *         A copy of the first K eigenvalues which will be used by
    *         SLAED3 to form the secular equation.
    *
    *  Q2     (output) REAL array, dimension (LDQ2,N)
    *         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
    *         a copy of the first K eigenvectors which will be used by
    *         SLAED7 in a matrix multiply (SGEMM) to update the new
    *         eigenvectors.
    *
    *  LDQ2   (input) INTEGER
    *         The leading dimension of the array Q2.  LDQ2 >= max(1,N).
    *
    *  W      (output) REAL array, dimension (N)
    *         The first k values of the final deflation-altered z-vector and
    *         will be passed to SLAED3.
    *
    *  PERM   (output) INTEGER array, dimension (N)
    *         The permutations (from deflation and sorting) to be applied
    *         to each eigenblock.
    *
    *  GIVPTR (output) INTEGER
    *         The number of Givens rotations which took place in this
    *         subproblem.
    *
    *  GIVCOL (output) INTEGER array, dimension (2, N)
    *         Each pair of numbers indicates a pair of columns to take place
    *         in a Givens rotation.
    *
    *  GIVNUM (output) REAL array, dimension (2, N)
    *         Each number indicates the S value to be used in the
    *         corresponding Givens rotation.
    *
    *  INDXP  (workspace) INTEGER array, dimension (N)
    *         The permutation used to place deflated values of D at the end
    *         of the array.  INDXP(1:K) points to the nondeflated D-values
    *         and INDXP(K+1:N) points to the deflated eigenvalues.
    *
    *  INDX   (workspace) INTEGER array, dimension (N)
    *         The permutation used to sort the contents of D into ascending
    *         order.
    *
    *  INFO   (output) INTEGER
    *          = 0:  successful exit.
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Jeff Rutter, Computer Science Division, University of California
    *     at Berkeley, USA
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               MONE, ZERO, ONE, TWO, EIGHT
          PARAMETER          ( MONE = -1.0E0, ZERO = 0.0E0, ONE = 1.0E0,
         $                   TWO = 2.0E0, EIGHT = 8.0E0 )
    *     ..
    *     .. Local Scalars ..
    *
          INTEGER            I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
          REAL               C, EPS, S, T, TAU, TOL
    *     ..
    *     .. External Functions ..
          INTEGER            ISAMAX
          REAL               SLAMCH, SLAPY2
          EXTERNAL           ISAMAX, SLAMCH, SLAPY2
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SCOPY, SLACPY, SLAMRG, SROT, SSCAL, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX, MIN, SQRT
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
    *
          IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
             INFO = -4
          ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
             INFO = -7
          ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN
             INFO = -10
          ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN
             INFO = -14
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'SLAED8', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
          N1 = CUTPNT
          N2 = N - N1
          N1P1 = N1 + 1
    *
          IF( RHO.LT.ZERO ) THEN
             CALL SSCAL( N2, MONE, Z( N1P1 ), 1 )
          END IF
    *
    *     Normalize z so that norm(z) = 1
    *
          T = ONE / SQRT( TWO )
          DO 10 J = 1, N
             INDX( J ) = J
       10 CONTINUE
          CALL SSCAL( N, T, Z, 1 )
          RHO = ABS( TWO*RHO )
    *
    *     Sort the eigenvalues into increasing order
    *
          DO 20 I = CUTPNT + 1, N
             INDXQ( I ) = INDXQ( I ) + CUTPNT
       20 CONTINUE
          DO 30 I = 1, N
             DLAMDA( I ) = D( INDXQ( I ) )
             W( I ) = Z( INDXQ( I ) )
       30 CONTINUE
          I = 1
          J = CUTPNT + 1
          CALL SLAMRG( N1, N2, DLAMDA, 1, 1, INDX )
          DO 40 I = 1, N
             D( I ) = DLAMDA( INDX( I ) )
             Z( I ) = W( INDX( I ) )
       40 CONTINUE
    *
    *     Calculate the allowable deflation tolerence
    *
          IMAX = ISAMAX( N, Z, 1 )
          JMAX = ISAMAX( N, D, 1 )
          EPS = SLAMCH( 'Epsilon' )
          TOL = EIGHT*EPS*ABS( D( JMAX ) )
    *
    *     If the rank-1 modifier is small enough, no more needs to be done
    *     except to reorganize Q so that its columns correspond with the
    *     elements in D.
    *
          IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
             K = 0
             IF( ICOMPQ.EQ.0 ) THEN
                DO 50 J = 1, N
                   PERM( J ) = INDXQ( INDX( J ) )
       50       CONTINUE
             ELSE
                DO 60 J = 1, N
                   PERM( J ) = INDXQ( INDX( J ) )
                   CALL SCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
       60       CONTINUE
                CALL SLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ),
         $                   LDQ )
             END IF
             RETURN
          END IF
    *
    *     If there are multiple eigenvalues then the problem deflates.  Here
    *     the number of equal eigenvalues are found.  As each equal
    *     eigenvalue is found, an elementary reflector is computed to rotate
    *     the corresponding eigensubspace so that the corresponding
    *     components of Z are zero in this new basis.
    *
          K = 0
          GIVPTR = 0
          K2 = N + 1
          DO 70 J = 1, N
             IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
    *
    *           Deflate due to small z component.
    *
                K2 = K2 - 1
                INDXP( K2 ) = J
                IF( J.EQ.N )
         $         GO TO 110
             ELSE
                JLAM = J
                GO TO 80
             END IF
       70 CONTINUE
       80 CONTINUE
          J = J + 1
          IF( J.GT.N )
         $   GO TO 100
          IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
    *
    *        Deflate due to small z component.
    *
             K2 = K2 - 1
             INDXP( K2 ) = J
          ELSE
    *
    *        Check if eigenvalues are close enough to allow deflation.
    *
             S = Z( JLAM )
             C = Z( J )
    *
    *        Find sqrt(a**2+b**2) without overflow or
    *        destructive underflow.
    *
             TAU = SLAPY2( C, S )
             T = D( J ) - D( JLAM )
             C = C / TAU
             S = -S / TAU
             IF( ABS( T*C*S ).LE.TOL ) THEN
    *
    *           Deflation is possible.
    *
                Z( J ) = TAU
                Z( JLAM ) = ZERO
    *
    *           Record the appropriate Givens rotation
    *
                GIVPTR = GIVPTR + 1
                GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) )
                GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) )
                GIVNUM( 1, GIVPTR ) = C
                GIVNUM( 2, GIVPTR ) = S
                IF( ICOMPQ.EQ.1 ) THEN
                   CALL SROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
         $                    Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
                END IF
                T = D( JLAM )*C*C + D( J )*S*S
                D( J ) = D( JLAM )*S*S + D( J )*C*C
                D( JLAM ) = T
                K2 = K2 - 1
                I = 1
       90       CONTINUE
                IF( K2+I.LE.N ) THEN
                   IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN
                      INDXP( K2+I-1 ) = INDXP( K2+I )
                      INDXP( K2+I ) = JLAM
                      I = I + 1
                      GO TO 90
                   ELSE
                      INDXP( K2+I-1 ) = JLAM
                   END IF
                ELSE
                   INDXP( K2+I-1 ) = JLAM
                END IF
                JLAM = J
             ELSE
                K = K + 1
                W( K ) = Z( JLAM )
                DLAMDA( K ) = D( JLAM )
                INDXP( K ) = JLAM
                JLAM = J
             END IF
          END IF
          GO TO 80
      100 CONTINUE
    *
    *     Record the last eigenvalue.
    *
          K = K + 1
          W( K ) = Z( JLAM )
          DLAMDA( K ) = D( JLAM )
          INDXP( K ) = JLAM
    *
      110 CONTINUE
    *
    *     Sort the eigenvalues and corresponding eigenvectors into DLAMDA
    *     and Q2 respectively.  The eigenvalues/vectors which were not
    *     deflated go into the first K slots of DLAMDA and Q2 respectively,
    *     while those which were deflated go into the last N - K slots.
    *
          IF( ICOMPQ.EQ.0 ) THEN
             DO 120 J = 1, N
                JP = INDXP( J )
                DLAMDA( J ) = D( JP )
                PERM( J ) = INDXQ( INDX( JP ) )
      120    CONTINUE
          ELSE
             DO 130 J = 1, N
                JP = INDXP( J )
                DLAMDA( J ) = D( JP )
                PERM( J ) = INDXQ( INDX( JP ) )
                CALL SCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
      130    CONTINUE
          END IF
    *
    *     The deflated eigenvalues and their corresponding vectors go back
    *     into the last N - K slots of D and Q respectively.
    *
          IF( K.LT.N ) THEN
             IF( ICOMPQ.EQ.0 ) THEN
                CALL SCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
             ELSE
                CALL SCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
                CALL SLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2,
         $                   Q( 1, K+1 ), LDQ )
             END IF
          END IF
    *
          RETURN
    *
    *     End of SLAED8
    *
          END