```      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

```