```      SUBROUTINE SLAED9( 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
REAL               RHO
*     ..
*     .. Array Arguments ..
REAL               D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
\$                   W( * )
*     ..
*
*  Purpose
*  =======
*
*  SLAED9 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 SLAED4 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
*          SLAED4.  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) REAL array, dimension (N)
*          D(I) contains the updated eigenvalues
*          for KSTART <= I <= KSTOP.
*
*  Q       (workspace) REAL array, dimension (LDQ,N)
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.  LDQ >= max( 1, N ).
*
*  RHO     (input) REAL
*          The value of the parameter in the rank one update equation.
*          RHO >= 0 required.
*
*  DLAMDA  (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.
*
*  W       (input) REAL array, dimension (K)
*          The first K elements of this array contain the components
*          of the deflation-adjusted updating vector.
*
*  S       (output) REAL 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
REAL               TEMP
*     ..
*     .. External Functions ..
REAL               SLAMC3, SNRM2
EXTERNAL           SLAMC3, SNRM2
*     ..
*     .. External Subroutines ..
EXTERNAL           SCOPY, SLAED4, 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( 'SLAED9', -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 ) = SLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I )
10 CONTINUE
*
DO 20 J = KSTART, KSTOP
CALL SLAED4( 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 SCOPY( K, W, 1, S, 1 )
*
*     Initialize W(I) = Q(I,I)
*
CALL SCOPY( 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 = SNRM2( 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 SLAED9
*
END

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