DLAED9  -  find  the  roots  of  the  secular equation, as
       defined by the values in D, Z, and RHO, between KSTART and
       KSTOP


SYNOPSIS

       SUBROUTINE DLAED9( K,  KSTART,  KSTOP,  N, D, Q, LDQ, RHO,
                          DLAMDA, W, S, LDS, INFO )

           INTEGER        INFO, K, KSTART, KSTOP, LDQ, LDS, N

           DOUBLE         PRECISION RHO

           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 com­
               puted.  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.

               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 vec­
               tor.

       S       (output) DOUBLE PRECISION array, dimension (LDS,
               K)
               Will contain  the  eigenvectors  of  the  repaired
               matrix  which will be stored for subsequent Z vec­
               tor 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 ille­
               gal 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


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