SLAED9  -  find  the  roots  of  the  secular equation, as
       defined by the values in D, Z, and RHO, between KSTART and
       KSTOP


SYNOPSIS

       SUBROUTINE SLAED9( K,  KSTART,  KSTOP,  N, D, Q, LDQ, RHO,
                          DLAMDA, W, S, LDS, INFO )

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

           REAL           RHO

           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 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) 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

               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 vec­
               tor.

       S       (output) REAL 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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