SLAED3  -  find  the  roots  of  the  secular equation, as
       defined by the values in D, W, and RHO, between 1 and K


SYNOPSIS

       SUBROUTINE SLAED3( K, N, N1, D, Q, LDQ, RHO,  DLAMDA,  Q2,
                          INDX, CTOT, W, S, INFO )

           INTEGER        INFO, K, LDQ, N, N1

           REAL           RHO

           INTEGER        CTOT( * ), INDX( * )

           REAL           D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( *
                          ), S( * ), W( * )


PURPOSE

       SLAED3 finds the roots of the secular equation, as defined
       by  the values in D, W, and RHO, between 1 and K. It makes
       the appropriate calls  to  SLAED4  and  then  updates  the
       eigenvectors  by multiplying the matrix of eigenvectors of
       the pair of eigensystems being combined by the  matrix  of
       eigenvectors of the K-by-K system which is solved here.

       This code makes very mild assumptions about floating point
       arithmetic. It will work on machines with a guard digit in
       add/subtract,  or  on  those binary machines without guard
       digits which subtract like the Cray X-MP, Cray Y-MP,  Cray
       C-90, or Cray-2.  It could conceivably fail on hexadecimal
       or decimal machines without guard digits, but we  know  of
       none.


ARGUMENTS

       K       (input) INTEGER
               The number of terms in the rational function to be
               solved by SLAED4.  K >= 0.

       N       (input) INTEGER
               The number of rows and columns in the Q matrix.  N
               >= K (deflation may result in N>K).

       N1      (input) INTEGER
               The location of the last eigenvalue in the leading
               submatrix.  min(1,N) <= N1 <= N/2.

       D       (output) REAL array, dimension (N)
               D(I) contains the updated eigenvalues for 1  <=  I
               <= K.

       Q       (output) REAL array, dimension (LDQ,N)
               Initially   the   first  K  columns  are  used  as
               workspace.  On output the columns 1 to  K  contain


       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/output) 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. May be changed
               on output by having lowest order bit set  to  zero
               on  Cray X-MP, Cray Y-MP, Cray-2, or Cray C-90, as
               described above.

       Q2      (input) REAL array, dimension (LDQ2, N)
               The first K columns of  this  matrix  contain  the
               non-deflated eigenvectors for the split problem.

       INDX    (input) INTEGER array, dimension (N)
               The permutation used to arrange the columns of the
               deflated Q matrix into three groups (see  SLAED2).
               The  rows of the eigenvectors found by SLAED4 must
               be likewise permuted before  the  matrix  multiply
               can take place.

       CTOT    (input) INTEGER array, dimension (4)
               A  count  of the total number of the various types
               of columns in Q, as described in INDX.  The fourth
               column type is any column which has been deflated.

       W       (input/output) REAL array, dimension (K)
               The first K elements of  this  array  contain  the
               components of the deflation-adjusted updating vec­
               tor. Destroyed on output.

       S       (workspace) REAL array, dimension (N1 + 1)*K
               Will contain  the  eigenvectors  of  the  repaired
               matrix  which will be 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

       Based on contributions by
          Jeff  Rutter,  Computer Science Division, University of
       California
          at Berkeley, USA
       Modified by Francoise Tisseur, University of Tennessee.


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