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NAME
     slaed3 - find the roots of the secular equation, as  defined
     by the values in D, W, and RHO, between KSTART and KSTOP

SYNOPSIS
     SUBROUTINE SLAED3( K, KSTART, KSTOP,  N,  D,  Q,  LDQ,  RHO,
               CUTPNT,  DLAMDA, Q2, LDQ2, INDXC, CTOT, W, S, LDS,
               INFO )

     INTEGER CUTPNT, INFO, K, KSTART, KSTOP, LDQ, LDQ2, LDS, N

     REAL RHO

     INTEGER CTOT( * ), INDXC( * )

     REAL D( * ), DLAMDA( * ), Q( LDQ, * ), Q2(  LDQ2,  *  ),  S(
               LDS, * ), W( * )



     #include <sunperf.h>

     void slaed3(int k, int kstart, int kstop, int n,  float  *d,
               float  *q,  int ldq, float srho, int cutpnt, float
               *dlamda, float *q2,  int  ldq2,  int  *indxc,  int
               *ctot, float *w, float *s, int lds, int *info) ;

PURPOSE
     SLAED3 finds the roots of the secular equation,  as  defined
     by  the  values  in D, W, and RHO, between KSTART and KSTOP.
     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.

     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 (deflation may result in N>K).

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

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

     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.

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

     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.

     LDQ2      (input) INTEGER
               The leading dimension of the array  Q2.   LDQ2  >=
               max(1,N).

     INDXC     (input) INTEGER array, dimension (N)
               The permutation used to arrange the columns of the
               deflated  Q  matrix  into three groups:  the first
               group contains non-zero elements only at and above
               CUTPNT, the second contains non-zero elements only
               below CUTPNT, and the third is dense.  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 INDXC.  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 (LDS, 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

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