slaed3(3F)

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( * )

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 vector. 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 illegal value.
             > 0:  if INFO = 1, an eigenvalue did not converge