SLAED3 (3lapack)

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