SLAED8 (3lapack)

merge the two sets of eigenvalues together into a single sorted set

SYNOPSIS

  SUBROUTINE SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT, Z,
                     DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL, GIVNUM,
                     INDXP, INDX, INFO )

      INTEGER        CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, QSIZ

      REAL           RHO

      INTEGER        GIVCOL( 2, * ), INDX( * ), INDXP( * ), INDXQ( * ), PERM(
                     * )

      REAL           D( * ), DLAMDA( * ), GIVNUM( 2, * ), Q( LDQ, * ), Q2(
                     LDQ2, * ), W( * ), Z( * )

PURPOSE

  SLAED8 merges the two sets of eigenvalues together into a single sorted
  set.  Then it tries to deflate the size of the problem.  There are two ways
  in which deflation can occur:  when two or more eigenvalues are close
  together or if there is a tiny element in the Z vector.  For each such
  occurrence the order of the related secular equation problem is reduced by
  one.

ARGUMENTS

  ICOMPQ  (input) INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix also.
          On entry, Q contains the orthogonal matrix used to reduce the
          original matrix to tridiagonal form.

  K      (output) INTEGER
         The number of non-deflated eigenvalues, and the order of the related
         secular equation.

  N      (input) INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

  QSIZ   (input) INTEGER
         The dimension of the orthogonal matrix used to reduce the full
         matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

  D      (input/output) REAL array, dimension (N)
         On entry, the eigenvalues of the two submatrices to be combined.  On
         exit, the trailing (N-K) updated eigenvalues (those which were
         deflated) sorted into increasing order.

  Q      (input/output) REAL array, dimension (LDQ,N)
         If ICOMPQ = 0, Q is not referenced.  Otherwise, on entry, Q contains
         the eigenvectors of the partially solved system which has been
         previously updated in matrix multiplies with other partially solved
         eigensystems.  On exit, Q contains the trailing (N-K) updated
         eigenvectors (those which were deflated) in its last N-K columns.

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

  INDXQ  (input) INTEGER array, dimension (N)
         The permutation which separately sorts the two sub-problems in D
         into ascending order.  Note that elements in the second half of this
         permutation must first have CUTPNT added to their values in order to
         be accurate.

  RHO    (input/output) REAL
         On entry, the off-diagonal element associated with the rank-1 cut
         which originally split the two submatrices which are now being
         recombined.  On exit, RHO has been modified to the value required by
         SLAED3.

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

  Z      (input) REAL array, dimension (N)
         On entry, Z contains the updating vector (the last row of the first
         sub-eigenvector matrix and the first row of the second sub-
         eigenvector matrix).  On exit, the contents of Z are destroyed by
         the updating process.

         DLAMDA (output) REAL array, dimension (N) A copy of the first K
         eigenvalues which will be used by SLAED3 to form the secular
         equation.

  Q2     (output) REAL array, dimension (LDQ2,N)
         If ICOMPQ = 0, Q2 is not referenced.  Otherwise, a copy of the first
         K eigenvectors which will be used by SLAED7 in a matrix multiply
         (SGEMM) to update the new eigenvectors.

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

  W      (output) REAL array, dimension (N)
         The first k values of the final deflation-altered z-vector and will
         be passed to SLAED3.

  PERM   (output) INTEGER array, dimension (N)
         The permutations (from deflation and sorting) to be applied to each
         eigenblock.

         GIVPTR (output) INTEGER The number of Givens rotations which took
         place in this subproblem.

         GIVCOL (output) INTEGER array, dimension (2, N) Each pair of numbers
         indicates a pair of columns to take place in a Givens rotation.

         GIVNUM (output) REAL array, dimension (2, N) Each number indicates
         the S value to be used in the corresponding Givens rotation.

  INDXP  (workspace) INTEGER array, dimension (N)
         The permutation used to place deflated values of D at the end of the
         array.  INDXP(1:K) points to the nondeflated D-values
         and INDXP(K+1:N) points to the deflated eigenvalues.

  INDX   (workspace) INTEGER array, dimension (N)
         The permutation used to sort the contents of D into ascending order.

  INFO   (output) INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.