ZLAED8 (3lapack)

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

SYNOPSIS

  SUBROUTINE ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2,
                     W, INDXP, INDX, INDXQ, PERM, GIVPTR, GIVCOL, GIVNUM,
                     INFO )

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

      DOUBLE         PRECISION RHO

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

      DOUBLE         PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
                     Z( * )

      COMPLEX*16     Q( LDQ, * ), Q2( LDQ2, * )

PURPOSE

  ZLAED8 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

  K      (output) INTEGER
         Contains the number of non-deflated eigenvalues.  This is 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 unitary matrix used to reduce the dense or band
         matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

  Q      (input/output) COMPLEX*16 array, dimension (LDQ,N)
         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 ).

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

  RHO    (input/output) DOUBLE PRECISION
         Contains the off diagonal element associated with the rank-1 cut
         which originally split the two submatrices which are now being
         recombined. RHO is modified during the computation to the value
         required by DLAED3.

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

  Z      (input) DOUBLE PRECISION array, dimension (N)
         On input this vector contains the updating vector (the last row of
         the first sub-eigenvector matrix and the first row of the second
         sub-eigenvector matrix).  The contents of Z are destroyed during the
         updating process.

         DLAMDA (output) DOUBLE PRECISION array, dimension (N) Contains a
         copy of the first K eigenvalues which will be used by DLAED3 to form
         the secular equation.

  Q2     (output) COMPLEX*16 array, dimension (LDQ2,N)
         If ICOMPQ = 0, Q2 is not referenced.  Otherwise, Contains a copy of
         the first K eigenvectors which will be used by DLAED7 in a matrix
         multiply (DGEMM) to update the new eigenvectors.

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

  W      (output) DOUBLE PRECISION array, dimension (N)
         This will hold the first k values of the final deflation-altered z-
         vector and will be passed to DLAED3.

  INDXP  (workspace) INTEGER array, dimension (N)
         This will contain the permutation used to place deflated values of D
         at the end of the array. On output 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)
         This will contain the permutation used to sort the contents of D
         into ascending order.

  INDXQ  (input) INTEGER array, dimension (N)
         This contains 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.

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

         GIVPTR (output) INTEGER Contains 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) DOUBLE PRECISION array, dimension (2, N) Each number
         indicates the S value to be used in the corresponding Givens
         rotation.

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