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## claed8()

subroutine claed8 | ( | integer | k,
|  | integer | n,
|  | integer | qsiz,
|  | complex, dimension( ldq, * ) | q,
|  | integer | ldq,
|  | real, dimension( * ) | d,
|  | real | rho,
|  | integer | cutpnt,
|  | real, dimension( * ) | z,
|  | real, dimension( * ) | dlambda,
|  | complex, dimension( ldq2, * ) | q2,
|  | integer | ldq2,
|  | real, dimension( * ) | w,
|  | integer, dimension( * ) | indxp,
|  | integer, dimension( * ) | indx,
|  | integer, dimension( * ) | indxq,
|  | integer, dimension( * ) | perm,
|  | integer | givptr,
|  | integer, dimension( 2, * ) | givcol,
|  | real, dimension( 2, * ) | givnum,
|  | integer | info )

CLAED8 used by CSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Download CLAED8 + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> CLAED8 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.
      !>

Parameters

  [out]

  K

      !>          K is INTEGER
      !>         Contains the number of non-deflated eigenvalues.
      !>         This is the order of the related secular equation.
      !>

  [in]

  N

      !>          N is INTEGER
      !>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
      !>

  [in]

  QSIZ

      !>          QSIZ is INTEGER
      !>         The dimension of the unitary matrix used to reduce
      !>         the dense or band matrix to tridiagonal form.
      !>         QSIZ >= N if ICOMPQ = 1.
      !>

  [in,out]

  Q

      !>          Q is COMPLEX 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.
      !>

  [in]

  LDQ

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

  [in,out]

  D

      !>          D is REAL 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.
      !>

  [in,out]

  RHO

      !>          RHO is REAL
      !>         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 SLAED3.
      !>

  [in]

  CUTPNT

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

  [in]

  Z

      !>          Z is REAL 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.
      !>

  [out]

  DLAMBDA

      !>          DLAMBDA is REAL array, dimension (N)
      !>         Contains a copy of the first K eigenvalues which will be used
      !>         by SLAED3 to form the secular equation.
      !>

  [out]

  Q2

      !>          Q2 is COMPLEX 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 SLAED7 in a matrix multiply (SGEMM) to update the new
      !>         eigenvectors.
      !>

  [in]

  LDQ2

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

  [out]

  W

      !>          W is REAL array, dimension (N)
      !>         This will hold the first k values of the final
      !>         deflation-altered z-vector and will be passed to SLAED3.
      !>

  [out]

  INDXP

      !>          INDXP is 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.
      !>

  [out]

  INDX

      !>          INDX is INTEGER array, dimension (N)
      !>         This will contain the permutation used to sort the contents of
      !>         D into ascending order.
      !>

  [in]

  INDXQ

      !>          INDXQ is 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.
      !>

  [out]

  PERM

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

  [out]

  GIVPTR

      !>          GIVPTR is INTEGER
      !>         Contains the number of Givens rotations which took place in
      !>         this subproblem.
      !>

  [out]

  GIVCOL

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

  [out]

  GIVNUM

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

  [out]

  INFO

      !>          INFO is INTEGER
      !>          = 0:  successful exit.
      !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.