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## slaed8()

subroutine slaed8 | ( | integer | icompq,
|  | integer | k,
|  | integer | n,
|  | integer | qsiz,
|  | real, dimension( * ) | d,
|  | real, dimension( ldq, * ) | q,
|  | integer | ldq,
|  | integer, dimension( * ) | indxq,
|  | real | rho,
|  | integer | cutpnt,
|  | real, dimension( * ) | z,
|  | real, dimension( * ) | dlambda,
|  | real, dimension( ldq2, * ) | q2,
|  | integer | ldq2,
|  | real, dimension( * ) | w,
|  | integer, dimension( * ) | perm,
|  | integer | givptr,
|  | integer, dimension( 2, * ) | givcol,
|  | real, dimension( 2, * ) | givnum,
|  | integer, dimension( * ) | indxp,
|  | integer, dimension( * ) | indx,
|  | integer | info )

SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Download SLAED8 + dependencies [TGZ] [ZIP] [TXT]

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.
      !>

Parameters

  [in]

  ICOMPQ

      !>          ICOMPQ is 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.
      !>

  [out]

  K

      !>          K is INTEGER
      !>         The number of non-deflated eigenvalues, and 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 orthogonal matrix used to reduce
      !>         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
      !>

  [in,out]

  D

      !>          D is 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.
      !>

  [in,out]

  Q

      !>          Q is 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.
      !>

  [in]

  LDQ

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

  [in]

  INDXQ

      !>          INDXQ is 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.
      !>

  [in,out]

  RHO

      !>          RHO is 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.
      !>

  [in]

  CUTPNT

      !>          CUTPNT is INTEGER
      !>         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 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.
      !>

  [out]

  DLAMBDA

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

  [out]

  Q2

      !>          Q2 is 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.
      !>

  [in]

  LDQ2

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

  [out]

  W

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

  [out]

  PERM

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

  [out]

  GIVPTR

      !>          GIVPTR is INTEGER
      !>         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]

  INDXP

      !>          INDXP is 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.
      !>

  [out]

  INDX

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

  [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.

Contributors:
  Jeff Rutter, Computer Science Division, University of California at Berkeley, USA