SLAED2 (3lapack)

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

SYNOPSIS

  SUBROUTINE SLAED2( K, N, D, Q, LDQ, INDXQ, RHO, CUTPNT, Z, DLAMDA, Q2,
                     LDQ2, INDXC, W, INDXP, INDX, COLTYP, INFO )

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

      REAL           RHO

      INTEGER        COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), INDXQ( *
                     )

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

PURPOSE

  SLAED2 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 entry in the Z vector.  For each such
  occurrence the order of the related secular equation problem is reduced by
  one.

ARGUMENTS

  K      (output) INTEGER
         The number of non-deflated eigenvalues, and the order of the related
         secular equation. 0 <= K <=N.

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

  D      (input/output) 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.

  Q      (input/output) REAL array, dimension (LDQ, N)
         On entry, Q contains the eigenvectors of two submatrices in the two
         square blocks with corners at (1,1), (CUTPNT,CUTPNT) and (CUTPNT+1,
         CUTPNT+1), (N,N).  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/output) 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. Destroyed
         on exit.

  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 have been 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)
         A copy of the first K eigenvectors which will be used by SLAED3 in a
         matrix multiply (SGEMM) to solve for the new eigenvectors.   Q2 is
         arranged into three blocks.  The first block contains non-zero
         elements only at and above CUTPNT, the second contains non-zero
         elements only below CUTPNT, and the third is dense.

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

  INDXC  (output) 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.

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

  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.

         COLTYP (workspace/output) INTEGER array, dimension (N) During
         execution, a label which will indicate which of the following types
         a column in the Q2 matrix is:
         1 : non-zero in the upper half only;
         2 : non-zero in the lower half only;
         3 : dense;
         4 : deflated.  On exit, COLTYP(i) is the number of columns of type
         i, for i=1 to 4 only.

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