CXML
        	    
LAPACK version 3.0
dlaed8(3)

PURPOSE
  DLAED8 - merge the two sets of eigenvalues together into a single sorted
  set

SYNTAX
  SUBROUTINE DLAED8( 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

      DOUBLE	     PRECISION RHO

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

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

DESCRIPTION
  DLAED8 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
	 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
	 DLAED3.

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

  Z	 (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N) A copy of the
	 first K eigenvalues which will be used by DLAED3 to form the secular
	 equation.

  Q2	 (output) DOUBLE PRECISION array, dimension (LDQ2,N)
	 If ICOMPQ = 0, Q2 is not referenced.  Otherwise, 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)
	 The first k values of the final deflation-altered z-vector and will
	 be passed to DLAED3.

  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) DOUBLE PRECISION 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.

FURTHER DETAILS
  Based on contributions by
     Jeff Rutter, Computer Science Division, University of California
     at Berkeley, USA