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DLAED8(1)		 LAPACK routine (version 3.2)		     DLAED8(1)

NAME
       DLAED8  -  merges  the  two  sets of eigenvalues together into a single
       sorted set

SYNOPSIS
       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( * )

PURPOSE
       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 con‐
	      tains the eigenvectors of the partially solved system which  has
	      been  previously	updated	 in  matrix multiplies with other par‐
	      tially 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 PRE‐
	      CISION 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

 LAPACK routine (version 3.2)	 November 2008			     DLAED8(1)
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