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version 1.8, 2010/12/21 13:53:29 version 1.9, 2011/11/21 20:42:54
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   *> \brief \b DLAED8
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLAED8 + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed8.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed8.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed8.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
   *                          CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
   *                          GIVCOL, GIVNUM, INDXP, INDX, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
   *      $                   QSIZ
   *       DOUBLE PRECISION   RHO
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            GIVCOL( 2, * ), INDX( * ), INDXP( * ),
   *      $                   INDXQ( * ), PERM( * )
   *       DOUBLE PRECISION   D( * ), DLAMDA( * ), GIVNUM( 2, * ),
   *      $                   Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> 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.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] ICOMPQ
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] K
   *> \verbatim
   *>          K is INTEGER
   *>         The number of non-deflated eigenvalues, and the order of the
   *>         related secular equation.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] QSIZ
   *> \verbatim
   *>          QSIZ is INTEGER
   *>         The dimension of the orthogonal matrix used to reduce
   *>         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
   *> \endverbatim
   *>
   *> \param[in,out] D
   *> \verbatim
   *>          D is 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.
   *> \endverbatim
   *>
   *> \param[in,out] Q
   *> \verbatim
   *>          Q is 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.
   *> \endverbatim
   *>
   *> \param[in] LDQ
   *> \verbatim
   *>          LDQ is INTEGER
   *>         The leading dimension of the array Q.  LDQ >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in] INDXQ
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in,out] RHO
   *> \verbatim
   *>          RHO is 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.
   *> \endverbatim
   *>
   *> \param[in] CUTPNT
   *> \verbatim
   *>          CUTPNT is INTEGER
   *>         The location of the last eigenvalue in the leading
   *>         sub-matrix.  min(1,N) <= CUTPNT <= N.
   *> \endverbatim
   *>
   *> \param[in] Z
   *> \verbatim
   *>          Z is 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.
   *> \endverbatim
   *>
   *> \param[out] DLAMDA
   *> \verbatim
   *>          DLAMDA is DOUBLE PRECISION array, dimension (N)
   *>         A copy of the first K eigenvalues which will be used by
   *>         DLAED3 to form the secular equation.
   *> \endverbatim
   *>
   *> \param[out] Q2
   *> \verbatim
   *>          Q2 is 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.
   *> \endverbatim
   *>
   *> \param[in] LDQ2
   *> \verbatim
   *>          LDQ2 is INTEGER
   *>         The leading dimension of the array Q2.  LDQ2 >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] W
   *> \verbatim
   *>          W is DOUBLE PRECISION array, dimension (N)
   *>         The first k values of the final deflation-altered z-vector and
   *>         will be passed to DLAED3.
   *> \endverbatim
   *>
   *> \param[out] PERM
   *> \verbatim
   *>          PERM is INTEGER array, dimension (N)
   *>         The permutations (from deflation and sorting) to be applied
   *>         to each eigenblock.
   *> \endverbatim
   *>
   *> \param[out] GIVPTR
   *> \verbatim
   *>          GIVPTR is INTEGER
   *>         The number of Givens rotations which took place in this
   *>         subproblem.
   *> \endverbatim
   *>
   *> \param[out] GIVCOL
   *> \verbatim
   *>          GIVCOL is INTEGER array, dimension (2, N)
   *>         Each pair of numbers indicates a pair of columns to take place
   *>         in a Givens rotation.
   *> \endverbatim
   *>
   *> \param[out] GIVNUM
   *> \verbatim
   *>          GIVNUM is DOUBLE PRECISION array, dimension (2, N)
   *>         Each number indicates the S value to be used in the
   *>         corresponding Givens rotation.
   *> \endverbatim
   *>
   *> \param[out] INDXP
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] INDX
   *> \verbatim
   *>          INDX is INTEGER array, dimension (N)
   *>         The permutation used to sort the contents of D into ascending
   *>         order.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit.
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup auxOTHERcomputational
   *
   *> \par Contributors:
   *  ==================
   *>
   *> Jeff Rutter, Computer Science Division, University of California
   *> at Berkeley, USA
   *
   *  =====================================================================
       SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,        SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
      $                   CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,       $                   CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
      $                   GIVCOL, GIVNUM, INDXP, INDX, INFO )       $                   GIVCOL, GIVNUM, INDXP, INDX, INFO )
 *  *
 *  -- LAPACK routine (version 3.2.2) --  *  -- LAPACK computational routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     June 2010  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,        INTEGER            CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
Line 19 Line 260
      $                   Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )       $                   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 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  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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