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version 1.1.1.1, 2010/01/26 15:22:45 version 1.19, 2023/08/07 08:39:28
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   *> \brief \b ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZLAED8 + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaed8.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaed8.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaed8.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
   *                          Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,
   *                          GIVCOL, GIVNUM, INFO )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ
   *       DOUBLE PRECISION   RHO
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            GIVCOL( 2, * ), INDX( * ), INDXP( * ),
   *      $                   INDXQ( * ), PERM( * )
   *       DOUBLE PRECISION   D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
   *      $                   Z( * )
   *       COMPLEX*16         Q( LDQ, * ), Q2( LDQ2, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZLAED8 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[out] K
   *> \verbatim
   *>          K is INTEGER
   *>         Contains the number of non-deflated eigenvalues.
   *>         This is 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 unitary matrix used to reduce
   *>         the dense or band matrix to tridiagonal form.
   *>         QSIZ >= N if ICOMPQ = 1.
   *> \endverbatim
   *>
   *> \param[in,out] Q
   *> \verbatim
   *>          Q is COMPLEX*16 array, dimension (LDQ,N)
   *>         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,out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION 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.
   *> \endverbatim
   *>
   *> \param[in,out] RHO
   *> \verbatim
   *>          RHO is DOUBLE PRECISION
   *>         Contains the off diagonal element associated with the rank-1
   *>         cut which originally split the two submatrices which are now
   *>         being recombined. RHO is modified during the computation to
   *>         the value required by DLAED3.
   *> \endverbatim
   *>
   *> \param[in] CUTPNT
   *> \verbatim
   *>          CUTPNT is INTEGER
   *>         Contains 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 input this vector contains the updating vector (the last
   *>         row of the first sub-eigenvector matrix and the first row of
   *>         the second sub-eigenvector matrix).  The contents of Z are
   *>         destroyed during the updating process.
   *> \endverbatim
   *>
   *> \param[out] DLAMDA
   *> \verbatim
   *>          DLAMDA is DOUBLE PRECISION array, dimension (N)
   *>         Contains 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 COMPLEX*16 array, dimension (LDQ2,N)
   *>         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
   *>         Contains 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)
   *>         This will hold the first k values of the final
   *>         deflation-altered z-vector and will be passed to DLAED3.
   *> \endverbatim
   *>
   *> \param[out] INDXP
   *> \verbatim
   *>          INDXP is INTEGER array, dimension (N)
   *>         This will contain the permutation used to place deflated
   *>         values of D at the end of the array. On output 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)
   *>         This will contain the permutation used to sort the contents of
   *>         D into ascending order.
   *> \endverbatim
   *>
   *> \param[in] INDXQ
   *> \verbatim
   *>          INDXQ is INTEGER array, dimension (N)
   *>         This contains 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[out] PERM
   *> \verbatim
   *>          PERM is INTEGER array, dimension (N)
   *>         Contains the permutations (from deflation and sorting) to be
   *>         applied to each eigenblock.
   *> \endverbatim
   *>
   *> \param[out] GIVPTR
   *> \verbatim
   *>          GIVPTR is INTEGER
   *>         Contains 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] 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.
   *
   *> \ingroup complex16OTHERcomputational
   *
   *  =====================================================================
       SUBROUTINE ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,        SUBROUTINE ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
      $                   Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,       $                   Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,
      $                   GIVCOL, GIVNUM, INFO )       $                   GIVCOL, GIVNUM, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational routine --
 *  -- 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..--
 *     November 2006  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ        INTEGER            CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ
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       COMPLEX*16         Q( LDQ, * ), Q2( LDQ2, * )        COMPLEX*16         Q( LDQ, * ), Q2( LDQ2, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZLAED8 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  
 *  =========  
 *  
 *  K      (output) INTEGER  
 *         Contains the number of non-deflated eigenvalues.  
 *         This is 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 unitary matrix used to reduce  
 *         the dense or band matrix to tridiagonal form.  
 *         QSIZ >= N if ICOMPQ = 1.  
 *  
 *  Q      (input/output) COMPLEX*16 array, dimension (LDQ,N)  
 *         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 ).  
 *  
 *  D      (input/output) DOUBLE PRECISION 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.  
 *  
 *  RHO    (input/output) DOUBLE PRECISION  
 *         Contains the off diagonal element associated with the rank-1  
 *         cut which originally split the two submatrices which are now  
 *         being recombined. RHO is modified during the computation to  
 *         the value required by DLAED3.  
 *  
 *  CUTPNT (input) INTEGER  
 *         Contains the location of the last eigenvalue in the leading  
 *         sub-matrix.  MIN(1,N) <= CUTPNT <= N.  
 *  
 *  Z      (input) DOUBLE PRECISION array, dimension (N)  
 *         On input this vector contains the updating vector (the last  
 *         row of the first sub-eigenvector matrix and the first row of  
 *         the second sub-eigenvector matrix).  The contents of Z are  
 *         destroyed during the updating process.  
 *  
 *  DLAMDA (output) DOUBLE PRECISION array, dimension (N)  
 *         Contains a copy of the first K eigenvalues which will be used  
 *         by DLAED3 to form the secular equation.  
 *  
 *  Q2     (output) COMPLEX*16 array, dimension (LDQ2,N)  
 *         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,  
 *         Contains 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)  
 *         This will hold the first k values of the final  
 *         deflation-altered z-vector and will be passed to DLAED3.  
 *  
 *  INDXP  (workspace) INTEGER array, dimension (N)  
 *         This will contain the permutation used to place deflated  
 *         values of D at the end of the array. On output 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)  
 *         This will contain the permutation used to sort the contents of  
 *         D into ascending order.  
 *  
 *  INDXQ  (input) INTEGER array, dimension (N)  
 *         This contains 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.  
 *  
 *  PERM   (output) INTEGER array, dimension (N)  
 *         Contains the permutations (from deflation and sorting) to be  
 *         applied to each eigenblock.  
 *  
 *  GIVPTR (output) INTEGER  
 *         Contains 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.  
 *  
 *  INFO   (output) INTEGER  
 *          = 0:  successful exit.  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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          RETURN           RETURN
       END IF        END IF
 *  *
   *     Need to initialize GIVPTR to O here in case of quick exit
   *     to prevent an unspecified code behavior (usually sigfault)
   *     when IWORK array on entry to *stedc is not zeroed
   *     (or at least some IWORK entries which used in *laed7 for GIVPTR).
   *
         GIVPTR = 0
   *
 *     Quick return if possible  *     Quick return if possible
 *  *
       IF( N.EQ.0 )        IF( N.EQ.0 )
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 *     components of Z are zero in this new basis.  *     components of Z are zero in this new basis.
 *  *
       K = 0        K = 0
       GIVPTR = 0  
       K2 = N + 1        K2 = N + 1
       DO 60 J = 1, N        DO 60 J = 1, N
          IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN           IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
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  Added in v.1.19

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