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version 1.7, 2010/12/21 13:53:28 version 1.8, 2011/11/21 20:42:54
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   *> \brief \b DLAED2
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLAED2 + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed2.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed2.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed2.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
   *                          Q2, INDX, INDXC, INDXP, COLTYP, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, K, LDQ, N, N1
   *       DOUBLE PRECISION   RHO
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
   *      $                   INDXQ( * )
   *       DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
   *      $                   W( * ), Z( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLAED2 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.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[out] K
   *> \verbatim
   *>          K is INTEGER
   *>         The number of non-deflated eigenvalues, and the order of the
   *>         related secular equation. 0 <= K <=N.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] N1
   *> \verbatim
   *>          N1 is INTEGER
   *>         The location of the last eigenvalue in the leading sub-matrix.
   *>         min(1,N) <= N1 <= N/2.
   *> \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] Q
   *> \verbatim
   *>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
   *>         On entry, Q contains the eigenvectors of two submatrices in
   *>         the two square blocks with corners at (1,1), (N1,N1)
   *>         and (N1+1, N1+1), (N,N).
   *>         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] 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 N1 added to their
   *>         values. Destroyed on exit.
   *> \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] 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 have been 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] W
   *> \verbatim
   *>          W is DOUBLE PRECISION array, dimension (N)
   *>         The first k values of the final deflation-altered z-vector
   *>         which will be passed to DLAED3.
   *> \endverbatim
   *>
   *> \param[out] Q2
   *> \verbatim
   *>          Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
   *>         A copy of the first K eigenvectors which will be used by
   *>         DLAED3 in a matrix multiply (DGEMM) to solve for the new
   *>         eigenvectors.
   *> \endverbatim
   *>
   *> \param[out] INDX
   *> \verbatim
   *>          INDX is INTEGER array, dimension (N)
   *>         The permutation used to sort the contents of DLAMDA into
   *>         ascending order.
   *> \endverbatim
   *>
   *> \param[out] INDXC
   *> \verbatim
   *>          INDXC is 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 N1, the second contains
   *>         non-zero elements only below N1, and the third is dense.
   *> \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] COLTYP
   *> \verbatim
   *>          COLTYP is 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 : dense;
   *>         3 : non-zero in the lower half only;
   *>         4 : deflated.
   *>         On exit, COLTYP(i) is the number of columns of type i,
   *>         for i=1 to 4 only.
   *> \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 \n
   *>  Modified by Francoise Tisseur, University of Tennessee
   *>
   *  =====================================================================
       SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,        SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
      $                   Q2, INDX, INDXC, INDXP, COLTYP, INFO )       $                   Q2, INDX, INDXC, INDXP, COLTYP, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- 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..--
 *     November 2006  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, K, LDQ, N, N1        INTEGER            INFO, K, LDQ, N, N1
Line 17 Line 228
      $                   W( * ), Z( * )       $                   W( * ), Z( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLAED2 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.  
 *  
 *  N1     (input) INTEGER  
 *         The location of the last eigenvalue in the leading sub-matrix.  
 *         min(1,N) <= N1 <= N/2.  
 *  
 *  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.  
 *  
 *  Q      (input/output) DOUBLE PRECISION array, dimension (LDQ, N)  
 *         On entry, Q contains the eigenvectors of two submatrices in  
 *         the two square blocks with corners at (1,1), (N1,N1)  
 *         and (N1+1, N1+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 N1 added to their  
 *         values. Destroyed on exit.  
 *  
 *  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.  
 *  
 *  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 have been 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.  
 *  
 *  W      (output) DOUBLE PRECISION array, dimension (N)  
 *         The first k values of the final deflation-altered z-vector  
 *         which will be passed to DLAED3.  
 *  
 *  Q2     (output) DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)  
 *         A copy of the first K eigenvectors which will be used by  
 *         DLAED3 in a matrix multiply (DGEMM) to solve for the new  
 *         eigenvectors.  
 *  
 *  INDX   (workspace) INTEGER array, dimension (N)  
 *         The permutation used to sort the contents of DLAMDA into  
 *         ascending order.  
 *  
 *  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 N1, the second contains  
 *         non-zero elements only below N1, and the third is dense.  
 *  
 *  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.  
 *  
 *  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 : dense;  
 *         3 : non-zero in the lower half only;  
 *         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.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *     Jeff Rutter, Computer Science Division, University of California  
 *     at Berkeley, USA  
 *  Modified by Francoise Tisseur, University of Tennessee.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 418 Line 519
 *     The deflated eigenvalues and their corresponding vectors go back  *     The deflated eigenvalues and their corresponding vectors go back
 *     into the last N - K slots of D and Q respectively.  *     into the last N - K slots of D and Q respectively.
 *  *
       CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N, Q( 1, K+1 ), LDQ )        IF( K.LT.N ) THEN
       CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 )           CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N, 
        $                Q( 1, K+1 ), LDQ )
            CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 )
         END IF         
 *  *
 *     Copy CTOT into COLTYP for referencing in DLAED3.  *     Copy CTOT into COLTYP for referencing in DLAED3.
 *  *
Removed from v.1.7  
changed lines
  Added in v.1.8

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