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version 1.8, 2010/12/21 13:53:43 version 1.9, 2011/11/21 20:43:08
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   *> \brief <b> ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZGEESX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgeesx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgeesx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgeesx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
   *                          VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
   *                          BWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBVS, SENSE, SORT
   *       INTEGER            INFO, LDA, LDVS, LWORK, N, SDIM
   *       DOUBLE PRECISION   RCONDE, RCONDV
   *       ..
   *       .. Array Arguments ..
   *       LOGICAL            BWORK( * )
   *       DOUBLE PRECISION   RWORK( * )
   *       COMPLEX*16         A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
   *       ..
   *       .. Function Arguments ..
   *       LOGICAL            SELECT
   *       EXTERNAL           SELECT
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGEESX computes for an N-by-N complex nonsymmetric matrix A, the
   *> eigenvalues, the Schur form T, and, optionally, the matrix of Schur
   *> vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).
   *>
   *> Optionally, it also orders the eigenvalues on the diagonal of the
   *> Schur form so that selected eigenvalues are at the top left;
   *> computes a reciprocal condition number for the average of the
   *> selected eigenvalues (RCONDE); and computes a reciprocal condition
   *> number for the right invariant subspace corresponding to the
   *> selected eigenvalues (RCONDV).  The leading columns of Z form an
   *> orthonormal basis for this invariant subspace.
   *>
   *> For further explanation of the reciprocal condition numbers RCONDE
   *> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
   *> these quantities are called s and sep respectively).
   *>
   *> A complex matrix is in Schur form if it is upper triangular.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBVS
   *> \verbatim
   *>          JOBVS is CHARACTER*1
   *>          = 'N': Schur vectors are not computed;
   *>          = 'V': Schur vectors are computed.
   *> \endverbatim
   *>
   *> \param[in] SORT
   *> \verbatim
   *>          SORT is CHARACTER*1
   *>          Specifies whether or not to order the eigenvalues on the
   *>          diagonal of the Schur form.
   *>          = 'N': Eigenvalues are not ordered;
   *>          = 'S': Eigenvalues are ordered (see SELECT).
   *> \endverbatim
   *>
   *> \param[in] SELECT
   *> \verbatim
   *>          SELECT is procedure) LOGICAL FUNCTION of one COMPLEX*16 argument
   *>          SELECT must be declared EXTERNAL in the calling subroutine.
   *>          If SORT = 'S', SELECT is used to select eigenvalues to order
   *>          to the top left of the Schur form.
   *>          If SORT = 'N', SELECT is not referenced.
   *>          An eigenvalue W(j) is selected if SELECT(W(j)) is true.
   *> \endverbatim
   *>
   *> \param[in] SENSE
   *> \verbatim
   *>          SENSE is CHARACTER*1
   *>          Determines which reciprocal condition numbers are computed.
   *>          = 'N': None are computed;
   *>          = 'E': Computed for average of selected eigenvalues only;
   *>          = 'V': Computed for selected right invariant subspace only;
   *>          = 'B': Computed for both.
   *>          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix A. N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is COMPLEX*16 array, dimension (LDA, N)
   *>          On entry, the N-by-N matrix A.
   *>          On exit, A is overwritten by its Schur form T.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[out] SDIM
   *> \verbatim
   *>          SDIM is INTEGER
   *>          If SORT = 'N', SDIM = 0.
   *>          If SORT = 'S', SDIM = number of eigenvalues for which
   *>                         SELECT is true.
   *> \endverbatim
   *>
   *> \param[out] W
   *> \verbatim
   *>          W is COMPLEX*16 array, dimension (N)
   *>          W contains the computed eigenvalues, in the same order
   *>          that they appear on the diagonal of the output Schur form T.
   *> \endverbatim
   *>
   *> \param[out] VS
   *> \verbatim
   *>          VS is COMPLEX*16 array, dimension (LDVS,N)
   *>          If JOBVS = 'V', VS contains the unitary matrix Z of Schur
   *>          vectors.
   *>          If JOBVS = 'N', VS is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDVS
   *> \verbatim
   *>          LDVS is INTEGER
   *>          The leading dimension of the array VS.  LDVS >= 1, and if
   *>          JOBVS = 'V', LDVS >= N.
   *> \endverbatim
   *>
   *> \param[out] RCONDE
   *> \verbatim
   *>          RCONDE is DOUBLE PRECISION
   *>          If SENSE = 'E' or 'B', RCONDE contains the reciprocal
   *>          condition number for the average of the selected eigenvalues.
   *>          Not referenced if SENSE = 'N' or 'V'.
   *> \endverbatim
   *>
   *> \param[out] RCONDV
   *> \verbatim
   *>          RCONDV is DOUBLE PRECISION
   *>          If SENSE = 'V' or 'B', RCONDV contains the reciprocal
   *>          condition number for the selected right invariant subspace.
   *>          Not referenced if SENSE = 'N' or 'E'.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
   *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is INTEGER
   *>          The dimension of the array WORK.  LWORK >= max(1,2*N).
   *>          Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
   *>          where SDIM is the number of selected eigenvalues computed by
   *>          this routine.  Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
   *>          that an error is only returned if LWORK < max(1,2*N), but if
   *>          SENSE = 'E' or 'V' or 'B' this may not be large enough.
   *>          For good performance, LWORK must generally be larger.
   *>
   *>          If LWORK = -1, then a workspace query is assumed; the routine
   *>          only calculates upper bound on the optimal size of the
   *>          array WORK, returns this value as the first entry of the WORK
   *>          array, and no error message related to LWORK is issued by
   *>          XERBLA.
   *> \endverbatim
   *>
   *> \param[out] RWORK
   *> \verbatim
   *>          RWORK is DOUBLE PRECISION array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] BWORK
   *> \verbatim
   *>          BWORK is LOGICAL array, dimension (N)
   *>          Not referenced if SORT = 'N'.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0: successful exit
   *>          < 0: if INFO = -i, the i-th argument had an illegal value.
   *>          > 0: if INFO = i, and i is
   *>             <= N: the QR algorithm failed to compute all the
   *>                   eigenvalues; elements 1:ILO-1 and i+1:N of W
   *>                   contain those eigenvalues which have converged; if
   *>                   JOBVS = 'V', VS contains the transformation which
   *>                   reduces A to its partially converged Schur form.
   *>             = N+1: the eigenvalues could not be reordered because some
   *>                   eigenvalues were too close to separate (the problem
   *>                   is very ill-conditioned);
   *>             = N+2: after reordering, roundoff changed values of some
   *>                   complex eigenvalues so that leading eigenvalues in
   *>                   the Schur form no longer satisfy SELECT=.TRUE.  This
   *>                   could also be caused by underflow due to scaling.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup complex16GEeigen
   *
   *  =====================================================================
       SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,        SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
      $                   VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,       $                   VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
      $                   BWORK, INFO )       $                   BWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2.2) --  *  -- LAPACK driver 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 ..
       CHARACTER          JOBVS, SENSE, SORT        CHARACTER          JOBVS, SENSE, SORT
Line 22 Line 259
       EXTERNAL           SELECT        EXTERNAL           SELECT
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGEESX computes for an N-by-N complex nonsymmetric matrix A, the  
 *  eigenvalues, the Schur form T, and, optionally, the matrix of Schur  
 *  vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).  
 *  
 *  Optionally, it also orders the eigenvalues on the diagonal of the  
 *  Schur form so that selected eigenvalues are at the top left;  
 *  computes a reciprocal condition number for the average of the  
 *  selected eigenvalues (RCONDE); and computes a reciprocal condition  
 *  number for the right invariant subspace corresponding to the  
 *  selected eigenvalues (RCONDV).  The leading columns of Z form an  
 *  orthonormal basis for this invariant subspace.  
 *  
 *  For further explanation of the reciprocal condition numbers RCONDE  
 *  and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where  
 *  these quantities are called s and sep respectively).  
 *  
 *  A complex matrix is in Schur form if it is upper triangular.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOBVS   (input) CHARACTER*1  
 *          = 'N': Schur vectors are not computed;  
 *          = 'V': Schur vectors are computed.  
 *  
 *  SORT    (input) CHARACTER*1  
 *          Specifies whether or not to order the eigenvalues on the  
 *          diagonal of the Schur form.  
 *          = 'N': Eigenvalues are not ordered;  
 *          = 'S': Eigenvalues are ordered (see SELECT).  
 *  
 *  SELECT  (external procedure) LOGICAL FUNCTION of one COMPLEX*16 argument  
 *          SELECT must be declared EXTERNAL in the calling subroutine.  
 *          If SORT = 'S', SELECT is used to select eigenvalues to order  
 *          to the top left of the Schur form.  
 *          If SORT = 'N', SELECT is not referenced.  
 *          An eigenvalue W(j) is selected if SELECT(W(j)) is true.  
 *  
 *  SENSE   (input) CHARACTER*1  
 *          Determines which reciprocal condition numbers are computed.  
 *          = 'N': None are computed;  
 *          = 'E': Computed for average of selected eigenvalues only;  
 *          = 'V': Computed for selected right invariant subspace only;  
 *          = 'B': Computed for both.  
 *          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix A. N >= 0.  
 *  
 *  A       (input/output) COMPLEX*16 array, dimension (LDA, N)  
 *          On entry, the N-by-N matrix A.  
 *          On exit, A is overwritten by its Schur form T.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *  SDIM    (output) INTEGER  
 *          If SORT = 'N', SDIM = 0.  
 *          If SORT = 'S', SDIM = number of eigenvalues for which  
 *                         SELECT is true.  
 *  
 *  W       (output) COMPLEX*16 array, dimension (N)  
 *          W contains the computed eigenvalues, in the same order  
 *          that they appear on the diagonal of the output Schur form T.  
 *  
 *  VS      (output) COMPLEX*16 array, dimension (LDVS,N)  
 *          If JOBVS = 'V', VS contains the unitary matrix Z of Schur  
 *          vectors.  
 *          If JOBVS = 'N', VS is not referenced.  
 *  
 *  LDVS    (input) INTEGER  
 *          The leading dimension of the array VS.  LDVS >= 1, and if  
 *          JOBVS = 'V', LDVS >= N.  
 *  
 *  RCONDE  (output) DOUBLE PRECISION  
 *          If SENSE = 'E' or 'B', RCONDE contains the reciprocal  
 *          condition number for the average of the selected eigenvalues.  
 *          Not referenced if SENSE = 'N' or 'V'.  
 *  
 *  RCONDV  (output) DOUBLE PRECISION  
 *          If SENSE = 'V' or 'B', RCONDV contains the reciprocal  
 *          condition number for the selected right invariant subspace.  
 *          Not referenced if SENSE = 'N' or 'E'.  
 *  
 *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))  
 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.  
 *  
 *  LWORK   (input) INTEGER  
 *          The dimension of the array WORK.  LWORK >= max(1,2*N).  
 *          Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),  
 *          where SDIM is the number of selected eigenvalues computed by  
 *          this routine.  Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also  
 *          that an error is only returned if LWORK < max(1,2*N), but if  
 *          SENSE = 'E' or 'V' or 'B' this may not be large enough.  
 *          For good performance, LWORK must generally be larger.  
 *  
 *          If LWORK = -1, then a workspace query is assumed; the routine  
 *          only calculates upper bound on the optimal size of the  
 *          array WORK, returns this value as the first entry of the WORK  
 *          array, and no error message related to LWORK is issued by  
 *          XERBLA.  
 *  
 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)  
 *  
 *  BWORK   (workspace) LOGICAL array, dimension (N)  
 *          Not referenced if SORT = 'N'.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0: successful exit  
 *          < 0: if INFO = -i, the i-th argument had an illegal value.  
 *          > 0: if INFO = i, and i is  
 *             <= N: the QR algorithm failed to compute all the  
 *                   eigenvalues; elements 1:ILO-1 and i+1:N of W  
 *                   contain those eigenvalues which have converged; if  
 *                   JOBVS = 'V', VS contains the transformation which  
 *                   reduces A to its partially converged Schur form.  
 *             = N+1: the eigenvalues could not be reordered because some  
 *                   eigenvalues were too close to separate (the problem  
 *                   is very ill-conditioned);  
 *             = N+2: after reordering, roundoff changed values of some  
 *                   complex eigenvalues so that leading eigenvalues in  
 *                   the Schur form no longer satisfy SELECT=.TRUE.  This  
 *                   could also be caused by underflow due to scaling.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.8  
changed lines
  Added in v.1.9

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