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   *> \brief <b> DGEESX 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 DGEESX + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeesx.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeesx.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeesx.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
   *                          WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
   *                          IWORK, LIWORK, BWORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBVS, SENSE, SORT
   *       INTEGER            INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
   *       DOUBLE PRECISION   RCONDE, RCONDV
   *       ..
   *       .. Array Arguments ..
   *       LOGICAL            BWORK( * )
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
   *      $                   WR( * )
   *       ..
   *       .. Function Arguments ..
   *       LOGICAL            SELECT
   *       EXTERNAL           SELECT
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGEESX computes for an N-by-N real nonsymmetric matrix A, the
   *> eigenvalues, the real Schur form T, and, optionally, the matrix of
   *> Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).
   *>
   *> Optionally, it also orders the eigenvalues on the diagonal of the
   *> real 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 real matrix is in real Schur form if it is upper quasi-triangular
   *> with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
   *> the form
   *>           [  a  b  ]
   *>           [  c  a  ]
   *>
   *> where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
   *> \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 a LOGICAL FUNCTION of two DOUBLE PRECISION arguments
   *>          SELECT must be declared EXTERNAL in the calling subroutine.
   *>          If SORT = 'S', SELECT is used to select eigenvalues to sort
   *>          to the top left of the Schur form.
   *>          If SORT = 'N', SELECT is not referenced.
   *>          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
   *>          SELECT(WR(j),WI(j)) is true; i.e., if either one of a
   *>          complex conjugate pair of eigenvalues is selected, then both
   *>          are.  Note that a selected complex eigenvalue may no longer
   *>          satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
   *>          ordering may change the value of complex eigenvalues
   *>          (especially if the eigenvalue is ill-conditioned); in this
   *>          case INFO may be set to N+3 (see INFO below).
   *> \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 DOUBLE PRECISION array, dimension (LDA, N)
   *>          On entry, the N-by-N matrix A.
   *>          On exit, A is overwritten by its real 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 (after sorting)
   *>                         for which SELECT is true. (Complex conjugate
   *>                         pairs for which SELECT is true for either
   *>                         eigenvalue count as 2.)
   *> \endverbatim
   *>
   *> \param[out] WR
   *> \verbatim
   *>          WR is DOUBLE PRECISION array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] WI
   *> \verbatim
   *>          WI is DOUBLE PRECISION array, dimension (N)
   *>          WR and WI contain the real and imaginary parts, respectively,
   *>          of the computed eigenvalues, in the same order that they
   *>          appear on the diagonal of the output Schur form T.  Complex
   *>          conjugate pairs of eigenvalues appear consecutively with the
   *>          eigenvalue having the positive imaginary part first.
   *> \endverbatim
   *>
   *> \param[out] VS
   *> \verbatim
   *>          VS is DOUBLE PRECISION array, dimension (LDVS,N)
   *>          If JOBVS = 'V', VS contains the orthogonal 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 DOUBLE PRECISION 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(1,3*N).
   *>          Also, if SENSE = 'E' or 'V' or 'B',
   *>          LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
   *>          selected eigenvalues computed by this routine.  Note that
   *>          N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
   *>          returned if LWORK < max(1,3*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 bounds on the optimal sizes of the
   *>          arrays WORK and IWORK, returns these values as the first
   *>          entries of the WORK and IWORK arrays, and no error messages
   *>          related to LWORK or LIWORK are issued by XERBLA.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
   *>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
   *> \endverbatim
   *>
   *> \param[in] LIWORK
   *> \verbatim
   *>          LIWORK is INTEGER
   *>          The dimension of the array IWORK.
   *>          LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
   *>          Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
   *>          only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
   *>          may not be large enough.
   *>
   *>          If LIWORK = -1, then a workspace query is assumed; the
   *>          routine only calculates upper bounds on the optimal sizes of
   *>          the arrays WORK and IWORK, returns these values as the first
   *>          entries of the WORK and IWORK arrays, and no error messages
   *>          related to LWORK or LIWORK are issued by XERBLA.
   *> \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 WR and WI
   *>                   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.
   *
   *> \ingroup doubleGEeigen
   *
   *  =====================================================================
       SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,        SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
      $                   WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,       $                   WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
      $                   IWORK, LIWORK, BWORK, INFO )       $                   IWORK, LIWORK, BWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2.2) --  *  -- LAPACK driver 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..--
 *     June 2010  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBVS, SENSE, SORT        CHARACTER          JOBVS, SENSE, SORT
Line 23 Line 299
       EXTERNAL           SELECT        EXTERNAL           SELECT
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DGEESX computes for an N-by-N real nonsymmetric matrix A, the  
 *  eigenvalues, the real Schur form T, and, optionally, the matrix of  
 *  Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).  
 *  
 *  Optionally, it also orders the eigenvalues on the diagonal of the  
 *  real 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 real matrix is in real Schur form if it is upper quasi-triangular  
 *  with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in  
 *  the form  
 *            [  a  b  ]  
 *            [  c  a  ]  
 *  
 *  where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).  
 *  
 *  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 two DOUBLE PRECISION arguments  
 *          SELECT must be declared EXTERNAL in the calling subroutine.  
 *          If SORT = 'S', SELECT is used to select eigenvalues to sort  
 *          to the top left of the Schur form.  
 *          If SORT = 'N', SELECT is not referenced.  
 *          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if  
 *          SELECT(WR(j),WI(j)) is true; i.e., if either one of a  
 *          complex conjugate pair of eigenvalues is selected, then both  
 *          are.  Note that a selected complex eigenvalue may no longer  
 *          satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since  
 *          ordering may change the value of complex eigenvalues  
 *          (especially if the eigenvalue is ill-conditioned); in this  
 *          case INFO may be set to N+3 (see INFO below).  
 *  
 *  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) DOUBLE PRECISION array, dimension (LDA, N)  
 *          On entry, the N-by-N matrix A.  
 *          On exit, A is overwritten by its real 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 (after sorting)  
 *                         for which SELECT is true. (Complex conjugate  
 *                         pairs for which SELECT is true for either  
 *                         eigenvalue count as 2.)  
 *  
 *  WR      (output) DOUBLE PRECISION array, dimension (N)  
 *  WI      (output) DOUBLE PRECISION array, dimension (N)  
 *          WR and WI contain the real and imaginary parts, respectively,  
 *          of the computed eigenvalues, in the same order that they  
 *          appear on the diagonal of the output Schur form T.  Complex  
 *          conjugate pairs of eigenvalues appear consecutively with the  
 *          eigenvalue having the positive imaginary part first.  
 *  
 *  VS      (output) DOUBLE PRECISION array, dimension (LDVS,N)  
 *          If JOBVS = 'V', VS contains the orthogonal 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) DOUBLE PRECISION 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(1,3*N).  
 *          Also, if SENSE = 'E' or 'V' or 'B',  
 *          LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of  
 *          selected eigenvalues computed by this routine.  Note that  
 *          N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only  
 *          returned if LWORK < max(1,3*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 bounds on the optimal sizes of the  
 *          arrays WORK and IWORK, returns these values as the first  
 *          entries of the WORK and IWORK arrays, and no error messages  
 *          related to LWORK or LIWORK are issued by XERBLA.  
 *  
 *  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))  
 *          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.  
 *  
 *  LIWORK  (input) INTEGER  
 *          The dimension of the array IWORK.  
 *          LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).  
 *          Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is  
 *          only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this  
 *          may not be large enough.  
 *  
 *          If LIWORK = -1, then a workspace query is assumed; the  
 *          routine only calculates upper bounds on the optimal sizes of  
 *          the arrays WORK and IWORK, returns these values as the first  
 *          entries of the WORK and IWORK arrays, and no error messages  
 *          related to LWORK or LIWORK are issued by XERBLA.  
 *  
 *  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 WR and WI  
 *                   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 ..
Line 267 Line 382
 *  *
             CALL DHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS, LDVS,              CALL DHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS, LDVS,
      $             WORK, -1, IEVAL )       $             WORK, -1, IEVAL )
             HSWORK = WORK( 1 )              HSWORK = INT( WORK( 1 ) )
 *  *
             IF( .NOT.WANTVS ) THEN              IF( .NOT.WANTVS ) THEN
                MAXWRK = MAX( MAXWRK, N + HSWORK )                 MAXWRK = MAX( MAXWRK, N + HSWORK )
Line 465 Line 580
                      IF( N.GT.I+1 )                       IF( N.GT.I+1 )
      $                  CALL DSWAP( N-I-1, A( I, I+2 ), LDA,       $                  CALL DSWAP( N-I-1, A( I, I+2 ), LDA,
      $                              A( I+1, I+2 ), LDA )       $                              A( I+1, I+2 ), LDA )
                      CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 )                       IF( WANTVS ) THEN
                          CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 )
                        END IF
                      A( I, I+1 ) = A( I+1, I )                       A( I, I+1 ) = A( I+1, I )
                      A( I+1, I ) = ZERO                       A( I+1, I ) = ZERO
                   END IF                    END IF
Removed from v.1.8  
changed lines
  Added in v.1.20

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