dgeesx.f

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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/
*
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*> \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.
*
*> \date June 2016
*
*> \ingroup doubleGEeigen
*
*  =====================================================================
      SUBROUTINE dgeesx( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
     $                   WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
     $                   IWORK, LIWORK, BWORK, INFO )
*
*  -- LAPACK driver routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. 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
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d0, one = 1.0d0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTSB,
     $                   WANTSE, WANTSN, WANTST, WANTSV, WANTVS
      INTEGER            HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL,
     $                   IHI, ILO, INXT, IP, ITAU, IWRK, LIWRK, LWRK,
     $                   maxwrk, minwrk
      DOUBLE PRECISION   ANRM, BIGNUM, CSCALE, EPS, SMLNUM
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           dcopy, dgebak, dgebal, dgehrd, dhseqr, dlacpy,
     $                   dlascl, dorghr, dswap, dtrsen, xerbla
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           lsame, ilaenv, dlabad, dlamch, dlange
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      wantvs = lsame( jobvs, 'V' )
      wantst = lsame( sort, 'S' )
      wantsn = lsame( sense, 'N' )
      wantse = lsame( sense, 'E' )
      wantsv = lsame( sense, 'V' )
      wantsb = lsame( sense, 'B' )
      lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
*
      IF( ( .NOT.wantvs ) .AND. ( .NOT.lsame( jobvs, 'N' ) ) ) THEN
         info = -1
      ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
         info = -2
      ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsv .OR. wantsb ) .OR.
     $         ( .NOT.wantst .AND. .NOT.wantsn ) ) THEN
         info = -4
      ELSE IF( n.LT.0 ) THEN
         info = -5
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -7
      ELSE IF( ldvs.LT.1 .OR. ( wantvs .AND. ldvs.LT.n ) ) THEN
         info = -12
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "RWorkspace:" describe the
*       minimal amount of real workspace needed at that point in the
*       code, as well as the preferred amount for good performance.
*       IWorkspace refers to integer workspace.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.
*       HSWORK refers to the workspace preferred by DHSEQR, as
*       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
*       the worst case.
*       If SENSE = 'E', 'V' or 'B', then the amount of workspace needed
*       depends on SDIM, which is computed by the routine DTRSEN later
*       in the code.)
*
      IF( info.EQ.0 ) THEN
         liwrk = 1
         IF( n.EQ.0 ) THEN
            minwrk = 1
            lwrk = 1
         ELSE
            maxwrk = 2*n + n*ilaenv( 1, 'DGEHRD', ' ', n, 1, n, 0 )
            minwrk = 3*n
*
            CALL dhseqr( 'S', jobvs, n, 1, n, a, lda, wr, wi, vs, ldvs,
     $             work, -1, ieval )
            hswork = work( 1 )
*
            IF( .NOT.wantvs ) THEN
               maxwrk = max( maxwrk, n + hswork )
            ELSE
               maxwrk = max( maxwrk, 2*n + ( n - 1 )*ilaenv( 1,
     $                       'DORGHR', ' ', n, 1, n, -1 ) )
               maxwrk = max( maxwrk, n + hswork )
            END IF
            lwrk = maxwrk
            IF( .NOT.wantsn )
     $         lwrk = max( lwrk, n + ( n*n )/2 )
            IF( wantsv .OR. wantsb )
     $         liwrk = ( n*n )/4
         END IF
         iwork( 1 ) = liwrk
         work( 1 ) = lwrk
*
         IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
            info = -16
         ELSE IF( liwork.LT.1 .AND. .NOT.lquery ) THEN
            info = -18
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DGEESX', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 ) THEN
         sdim = 0
         RETURN
      END IF
*
*     Get machine constants
*
      eps = dlamch( 'P' )
      smlnum = dlamch( 'S' )
      bignum = one / smlnum
      CALL dlabad( smlnum, bignum )
      smlnum = sqrt( smlnum ) / eps
      bignum = one / smlnum
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      anrm = dlange( 'M', n, n, a, lda, dum )
      scalea = .false.
      IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
         scalea = .true.
         cscale = smlnum
      ELSE IF( anrm.GT.bignum ) THEN
         scalea = .true.
         cscale = bignum
      END IF
      IF( scalea )
     $   CALL dlascl( 'G', 0, 0, anrm, cscale, n, n, a, lda, ierr )
*
*     Permute the matrix to make it more nearly triangular
*     (RWorkspace: need N)
*
      ibal = 1
      CALL dgebal( 'P', n, a, lda, ilo, ihi, work( ibal ), ierr )
*
*     Reduce to upper Hessenberg form
*     (RWorkspace: need 3*N, prefer 2*N+N*NB)
*
      itau = n + ibal
      iwrk = n + itau
      CALL dgehrd( n, ilo, ihi, a, lda, work( itau ), work( iwrk ),
     $             lwork-iwrk+1, ierr )
*
      IF( wantvs ) THEN
*
*        Copy Householder vectors to VS
*
         CALL dlacpy( 'L', n, n, a, lda, vs, ldvs )
*
*        Generate orthogonal matrix in VS
*        (RWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
         CALL dorghr( n, ilo, ihi, vs, ldvs, work( itau ), work( iwrk ),
     $                lwork-iwrk+1, ierr )
      END IF
*
      sdim = 0
*
*     Perform QR iteration, accumulating Schur vectors in VS if desired
*     (RWorkspace: need N+1, prefer N+HSWORK (see comments) )
*
      iwrk = itau
      CALL dhseqr( 'S', jobvs, n, ilo, ihi, a, lda, wr, wi, vs, ldvs,
     $             work( iwrk ), lwork-iwrk+1, ieval )
      IF( ieval.GT.0 )
     $   info = ieval
*
*     Sort eigenvalues if desired
*
      IF( wantst .AND. info.EQ.0 ) THEN
         IF( scalea ) THEN
            CALL dlascl( 'G', 0, 0, cscale, anrm, n, 1, wr, n, ierr )
            CALL dlascl( 'G', 0, 0, cscale, anrm, n, 1, wi, n, ierr )
         END IF
         DO 10 i = 1, n
            bwork( i ) = SELECT( wr( i ), wi( i ) )
   10    CONTINUE
*
*        Reorder eigenvalues, transform Schur vectors, and compute
*        reciprocal condition numbers
*        (RWorkspace: if SENSE is not 'N', need N+2*SDIM*(N-SDIM)
*                     otherwise, need N )
*        (IWorkspace: if SENSE is 'V' or 'B', need SDIM*(N-SDIM)
*                     otherwise, need 0 )
*
         CALL dtrsen( sense, jobvs, bwork, n, a, lda, vs, ldvs, wr, wi,
     $                sdim, rconde, rcondv, work( iwrk ), lwork-iwrk+1,
     $                iwork, liwork, icond )
         IF( .NOT.wantsn )
     $      maxwrk = max( maxwrk, n+2*sdim*( n-sdim ) )
         IF( icond.EQ.-15 ) THEN
*
*           Not enough real workspace
*
            info = -16
         ELSE IF( icond.EQ.-17 ) THEN
*
*           Not enough integer workspace
*
            info = -18
         ELSE IF( icond.GT.0 ) THEN
*
*           DTRSEN failed to reorder or to restore standard Schur form
*
            info = icond + n
         END IF
      END IF
*
      IF( wantvs ) THEN
*
*        Undo balancing
*        (RWorkspace: need N)
*
         CALL dgebak( 'P', 'R', n, ilo, ihi, work( ibal ), n, vs, ldvs,
     $                ierr )
      END IF
*
      IF( scalea ) THEN
*
*        Undo scaling for the Schur form of A
*
         CALL dlascl( 'H', 0, 0, cscale, anrm, n, n, a, lda, ierr )
         CALL dcopy( n, a, lda+1, wr, 1 )
         IF( ( wantsv .OR. wantsb ) .AND. info.EQ.0 ) THEN
            dum( 1 ) = rcondv
            CALL dlascl( 'G', 0, 0, cscale, anrm, 1, 1, dum, 1, ierr )
            rcondv = dum( 1 )
         END IF
         IF( cscale.EQ.smlnum ) THEN
*
*           If scaling back towards underflow, adjust WI if an
*           offdiagonal element of a 2-by-2 block in the Schur form
*           underflows.
*
            IF( ieval.GT.0 ) THEN
               i1 = ieval + 1
               i2 = ihi - 1
               CALL dlascl( 'G', 0, 0, cscale, anrm, ilo-1, 1, wi, n,
     $                      ierr )
            ELSE IF( wantst ) THEN
               i1 = 1
               i2 = n - 1
            ELSE
               i1 = ilo
               i2 = ihi - 1
            END IF
            inxt = i1 - 1
            DO 20 i = i1, i2
               IF( i.LT.inxt )
     $            GO TO 20
               IF( wi( i ).EQ.zero ) THEN
                  inxt = i + 1
               ELSE
                  IF( a( i+1, i ).EQ.zero ) THEN
                     wi( i ) = zero
                     wi( i+1 ) = zero
                  ELSE IF( a( i+1, i ).NE.zero .AND. a( i, i+1 ).EQ.
     $                     zero ) THEN
                     wi( i ) = zero
                     wi( i+1 ) = zero
                     IF( i.GT.1 )
     $                  CALL dswap( i-1, a( 1, i ), 1, a( 1, i+1 ), 1 )
                     IF( n.GT.i+1 )
     $                  CALL dswap( n-i-1, a( i, i+2 ), lda,
     $                              a( i+1, i+2 ), lda )
                     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+1, i ) = zero
                  END IF
                  inxt = i + 2
               END IF
   20       CONTINUE
         END IF
         CALL dlascl( 'G', 0, 0, cscale, anrm, n-ieval, 1,
     $                wi( ieval+1 ), max( n-ieval, 1 ), ierr )
      END IF
*
      IF( wantst .AND. info.EQ.0 ) THEN
*
*        Check if reordering successful
*
         lastsl = .true.
         lst2sl = .true.
         sdim = 0
         ip = 0
         DO 30 i = 1, n
            cursl = SELECT( wr( i ), wi( i ) )
            IF( wi( i ).EQ.zero ) THEN
               IF( cursl )
     $            sdim = sdim + 1
               ip = 0
               IF( cursl .AND. .NOT.lastsl )
     $            info = n + 2
            ELSE
               IF( ip.EQ.1 ) THEN
*
*                 Last eigenvalue of conjugate pair
*
                  cursl = cursl .OR. lastsl
                  lastsl = cursl
                  IF( cursl )
     $               sdim = sdim + 2
                  ip = -1
                  IF( cursl .AND. .NOT.lst2sl )
     $               info = n + 2
               ELSE
*
*                 First eigenvalue of conjugate pair
*
                  ip = 1
               END IF
            END IF
            lst2sl = lastsl
            lastsl = cursl
   30    CONTINUE
      END IF
*
      work( 1 ) = maxwrk
      IF( wantsv .OR. wantsb ) THEN
         iwork( 1 ) = max( 1, sdim*( n-sdim ) )
      ELSE
         iwork( 1 ) = 1
      END IF
*
      RETURN
*
*     End of DGEESX
*
      END