```      SUBROUTINE CGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
\$                   VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
\$                   BWORK, INFO )
*
*  -- LAPACK driver routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
CHARACTER          JOBVS, SENSE, SORT
INTEGER            INFO, LDA, LDVS, LWORK, N, SDIM
REAL               RCONDE, RCONDV
*     ..
*     .. Array Arguments ..
LOGICAL            BWORK( * )
REAL               RWORK( * )
COMPLEX            A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
*     ..
*     .. Function Arguments ..
LOGICAL            SELECT
EXTERNAL           SELECT
*     ..
*
*  Purpose
*  =======
*
*  CGEESX 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 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 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 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 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) REAL
*          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) REAL
*          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 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) REAL 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 ..
REAL               ZERO, ONE
PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
*     ..
*     .. Local Scalars ..
LOGICAL            SCALEA, WANTSB, WANTSE, WANTSN, WANTST,
\$                   WANTSV, WANTVS
INTEGER            HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO,
\$                   ITAU, IWRK, LWRK, MAXWRK, MINWRK
REAL               ANRM, BIGNUM, CSCALE, EPS, SMLNUM
*     ..
*     .. Local Arrays ..
REAL               DUM( 1 )
*     ..
*     .. External Subroutines ..
EXTERNAL           CCOPY, CGEBAK, CGEBAL, CGEHRD, CHSEQR, CLACPY,
\$                   CLASCL, CTRSEN, CUNGHR, SLABAD, SLASCL, XERBLA
*     ..
*     .. External Functions ..
LOGICAL            LSAME
INTEGER            ILAENV
REAL               CLANGE, SLAMCH
EXTERNAL           LSAME, ILAENV, CLANGE, SLAMCH
*     ..
*     .. 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' )
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 = -11
END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of real workspace needed at that point in the
*       code, as well as the preferred amount for good performance.
*       CWorkspace refers to complex workspace, and RWorkspace to real
*       workspace. NB refers to the optimal block size for the
*       immediately following subroutine, as returned by ILAENV.
*       HSWORK refers to the workspace preferred by CHSEQR, 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 CTRSEN later
*       in the code.)
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
MINWRK = 1
LWRK = 1
ELSE
MAXWRK = N + N*ILAENV( 1, 'CGEHRD', ' ', N, 1, N, 0 )
MINWRK = 2*N
*
CALL CHSEQR( 'S', JOBVS, N, 1, N, A, LDA, W, VS, LDVS,
\$             WORK, -1, IEVAL )
HSWORK = WORK( 1 )
*
IF( .NOT.WANTVS ) THEN
MAXWRK = MAX( MAXWRK, HSWORK )
ELSE
MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'CUNGHR',
\$                       ' ', N, 1, N, -1 ) )
MAXWRK = MAX( MAXWRK, HSWORK )
END IF
LWRK = MAXWRK
IF( .NOT.WANTSN )
\$         LWRK = MAX( LWRK, ( N*N )/2 )
END IF
WORK( 1 ) = LWRK
*
IF( LWORK.LT.MINWRK ) THEN
INFO = -15
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGEESX', -INFO )
RETURN
END IF
*
*     Quick return if possible
*
IF( N.EQ.0 ) THEN
SDIM = 0
RETURN
END IF
*
*     Get machine constants
*
EPS = SLAMCH( 'P' )
SMLNUM = SLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = CLANGE( '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 CLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
*
*     Permute the matrix to make it more nearly triangular
*     (CWorkspace: none)
*     (RWorkspace: need N)
*
IBAL = 1
CALL CGEBAL( 'P', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR )
*
*     Reduce to upper Hessenberg form
*     (CWorkspace: need 2*N, prefer N+N*NB)
*     (RWorkspace: none)
*
ITAU = 1
IWRK = N + ITAU
CALL CGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
\$             LWORK-IWRK+1, IERR )
*
IF( WANTVS ) THEN
*
*        Copy Householder vectors to VS
*
CALL CLACPY( 'L', N, N, A, LDA, VS, LDVS )
*
*        Generate unitary matrix in VS
*        (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
*        (RWorkspace: none)
*
CALL CUNGHR( 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
*     (CWorkspace: need 1, prefer HSWORK (see comments) )
*     (RWorkspace: none)
*
IWRK = ITAU
CALL CHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, W, 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 )
\$      CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, W, N, IERR )
DO 10 I = 1, N
BWORK( I ) = SELECT( W( I ) )
10    CONTINUE
*
*        Reorder eigenvalues, transform Schur vectors, and compute
*        reciprocal condition numbers
*        (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM)
*                     otherwise, need none )
*        (RWorkspace: none)
*
CALL CTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, W, SDIM,
\$                RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1,
\$                ICOND )
IF( .NOT.WANTSN )
\$      MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) )
IF( ICOND.EQ.-14 ) THEN
*
*           Not enough complex workspace
*
INFO = -15
END IF
END IF
*
IF( WANTVS ) THEN
*
*        Undo balancing
*        (CWorkspace: none)
*        (RWorkspace: need N)
*
CALL CGEBAK( 'P', 'R', N, ILO, IHI, RWORK( IBAL ), N, VS, LDVS,
\$                IERR )
END IF
*
IF( SCALEA ) THEN
*
*        Undo scaling for the Schur form of A
*
CALL CLASCL( 'U', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
CALL CCOPY( N, A, LDA+1, W, 1 )
IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN
DUM( 1 ) = RCONDV
CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
RCONDV = DUM( 1 )
END IF
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
*     End of CGEESX
*
END

```