```      SUBROUTINE SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
\$                   WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
\$                   IWORK, LIWORK, BWORK, INFO )
*
*  -- LAPACK driver routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
CHARACTER          JOBVS, SENSE, SORT
INTEGER            INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
REAL               RCONDE, RCONDV
*     ..
*     .. Array Arguments ..
LOGICAL            BWORK( * )
INTEGER            IWORK( * )
REAL               A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
\$                   WR( * )
*     ..
*     .. Function Arguments ..
LOGICAL            SELECT
EXTERNAL           SELECT
*     ..
*
*  Purpose
*  =======
*
*  SGEESX 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 REAL 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) REAL 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) REAL array, dimension (N)
*  WI      (output) REAL 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) REAL 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) 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) REAL 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 ..
REAL               ZERO, ONE
PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
*     ..
*     .. 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, LWRK, LIWRK,
\$                   MAXWRK, MINWRK
REAL               ANRM, BIGNUM, CSCALE, EPS, SMLNUM
*     ..
*     .. Local Arrays ..
REAL               DUM( 1 )
*     ..
*     .. External Subroutines ..
EXTERNAL           SCOPY, SGEBAK, SGEBAL, SGEHRD, SHSEQR, SLABAD,
\$                   SLACPY, SLASCL, SORGHR, SSWAP, STRSEN, XERBLA
*     ..
*     .. External Functions ..
LOGICAL            LSAME
INTEGER            ILAENV
REAL               SLAMCH, SLANGE
EXTERNAL           LSAME, ILAENV, SLAMCH, SLANGE
*     ..
*     .. 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 SHSEQR, 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 STRSEN 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, 'SGEHRD', ' ', N, 1, N, 0 )
MINWRK = 3*N
*
CALL SHSEQR( '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,
\$                       'SORGHR', ' ', 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( 'SGEESX', -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 = SLANGE( '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 SLASCL( '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 SGEBAL( '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 SGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
\$             LWORK-IWRK+1, IERR )
*
IF( WANTVS ) THEN
*
*        Copy Householder vectors to VS
*
CALL SLACPY( 'L', N, N, A, LDA, VS, LDVS )
*
*        Generate orthogonal matrix in VS
*        (RWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL SORGHR( 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 SHSEQR( '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 SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR )
CALL SLASCL( '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 STRSEN( 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
*
*           STRSEN 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 SGEBAK( '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 SLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
CALL SCOPY( N, A, LDA+1, WR, 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
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 SLASCL( '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 SSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ), 1 )
IF( N.GT.I+1 )
\$                  CALL SSWAP( N-I-1, A( I, I+2 ), LDA,
\$                              A( I+1, I+2 ), LDA )
CALL SSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 )
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 SLASCL( '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 ) = SDIM*(N-SDIM)
ELSE
IWORK( 1 ) = 1
END IF
*
RETURN
*
*     End of SGEESX
*
END

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