```      SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
\$                   M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
CHARACTER          COMPQ, JOB
INTEGER            INFO, LDQ, LDT, LIWORK, LWORK, M, N
DOUBLE PRECISION   S, SEP
*     ..
*     .. Array Arguments ..
LOGICAL            SELECT( * )
INTEGER            IWORK( * )
DOUBLE PRECISION   Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
\$                   WR( * )
*     ..
*
*  Purpose
*  =======
*
*  DTRSEN reorders the real Schur factorization of a real matrix
*  A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
*  the leading diagonal blocks of the upper quasi-triangular matrix T,
*  and the leading columns of Q form an orthonormal basis of the
*  corresponding right invariant subspace.
*
*  Optionally the routine computes the reciprocal condition numbers of
*  the cluster of eigenvalues and/or the invariant subspace.
*
*  T must be in Schur canonical form (as returned by DHSEQR), that is,
*  block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
*  2-by-2 diagonal block has its diagonal elemnts equal and its
*  off-diagonal elements of opposite sign.
*
*  Arguments
*  =========
*
*  JOB     (input) CHARACTER*1
*          Specifies whether condition numbers are required for the
*          cluster of eigenvalues (S) or the invariant subspace (SEP):
*          = 'N': none;
*          = 'E': for eigenvalues only (S);
*          = 'V': for invariant subspace only (SEP);
*          = 'B': for both eigenvalues and invariant subspace (S and
*                 SEP).
*
*  COMPQ   (input) CHARACTER*1
*          = 'V': update the matrix Q of Schur vectors;
*          = 'N': do not update Q.
*
*  SELECT  (input) LOGICAL array, dimension (N)
*          SELECT specifies the eigenvalues in the selected cluster. To
*          select a real eigenvalue w(j), SELECT(j) must be set to
*          .TRUE.. To select a complex conjugate pair of eigenvalues
*          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
*          either SELECT(j) or SELECT(j+1) or both must be set to
*          .TRUE.; a complex conjugate pair of eigenvalues must be
*          either both included in the cluster or both excluded.
*
*  N       (input) INTEGER
*          The order of the matrix T. N >= 0.
*
*  T       (input/output) DOUBLE PRECISION array, dimension (LDT,N)
*          On entry, the upper quasi-triangular matrix T, in Schur
*          canonical form.
*          On exit, T is overwritten by the reordered matrix T, again in
*          Schur canonical form, with the selected eigenvalues in the
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T. LDT >= max(1,N).
*
*  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
*          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
*          On exit, if COMPQ = 'V', Q has been postmultiplied by the
*          orthogonal transformation matrix which reorders T; the
*          leading M columns of Q form an orthonormal basis for the
*          specified invariant subspace.
*          If COMPQ = 'N', Q is not referenced.
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.
*          LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
*
*  WR      (output) DOUBLE PRECISION array, dimension (N)
*  WI      (output) DOUBLE PRECISION array, dimension (N)
*          The real and imaginary parts, respectively, of the reordered
*          eigenvalues of T. The eigenvalues are stored in the same
*          order as on the diagonal of T, with WR(i) = T(i,i) and, if
*          T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
*          WI(i+1) = -WI(i). Note that if a complex eigenvalue is
*          sufficiently ill-conditioned, then its value may differ
*          significantly from its value before reordering.
*
*  M       (output) INTEGER
*          The dimension of the specified invariant subspace.
*          0 < = M <= N.
*
*  S       (output) DOUBLE PRECISION
*          If JOB = 'E' or 'B', S is a lower bound on the reciprocal
*          condition number for the selected cluster of eigenvalues.
*          S cannot underestimate the true reciprocal condition number
*          by more than a factor of sqrt(N). If M = 0 or N, S = 1.
*          If JOB = 'N' or 'V', S is not referenced.
*
*  SEP     (output) DOUBLE PRECISION
*          If JOB = 'V' or 'B', SEP is the estimated reciprocal
*          condition number of the specified invariant subspace. If
*          M = 0 or N, SEP = norm(T).
*          If JOB = 'N' or 'E', SEP is not referenced.
*
*  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.
*          If JOB = 'N', LWORK >= max(1,N);
*          if JOB = 'E', LWORK >= max(1,M*(N-M));
*          if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  IWORK   (workspace) 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.
*          If JOB = 'N' or 'E', LIWORK >= 1;
*          if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
*
*          If LIWORK = -1, then a workspace query is assumed; the
*          routine only calculates the optimal size of the IWORK array,
*          returns this value as the first entry of the IWORK array, and
*          no error message related to LIWORK is issued by XERBLA.
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*          = 1: reordering of T failed because some eigenvalues are too
*               close to separate (the problem is very ill-conditioned);
*               T may have been partially reordered, and WR and WI
*               contain the eigenvalues in the same order as in T; S and
*               SEP (if requested) are set to zero.
*
*  Further Details
*  ===============
*
*  DTRSEN first collects the selected eigenvalues by computing an
*  orthogonal transformation Z to move them to the top left corner of T.
*  In other words, the selected eigenvalues are the eigenvalues of T11
*  in:
*
*                Z'*T*Z = ( T11 T12 ) n1
*                         (  0  T22 ) n2
*                            n1  n2
*
*  where N = n1+n2 and Z' means the transpose of Z. The first n1 columns
*  of Z span the specified invariant subspace of T.
*
*  If T has been obtained from the real Schur factorization of a matrix
*  A = Q*T*Q', then the reordered real Schur factorization of A is given
*  by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span
*  the corresponding invariant subspace of A.
*
*  The reciprocal condition number of the average of the eigenvalues of
*  T11 may be returned in S. S lies between 0 (very badly conditioned)
*  and 1 (very well conditioned). It is computed as follows. First we
*  compute R so that
*
*                         P = ( I  R ) n1
*                             ( 0  0 ) n2
*                               n1 n2
*
*  is the projector on the invariant subspace associated with T11.
*  R is the solution of the Sylvester equation:
*
*                        T11*R - R*T22 = T12.
*
*  Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
*  the two-norm of M. Then S is computed as the lower bound
*
*                      (1 + F-norm(R)**2)**(-1/2)
*
*  on the reciprocal of 2-norm(P), the true reciprocal condition number.
*  S cannot underestimate 1 / 2-norm(P) by more than a factor of
*  sqrt(N).
*
*  An approximate error bound for the computed average of the
*  eigenvalues of T11 is
*
*                         EPS * norm(T) / S
*
*  where EPS is the machine precision.
*
*  The reciprocal condition number of the right invariant subspace
*  spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
*  SEP is defined as the separation of T11 and T22:
*
*                     sep( T11, T22 ) = sigma-min( C )
*
*  where sigma-min(C) is the smallest singular value of the
*  n1*n2-by-n1*n2 matrix
*
*     C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
*
*  I(m) is an m by m identity matrix, and kprod denotes the Kronecker
*  product. We estimate sigma-min(C) by the reciprocal of an estimate of
*  the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
*  cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
*
*  When SEP is small, small changes in T can cause large changes in
*  the invariant subspace. An approximate bound on the maximum angular
*  error in the computed right invariant subspace is
*
*                      EPS * norm(T) / SEP
*
*  =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ZERO, ONE
PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
LOGICAL            LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
\$                   WANTSP
INTEGER            IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
\$                   NN
DOUBLE PRECISION   EST, RNORM, SCALE
*     ..
*     .. Local Arrays ..
INTEGER            ISAVE( 3 )
*     ..
*     .. External Functions ..
LOGICAL            LSAME
DOUBLE PRECISION   DLANGE
EXTERNAL           LSAME, DLANGE
*     ..
*     .. External Subroutines ..
EXTERNAL           DLACN2, DLACPY, DTREXC, DTRSYL, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
WANTBH = LSAME( JOB, 'B' )
WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
WANTQ = LSAME( COMPQ, 'V' )
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
\$     THEN
INFO = -1
ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -8
ELSE
*
*        Set M to the dimension of the specified invariant subspace,
*        and test LWORK and LIWORK.
*
M = 0
PAIR = .FALSE.
DO 10 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
IF( K.LT.N ) THEN
IF( T( K+1, K ).EQ.ZERO ) THEN
IF( SELECT( K ) )
\$                  M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( K ) .OR. SELECT( K+1 ) )
\$                  M = M + 2
END IF
ELSE
IF( SELECT( N ) )
\$               M = M + 1
END IF
END IF
10    CONTINUE
*
N1 = M
N2 = N - M
NN = N1*N2
*
IF( WANTSP ) THEN
LWMIN = MAX( 1, 2*NN )
LIWMIN = MAX( 1, NN )
ELSE IF( LSAME( JOB, 'N' ) ) THEN
LWMIN = MAX( 1, N )
LIWMIN = 1
ELSE IF( LSAME( JOB, 'E' ) ) THEN
LWMIN = MAX( 1, NN )
LIWMIN = 1
END IF
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -17
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRSEN', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
*     Quick return if possible.
*
IF( M.EQ.N .OR. M.EQ.0 ) THEN
IF( WANTS )
\$      S = ONE
IF( WANTSP )
\$      SEP = DLANGE( '1', N, N, T, LDT, WORK )
GO TO 40
END IF
*
*     Collect the selected blocks at the top-left corner of T.
*
KS = 0
PAIR = .FALSE.
DO 20 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
SWAP = SELECT( K )
IF( K.LT.N ) THEN
IF( T( K+1, K ).NE.ZERO ) THEN
PAIR = .TRUE.
SWAP = SWAP .OR. SELECT( K+1 )
END IF
END IF
IF( SWAP ) THEN
KS = KS + 1
*
*              Swap the K-th block to position KS.
*
IERR = 0
KK = K
IF( K.NE.KS )
\$            CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK,
\$                         IERR )
IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
*
*                 Blocks too close to swap: exit.
*
INFO = 1
IF( WANTS )
\$               S = ZERO
IF( WANTSP )
\$               SEP = ZERO
GO TO 40
END IF
IF( PAIR )
\$            KS = KS + 1
END IF
END IF
20 CONTINUE
*
IF( WANTS ) THEN
*
*        Solve Sylvester equation for R:
*
*           T11*R - R*T22 = scale*T12
*
CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
\$                LDT, WORK, N1, SCALE, IERR )
*
*        Estimate the reciprocal of the condition number of the cluster
*        of eigenvalues.
*
RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK )
IF( RNORM.EQ.ZERO ) THEN
S = ONE
ELSE
S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
\$          SQRT( RNORM ) )
END IF
END IF
*
IF( WANTSP ) THEN
*
*        Estimate sep(T11,T22).
*
EST = ZERO
KASE = 0
30    CONTINUE
CALL DLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
*              Solve  T11*R - R*T22 = scale*X.
*
CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
\$                      T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
\$                      IERR )
ELSE
*
*              Solve  T11'*R - R*T22' = scale*X.
*
CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT,
\$                      T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
\$                      IERR )
END IF
GO TO 30
END IF
*
SEP = SCALE / EST
END IF
*
40 CONTINUE
*
*     Store the output eigenvalues in WR and WI.
*
DO 50 K = 1, N
WR( K ) = T( K, K )
WI( K ) = ZERO
50 CONTINUE
DO 60 K = 1, N - 1
IF( T( K+1, K ).NE.ZERO ) THEN
WI( K ) = SQRT( ABS( T( K, K+1 ) ) )*
\$                SQRT( ABS( T( K+1, K ) ) )
WI( K+1 ) = -WI( K )
END IF
60 CONTINUE
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
*     End of DTRSEN
*
END

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