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      SUBROUTINE <a name="CTRSEN.1"></a><a href="ctrsen.f.html#CTRSEN.1">CTRSEN</a>( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
     $                   SEP, WORK, LWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Modified to call <a name="CLACN2.8"></a><a href="clacn2.f.html#CLACN2.1">CLACN2</a> in place of <a name="CLACON.8"></a><a href="clacon.f.html#CLACON.1">CLACON</a>, 10 Feb 03, SJH.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      CHARACTER          COMPQ, JOB
      INTEGER            INFO, LDQ, LDT, LWORK, M, N
      REAL               S, SEP
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      LOGICAL            SELECT( * )
      COMPLEX            Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="CTRSEN.23"></a><a href="ctrsen.f.html#CTRSEN.1">CTRSEN</a> reorders the Schur factorization of a complex matrix
</span><span class="comment">*</span><span class="comment">  A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
</span><span class="comment">*</span><span class="comment">  the leading positions on the diagonal of the upper triangular matrix
</span><span class="comment">*</span><span class="comment">  T, and the leading columns of Q form an orthonormal basis of the
</span><span class="comment">*</span><span class="comment">  corresponding right invariant subspace.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Optionally the routine computes the reciprocal condition numbers of
</span><span class="comment">*</span><span class="comment">  the cluster of eigenvalues and/or the invariant subspace.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  JOB     (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          Specifies whether condition numbers are required for the
</span><span class="comment">*</span><span class="comment">          cluster of eigenvalues (S) or the invariant subspace (SEP):
</span><span class="comment">*</span><span class="comment">          = 'N': none;
</span><span class="comment">*</span><span class="comment">          = 'E': for eigenvalues only (S);
</span><span class="comment">*</span><span class="comment">          = 'V': for invariant subspace only (SEP);
</span><span class="comment">*</span><span class="comment">          = 'B': for both eigenvalues and invariant subspace (S and
</span><span class="comment">*</span><span class="comment">                 SEP).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  COMPQ   (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          = 'V': update the matrix Q of Schur vectors;
</span><span class="comment">*</span><span class="comment">          = 'N': do not update Q.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SELECT  (input) LOGICAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">          SELECT specifies the eigenvalues in the selected cluster. To
</span><span class="comment">*</span><span class="comment">          select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrix T. N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  T       (input/output) COMPLEX array, dimension (LDT,N)
</span><span class="comment">*</span><span class="comment">          On entry, the upper triangular matrix T.
</span><span class="comment">*</span><span class="comment">          On exit, T is overwritten by the reordered matrix T, with the
</span><span class="comment">*</span><span class="comment">          selected eigenvalues as the leading diagonal elements.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDT     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array T. LDT &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Q       (input/output) COMPLEX array, dimension (LDQ,N)
</span><span class="comment">*</span><span class="comment">          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
</span><span class="comment">*</span><span class="comment">          On exit, if COMPQ = 'V', Q has been postmultiplied by the
</span><span class="comment">*</span><span class="comment">          unitary transformation matrix which reorders T; the leading M
</span><span class="comment">*</span><span class="comment">          columns of Q form an orthonormal basis for the specified
</span><span class="comment">*</span><span class="comment">          invariant subspace.
</span><span class="comment">*</span><span class="comment">          If COMPQ = 'N', Q is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDQ     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array Q.
</span><span class="comment">*</span><span class="comment">          LDQ &gt;= 1; and if COMPQ = 'V', LDQ &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  W       (output) COMPLEX array, dimension (N)
</span><span class="comment">*</span><span class="comment">          The reordered eigenvalues of T, in the same order as they
</span><span class="comment">*</span><span class="comment">          appear on the diagonal of T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  M       (output) INTEGER
</span><span class="comment">*</span><span class="comment">          The dimension of the specified invariant subspace.
</span><span class="comment">*</span><span class="comment">          0 &lt;= M &lt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  S       (output) REAL
</span><span class="comment">*</span><span class="comment">          If JOB = 'E' or 'B', S is a lower bound on the reciprocal
</span><span class="comment">*</span><span class="comment">          condition number for the selected cluster of eigenvalues.
</span><span class="comment">*</span><span class="comment">          S cannot underestimate the true reciprocal condition number
</span><span class="comment">*</span><span class="comment">          by more than a factor of sqrt(N). If M = 0 or N, S = 1.
</span><span class="comment">*</span><span class="comment">          If JOB = 'N' or 'V', S is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SEP     (output) REAL
</span><span class="comment">*</span><span class="comment">          If JOB = 'V' or 'B', SEP is the estimated reciprocal
</span><span class="comment">*</span><span class="comment">          condition number of the specified invariant subspace. If
</span><span class="comment">*</span><span class="comment">          M = 0 or N, SEP = norm(T).
</span><span class="comment">*</span><span class="comment">          If JOB = 'N' or 'E', SEP is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
</span><span class="comment">*</span><span class="comment">          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LWORK   (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The dimension of the array WORK.
</span><span class="comment">*</span><span class="comment">          If JOB = 'N', LWORK &gt;= 1;
</span><span class="comment">*</span><span class="comment">          if JOB = 'E', LWORK = max(1,M*(N-M));
</span><span class="comment">*</span><span class="comment">          if JOB = 'V' or 'B', LWORK &gt;= max(1,2*M*(N-M)).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">          If LWORK = -1, then a workspace query is assumed; the routine
</span><span class="comment">*</span><span class="comment">          only calculates the optimal size of the WORK array, returns
</span><span class="comment">*</span><span class="comment">          this value as the first entry of the WORK array, and no error
</span><span class="comment">*</span><span class="comment">          message related to LWORK is issued by <a name="XERBLA.108"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          = 0:  successful exit
</span><span class="comment">*</span><span class="comment">          &lt; 0:  if INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="CTRSEN.117"></a><a href="ctrsen.f.html#CTRSEN.1">CTRSEN</a> first collects the selected eigenvalues by computing a unitary
</span><span class="comment">*</span><span class="comment">  transformation Z to move them to the top left corner of T. In other
</span><span class="comment">*</span><span class="comment">  words, the selected eigenvalues are the eigenvalues of T11 in:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                Z'*T*Z = ( T11 T12 ) n1
</span><span class="comment">*</span><span class="comment">                         (  0  T22 ) n2
</span><span class="comment">*</span><span class="comment">                            n1  n2
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where N = n1+n2 and Z' means the conjugate transpose of Z. The first
</span><span class="comment">*</span><span class="comment">  n1 columns of Z span the specified invariant subspace of T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  If T has been obtained from the Schur factorization of a matrix
</span><span class="comment">*</span><span class="comment">  A = Q*T*Q', then the reordered Schur factorization of A is given by
</span><span class="comment">*</span><span class="comment">  A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span the
</span><span class="comment">*</span><span class="comment">  corresponding invariant subspace of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The reciprocal condition number of the average of the eigenvalues of
</span><span class="comment">*</span><span class="comment">  T11 may be returned in S. S lies between 0 (very badly conditioned)
</span><span class="comment">*</span><span class="comment">  and 1 (very well conditioned). It is computed as follows. First we
</span><span class="comment">*</span><span class="comment">  compute R so that
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                         P = ( I  R ) n1
</span><span class="comment">*</span><span class="comment">                             ( 0  0 ) n2
</span><span class="comment">*</span><span class="comment">                               n1 n2
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  is the projector on the invariant subspace associated with T11.
</span><span class="comment">*</span><span class="comment">  R is the solution of the Sylvester equation:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                        T11*R - R*T22 = T12.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
</span><span class="comment">*</span><span class="comment">  the two-norm of M. Then S is computed as the lower bound
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                      (1 + F-norm(R)**2)**(-1/2)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  on the reciprocal of 2-norm(P), the true reciprocal condition number.
</span><span class="comment">*</span><span class="comment">  S cannot underestimate 1 / 2-norm(P) by more than a factor of
</span><span class="comment">*</span><span class="comment">  sqrt(N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  An approximate error bound for the computed average of the
</span><span class="comment">*</span><span class="comment">  eigenvalues of T11 is
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                         EPS * norm(T) / S
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where EPS is the machine precision.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The reciprocal condition number of the right invariant subspace
</span><span class="comment">*</span><span class="comment">  spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
</span><span class="comment">*</span><span class="comment">  SEP is defined as the separation of T11 and T22:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                     sep( T11, T22 ) = sigma-min( C )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where sigma-min(C) is the smallest singular value of the
</span><span class="comment">*</span><span class="comment">  n1*n2-by-n1*n2 matrix
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
</span><span class="comment">*</span><span class="comment">
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