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ctrsen


 NAME
      CTRSEN - reorder the Schur factorization of a complex matrix
      A = Q*T*Q**H, so that a selected cluster of eigenvalues
      appears in the leading positions on the diagonal of the
      upper triangular matrix T, and the leading columns of Q form
      an orthonormal basis of the corresponding right invariant
      subspace

 SYNOPSIS
      SUBROUTINE CTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W,
                         M, S, SEP, WORK, LWORK, INFO )

          CHARACTER      COMPQ, JOB

          INTEGER        INFO, LDQ, LDT, LWORK, M, N

          REAL           S, SEP

          LOGICAL        SELECT( * )

          COMPLEX        Q( LDQ, * ), T( LDT, * ), W( * ), WORK( *
                         )

 PURPOSE
      CTRSEN reorders the Schur factorization of a complex matrix
      A = Q*T*Q**H, so that a selected cluster of eigenvalues
      appears in the leading positions on the diagonal of the
      upper 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.

 ARGUMENTS
      JOB     (input) CHARACTER*1
              Specifies whether condition numbers are required for
              the cluster of eigenvalues (S) or the invariant sub-
              space (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 the j-th eigenvalue, SELECT(j)
              must be set to .TRUE..

      N       (input) INTEGER
              The order of the matrix T. N >= 0.

      T       (input/output) COMPLEX array, dimension(LDT,N)
              On entry, the upper triangular matrix T.  On exit, T
              is overwritten by the reordered matrix T, with the
              selected eigenvalues as the leading diagonal ele-
              ments.

      LDT     (input) INTEGER
              The leading dimension of the array T. LDT >=
              max(1,N).

      Q       (input/output) COMPLEX array, dimension (LDQ,N)
              On entry, if COMPQ = 'V', the matrix Q of Schur vec-
              tors.  On exit, if COMPQ = 'V', Q has been postmul-
              tiplied by the unitary transformation matrix which
              reorders T; the leading M columns of Q form an
              orthonormal basis for the specified invariant sub-
              space.  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.

      W       (output) COMPLEX
              The reordered eigenvalues of T, in the same order as
              they appear on the diagonal of T.

      M       (output) INTEGER
              The dimension of the specified invariant subspace.
              0 <= M <= N.

      S       (output) REAL
              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) REAL
              If JOB = 'V' or 'B', SEP is the estimated reciprocal
              condition number of the specified invariant sub-
              space. If M = 0 or N, SEP = norm(T).  If JOB = 'N'
              or 'E', SEP is not referenced.

      WORK    (workspace) COMPLEX array, dimension (LWORK)

              If JOB = 'N', WORK is not referenced.

      LWORK   (input) INTEGER
              The dimension of the array WORK.  If JOB = 'N',
              LWORK >= 1; if JOB = 'E', LWORK = M*(N-M); if JOB =
              'V' or 'B', LWORK >= 2*M*(N-M).

      INFO    (output) INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal
              value

 FURTHER DETAILS
      CTRSEN first collects the selected eigenvalues by computing
      a unitary 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 conjugate transpose of Z.
      The first n1 columns of Z span the specified invariant sub-
      space of T.

      If T has been obtained from the Schur factorization of a
      matrix A = Q*T*Q', then the reordered 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 eigen-
      values of T11 may be returned in S. S lies between 0 (very
      badly conditioned) and 1 (very well conditioned). It is com-
      puted 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 condi-
      tion 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 sub-
      space 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 recipro-
      cal 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