CTRSEN (3lapack)

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 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
          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 elements.

  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 vectors.  On exit,
          if COMPQ = 'V', Q has been postmultiplied by the unitary
          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.

  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 subspace. 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 subspace 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 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