CXML
        	    
LAPACK version 3.0
ztrsen(3)

PURPOSE
  ZTRSEN - 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

SYNTAX
  SUBROUTINE ZTRSEN( 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

      DOUBLE	     PRECISION S, SEP

      LOGICAL	     SELECT( * )

      COMPLEX*16     Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )

DESCRIPTION
  ZTRSEN 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*16 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*16 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*16 array, dimension (N)
	  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) 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) COMPLEX*16 array, dimension (LWORK)
	  If JOB = 'N', WORK is not referenced.	 Otherwise, on exit, if INFO
	  = 0, WORK(1) returns the optimal LWORK.

  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).

	  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.

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

FURTHER DETAILS
  ZTRSEN 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