ZTRSEN(l)		LAPACK routine (version	1.1)		    ZTRSEN(l)

NAME
  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 sub-
  space

SYNOPSIS

  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( * )

PURPOSE
  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 sub-
  space.

  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 transfor-
	  mation 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
	  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 underes-
	  timate 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) COMPLEX*16 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
  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 correspond-
  ing 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