STRSEN(l)		LAPACK routine (version	1.1)		    STRSEN(l)

NAME
  STRSEN - reorder the real Schur factorization	of a real matrix A =
  Q*T*Q**T, so that a selected cluster of eigenvalues appears in the leading
  diagonal blocks of the upper quasi-triangular	matrix T,

SYNOPSIS

  SUBROUTINE STRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR,	WI, M, S,
		     SEP, WORK,	LWORK, IWORK, LIWORK, INFO )

      CHARACTER	     COMPQ, JOB

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

      REAL	     S,	SEP

      LOGICAL	     SELECT( * )

      INTEGER	     IWORK( * )

      REAL	     Q(	LDQ, * ), T( LDT, * ), WI( * ),	WORK( *	), WR( * )

PURPOSE
  STRSEN reorders the real Schur factorization of a real matrix	A = Q*T*Q**T,
  so that a selected cluster of	eigenvalues appears in the leading diagonal
  blocks of the	upper quasi-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.

  T must be in Schur canonical form (as	returned by SHSEQR), that is, block
  upper	triangular with	1-by-1 and 2-by-2 diagonal blocks; each	2-by-2 diago-
  nal block has	its diagonal elemnts equal and its off-diagonal	elements of
  opposite sign.

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
	  a real eigenvalue w(j), SELECT(j) must be set	to w(j)	and w(j+1),
	  corresponding	to a 2-by-2 diagonal block, either SELECT(j) or
	  SELECT(j+1) or both must be set to either both included in the
	  cluster or both excluded.

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

  T	  (input/output) REAL array, dimension(LDT,N)
	  On entry, the	upper quasi-triangular matrix T, in Schur canonical
	  form.	 On exit, T is overwritten by the reordered matrix T, again
	  in Schur canonical form, with	the selected eigenvalues in the	lead-
	  ing diagonal blocks.

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

  Q	  (input/output) REAL 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 orthogonal
	  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.

  WR	  (output) REAL	array, dimension (N)
	  WI	  (output) REAL	array, dimension (N) The real and imaginary
	  parts, respectively, of the reordered	eigenvalues of T. The eigen-
	  values are stored in the same	order as on the	diagonal of T, with
	  WR(i)	= T(i,i) and, if T(i:i+1,i:i+1)	is a 2-by-2 diagonal block,
	  WI(i)	> 0 and	WI(i+1)	= -WI(i). Note that if a complex eigenvalue
	  is sufficiently ill-conditioned, then	its value may differ signifi-
	  cantly from its value	before reordering.

  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 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) 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) REAL array, dimension (LWORK)

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

  IWORK	  (workspace) INTEGER
	  IF JOB = 'N' or 'E', IWORK is	not referenced.

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

  INFO	  (output) INTEGER
	  = 0: successful exit
	  < 0: if INFO = -i, the i-th argument had an illegal value
	  = 1: reordering of T failed because some eigenvalues are too close
	  to separate (the problem is very ill-conditioned); T may have	been
	  partially reordered, and WR and WI contain the eigenvalues in	the
	  same order as	in T; S	and SEP	(if requested) are set to zero.

FURTHER	DETAILS
  STRSEN first collects	the selected eigenvalues by computing an orthogonal
  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 transpose of	Z. The first n1	columns	of Z
  span the specified invariant subspace	of T.

  If T has been	obtained from the real Schur factorization of a	matrix A =
  Q*T*Q', then the reordered real 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