CTRSEN(l)								 )								 CTRSEN(l)

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 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 trans-
	       formation 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 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) 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/output) COMPLEX 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

       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

LAPACK version 3.0						   15 June 2000 							 CTRSEN(l)