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DTRSEN(l)			       )			     DTRSEN(l)

NAME
       DTRSEN  -  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 DTRSEN( 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

	   DOUBLE	  PRECISION S, SEP

	   LOGICAL	  SELECT( * )

	   INTEGER	  IWORK( * )

	   DOUBLE	  PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( *
			  ), WR( * )

PURPOSE
       DTRSEN  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  DHSEQR),  that  is,
       block  upper  triangular	 with  1-by-1 and 2-by-2 diagonal blocks; each
       2-by-2 diagonal block has its diagonal elemnts equal and its off-diago‐
       nal elements of opposite sign.

ARGUMENTS
       JOB     (input) CHARACTER*1
	       Specifies  whether condition numbers are required for the clus‐
	       ter 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) DOUBLE PRECISION array, dimension (LDT,N)
	       On entry, the upper quasi-triangular matrix T, in Schur canoni‐
	       cal form.  On exit, T is overwritten by the reordered matrix T,
	       again in Schur canonical form, with the selected eigenvalues in
	       the leading diagonal blocks.

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

       Q       (input/output) DOUBLE PRECISION 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 orthogo‐
	       nal  transformation matrix which reorders T; the leading M col‐
	       umns 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) DOUBLE PRECISION array, dimension (N)
	       WI      (output) DOUBLE PRECISION array, dimension (N) The real
	       and imaginary parts, respectively, of the reordered eigenvalues
	       of T. The eigenvalues 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 significantly from its	 value	before
	       reordering.

       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  con‐
	       dition  number for the selected cluster of eigenvalues.	S can‐
	       not 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) DOUBLE PRECISION array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal 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).

	       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.

       IWORK   (workspace) INTEGER array, dimension (LIWORK)
	       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).

	       If LIWORK = -1, then a workspace query is assumed; the  routine
	       only  calculates	 the  optimal size of the IWORK array, returns
	       this value as the first entry of the IWORK array, and no	 error
	       message related to LIWORK is issued by XERBLA.

       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 eigen‐
	       values in the same order as in T; S and SEP (if requested)  are
	       set to zero.

FURTHER DETAILS
       DTRSEN first collects the selected eigenvalues by computing an orthogo‐
       nal 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
       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) can‐
       not 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			     DTRSEN(l)
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