dtrsen man page on Scientific

```DTRSEN(1)		 LAPACK routine (version 3.2)		     DTRSEN(1)

NAME
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,

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 tri‐
angular with 1-by-1 and 2-by-2 diagonal blocks;	each  2-by-2  diagonal
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	 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 .TRUE..
To  select  a  complex  conjugate  pair of eigenvalues 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 .TRUE.; a com‐
plex conjugate pair of eigenvalues must be 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

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
(MAX(1,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 >= max(1,M*(N-M)); if JOB = 'V'
or 'B', LWORK >= max(1,2*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 (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK  (input) INTEGER
The dimension of the array IWORK.  If JOB = 'N' or 'E',	LIWORK
>= 1; if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).  If LIWORK
= -1, then a workspace query is assumed; the routine only  cal‐
culates 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) 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 routine (version 3.2)	 November 2008			     DTRSEN(1)
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