
STRSEN(l) ) 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 corre
sponding right invariant subspace.
Optionally the routine computes the reciprocal condition numbers of the cluster of eigen
values and/or the invariant subspace.
T must be in Schur canonical form (as returned by SHSEQR), that is, block upper triangular
with 1by1 and 2by2 diagonal blocks; each 2by2 diagonal block has its diagonal elem
nts equal and its offdiagonal 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 eigen
value w(j), SELECT(j) must be set to w(j) and w(j+1), corresponding to a 2by2
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 quasitriangular 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 leading 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 invari
ant 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, respec
tively, 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
2by2 diagonal block, WI(i) > 0 and WI(i+1) = WI(i). Note that if a complex ei
genvalue is sufficiently illconditioned, 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) 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 con
dition 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 spec
ified invariant subspace. If M = 0 or N, SEP = norm(T). If JOB = 'N' or 'E', SEP
is not referenced.
WORK (workspace/output) REAL 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*(NM); if JOB = 'V' or 'B', LWORK >= 2*M*(NM).
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*(NM).
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 ith argument had an illegal value
= 1: reordering of T failed because some eigenvalues are too close to separate
(the problem is very illconditioned); 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 speci
fied 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 com
puted 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 Fnorm(M) denote the Frobeniusnorm of M and 2norm(M) denote the twonorm of M. Then
S is computed as the lower bound
(1 + Fnorm(R)**2)**(1/2)
on the reciprocal of 2norm(P), the true reciprocal condition number. S cannot underesti
mate 1 / 2norm(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 ) = sigmamin( C )
where sigmamin(C) is the smallest singular value of the
n1*n2byn1*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
sigmamin(C) by the reciprocal of an estimate of the 1norm of inverse(C). The true recip
rocal 1norm of inverse(C) cannot differ from sigmamin(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 STRSEN(l) 
