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NAME
STRSEN - reorder the real Schur factorization of a real
matrix A = Q*T*Q**T, so that a selected cluster of eigen-
values 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 eigen-
values 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 diag-
onal blocks; each 2-by-2 diagonal block has its diagonal
elements 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 sub-
space (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 complex
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) 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 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 vec-
tors. On exit, if COMPQ = 'V', Q has been postmul-
tiplied by the orthogonal transformation matrix
which reorders T; the leading M columns of Q form an
orthonormal basis for the specified invariant sub-
space. 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 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) 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 sub-
space. 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 factori-
zation 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 eigen-
values 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 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 condi-
tion 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 sub-
space 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 recipro-
cal 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