ZTRSEN - reorder the Schur factorization of a complex
matrix A = Q*T*Q**H, so that a selected cluster of eigen
values 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 ZTRSEN( 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
DOUBLE PRECISION S, SEP
LOGICAL SELECT( * )
COMPLEX*16 Q( LDQ, * ), T( LDT, * ), W( * ), WORK(
* )
PURPOSE
ZTRSEN reorders the Schur factorization of a complex
matrix A = Q*T*Q**H, so that a selected cluster of eigen
values 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 invari
ant 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..
The order of the matrix T. N >= 0.
T (input/output) COMPLEX*16 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*16 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 transformation
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 ref
erenced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1;
and if COMPQ = 'V', LDQ >= N.
W (output) COMPLEX*16 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) DOUBLE PRECISION
If JOB = 'E' or 'B', S is a lower bound on the
reciprocal condition number for the selected clus
ter 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) DOUBLE PRECISION
If JOB = 'V' or 'B', SEP is the estimated recipro
cal 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*16 array, dimension
(LWORK)
If JOB = 'N', WORK is not referenced. Otherwise,
on exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
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 ille
gal value
FURTHER DETAILS
ZTRSEN first collects the selected eigenvalues by comput
ing 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 sub
space 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
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
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 recip
rocal 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
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