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# ctrsen

NAME
CTRSEN - reorder the Schur factorization of a complex matrix
A = Q*T*Q**H, so that a selected cluster of eigenvalues
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 CTRSEN( 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

REAL           S, SEP

LOGICAL        SELECT( * )

COMPLEX        Q( LDQ, * ), T( LDT, * ), W( * ), WORK( *
)

PURPOSE
CTRSEN reorders the Schur factorization of a complex matrix
A = Q*T*Q**H, so that a selected cluster of eigenvalues
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 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 the j-th eigenvalue, SELECT(j)
must be set to .TRUE..

N       (input) INTEGER
The order of the matrix T. N >= 0.

T       (input/output) COMPLEX 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 ele-
ments.

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

Q       (input/output) COMPLEX 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 unitary 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.

W       (output) COMPLEX
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) 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) COMPLEX array, dimension (LWORK)

If JOB = 'N', WORK is not referenced.

LWORK   (input) INTEGER
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).

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value

FURTHER DETAILS
CTRSEN first collects the selected eigenvalues by computing
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 sub-
space 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 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