
CTRSEN(l) ) CTRSEN(l)
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 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 jth ei
genvalue, 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 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 referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
W (output) COMPLEX 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) 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) COMPLEX array, dimension (LWORK)
If JOB = 'N', WORK is not referenced. Otherwise, on exit, if INFO = 0, WORK(1)
returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. If JOB = 'N', LWORK >= 1; 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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith 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 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 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 CTRSEN(l) 
