CTRSEN(3) MathKeisan LAPACK routine CTRSEN(3)
NAME
CTRSEN - 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 clus-
ter 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 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 over-
written by the reordered matrix T, with the selected eigenval-
ues as the leading diagonal elements.
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 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 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 con-
dition number for the selected cluster of eigenvalues. S can-
not 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 subspace. If M = 0 or N, SEP
= norm(T). If JOB = 'N' or 'E', SEP is not referenced.
WORK (workspace/output) COMPLEX array, dimension (MAX(1,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 >= 1; if
JOB = 'E', LWORK = max(1,M*(N-M)); if JOB = 'V' or 'B', LWORK
>= max(1,2*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 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 corre-
sponding 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 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
(1 + F-norm(R)**2)**(-1/2)
on the reciprocal of 2-norm(P), the true reciprocal condition 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 reciprocal of an estimate of
the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) can-
not 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
LAPACK routine (version 3.1) November 2006 CTRSEN(3)