ctrsen(3P) Sun Performance Library ctrsen(3P)NAMEctrsen - 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 * 1 JOB, COMPQ
COMPLEX T(LDT,*), Q(LDQ,*), W(*), WORK(*)
INTEGER N, LDT, LDQ, M, LWORK, INFO
LOGICAL SELECT(*)
REAL S, SEP
SUBROUTINE CTRSEN_64(JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
SEP, WORK, LWORK, INFO)
CHARACTER * 1 JOB, COMPQ
COMPLEX T(LDT,*), Q(LDQ,*), W(*), WORK(*)
INTEGER*8 N, LDT, LDQ, M, LWORK, INFO
LOGICAL*8 SELECT(*)
REAL S, SEP
F95 INTERFACE
SUBROUTINE TRSEN(JOB, COMPQ, SELECT, [N], T, [LDT], Q, [LDQ], W, M,
S, SEP, [WORK], [LWORK], [INFO])
CHARACTER(LEN=1) :: JOB, COMPQ
COMPLEX, DIMENSION(:) :: W, WORK
COMPLEX, DIMENSION(:,:) :: T, Q
INTEGER :: N, LDT, LDQ, M, LWORK, INFO
LOGICAL, DIMENSION(:) :: SELECT
REAL :: S, SEP
SUBROUTINE TRSEN_64(JOB, COMPQ, SELECT, [N], T, [LDT], Q, [LDQ], W,
M, S, SEP, [WORK], [LWORK], [INFO])
CHARACTER(LEN=1) :: JOB, COMPQ
COMPLEX, DIMENSION(:) :: W, WORK
COMPLEX, DIMENSION(:,:) :: T, Q
INTEGER(8) :: N, LDT, LDQ, M, LWORK, INFO
LOGICAL(8), DIMENSION(:) :: SELECT
REAL :: S, SEP
C INTERFACE
#include <sunperf.h>
void ctrsen(char job, char compq, int *select, int n, complex *t, int
ldt, complex *q, int ldq, complex *w, int *m, float *s, float
*sep, int *info);
void ctrsen_64(char job, char compq, long *select, long n, complex *t,
long ldt, complex *q, long ldq, complex *w, long *m, float
*s, float *sep, long *info);
PURPOSEctrsen 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)
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)
= 'V': update the matrix Q of Schur vectors;
= 'N': do not update Q.
SELECT (input)
SELECT specifies the eigenvalues in the selected cluster. To
select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
N (input) The order of the matrix T. N >= 0.
T (input/output)
On entry, the upper triangular matrix T. On exit, T is over‐
written by the reordered matrix T, with the selected eigen‐
values as the leading diagonal elements.
LDT (input)
The leading dimension of the array T. LDT >= max(1,N).
Q (input) On entry, if COMPQ = 'V', the matrix Q of Schur vectors. On
exit, if COMPQ = 'V', Q has been postmultiplied by the uni‐
tary 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)
The leading dimension of the array Q. LDQ >= 1; and if COMPQ
= 'V', LDQ >= N.
W (output)
The reordered eigenvalues of T, in the same order as they
appear on the diagonal of T.
M (output)
The dimension of the specified invariant subspace. 0 <= M <=
N.
S (output)
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)
If JOB = 'V' or 'B', SEP is the estimated reciprocal condi‐
tion number of the specified invariant subspace. If M = 0 or
N, SEP = norm(T). If JOB = 'N' or 'E', SEP is not refer‐
enced.
WORK (workspace)
If JOB = 'N', WORK is not referenced. Otherwise, on exit, if
INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
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)
= 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
6 Mar 2009 ctrsen(3P)
