Contents
ztrsen - 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
SUBROUTINE ZTRSEN(JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
SEP, WORK, LWORK, INFO)
CHARACTER * 1 JOB, COMPQ
DOUBLE COMPLEX T(LDT,*), Q(LDQ,*), W(*), WORK(*)
INTEGER N, LDT, LDQ, M, LWORK, INFO
LOGICAL SELECT(*)
DOUBLE PRECISION S, SEP
SUBROUTINE ZTRSEN_64(JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
SEP, WORK, LWORK, INFO)
CHARACTER * 1 JOB, COMPQ
DOUBLE COMPLEX T(LDT,*), Q(LDQ,*), W(*), WORK(*)
INTEGER*8 N, LDT, LDQ, M, LWORK, INFO
LOGICAL*8 SELECT(*)
DOUBLE PRECISION 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(8), DIMENSION(:) :: W, WORK
COMPLEX(8), DIMENSION(:,:) :: T, Q
INTEGER :: N, LDT, LDQ, M, LWORK, INFO
LOGICAL, DIMENSION(:) :: SELECT
REAL(8) :: 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(8), DIMENSION(:) :: W, WORK
COMPLEX(8), DIMENSION(:,:) :: T, Q
INTEGER(8) :: N, LDT, LDQ, M, LWORK, INFO
LOGICAL(8), DIMENSION(:) :: SELECT
REAL(8) :: S, SEP
C INTERFACE
#include <sunperf.h>
void ztrsen(char job, char compq, int *select, int n, doub-
lecomplex *t, int ldt, doublecomplex *q, int ldq,
doublecomplex *w, int *m, double *s, double *sep,
int *info);
void ztrsen_64(char job, char compq, long *select, long n,
doublecomplex *t, long ldt, doublecomplex *q, long
ldq, doublecomplex *w, long *m, double *s, double
*sep, long *info);
ztrsen 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.
JOB (input)
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)
= '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 overwritten by the reordered matrix T, with
the selected eigenvalues 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 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)
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 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)
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)
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 ille-
gal value
ZTRSEN 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