dtrsen(3P) Sun Performance Library dtrsen(3P)NAMEdtrsen - reorder the real Schur factorization of a real matrix A =
Q*T*Q**T, so that a selected cluster of eigenvalues appears in the
leading diagonal blocks of the upper quasi-triangular matrix T,
SYNOPSIS
SUBROUTINE DTRSEN(JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M,
S, SEP, WORK, LWORK, IWORK, LIWORK, INFO)
CHARACTER * 1 JOB, COMPQ
INTEGER N, LDT, LDQ, M, LWORK, LIWORK, INFO
INTEGER IWORK(*)
LOGICAL SELECT(*)
DOUBLE PRECISION S, SEP
DOUBLE PRECISION T(LDT,*), Q(LDQ,*), WR(*), WI(*), WORK(*)
SUBROUTINE DTRSEN_64(JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO)
CHARACTER * 1 JOB, COMPQ
INTEGER*8 N, LDT, LDQ, M, LWORK, LIWORK, INFO
INTEGER*8 IWORK(*)
LOGICAL*8 SELECT(*)
DOUBLE PRECISION S, SEP
DOUBLE PRECISION T(LDT,*), Q(LDQ,*), WR(*), WI(*), WORK(*)
F95 INTERFACE
SUBROUTINE TRSEN(JOB, COMPQ, SELECT, N, T, [LDT], Q, [LDQ], WR, WI,
M, S, SEP, [WORK], [LWORK], [IWORK], [LIWORK], [INFO])
CHARACTER(LEN=1) :: JOB, COMPQ
INTEGER :: N, LDT, LDQ, M, LWORK, LIWORK, INFO
INTEGER, DIMENSION(:) :: IWORK
LOGICAL, DIMENSION(:) :: SELECT
REAL(8) :: S, SEP
REAL(8), DIMENSION(:) :: WR, WI, WORK
REAL(8), DIMENSION(:,:) :: T, Q
SUBROUTINE TRSEN_64(JOB, COMPQ, SELECT, N, T, [LDT], Q, [LDQ], WR,
WI, M, S, SEP, [WORK], [LWORK], [IWORK], [LIWORK], [INFO])
CHARACTER(LEN=1) :: JOB, COMPQ
INTEGER(8) :: N, LDT, LDQ, M, LWORK, LIWORK, INFO
INTEGER(8), DIMENSION(:) :: IWORK
LOGICAL(8), DIMENSION(:) :: SELECT
REAL(8) :: S, SEP
REAL(8), DIMENSION(:) :: WR, WI, WORK
REAL(8), DIMENSION(:,:) :: T, Q
C INTERFACE
#include <sunperf.h>
void dtrsen(char job, char compq, int *select, int n, double *t, int
ldt, double *q, int ldq, double *wr, double *wi, int *m, dou‐
ble *s, double *sep, int *info);
void dtrsen_64(char job, char compq, long *select, long n, double *t,
long ldt, double *q, long ldq, double *wr, double *wi, long
*m, double *s, double *sep, long *info);
PURPOSEdtrsen reorders the real Schur factorization of a real matrix A =
Q*T*Q**T, so that a selected cluster of eigenvalues appears in the
leading diagonal blocks of the upper quasi-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.
T must be in Schur canonical form (as returned by SHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elemnts equal and its off-diago‐
nal elements of opposite sign.
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 a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
N (input) The order of the matrix T. N >= 0.
T (input/output)
On entry, the upper quasi-triangular matrix T, in Schur
canonical form. On exit, T is overwritten by the reordered
matrix T, again in Schur canonical form, with the selected
eigenvalues in the leading diagonal blocks.
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
orthogonal transformation matrix which reorders T; the lead‐
ing M columns of Q form an orthonormal basis for the speci‐
fied invariant subspace. If COMPQ = 'N', Q is not refer‐
enced.
LDQ (input)
The leading dimension of the array Q. LDQ >= 1; and if COMPQ
= 'V', LDQ >= N.
WR (output)
The real and imaginary parts, respectively, of the reordered
eigenvalues of T. The eigenvalues are stored in the same
order as on the diagonal of T, with WR(i) = T(i,i) and, if
T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
WI(i+1) = -WI(i). Note that if a complex eigenvalue is suffi‐
ciently ill-conditioned, then its value may differ signifi‐
cantly from its value before reordering.
WI (output)
See the description of WR.
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)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. If JOB = 'N', LWORK >=
max(1,N); 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.
IWORK (workspace/output)
If JOB = 'N' or 'E', IWORK is not referenced.
LIWORK (input)
The dimension of the array IWORK. If JOB = 'N' or 'E',
LIWORK >= 1; if JOB = 'V' or 'B', LIWORK >= M*(N-M).
If LIWORK = -1, then a workspace query is assumed; the rou‐
tine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: reordering of T failed because some eigenvalues are too
close to separate (the problem is very ill-conditioned); T
may have been partially reordered, and WR and WI contain the
eigenvalues in the same order as in T; S and SEP (if
requested) are set to zero.
FURTHER DETAILS
DTRSEN first collects the selected eigenvalues by computing an orthogo‐
nal 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 transpose of Z. The first n1 columns
of Z span the specified invariant subspace of T.
If T has been obtained from the real Schur factorization of a matrix A
= Q*T*Q', then the reordered real 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 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 dtrsen(3P)
