
dtrsen.f(3) LAPACK dtrsen.f(3)
NAME
dtrsen.f 
SYNOPSIS
Functions/Subroutines
subroutine dtrsen (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S, SEP, WORK, LWORK,
IWORK, LIWORK, INFO)
DTRSEN
Function/Subroutine Documentation
subroutine dtrsen (characterJOB, characterCOMPQ, logical, dimension( * )SELECT, integerN,
double precision, dimension( ldt, * )T, integerLDT, double precision, dimension( ldq, *
)Q, integerLDQ, double precision, dimension( * )WR, double precision, dimension( * )WI,
integerM, double precisionS, double precisionSEP, double precision, dimension( * )WORK,
integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)
DTRSEN
Purpose:
DTRSEN 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 quasitriangular 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 DHSEQR), that is,
block upper triangular with 1by1 and 2by2 diagonal blocks; each
2by2 diagonal block has its diagonal elements equal and its
offdiagonal elements of opposite sign.
Parameters:
JOB
JOB is 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
COMPQ is CHARACTER*1
= 'V': update the matrix Q of Schur vectors;
= 'N': do not update Q.
SELECT
SELECT is LOGICAL array, dimension (N)
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 2by2 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
N is INTEGER
The order of the matrix T. N >= 0.
T
T is DOUBLE PRECISION array, dimension (LDT,N)
On entry, the upper quasitriangular 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
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q
Q is DOUBLE PRECISION 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
orthogonal 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
LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
WR
WR is DOUBLE PRECISION array, dimension (N)
WI
WI is DOUBLE PRECISION array, dimension (N)
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 2by2 diagonal block, WI(i) > 0 and
WI(i+1) = WI(i). Note that if a complex eigenvalue is
sufficiently illconditioned, then its value may differ
significantly from its value before reordering.
M
M is INTEGER
The dimension of the specified invariant subspace.
0 < = M <= N.
S
S is DOUBLE PRECISION
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
SEP is DOUBLE PRECISION
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
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
If JOB = 'N', LWORK >= max(1,N);
if JOB = 'E', LWORK >= max(1,M*(NM));
if JOB = 'V' or 'B', LWORK >= max(1,2*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.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
If JOB = 'N' or 'E', LIWORK >= 1;
if JOB = 'V' or 'B', LIWORK >= max(1,M*(NM)).
If LIWORK = 1, then a workspace query is assumed; the
routine 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
INFO is INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
= 1: reordering of T failed because some eigenvalues are too
close to separate (the problem is very illconditioned);
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.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
April 2012
Further Details:
DTRSEN first collects the selected eigenvalues by computing an
orthogonal 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 * T * Z = ( T11 T12 ) n1
( 0 T22 ) n2
n1 n2
where N = n1+n2 and Z**T 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**T, then the reordered real Schur factorization of A is given
by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, 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 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 underestimate 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 reciprocal 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
Definition at line 313 of file dtrsen.f.
Author
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Version 3.4.2 Tue Sep 25 2012 dtrsen.f(3) 
