STRSEN(3F) STRSEN(3F)
STRSEN - 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,
SUBROUTINE STRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S,
SEP, WORK, LWORK, IWORK, LIWORK, INFO )
CHARACTER COMPQ, JOB
INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
REAL S, SEP
LOGICAL SELECT( * )
INTEGER IWORK( * )
REAL Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), WR( * )
STRSEN 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-diagonal
elements of opposite sign.
JOB (input) 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 (input) CHARACTER*1
= 'V': update the matrix Q of Schur vectors;
= 'N': do not update Q.
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STRSEN(3F) STRSEN(3F)
SELECT (input) 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 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 either both
included in the cluster or both excluded.
N (input) INTEGER
The order of the matrix T. N >= 0.
T (input/output) REAL array, dimension (LDT,N)
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) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q (input/output) REAL 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 (input) INTEGER
The leading dimension of the array Q. LDQ >= 1; and if COMPQ =
'V', LDQ >= N.
WR (output) REAL array, dimension (N)
WI (output) REAL 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 2-by-2 diagonal
block, WI(i) > 0 and WI(i+1) = -WI(i). Note that if a complex
eigenvalue is sufficiently ill-conditioned, then its value may
differ significantly from its value before reordering.
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
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) REAL
If JOB = 'V' or 'B', SEP is the estimated reciprocal condition
number of the specified invariant subspace. If M = 0 or N, SEP =
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STRSEN(3F) STRSEN(3F)
norm(T). If JOB = 'N' or 'E', SEP is not referenced.
WORK (workspace) REAL array, dimension (LWORK)
LWORK (input) INTEGER
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).
IWORK (workspace) INTEGER array, dimension (LIWORK)
IF JOB = 'N' or 'E', IWORK is not referenced.
LIWORK (input) INTEGER
The dimension of the array IWORK. If JOB = 'N' or 'E', LIWORK >=
1; if JOB = 'V' or 'B', LIWORK >= M*(N-M).
INFO (output) INTEGER
= 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
STRSEN 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*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
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STRSEN(3F) STRSEN(3F)
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) 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
STRSEN(3F) STRSEN(3F)
STRSEN - 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,
SUBROUTINE STRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S,
SEP, WORK, LWORK, IWORK, LIWORK, INFO )
CHARACTER COMPQ, JOB
INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
REAL S, SEP
LOGICAL SELECT( * )
INTEGER IWORK( * )
REAL Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), WR( * )
STRSEN 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-diagonal
elements of opposite sign.
JOB (input) 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 (input) CHARACTER*1
= 'V': update the matrix Q of Schur vectors;
= 'N': do not update Q.
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STRSEN(3F) STRSEN(3F)
SELECT (input) 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 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 either both
included in the cluster or both excluded.
N (input) INTEGER
The order of the matrix T. N >= 0.
T (input/output) REAL array, dimension (LDT,N)
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) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q (input/output) REAL 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 (input) INTEGER
The leading dimension of the array Q. LDQ >= 1; and if COMPQ =
'V', LDQ >= N.
WR (output) REAL array, dimension (N)
WI (output) REAL 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 2-by-2 diagonal
block, WI(i) > 0 and WI(i+1) = -WI(i). Note that if a complex
eigenvalue is sufficiently ill-conditioned, then its value may
differ significantly from its value before reordering.
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
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) REAL
If JOB = 'V' or 'B', SEP is the estimated reciprocal condition
number of the specified invariant subspace. If M = 0 or N, SEP =
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STRSEN(3F) STRSEN(3F)
norm(T). If JOB = 'N' or 'E', SEP is not referenced.
WORK (workspace) REAL array, dimension (LWORK)
LWORK (input) INTEGER
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).
IWORK (workspace) INTEGER array, dimension (LIWORK)
IF JOB = 'N' or 'E', IWORK is not referenced.
LIWORK (input) INTEGER
The dimension of the array IWORK. If JOB = 'N' or 'E', LIWORK >=
1; if JOB = 'V' or 'B', LIWORK >= M*(N-M).
INFO (output) INTEGER
= 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
STRSEN 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*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
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STRSEN(3F) STRSEN(3F)
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) 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
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