ZTRSEN(3F) ZTRSEN(3F)
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 COMPQ, JOB
INTEGER INFO, LDQ, LDT, LWORK, M, N
DOUBLE PRECISION S, SEP
LOGICAL SELECT( * )
COMPLEX*16 Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
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) 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.
SELECT (input) LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To
select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
N (input) INTEGER
The order of the matrix T. N >= 0.
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ZTRSEN(3F) ZTRSEN(3F)
T (input/output) COMPLEX*16 array, dimension (LDT,N)
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) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q (input/output) COMPLEX*16 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 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) INTEGER
The leading dimension of the array Q. LDQ >= 1; and if COMPQ =
'V', LDQ >= N.
W (output) COMPLEX*16
The reordered eigenvalues of T, in the same order as they appear
on the diagonal of T.
M (output) INTEGER
The dimension of the specified invariant subspace. 0 <= M <= N.
S (output) 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 (output) 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 (workspace) COMPLEX*16 array, dimension (LWORK)
If JOB = 'N', WORK is not referenced.
LWORK (input) INTEGER
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).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
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ZTRSEN(3F) ZTRSEN(3F)
FURTHER DETAILS
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 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
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 )
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ZTRSEN(3F) ZTRSEN(3F)
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
ZTRSEN(3F) ZTRSEN(3F)
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 COMPQ, JOB
INTEGER INFO, LDQ, LDT, LWORK, M, N
DOUBLE PRECISION S, SEP
LOGICAL SELECT( * )
COMPLEX*16 Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
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) 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.
SELECT (input) LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To
select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
N (input) INTEGER
The order of the matrix T. N >= 0.
Page 1
ZTRSEN(3F) ZTRSEN(3F)
T (input/output) COMPLEX*16 array, dimension (LDT,N)
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) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q (input/output) COMPLEX*16 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 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) INTEGER
The leading dimension of the array Q. LDQ >= 1; and if COMPQ =
'V', LDQ >= N.
W (output) COMPLEX*16
The reordered eigenvalues of T, in the same order as they appear
on the diagonal of T.
M (output) INTEGER
The dimension of the specified invariant subspace. 0 <= M <= N.
S (output) 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 (output) 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 (workspace) COMPLEX*16 array, dimension (LWORK)
If JOB = 'N', WORK is not referenced.
LWORK (input) INTEGER
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).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Page 2
ZTRSEN(3F) ZTRSEN(3F)
FURTHER DETAILS
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 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
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 )
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ZTRSEN(3F) ZTRSEN(3F)
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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