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In the case when the eigenvalue
has one or more other eigenvalues of
close by, in other words,
when
belongs to clustered eigenvalues,
as guaranteed by (4.54), the computed
is still accurate as long as
is tiny, but the computed
eigenvector
may be inaccurate because of the
appearance of the gap
in the denominator of (4.56).
It turns out that each individual
eigenvector associated with the clustered eigenvalues
is very sensitive to perturbations,
but the eigenspace spanned by all the eigenvectors associated with
the clustered eigenvalues is not. Thus, for the clustered eigenvalues,
we should instead compute the entire eigenspace.
A theory along the lines given above can be established, starting with
a residual matrix
where
is diagonal with diagonal entries consisting of
approximations to all the eigenvalues in the cluster, and the columns of
are the corresponding approximate eigenvectors.
Assume that
has orthonormal columns and that
is
closer to
, the diagonal matrix whose diagonal entries
consist of all the eigenvalues in the cluster. Let
be the eigenvector
matrix associated with
, and let
be the smallest difference
between any approximate eigenvalue in the diagonal of
and those eigenvalues of
not presented in the diagonal of
.
Then [101]
where
is diagonal with diagonal entries being
the arccosines of the singular values of
.
Because of the way it is defined, this gap is expected
to be big and thus
will be small as
long as
is.
Next: Remarks on Eigenvalue Computations
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Susan Blackford
2000-11-20