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Error Bound for Computed Eigenvalues.
From (7.105) and the
perturbation theory of eigenvalues and eigenvectors, it can shown that
up to the first order of
residual norms
and
,
there is an eigenvalue
of
satisfying
![\begin{displaymath}
\vert\lambda - \wtd\lambda \vert
\simle \frac{1}{\vert\wtd ...
... \max\left\{\Vert r\Vert _2, \Vert s^{\ast} \Vert _2 \right\},
\end{displaymath}](img2605.png) |
(212) |
where ``
'' denotes ``less than'' up to the first order of
and
.
Note that
the individual condition number
is defined as
and
and
are the corresponding eigenvector and left
eigenvector of
and are normalized so that
.
Since exact
and
usually are not known, for
all practical purposes, one could simply estimate this condition number
by
instead.
Notice that
, with equality if
is Hermitian.
If the approximate left eigenvector
is not available, bounds
directly on
do not exist in general.
The equation (7.104) would be the only thing available
to explain the accuracy of the
computed eigenvalues
and
.
Next: Error Bound for Computed
Up: Stability and Accuracy Assessments
Previous: Transfer Residual Errors to
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Susan Blackford
2000-11-20