Next: Error Bound for Computed
Up: Stability and Accuracy Assessments
Previous: Residual Vectors.
  Contents
  Index
Transfer Residual Errors to Backward Errors.
It can be shown that the computed
eigenvalue and eigenvector(s) are the exact ones of a nearby matrix pair,
i.e.,
and
if
is available,
where error matrices
and
are small relative to the
norms of
and
.
- Only
is available but
is not. Then
the optimal error matrix
(in both the 2-norm
and the Frobenius norm) for which
and
are
an exact eigenvalue and its corresponding eigenvector of the pair
satisfies
![\begin{displaymath}
\Vert(E,F)\Vert _2=\Vert(E,F)\Vert _{F}= \Vert r\Vert _2.
\end{displaymath}](img3283.png) |
(241) |
- Both
and
are available. Then
the optimal error matrices
(in the 2-norm) and
(in the Frobenius norm) for which
,
, and
are
an exact eigenvalue and its corresponding eigenvectors of
the pair
satisfy
![\begin{displaymath}
\Vert(E_2,F_2)\Vert _2= %%\frac{1}{\sqrt{\vert\wtd\alpha\ve...
...eta\vert^2}}
\max\{\Vert r\Vert _2,\Vert s^{\ast}\Vert _2\}
\end{displaymath}](img3285.png) |
(242) |
and
![\begin{displaymath}
\Vert(E_{F},F_{F})\Vert _{F}
= %%\frac{1}{\sqrt{\vert\wtd\...
...eta\wtd y^{\ast} A\wtd x-\wtd\alpha\wtd y^{\ast} B\wtd x)^2}.
\end{displaymath}](img3286.png) |
(243) |
See [256,431,473].
We say the algorithm
that delivers the approximate eigenpair
is
-backward stable
for the pair with respect to the norm
if it is an exact eigenpair for
with
; analogously
the algorithm that delivers the eigentriplet
is
-backward stable for the triplet with respect to the norm
if it is an exact eigentriplet for
with
. With these in mind,
statements can be made about the backward stability of the algorithm which
computes the eigenpair
or
the eigentriplet
.
Conventionally, an algorithm is called backward stable
if
.
Next: Error Bound for Computed
Up: Stability and Accuracy Assessments
Previous: Residual Vectors.
  Contents
  Index
Susan Blackford
2000-11-20