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Error Bounds for Computed Eigenvalues.
Rewrite
as
By (4.54) on p.
, we have
![\begin{displaymath}
\vert\wtd\lambda-\lambda\vert
\le \frac {\Vert G^{-1}r\Vert...
...}{\Vert\wtd x\Vert _B}
\le\Vert B^{-1}\Vert _2\Vert r\Vert _2
\end{displaymath}](img1638.png) |
(95) |
for some eigenvalue
of the pair
.
A good estimate to
is needed to use this bound.
With more information, a much better bound
can be obtained.
Let us assume that
is
an approximation of the eigenpair
of the pair.
The ``best''
corresponding to
is the Rayleigh quotient
,
so we assume that
has this value.
Suppose that
is closer
to
than any other eigenvalues of the pair, and
let
be the gap between
and any other eigenvalue of the pair. Then
![\begin{displaymath}
\vert\wtd\lambda-\lambda\vert
\le \frac 1{\delta}
\cdot\fr...
...
\le \Vert B^{-1}\Vert _2^2\frac {\Vert r\Vert _2^2}{\delta}.
\end{displaymath}](img1641.png) |
(96) |
This improves (5.31) if the gap
is
reasonably big. In practice we can always pick the better one.
This bound also needs information on
, besides the residual error
and
.
Usually such information
is available after a successful computation by,
e.g., the shift-and-invert Lanczos
algorithm, which usually delivers eigenvalues in the neighborhood
of a shift and consequently yields good information on the
. This comment also applies to the bounds in
(5.33) and (5.34)
below.
Next: Error Bounds for Computed
Up: Positive Definite
Previous: Transfer Residual Error to
  Contents
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Susan Blackford
2000-11-20