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Error Bounds for Computed Eigenvectors.
Keep the assignments to
,
, and
as above, and
let
be the eigenvector of
corresponding to
.
The error bound for the computed eigenvector
and the
true eigenvector
are
![\begin{displaymath}
\sin\theta_B(x,\wtd x)\le\frac 1{\delta}
\cdot\frac {\Vert r\Vert _{B^{-1}}}{\Vert\wtd x\Vert _B}.
\end{displaymath}](img1642.png) |
(97) |
This, (5.27), and (5.28)
yield
![\begin{displaymath}
\sin\theta(x,\wtd x)\le\Vert B^{-1}\Vert _2\frac {\sqrt{2\kappa(B)}}{\delta}\Vert r\Vert _2.
\end{displaymath}](img1643.png) |
(98) |
The factor
can be removed
by a more elaborate argument, but we shall omit it here.
Susan Blackford
2000-11-20