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Error Bound for Computed Eigenvectors.
Keep the assignments to
, let
be the eigenvector of
corresponding to
,
and let
be the smallest distance in chordal metric between
and all the other eigenvalues
of the pair. Then we have
![\begin{displaymath}
\sin\theta(x,\wtd x)
\le\frac{1}{\eta}\cdot\frac {\Vert r\Vert _2}{\gamma(A,B)}.
\end{displaymath}](img1682.png) |
(107) |
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
.
Susan Blackford
2000-11-20