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Next: Example 8.6.3. Up: Numerical Examples Previous: Example 8.6.1.   Contents   Index

Example 8.6.2.

Even when the initial vector is premultiplied by $H^{-1} B$, the Ritz vectors may be further improved by implementing the postprocessing step shown in (8.27). The following example represents the same shaft as in Example 8.6.1, but with a coarser grid so that the order of $\{A,B\}$ is 160. A shift of $\sigma =1.e5i$ was used and full orthogonality was maintained. The following table shows that the postprocessing step improves the direct residuals by up to four orders of magnitude.
Ritz value $ \Vert A\tilde{x} - \tilde{\lambda}B\tilde{x}\Vert$ $ \Vert A\tilde{x} - \tilde{\lambda}B\tilde{x}\Vert$
$\tilde{\lambda}$ (with postprocessing) (no postprocessing)
$-1.96e\!-\!04 + 1.03e\!+\!05i$ $5.04e\!-\!08$ $5.04e\!-\!08$
$-1.30e\!-\!02 + 1.06e\!+\!05i$ $4.11e\!-\!08$ $4.11e\!-\!08$
$-7.16e\!-\!03 + 9.28e\!+\!04i$ $4.15e\!-\!08$ $4.18e\!-\!08 $
$-7.60e\!-\!02 + 1.11e\!+\!05i$ $8.41e\!-\!08$ $8.51e\!-\!05 $
$-1.01e\!-\!01 + 8.66e\!+\!04i$ $1.09e\!-\!07$ $5.06e\!-\!03 $

Susan Blackford 2000-11-20