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Even when the initial vector is premultiplied by
, 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
is 160. A shift of
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$](img2984.png) |
![$ \Vert A\tilde{x} - \tilde{\lambda}B\tilde{x}\Vert$](img2984.png) |
![$\tilde{\lambda}$](img731.png) |
(with postprocessing) |
(no postprocessing) |
![$-1.96e\!-\!04 + 1.03e\!+\!05i$](img3004.png) |
![$5.04e\!-\!08$](img3005.png) |
![$5.04e\!-\!08$](img3005.png) |
![$-1.30e\!-\!02 + 1.06e\!+\!05i$](img3006.png) |
![$4.11e\!-\!08$](img3007.png) |
![$4.11e\!-\!08$](img3007.png) |
![$-7.16e\!-\!03 + 9.28e\!+\!04i$](img3008.png) |
![$4.15e\!-\!08$](img3009.png) |
![$4.18e\!-\!08 $](img3010.png) |
![$-7.60e\!-\!02 + 1.11e\!+\!05i$](img3011.png) |
![$8.41e\!-\!08$](img3012.png) |
![$8.51e\!-\!05 $](img3013.png) |
![$-1.01e\!-\!01 + 8.66e\!+\!04i$](img3014.png) |
![$1.09e\!-\!07$](img3015.png) |
![$5.06e\!-\!03 $](img3016.png) |
Susan Blackford
2000-11-20