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)
$-1.96e\!-\!04 + 1.03e\!+\!05i$	$5.04e\!-\!08$	$5.04e\!-\!08$
$-1.30e\!-\!02 + 1.06e\!+\!05i$	$4.11e\!-\!08$	$4.11e\!-\!08$
$-7.16e\!-\!03 + 9.28e\!+\!04i$	$4.15e\!-\!08$	$4.18e\!-\!08$
$-7.60e\!-\!02 + 1.11e\!+\!05i$	$8.41e\!-\!08$	$8.51e\!-\!05$
$-1.01e\!-\!01 + 8.66e\!+\!04i$	$1.09e\!-\!07$	$5.06e\!-\!03$