Example 8.6.1.

The first example demonstrates the importance of the choice of starting vector when the $B$ matrix is singular. The matrices used in this example come from the shaft model using a fine mesh, and the order of the matrices $\{A,B\}$ is 800. A shift of $\sigma=100i$ was used and complete reorthogonalization was employed. The following table illustrates one ill effect which may result when the Lanczos vectors do not belong to the range of $H^{-1} B$. The symmetric indefinite Lanczos procedure was executed twice with an initial vector of all ones. In the first run, the initial vector is preprocessed by multiplying it by $H^{-1} B$, while in the second run, no preprocessing is used. The preprocessing step dramatically improves the quality of the Ritz pairs, as measured by the residuals $\Vert A\tilde{x} - \tilde{\lambda}B\tilde{x}\Vert$, by ensuring that the Lanczos vectors, and hence the Ritz vectors, belong to the range of $H^{-1} B$.

Ritz value	$\Vert A\tilde{x} - \tilde{\lambda}B\tilde{x}\Vert$	$\Vert A\tilde{x} - \tilde{\lambda}B\tilde{x}\Vert$
$\tilde{\lambda}$	(preprocessed initial vector)	(no preprocessing)
$-4.09e\!-\!06 + 5.63e\!+\!01$	$1.88e\!-\!08$	$3.05e\!-\!03$
$-4.15e\!-\!06 - 5.63e\!+\!01$	$1.45e\!-\!07$	$1.10e\!-\!02$
$-1.30e\!-\!04 + 3.55e\!+\!02$	$6.09e\!-\!07$	$5.48e\!-\!02$
$-1.31e\!-\!04 - 3.55e\!+\!02$	$5.24e\!-\!07$	$9.72e\!-\!02$
$-8.58e\!-\!04 + 1.00e\!+\!03$	$8.57e\!-\!07$	$2.85e\!-\!01$
$-8.65e\!-\!04 - 1.00e\!+\!03$	$8.12e\!-\!07$	$3.50e\!-\!01$