Locking $\theta$.

The first instance to discuss is the locking of a single converged Ritz value. Assume that

$$Ty = y \theta, \ \ \Vert y\Vert = 1,$$

with $e_k^* y = \eta$, where $\vert\eta \vert \le \epsilon_D \Vert T \Vert$. Here, it is understood that $\epsilon_M \le \epsilon_D < 1$ is a specified relative accuracy tolerance between $\epsilon_M$ and $1$.

If $\theta$ is ``wanted,'' it is desirable to lock $\theta$. However, in order to accomplish this, it will be necessary to arrange a transformation of the current Lanczos factorization to one with a small subdiagonal to isolate $\theta$. This may be accomplished by constructing a $k \times k$ orthogonal matrix $Q = Q(y)$ using Algorithm 4.9:

$$Q e_1 = y \ \ \mbox{and} \ \ e_k^* Q = ( \eta, \tau e_{k-1}^*),$$

with $\eta^2 + \tau^2 = 1$.

The end result of these transformations is Av_1 &=& v_1 + r , where v_1^* r = 0,
A V_2 &=& V_2 T_2 + r e_k-1^* , where $[v_1 , V_2] = V Q$.

This means that subsequent implicit restart takes place as if

$$A V_2 = V_2 T_2 + r\tau e_{k-1}^*$$

with all the subsequent orthogonal transformations associated with implicit restart applied to $T_2$ and never disturbing the relation $Av_1 = v_1 \theta + r \eta$. In subsequent Lanczos steps, $v_1$ participates in the orthogonalization so that the selective orthogonalization recommended by Parlett and Scott [363,353] is accomplished automatically.