Locking $\theta$.

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

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

with $e_k^* y = \eta$ , where $\vert\eta \vert \le \epsilon_D \Vert H \Vert$. 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" then it is desirable to lock $\theta$. However, in order to accomplish this it will be necessary to arrange a transformation of the current Arnoldi 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 7.8:

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

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

Now, when we apply $AVQ = VQ (Q^*H Q) + f e_k^* Q$, we obtain

$$A [v_1 , V_2] = [v_1 , V_2] \left[ \ba{cc} \theta & h^* \\ 0 & H_2 \ea \right] + f (\eta, \tau e_{k-1}^*) .$$

Unfortunately, $H_2$ is not upper Hessenberg. Further work will have to be done to bring it to Hessenberg form. This must be done without disturbing the form of the residual term $f (\eta, \tau e_{k-1}^*)$. We need to construct an orthogonal $U$ such that $\widehat{H}_2 = U^*H_2U$ is upper Hessenberg and $e_{k-1}^* U = e_{k-1}^*$. This can be done with Householder transformations or with a variant of the transformations defined in Algorithm 7.8 working from the last row upwards. The following MATLAB code segment shows a slightly inefficient way to obtain such a $U$.

       rev = [k-1:-1:1];
       C = H_2';
       [U,H_2] = hess(C(rev,rev));
       U = U(rev,rev);
       H_2 = H_2(rev,rev)';

Once $U$ is constructed, replace $Q$ with $Q : Q [{1 \atop 0}\;{0 \atop U}]$, and the end result of these transformations will be Av_1 &=& v_1 + f , where v_1^* f = 0,
A V_2 &=& [v_1, V_2] [ c h^*,
H_2
] + f e_k-1^* , where $[v_1 , V_2] = V Q$ and $H_2$ is upper Hessenberg. After this, all subsequent implicit restarting takes place as if

$$A V_2 = V_2 H_2 + f\tau e_{k-1}^*.$$

All the subsequent orthogonal transformations associated with implicit restarting are applied to $V_2$ and $[{h^* \atop H_2}]$ while never disturbing the relation $Av_1 = v_1 \theta + f \eta$. In subsequent Arnoldi steps, $v_1$ participates in the orthogonalization so that the selective orthogonalization recommended by Parlett and Scott [363,353] is accomplished automatically.