Purging $\theta$.

If $\theta$ is ``unwanted" then we may wish to remove $\theta$ from the spectrum of the projected matrix $H$. However, the implicit restart strategy using exact shifts will sometimes fail to purge a converged unwanted Ritz value [294].

We shall use (7.21) to purge an unwanted but converged Ritz value. In this case, a left eigenvector $y$ is needed with

$$y^*h = \theta y^*, \ \ \Vert y\Vert = 1.$$

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

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

where $H_2$ is upper Hessenberg. Here, $\eta = e_k^*y$ as before, but there is no requirement that $\eta$ be small. The desired purging amounts to simply discarding the first column on both sides of this equation. We are then left with

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

No error other than an acceptable level of roundoff will be introduced through this purging process. Moreover, there is no requirement that $y$ be an accurate left eigenvector for $H$. It is only necessary that the residual $y^* H - \theta y^*$ be small.