Purging $\theta$.

If $\theta$ is ``unwanted,'' we may wish to remove $\theta$ from the spectrum of the projected matrix $T$. However, the implicit restart strategy using exact shifts will sometimes fail to purge a converged unwanted Ritz value [294].

In the symmetric case, the purging process is essentially the same as the locking process just described. However, instead of keeping the deflated Ritz vector and value, they are simply discarded. After this, we are left with a $(k-1)$-step factorization

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

with $[v_1 , V_2] = V Q$. No error other than an acceptable level of roundoff will be introduced through purging.

Observe that there is no requirement that $y$ be an accurate eigenvector for $T$. It is only necessary for the residual $y^* T Q = \theta e_1^*$ to meet a componentwise accuracy condition that we shall discuss in the next subsection. Moreover, there is no need for the last element $\eta$ of the eigenvector to be small in the case of purging. However, when it is not small, the implicit restart mechanism with exact shifts will suffice to purge $\theta$. Finally, this procedure is valid in complex arithmetic with minor notational modifications.