Restart.

Suppose that the generalized Schur form (8.12) is ordered with respect to $\tau$ such that

$$\vert T^A_{1,1}/T^B_{1,1}-\tau\vert\leq \vert T^A_{2,2}/T^B... ...t\leq\cdots \leq \vert T^A_{{m},{m}}/T^B_{{m},{m}}-\tau\vert,$$

where ${m}$ is the dimension of $\mbox{span}(V)$. Then, for $i<{m}$, the space $\mbox{span} ( Vs^R_1, \ldots, Vs^R_i )$ spanned by the first $i$ columns of $V S^R$ contains the $i$ most promising Petrov vectors. The corresponding test subspace is given by $\mbox{span}(Ws^L,\ldots,Ws^L_i)$. Therefore, in order to reduce the dimension of the subspaces (``implicit restart'') to ${m}_{\min}$, ${m}_{\min}<{m}$, the columns $v_{{m}_{\min}+1}$ through $v_{m}$ and $w_{{m}_{\min}+1}$ through $w_{m}$ can simply be discarded and the Jacobi-Davidson algorithm can be continued with

$$V=[Vs^R_1,\ldots,Vs^R_{{m}_{\min}}]\quad \mbox{and}\quad W=[Ws^L_1,\ldots,Ws^L_{{m}_{\min}}].$$