Deflation.

The partial generalized Schur form can be obtained in a number of successive steps. Suppose that we have the partial generalized Schur form $AQ_{k-1}={Z}_{k-1}R^A_{k-1}$ and $BQ_{k-1}={Z}_{k-1}R^B_{k-1}$. We want to expand this partial generalized Schur form with the new right Schur vector ${q}$ and the left Schur vector ${z}$ to

$$A\onebytwo{Q_{k-1}}{{q}}=\onebytwo{{Z}_{k-1}}{{z}} \left[\b... ...{@{\,}cc@{\,}} R^A_{k-1}& {a}\\ 0& \alpha\end{array}\right]$$

and

$$B\onebytwo{Q_{k-1}}{q}=\onebytwo{{Z}_{k-1}}{{z}} \left[\be... ...@{\,}cc@{\,}} R^B_{k-1}& {b}\\ 0& \beta \end{array}\right].$$

The new generalized Schur pair $((\alpha,\beta), {q})$ satisfies

$$Q^\ast_{k-1}{q}=0 \quad \mbox{and} \quad (\beta A-\alpha B){q}-{Z}_{k-1}(\beta {a} - \alpha {b})=0,$$

or, since $\beta {a} - \alpha {b}={Z}^\ast_{k-1}(\beta A-\alpha B)q$,

$$Q^\ast_{k-1}q=0\quad\mbox{and}\quad \left(I-{Z}_{k-1}{Z}_{k-... ...ight)(\beta A-\alpha B) \left(I-Q_{k-1}Q_{k-1}^\ast\right)q=0.$$

The vectors ${a}$ and ${b}$ can be computed from

$${a}={Z}_{k-1}^\ast Aq\quad\mbox{and}\quad {b}={Z}_{k-1}^\ast Bq.$$

Hence, the generalized Schur pair $((\alpha,\beta),q)$ is an eigenpair of the deflated matrix pair

$$\begin{array}{l} \left(\left(I-{Z}_{k-1}{Z}_{k-1}^\ast\rig... ...ight)B \left(I-Q_{k-1}Q_{k-1}^\ast\right)\right). \end{array}$$ (228)

This eigenproblem can be solved again with the Jacobi-Davidson process that we have outlined in §8.4.1. In that process we construct vectors $v_i$ that are orthogonal to $Q_{k-1}$ and vectors $w_i$ that are orthogonal to ${Z}_{k-1}$. This simplifies the computation of the interaction matrices $M^A$ and $M^B$, associated with the deflated operators:

$$\left\{\begin{array}{l} M^A\equiv W^\ast\left(I-{Z}_{k-1}{... ...\ast\right) V=W^\ast B V,\rule{0pt}{3.5ex} \end{array}\right.$$ (229)

and $M^A$ and $M^B$ can be simply computed as $W^\ast A V$ and $W^\ast B V$, respectively.