Stopping Criterion.

Suppose that $(s,\theta)$ is an eigenpair of $H_{[m]}.$ It follows easily from equation (7.29) that

$$A V_{[m]} s - \theta V_{[m]} s = F_m E^{\ast}_m s ,$$ (136)

and so $\Vert A V_{[m]} s - \theta V_{[m]} s \Vert = \Vert F_m \Vert \, \Vert E^{\ast}_m s \Vert = \Vert H_{m+1,m} \Vert \, \Vert E^T_m s \Vert.$ Thus, if the last $b$ components of $s$ are small relative to the size of $\Vert H_{m+1,m} \Vert$, then the Ritz pair $(z = V_{[m]} s,\theta)$ is an exact eigenpair for a matrix near $A.$ This follows since (7.30) may be rewritten as $(A - F_m E^{\ast}_m s z^\ast)z = \theta z.$

The iteration in BIRAM terminates at the value of $i$ when the $k$ wanted eigenvalues of $H_{[m]}$ satisfy (7.30). The eigenvalues are partitioned as in (7.31) so that the wanted ones correspond to the eigenvalues of $A$ desired.