Error Bound for Computed Eigenvectors.

Keep the assignments to $(\alpha,\beta)$, let $x$ be the eigenvector of $\{A,B\}$ corresponding to $(\alpha,\beta)$, and let $\eta$ be the smallest distance in chordal metric between $(\wtd\alpha,\wtd\beta)$ and all the other eigenvalues of the pair. Then we have

$$\sin\theta(x,\wtd x) \le\frac{1}{\eta}\cdot\frac {\Vert r\Vert _2}{\gamma(A,B)}.$$ (107)

This bound also needs information on $\eta$, besides the residual error $r$ and $\gamma(A,B)$. Usually such information is available after a successful computation by, e.g., the shift-and-invert Lanczos algorithm which usually delivers eigenvalues in the neighborhood of a shift and consequently yields good information on the $\eta$.