Error Bounds for Computed Eigenvalues.

Rewrite $r$ as

$$G^{-1}r=(G^{-1}AG^{-*})(G^*\wtd x)-\wtd\lambda(G^*\wtd x).$$

By (4.54) on p. , we have

$$\vert\wtd\lambda-\lambda\vert \le \frac {\Vert G^{-1}r\Vert... ...}{\Vert\wtd x\Vert _B} \le\Vert B^{-1}\Vert _2\Vert r\Vert _2$$ (95)

for some eigenvalue $\lambda$ of the pair $\{A,B\}$. A good estimate to $\Vert B^{-1}\Vert _2$ is needed to use this bound.

With more information, a much better bound can be obtained. Let us assume that $(\wtd\lambda,\wtd x)$ is an approximation of the eigenpair $(\lambda,x)$ of the pair. The ``best'' $\wtd\lambda$ corresponding to $\wtd x$ is the Rayleigh quotient $\wtd\lambda = \wtd x^{\ast} A \wtd x/\wtd x^{\ast} B \wtd x$, so we assume that $\wtd\lambda$ has this value. Suppose that $\wtd\lambda$ is closer to $\lambda$ than any other eigenvalues of the pair, and let $\delta$ be the gap between $\wtd\lambda$ and any other eigenvalue of the pair. Then

$$\vert\wtd\lambda-\lambda\vert \le \frac 1{\delta} \cdot\fr... ... \le \Vert B^{-1}\Vert _2^2\frac {\Vert r\Vert _2^2}{\delta}.$$ (96)

This improves (5.31) if the gap $\delta$ is reasonably big. In practice we can always pick the better one. This bound also needs information on $\delta$, besides the residual error $r$ and $\Vert B^{-1}\Vert _2$. 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 $\delta$. This comment also applies to the bounds in (5.33) and (5.34) below.