Error Bound for Computed Eigenvalues.

It can be proved [425] that up to the first order of $\Vert(E,F)\Vert _2$, there is an eigenvalue $(\alpha,\beta)$ of $A$ and $B$ satisfying

$$\chi((\alpha, \beta),(\wtd\alpha,\wtd\beta)) \simle c_{(\alpha,\beta)} \cdot \Vert(E_2,F_2)\Vert _2,$$ (244)

where

$$c_{(\alpha,\beta)} \equiv\frac {\Vert x\Vert _2\Vert y\Vert... ...\sqrt {\vert y^{\ast} Ax\vert^2 + \vert y^{\ast} Bx\vert^2}}.$$

It is called the individual condition number of $(\alpha,\beta)$, which in actual computations will be estimated with the exact eigenvectors $x$ and $y$ replaced by their approximations $\wtd x$ and $\wtd y$.