The error analysis of the driver routines for the generalized symmetric definite eigenproblem goes as follows. In all cases is the absolute gap between and the nearest other eigenvalue.
The angular difference
between the computed eigenvector
and a true eigenvector z_{i} is
The angular difference between the computed eigenvector
and a true eigenvector z_{i} is
The code fragments above replace p(n) by 1, and makes sure neither RCONDB nor RCONDZ is so small as to cause overflow when used as divisors in the expressions for error bounds.
These error bounds are large when B is ill-conditioned with respect to inversion ( is large). It is often the case that the eigenvalues and eigenvectors are much better conditioned than indicated here. We mention three ways to get tighter bounds. The first way is effective when the diagonal entries of B differ widely in magnitude^{4.1}:
The second way to get tighter bounds does not actually supply guaranteed
bounds, but its estimates are often better in practice. It is not guaranteed
because it assumes the algorithm is backward stable, which is not necessarily
true when B is ill-conditioned. It estimates the chordal distance between a true
eigenvalue
and a computed eigenvalue
:
Suppose a computed eigenvalue of is the exact eigenvalue of a perturbed problem . Let x_{i} be the unit eigenvector (|x_{i}|_{2}=1) for the exact eigenvalue . Then if |E| is small compared to |A|, and if |F| is small compared to |B|, we have
Thus is a condition number for eigenvalue .
The third way applies only to the first problem , and only when A is positive definite. We use a different algorithm:
Other yet more refined algorithms and error bounds are discussed in [14,95,103].