Stability and Accuracy Assessments
  Z. Bai and R. Li

The generalized eigenvalue problem for a Hermitian matrix pair $\{A,B\}$ with one of $A$ and $B$, or some linear combination of $A$ and $B$, being positive definite takes a unique position among all generalized eigenvalue problems for matrix pairs because it resembles in many ways the standard Hermitian eigenvalue problem discussed in Chapter 4. Matrix pairs as such are called Hermitian definite pairs. We shall consider separately

1. $B$ is definite and well-conditioned, meaning that $\kappa(B)\equiv \Vert B\Vert _2\Vert B^{-1}\Vert _2$ is not too large.
2. Some combination of $A$ and $B$ is definite and well-conditioned.

In this section, we only review some basic results that are readily applicable to assess how accurate computed eigenvalues and eigenvectors may be. We assume the availability of residual vectors which are usually available upon the exit of a successful computation. If not, they can be computed at marginal cost afterwards.

For the treatment of error estimation of computed eigenvalues and eigenvectors of dense generalized Hermitian eigenproblems, see Chapter 4 of the LAPACK Users' Guide [12].