Let us consider the case of having
an approximate eigenpair
to an exact pair
. Let be the Schur decomposition of
(see §2.5, p. ),
where is unitary and
For the small and dense eigenproblems, can be efficiently estimated. This is available in LAPACK . However, for large sparse eigenproblems, since is generally not available, the estimation of is out of the question. We can only get a gross sense about the quality of computed eigenvectors.
the separation of from the eigenvalues of . We
have to say ``roughly'' because
As summarized in , the separation of the eigenvalues has a bearing upon eigenvector sensitivity. Indeed, if is a nondefective, repeated eigenvalue, then there are an infinite number of possible eigenvector bases for the associated invariant subspace. The preceding analysis merely indicates that this indeterminacy begins to be felt as the eigenvalues coalesce. It is well known that each individual eigenvector associated with eigenvalue clusters is very sensitive to perturbations, and consequently such an eigenvector cannot be accurately computed in general. Fortunately, often for the case of an eigenvalue cluster, it is the entire associated eigenspace that is of practical importance, and that eigenspace can be computed with satisfactory accuracy. Detailed treatment is beyond the scope of this book and the interested reader is referred to the literature, for example, .