Specifying an Eigenproblem

The following eigenproblems are typical, both because they arise naturally in applications and because we have good algorithms for them:

1. Compute all the eigenvalues to some specified accuracy.
2. Compute eigenvalues $\lambda_i$ for some specified set of subscripts $i \in {\cal I} = \{1,2,\ldots,n\}$, including the special cases of the largest $m$ eigenvalues $\lambda_{n-m+1}$ through $\lambda_n$, and the smallest $m$ eigenvalues $\lambda_1$ through $\lambda_m$. Again, the desired accuracy may be specified.
3. Compute all the eigenvalues within a given subset of the real axis, such as the interval $[\alpha, \beta]$. Again, the desired accuracy may be specified.
4. Count all the eigenvalues in the interval $[\alpha, \beta]$. This does not require computing the eigenvalues in $[\alpha, \beta]$, and so can be much cheaper.
5. Compute a certain number of eigenvalues closest to a given value $\mu$.

For each of these possibilities (except 4) the user can also compute the corresponding eigenvectors. For the eigenvalues that are clustered together, the user may choose to compute the associated invariant subspace, since in this case the individual eigenvectors can be very ill-conditioned, while the invariant subspace may be less so. Finally, for any of these quantities, the user might also want to compute its condition number.

Even though we have effective algorithms for these problems, we cannot necessarily solve all large scale problems in an amount of time and space acceptable to all users.