Sturm Sequence

The Sturm sequence is a very nice algorithm found in most libraries. It finds all the eigenvalues in a given range, $\alpha_{\mbox{\scriptsize min}} < \alpha < \alpha_{\mbox{\scriptsize max}}$, and the corresponding eigenvectors. It is also able to find the number of such eigenvalues very quickly and will return a message if insufficient storage has been allocated for the eigenvectors. It does require a tridiagonalisation beforehand and is often combined with the Lanczos algorithm, to deal with sparse matrices.