**``#vec''**- gives how many vectors one needs to store. The meaning of 2 is obvious; ``Moderate'' means a multiple of the number of eigenvalues sought, say to . ``Few'' is smaller than moderate, say , and ``Many'' is larger.
**``Fact''**- indicates whether we need extra matrix storage. ``'' means a sparse Cholesky factorization, ``,'' a sparse symmetric indefinite Gaussian elimination, and ``ILU'' is an incomplete factorization. It is supposed to be more compact and need less arithmetic work than the other two.

We note that a task such as counting the
number of eigenvalues of that are smaller than a given real
number or are in a given interval
does not require computing the eigenvalues, and so can be much cheaper.
The key tool is the matrix inertia as presented in
§4.1 (p. ).
It can be extended easily to the case of assuming
positive definite. In summary, let

be the LDL factorizations of matrices and , respectively, where we assume that and are nonsingular diagonal matrices. We refer to §10.3 for the information on software availability for the LDL factorization. Then the number of eigenvalues of in equals , where denotes the number of negative diagonal elements.