The first example is the computation of some eigenmodes of an L-shaped membrane. The standard five-point finite difference approximation with a grid spacing gives a sparse symmetric positive definite matrix of order . It will have a very regular sparse band structure with at most five nonzeros elements in each row. Its eigenvalues will be in the interval , symmetrically distributed around and more densely distributed towards the ends of the spectrum.
The second test example is taken from an information retrieval application and involves a term document matrix, where each column stands for one document and each row for one term. The element is if term occurs in document , zero otherwise. The space of a set of leading singular values can be used to find connections between some of the documents. The specific matrix MEDLINE is rectangular of size and moderately sparse with 53,287 filled elements, or about elements in each row. It is not a good idea to form the product explicitly, since this matrix will be nearly full with 910,755 elements (or ) nonzero. We implement the product by first computing followed by .