Numerical Examples

We consider the Lanczos algorithm applied to two simple but typical examples using the LANSO software.

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 $h=1/64$ gives a sparse symmetric positive definite matrix of order $n=2945$. 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 $0<\lambda_i<8$, symmetrically distributed around $\lambda=4$ 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 $x_{i,j}$ is $1$ if term $i$ occurs in document $j$, 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 $7014\times 1033$ and moderately sparse with 53,287 filled elements, or about $8$ elements in each row. It is not a good idea to form the product $A=X^TX$ explicitly, since this matrix will be nearly full with 910,755 elements (or $85\%$) nonzero. We implement the product $y=Ax$ by first computing $z=Xx$ followed by $y=X^Tz$.