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##

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 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 .

**Subsections**

** Next:** Results for L-Shaped Membrane.
** Up:** Lanczos Method A.
** Previous:** Software Availability
** Contents**
** Index**
Susan Blackford
2000-11-20