Lanczos methods for the smallest eigenvalues of large matrices on distributed memory supercomputers

We present results on applying Lanczos methods to find some of the eigenvalues of dense matrices whose size renders reduction to fill tridiagonal or hessenberg form undesirable. The use of distributed memory supercomputers is an ideal match for many problem in physics and other applications, where a system is modeled by building a large matrix, whose entries must be computed and whose few smallest eigenvalues provide the desired information. Our motivating application starts with 2000 by 2000 systems.