Computing Eigenvalues and Eigenvectors of a Dense Real Symmetric Matrix on the Ncube 6400

Abstract

This report demonstrates parallel versions of the Eispack functions TRED2 and TQL2 for finding all eigenvalues and eigenvectors of a dense, real symmetric matrix on the Ncube 6400. There are several techniques for solving this problem. An efficient and accurate method is tridiagonalization of the original matrix A (TRED2), followed by application of the QR method on the resulting tridiagonal matrix (TQL2). Since the eigenvalues are the roots of a characteristic polynomial, bisection and inverse iteration can be used to compute all eigenvalues and eigenvectors. This method is significantly faster than the QR method provided the eigenvalues are well separated. This technical report will also describe the algorithms TREDs and TQL2 and their parallel counterparts in greater detail. The results of numerical experiments on large problems on a 1024 processors Ncube 6400 are also presented.