Solving Eigenproblems on Multicomputers: Two Different Approaches

Abstract

The authors analyze two direct methods for solving the symmetric eigenproblem of large and sparse matrices in terms of their parallel implementations. Their proposals provide efficient parallel solutions for problems with large computational requirements. The direct solutions were decomposed in the following phases: (1) structuring the input matrix — the Lanczos method with complete reorthogonalization was implemented for this stage; (2) solving the eigenproblem of a structured matrix — two methods were applied to solve this stage, bisection and inverse iteration methods, and the divide-and-conquer method; (3) computing the eigenvectors of the input matrix, carried out by a matrix–matrix product. These methods have been implemented on a multiprocessor system and their performance evaluations carried out on a Cray T3E system with up to 32 nodes. Results show tha t the management of the memory hierarchy improves substantially as the number of processors increases, and that this is one of the reasons why superlinear speedups are obtained.