Parallel implementation for large and sparse eigenproblems

Abstract

This paper analyses and evaluates the computational aspects of an efficient parallel implementation for the eigenproblem. This parallel implementation allows to solve the eigenproblem of symmetric, sparse and very large matrices. Mathematically, the algorithm is supported by the Lanczos and Divide and Conquer methods. The Lanczos method transforms the eigenproblem of a symmetric matrix into an eigenproblem of a tridiagonal matrix which is easier to be solved. The Divide and Conquer method provides the solution for the eigenproblem of a large tridiagonal matrix by decomposing it in a set of smaller subproblems. The method has been implemented for a distributed memory multiprocessor system with the PVM parallel interface. A Cray T3E system with up to 32 nodes has been used to evaluate the performance of our parallel implementation. Due to the super-lineal speed-up values obtained for all the studied matrices, a detailed analysis of the experimental results is carried out. It will be shown that the management of the memory hierarchy plays an important role in the performance of the parallel implementation.