A parallel implementation of a deflation algorithm for systems of linear equations

Abstract

A general deflation technique for solving arbitrary systems of linear equations was described in Burrage et al. (1994a). This technique can be used with any iterative scheme. As the iterations proceed information is obtained about the eigenvalues of the iteration matrix which either cause slow convergence or divergence. These eigenvalues (and associated eigenvectors) are then deflated into a stiff subspace. This then leads to a coupled iteration process between the underlying iteration on the nonstiff space and a Newton iteration on the stiff system. This process can rapidly accelerate the convergence of even very ill-conditioned systems. In this paper a parallel implementation of the algorithm is presented for a distributed memory MIMD environment. A number of numerical results are given showing the efficacy of this approach.