QMR-Based Projection Techniques for the Solution of Non-Hermitian Systems with Multiple Right-Hand Sides

In this work we consider the simultaneous solution of large linear systems of the form Ax(j) = b(j); j = 1; : : : ; K where A is sparse and non-Hermitian. Our single-seed approach uses QMR to solve the seed system j and generate biorthogonal Krylov subspaces. Approximate solutions to the non-seed systems are simulanteously generated by minimizing their appropriately projected residuals. After the initial seed system has converged, the process is repeated by choosing a new seed from among the remaining non-converged systems and using the previously generated approximate solutions as initial guesses for the new seed and non-seed systems. We give theory supporting our observation in practice of super-convergence of (non-initial) seed systems as compared to the usual QMR process. The computational advantage of our method over using QMR to solve each system individually is illustrated on two examples. Finally, we propose a block QMR variant which combines the advantages of this approach and those of the block QMR with de ation scheme of Freund and Malhotra. The computational savings of our block method are shown in examples.