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-Hermi tian. We describe single-seed and block-seed projection approaches to these multiple right-hand side problems that are based on the QMR and block QMR algorithms, respectively. We use (block) QMR to solve the (block) seed syst em and generate the relevant biorthogonal subspaces. Approximate solutions to the nonseed systems are simultaneously generated by minimizing their app ropriately projected (block) residuals. After the initial (block) seed has converged, the process is repeated by choosing a new (block) seed from amon g the remaining nonconverged systems and using the previously generated app roximate solutions as initial guesses for the new seed and nonseed systems. We give theory for the single-seed case that helps explain the convergence behavior under certain conditions. Implementation details for both the sin gle-seed and block-seed algorithms are discussed and advantages of the bloc k-seed algorithm in cache-based serial and parallel environments are noted. The computational savings of our methods over using QMR to solve each syst em individually are illustrated in two examples.