A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sides

Abstract

Many applications require the solution of multiple linear systems that have the same coefficient matrix, but differ in their right-hand sides. Instead of applying an iterative method to each of these systems individually, it is often more efficient to employ a block version of the method that generates iterates for all the systems simultaneously. In this paper, we propose a block version of Freund and Nachtigal's quasi-minimal residual (QMR) method for the iterative solution of non-Hermitian linear systems. The block QMR method uses a novel Lanczos-type process for multiple starting vectors, which was recently developed by Aliaga, Boley, Freund, and Hernández, to compute suitable basis vectors for the underlying block Krylov subspaces. We describe the basic block QMR method, and also give important implementation details. In particular, we show how to incorporate deflation to drop converged linear systems, and to delete linearly and almost linearly dependent vectors in the underlying block Krylov sequences. Numerical results are reported that illustrate typical features of the block QMR method.