Product Hybrid Block GMRES for Nonsymmetrical Linear Systems with Multiple Right-hand Sides

Abstract

Recently, the complementary behavior of restarted GMRES has been studied. We observed that successive \u00a0cycles \u00a0of \u00a0restarted \u00a0block \u00a0BGMRES \u00a0(BGMRES(m,s)) \u00a0can \u00a0also \u00a0complement \u00a0one \u00a0another harmoniously in reducing the iterative residual. In the present paper, this characterization of BGMRES(m,s) is exploited to form a hybrid block iterative scheme. In particular, a product hybrid block GMRES algorithm for \u00a0nonsymmetrical \u00a0systems \u00a0with \u00a0multiple \u00a0right-hand \u00a0sides \u00a0is \u00a0proposed. \u00a0The \u00a0new \u00a0algorithm \u00a0combines \u00a0the advantage \u00a0of \u00a0Simoncini\u2019s \u00a0Hybrid \u00a0Block \u00a0GMRES \u00a0and \u00a0Zhong\u2019s \u00a0Product \u00a0Hybrid \u00a0GMRES. \u00a0Numerical experiments are conducted to show that the new algorithm can offer significant improvement over the hybrid block GMRES.