The quaternion biconjugate gradient (QBiCG) method, as a novel variant of quaternion Lanczos-type methods for solving the non-Hermitian quaternion linear systems, does not yield a minimization property. This means that the method possesses a rather irregular convergence behavior, which leads to numerical instability. In this paper, we propose a new structure-preserving quaternion quasi-minimal residual method, based on the quaternion biconjugate orthonormalization procedure with coupled two-term recurrences, which overcomes the drawback of QBiCG. The computational cost and storage required by the proposed method are much less than the traditional QMR iterations for the real representation of quaternion linear systems. Some convergence properties of which are also established. Finally, we report the numerical results to show the robustness and effectiveness of the proposed method compared with QBiCG.

翻译：四元数双共轭梯度（QBiCG）方法作为求解非厄米四元数线性系统的四元数Lanczos型方法的一种新变体，不具备最小化性质。这意味着该方法具有相当不规则的收敛行为，从而导致了数值不稳定性。本文基于耦合双项递推的四元数双共轭正交化过程，提出了一种新的保持结构的四元数拟最小残量方法，克服了QBiCG的缺点。所提方法所需的计算量和存储量远小于传统QMR迭代方法对四元数线性系统实表示的处理。此外，本文还建立了该方法的一些收敛性质。最后，我们通过数值结果展示了所提方法相较于QBiCG的鲁棒性和有效性。