SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 616-634, January 2021.
The main aim of this paper is to develop the quaternion generalized minimal residual method (QGMRES) for solving quaternion linear systems. Quaternion linear systems arise from three-dimensional or color imaging filtering problems. The proposed quaternion Arnoldi procedure can preserve quaternion Hessenberg form during the iterations. The main advantage is that the storage of the proposed iterative method can be reduced by comparing with the Hessenberg form constructed by the classical GMRES iterations for the real representation of quaternion linear systems. The convergence of the proposed QGMRES is also established. Numerical examples are presented to demonstrate the effectiveness of the proposed QGMRES compared with the traditional GMRES in terms of storage and computing time.

中文翻译：

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结构保持四元数广义最小残差法

SIAM 矩阵分析与应用杂志，第 42 卷，第 2 期，第 616-634 页，2021 年 1 月。
本文的主要目的是开发用于求解四元数线性系统的四元数广义最小残差方法 (QGMRES)。四元数线性系统产生于三维或彩色成像过滤问题。建议的四元数 Arnoldi 过程可以在迭代期间保留四元数 Hessenberg 形式。主要优点是通过与经典的 GMRES 迭代构建的 Hessenberg 形式相比，可以减少所提出的迭代方法的存储量，用于四元数线性系统的实数表示。所提议的 QGMRES 的收敛也被建立。数值例子证明了所提出的 QGMRES 在存储和计算时间方面与传统 GMRES 相比的有效性。