In this paper, we propose the global quaternion full orthogonalization (Gl-QFOM) and global quaternion generalized minimum residual (Gl-QGMRES) methods, which are built upon global orthogonal and oblique projections onto a quaternion matrix Krylov subspace, for solving quaternion linear systems with multiple right-hand sides. We first develop the global quaternion Arnoldi procedure to preserve the quaternion Hessenberg form during the iterations. We then establish the convergence analysis of the proposed methods, and show how to apply them to solve the Sylvester quaternion matrix equation. Numerical examples are provided to illustrate the effectiveness of our methods compared with the traditional Gl-FOM and Gl-GMRES iterations for the real representations of the original linear systems.

翻译：摘要：本文提出了全局四元数完全正交化（Gl-QFOM）与全局四元数广义最小残差（Gl-QGMRES）方法，该方法基于四元数矩阵Krylov子空间上的全局正交投影和斜投影，用于求解多右端四元数线性系统。我们首先构建全局四元数Arnoldi过程，以在迭代过程中保持四元数Hessenberg形式。随后，建立了所提方法的收敛性分析，并展示了如何将其应用于Sylvester四元数矩阵方程的求解。数值算例表明，与原始线性系统实数表示的传统Gl-FOM和Gl-GMRES迭代方法相比，本文方法具有显著有效性。