Gl-QFOM and Gl-QGMRES: two efficient algorithms for quaternion linear systems with multiple right-hand sides

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.