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

TL;DR

This paper introduces two novel algorithms, Gl-QFOM and Gl-QGMRES, designed for efficiently solving quaternion linear systems with multiple right-hand sides, including convergence analysis and numerical validation.

Contribution

The paper develops the first global quaternion Krylov subspace methods with convergence analysis and applies them to quaternion matrix equations, improving computational efficiency.

Findings

Effective algorithms for quaternion systems with multiple RHS.

Numerical results show improved performance over traditional methods.

Convergence analysis supports the robustness of the proposed methods.

Abstract

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.