QQMR: A Structure-Preserving Quaternion Quasi-Minimal Residual Method for Non-Hermitian Quaternion Linear Systems

TL;DR

This paper introduces QQMR, a new quaternion quasi-minimal residual method that preserves structure, improves stability, and reduces computational cost for solving non-Hermitian quaternion linear systems.

Contribution

The paper proposes a structure-preserving quaternion QMR method based on biconjugate orthonormalization, addressing instability issues of previous methods.

Findings

QQMR demonstrates improved robustness over QBiCG.

The method reduces computational cost and storage.

Numerical results confirm effectiveness and stability.

Abstract

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.