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

Tao Li, Qing-Wen Wang, Xin-Fang Zhang

TL;DR

This work addresses solving non-Hermitian quaternion linear systems with structure preservation and residual minimization. It introduces two quaternion biconjugate orthonormalization (QBIO) schemes, including coupled two-term recurrences, to form biorthonormal quaternion Krylov bases and reduce storage costs. Building on QBIO, the Quaternion QMR (QQMR) methods provide quasi-minimal residual minimization via quaternion QR and generalized quaternion Givens rotations, with preconditioning to enhance spectral properties. Numerical experiments demonstrate that QQMR variants achieve smoother convergence and competitive or superior performance compared with QBiCG, highlighting their practical impact for quaternion-valued problems in signal processing and color-image restoration.

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.

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

TL;DR

This work addresses solving non-Hermitian quaternion linear systems with structure preservation and residual minimization. It introduces two quaternion biconjugate orthonormalization (QBIO) schemes, including coupled two-term recurrences, to form biorthonormal quaternion Krylov bases and reduce storage costs. Building on QBIO, the Quaternion QMR (QQMR) methods provide quasi-minimal residual minimization via quaternion QR and generalized quaternion Givens rotations, with preconditioning to enhance spectral properties. Numerical experiments demonstrate that QQMR variants achieve smoother convergence and competitive or superior performance compared with QBiCG, highlighting their practical impact for quaternion-valued problems in signal processing and color-image restoration.

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.
Paper Structure (6 sections, 5 theorems, 78 equations, 4 figures, 4 tables, 8 algorithms)

This paper contains 6 sections, 5 theorems, 78 equations, 4 figures, 4 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quaternion biconjugate orthonormalization method
  4. Quaternion quasi-minimal residual method
  5. Numerical Experiments
  6. Conclusions

Key Result

Theorem 3.2

\newlabeltheorem-20 Li1 Let $M$ be an $4n\times 4n$ JRS-symmetric matrix having the form of 2-1. Then there exist orthogonally JRS-symplectic matrices $V, W\in\mathbb{R}^{4n\times 4n}$, such that is a JRS-tridiagonal matrix.

Figures (4)

  • Figure 1: Convergence histories for Example \ref{['ex-1']} with $\bf{mcca}$ (left) and $\bf{pde2961}$ (right).
  • Figure 2: Convergence histories of Example \ref{['ex-2']} with $n=101$.
  • Figure 3: Left to Right: the original $100\times 100$ images, the blurred images with single channel, the deblurred images were restored by QBiCG, Algorithms \ref{['algorithm-5']} and \ref{['algorithm-6']}, respectively. \newlabelfig:30
  • Figure 4: Left to Right: the original $100\times 100$ images, the blurred images with multichannel, the deblurred images were restored by QBiCG, Algorithms \ref{['algorithm-5']} and \ref{['algorithm-6']}, respectively. \newlabelfig:40

Theorems & Definitions (13)

  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Proof 1
  • Remark 3.5
  • Proposition 4.1
  • Proof 2
  • Theorem 4.2
  • Proof 3
  • ...and 3 more