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<p>Coupled quaternion Sylvester matrix equations present considerable computational difficulties arising from their high dimensionality and the inherent noncommutativity of quaternion multiplication. To resolve this issue, a hierarchical identification principle-based quaternion gradient descent (HIP-QGD) algorithm is developed using generalized Hamilton–real (GHR) calculus. A primary advantage of HIP-QGD stems from its capability to preserve intrinsic quaternion structure and eliminate reliance on real-valued representations. Consequently,the HIP-QGD algorithm achieves greater directness and numerical efficiency compared to real-valued approaches when solving quaternion matrix equations. Convergence guarantees are established for both the standard implementation and its blockwise variant. Numerical experiments demonstrate stable convergence and accelerated rates for single and coupled quaternion Sylvester equations. In particular, the Block HIP-QGD is successfully applied to an image restoration task modeled by coupled quaternion matrix equations, achieving effective recovery of original images from degraded observations. The HIP-QGD framework therefore establishes a reliable and computationally efficient methodology for quaternion matrix computation.</p>

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Efficient iterative solvers for coupled quaternion Sylvester equations via hierarchical identification

  • 期刊论文

  • 发表日期：

    2026年5月2日

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  • Qiankun Diao,
  • Jinlan Liu,
  • Hongmei Shao,
  • Dongpo Xu

摘要

Coupled quaternion Sylvester matrix equations present considerable computational difficulties arising from their high dimensionality and the inherent noncommutativity of quaternion multiplication. To resolve this issue, a hierarchical identification principle-based quaternion gradient descent (HIP-QGD) algorithm is developed using generalized Hamilton–real (GHR) calculus. A primary advantage of HIP-QGD stems from its capability to preserve intrinsic quaternion structure and eliminate reliance on real-valued representations. Consequently,the HIP-QGD algorithm achieves greater directness and numerical efficiency compared to real-valued approaches when solving quaternion matrix equations. Convergence guarantees are established for both the standard implementation and its blockwise variant. Numerical experiments demonstrate stable convergence and accelerated rates for single and coupled quaternion Sylvester equations. In particular, the Block HIP-QGD is successfully applied to an image restoration task modeled by coupled quaternion matrix equations, achieving effective recovery of original images from degraded observations. The HIP-QGD framework therefore establishes a reliable and computationally efficient methodology for quaternion matrix computation.