计算分裂四元数矩阵Moore–Penrose逆的迭代方法及应用
Iterative Methods for Computing the Moore--Penrose Inverse of Split-Quaternion Matrices with Applications
AI总结：
该研究基于2×2实表示框架分析Newton–Schulz迭代，提出低次多项式初始化，将方法应用于分裂四元数矩阵的交叉与CUR近似，验证了方法的精度与初始化的优势。
AI中文摘要：
我们研究计算分裂四元数矩阵Moore–Penrose逆的迭代方法。首先，基于2×2实表示及相关的i共轭转置建立一致框架，该表示明确了Moore–Penrose逆的定义，澄清了非零零因子的处理方式。接着分析矩形与秩亏矩阵的Newton–Schulz迭代，利用实表示的薄奇异值分解推导收敛条件、投影残差的演化规律，以及残差与误差间的精确关系。所得分析既适用于嵌入实实现，也适用于原生分裂四元数迭代，二者在该表示下等价。还提出受Souriau逆递推启发的低次多项式初始化，该多项式通过逆Gram因子的最小二乘近似得到，结合谱接受测试与安全回退初始化。最后将所提方法应用于分裂四元数矩阵的交叉近似与CUR近似，刻画固定采样行和列时的最优中间因子，给出更经济的交叉因子实现精确重构的条件。数值实验验证了方法的精度及多项式初始化的实际优势。
英文摘要：
We study iterative methods for computing the Moore--Penrose inverse of split-quaternion matrices. We first establish a consistent framework based on a \(2\times2\) real representation and the associated \(i\)-conjugate transpose. This representation gives a direct definition of the Moore--Penrose inverse and clarifies the treatment of nonzero zero divisors. We then analyze Newton--Schulz iterations for rectangular and rank-deficient matrices. Using a thin singular value decomposition of the real representative, we derive the convergence conditions, the evolution of the projector residuals, and an exact relation between the residual and the error. The resulting analysis applies both to an embedded real implementation and to a native split-quaternion iteration, which are shown to be equivalent under the representation. We also propose a low-degree polynomial initialization inspired by Souriau's inverse recursion. The polynomial is obtained by a least-squares approximation of the inverse Gram factor and is combined with a spectral acceptance test and a safe fallback initialization. Finally, we apply the proposed methods to cross and CUR approximations of split-quaternion matrices. We characterize the optimal middle factor for fixed sampled rows and columns and give conditions under which the cheaper cross factor gives an exact reconstruction. Numerical experiments illustrate the accuracy of the methods and the practical benefit of the polynomial initialization.
