Halley方法用于矩形矩阵变量与矩阵Schwarzian导数
Halley's Method for Rectangular Matrix Variables and the Matrix Schwarzian Derivative
- Nagoya Mathematical and Information Science Research（名古屋数学与信息科学研究）
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AI总结：
本文通过信息几何构造矩阵Schwarzian导数，将Halley方法推广至矩形矩阵变量，证明无自伴假设下的局部三次收敛，并用无矩阵算法和Oja型系统案例验证理论，揭示特征值间隙与病态性对收敛的影响。
AI中文摘要：
Alefeld（1981）将Halley三次收敛重新表述为在$g=f/\sqrt{f'}$上的牛顿法，并与Schwarzian导数相关联。我们通过信息几何（$\alpha$-连接）解释的矩阵Schwarzian导数，将此推广到矩阵梯度场$F=\nabla\phi$（$X\in\R^{m\times n}$）。简化Palmore（1994）的工作，我们从牛顿映射的第三次Fréchet导数构造此无平方根算子导数。我们的主定理证明了局部三次收敛，并给出显式误差常数，无需自伴性、交换性或梯度场假设。一种无矩阵算法（仅需Hessian-向量乘积）验证了该理论。我们将其与幂-牛顿族（其朴素矩阵扩展失败）和矩阵Laguerre族（需要算子平方根）进行对比。对Oja型系统$\dot X=AXB-XBX^TAX$的案例研究表明，收敛依赖于目标特征值间隙，三次增益随病态性增加而增长，70位数字测试确认了精确的理论阶数以及工作精度预算。最后，我们注意到耦合公式主要在顺序收缩不适用时表现出色。
英文摘要：
Alefeld (1981) recast Halley's cubic convergence as Newton's method on $g=f/\sqrt{f'}$, linked to the Schwarzian derivative. We generalize this to matrix gradient fields $F=\nablaϕ$ ($X\in\R^{m\times n}$) via a matrix Schwarzian derivative interpreted through information geometry ($α$-connections). Simplifying Palmore (1994), we construct this operator square-root-free derivative from the third Fréchet derivative of the Newton map. Our main theorem proves local cubic convergence with an explicit error constant, without self-adjointness, commutativity, or gradient-field assumptions. A matrix-free algorithm (Hessian-vector products only) validates the theory. We contrast this with a power-Newton family (whose naive matrix extension fails) and the matrix Laguerre family (which requires an operator square root). A case study on the Oja-type system $\dot X=AXB-XBX^TAX$ reveals that convergence depends on target eigenvalue gaps, cubic gains grow with ill-conditioning, and 70-digit tests confirm exact theoretical orders alongside a working-precision accuracy budget. Finally, we note coupled formulations excel primarily when sequential deflation is inapplicable.
