Halley's Method for Rectangular Matrix Variables and the Matrix Schwarzian Derivative

Abstract

Alefeld (1981) recast Halley's cubic convergence as Newton's method on $g=f/\sqrt{f^{\prime }}$, linked to the Schwarzian derivative. We generalize this to matrix gradient fields $F=\nabla \varphi$ ($X\in {\mathbb{R}}^{m\times n}$) via a matrix Schwarzian derivative interpreted through information geometry ($\alpha$-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 $\dot{X}=AXB-XB{X}^{T}AX$ 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.

30 pages, 9 figures