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/√(f'), linked to the Schwarzian derivative. We generalize this to matrix gradient fields F=∇ϕ (X∈Rm× 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 Ẋ=AXB−XBXTAX 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.
Related
- The Schwarzian derivative in several complex variables univalence and geometry
- Third-order Halley-type iterative method with positive and bounded correction function
- Sharp Bounds for Schippers–Schwarzian Derivatives via the Schur-Parameter Method
- Addendum to: Mahler's method in several variables and finite automata
- Stein's functional method for weakly dependent random variables
BibTeX
@misc{yoshizawa2026halleys,
title = {{Halley's Method for Rectangular Matrix Variables and the Matrix Schwarzian Derivative}},
author = {Shintaro Yoshizawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06592},
eprint = {2609.06592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06592},
}
RIS
TY - UNPB
TI - Halley's Method for Rectangular Matrix Variables and the Matrix Schwarzian Derivative
AU - Shintaro Yoshizawa
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06592
UR - https://arxiv.org/abs/2609.06592
AB - Alefeld (1981) recast Halley's cubic convergence as Newton's method on g=f/√(f'), linked to the Schwarzian derivative. We generalize this to matrix gradient fields F=∇ϕ (X∈R^m× 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 Ẋ=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.
ER -