Root-finding: from Newton to Halley and beyond

Paper: arXiv 2312.12305

Abstract

We give a new improvement over Newton’s method for root-finding, when the function in question is doubly differentiable. It generally exhibits faster and more reliable convergence. It can be also be thought of as a correction to Halley’s method, as this can exhibit undesirable behaviour.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a novel mathematical improvement to root-finding algorithms with theoretical derivations from Lie group theory, but provides only minimal illustrative examples without any systematic backtesting, statistical metrics, or implementation-heavy validation.

Research Flowchart

  flowchart TD
  A[""Research Goal:
Improve upon Newton's Method for root-finding of
doubly differentiable functions""] --> B[""Data/Inputs:
Test Functions & Initial Guesses""]
  B --> C[""Key Methodology:
Develop 'Corrected' Halley-like Step""]
  C --> D[""Computational Process:
Iterative Calculation & Comparison""]
  D --> E[""Key Outcomes:
1. Faster/More reliable convergence than Newton
2. Stabilizes undesirable Halley behavior
3. New valid improvement established""]