Author: Wayne Baker
Manuscript date: June 30, 2026
Revised Zenodo deposit: September 22, 2026
DOI: 10.5281/zenodo.22896047
Status: Published preprint / source and reproducibility archive
This repository contains the paper, LaTeX source, and numerical verification script for:
A Prefactor Coefficient Law for the Gauss-Legendre AGM Algorithm
- Zenodo: 10.5281/zenodo.22896047
paper/agm_prefactor_coefficient_law.pdfpaper/agm_prefactor_coefficient_law.tex
For the lower Gauss-Legendre approximants (\pi_n), define the exact fitted correction coefficient
-\frac{\pi_{n+1}-\pi}{\pi_{n+1}-\pi_n}. ]
Then
[ \frac{\alpha_{n+1}}{\alpha_n^2} \longrightarrow \frac12. ]
The proof uses Brent's theta-function expansion. Writing
[ \varepsilon_n=\pi_n-\pi=A_n e^{-\pi 2^{n+1}}, ]
one obtains
[ A_n\sim-2^{n+4}\pi^2, \qquad \frac{A_{n+1}}{A_n}\longrightarrow2. ]
The doubly exponential terms cancel exactly in the quotient that determines (\alpha_{n+1}/\alpha_n^2), leaving the prefactor limit (1/2).
The script
scripts/verify_agm_coefficient_law.py
reconstructs the Gauss-Legendre sequence at arbitrary precision and reproduces the two numerical tables in the paper. It also compares the fitted-coefficient quotient with the direct error quotient
[ \frac{\varepsilon_{n+2}\varepsilon_n^2}{\varepsilon_{n+1}^3}. ]
Run from the repository root:
python scripts/verify_agm_coefficient_law.py --precision 500 --outdir outputsThe theorem concerns a fitted dynamic error coefficient for the lower Gauss-Legendre approximants. It does not assert a fixed recurrence of the form (\varepsilon_{n+1}\sim C\varepsilon_n^2) with nonzero constant (C).
The companion statement about the numerator-surgery coefficients (\beta_n) is explicitly numerical; it is not promoted to a theorem in this paper.
The verification script supports the reported computations. The proof of the main coefficient law is analytic and rests on the cited theta-function asymptotic.
Gauss-Legendre algorithm · arithmetic-geometric mean · operator coefficients · sequence acceleration · Aitken (\Delta^2) · prefactor asymptotics · Baker N-series · numerical analysis
.
├── README.md
├── CITATION.cff
├── REPRODUCIBILITY.md
├── LICENSE_NOTICE.md
├── CHANGELOG.md
├── SHA256SUMS.txt
├── .gitignore
├── paper/
│ ├── agm_prefactor_coefficient_law.tex
│ ├── agm_prefactor_coefficient_law.pdf
│ └── README.md
├── scripts/
│ ├── verify_agm_coefficient_law.py
│ └── README.md
└── docs/
├── abstract.md
├── repository_description.md
└── github_upload_checklist.md
The revised preprint archived at Zenodo is licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). Other repository files are governed by LICENSE_NOTICE.md.
Copyright © 2026 C. Wayne Baker.