bezier_shuttle

Reproducibility code for "A Quadratic Bézier Shuttle for Brent’s Root-Finder" (Applied Mathematics Letters 2025)

Quick start

git clone https://github.com/oswcad/bezier_shuttle.git
cd bezier_shuttle
pip install -r requirements.txt
python table1.py

Bézier Shuttle for Brent's Root Finder

Reproducibility code for the paper:

A Quadratic Bézier Shuttle for Brent’s Root-Finder
Oswaldo Cadenas
Applied Mathematics Letters, 2025 (submitted/accepted)

This repository reproduces Table 1 of the paper, comparing the original 1973 Brent method with the proposed two-step Bézier Shuttle + Brent hybrid.
The experiments confirm that the Bézier Shuttle safely reduces the iteration count for curved and oscillatory functions while preserving bracketing stability and final residual accuracy.

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📦 Requirements

• Python ≥ 3.9
• NumPy ≥ 1.23

You can install the dependencies with:

pip install -r requirements.txt

Usage

To reproduce Table 1:

python table1.py

Expected output

Function                             α      Brent    Hybrid   Gain        Root
--------------------------------------------------------------------------------
e^(-x+11)-2                          0.5       40       30     10    10.306853
0.1(x-2)(x-5)(x-8)                   0.5        1        1      0     5.000000
sin(x)+0.5x-2                        0.5       27       27      0     5.462807
(x-3.5)^(-1)+2                       0.5       27       21      6     3.000000
ln(x+1)-2                            0.5       39       28     11     6.389056
x^3-5x+1                        adaptive       25       13     12     2.128419
cos(x)-x                             0.5       11       13     -2     0.739085
e^(-x)-0.1                           0.5       37       35      2     2.302585

Background

•	Brent (1973): Algorithms for Minimization Without Derivatives — the classical hybrid bisection/secant/IQI method.
•	Bézier Shuttle (Cadenas 2025): a two-step quadratic Bézier contraction that precedes Brent’s landing phase. Each shuttle uses the midpoint ( x_c ) to curve the bracket toward the function’s shape before invoking Brent’s interpolation.

Citation

@article{cadenas2026bezier, title={A Bézier Shuttle for Accelerating Brent's Root-Finder}, author={Cadenas, Oswaldo}, journal={Applied Numerical Mathematics}, year={2025} }

License

MIT License © 2025 Oswaldo Cadenas

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🧰 2. requirements.txt

numpy>=1.23