hybridroots 1.0.1

Four novel multi-phase hybrid bracketing algorithms for numerical root finding: Opt.BF (mpbf), Opt.BFMS (mpbfms), Opt.TF (mptf), Opt.TFMS (mptfms). Combines classical methods with modified secant for superior convergence (e.g., mptfms averages 2.3 iterations). Reference DOI: 10.21608/joems.2026.440115.1078

HybridRoots

Multi-Phase Hybrid Bracketing Algorithms for Numerical Root Finding

A Python package implementing four novel root-finding algorithms that combine bisection/trisection, false position, and modified secant methods for efficient, reliable nonlinear equation solving.

Features

• Four Algorithms: Opt.BF, Opt.BFMS, Opt.TF, Opt.TFMS
• Pure Python: No external dependencies required
• SciPy Compatible: Same interface as scipy.optimize.brentq
• Deterministic: Guaranteed convergence for bracketed roots

Algorithm Summary

Algorithm	Method	Avg Iterations	NFE/iter
mpbfms	Bisection + False Position + Modified Secant	~2.8	3-4
mptfms	Trisection + False Position + Modified Secant	~2.4	4-5
mpbf	Bisection + False Position	~6.7	2
mptf	Trisection + False Position	~5.3	3

Installation

pip install hybridroots

For development with testing:

pip install hybridroots[test]

🔧 Usage

from hybridroots import mpbf, mpbfms, mptf, mptfms

# Define your function
def f(x):
    return x**3 - x - 2

# Find root in interval [1, 2]
root, info = mpbfms(f, 1, 2)
print(f"Root: {root}")  # Root: 1.5213797068045676
print(f"Iterations: {info['iterations']}")
print(f"Function calls: {info['function_calls']}")

# All algorithms have the same interface
root1, _ = mpbf(f, 1, 2)    # Opt.BF
root2, _ = mpbfms(f, 1, 2)  # Opt.BFMS
root3, _ = mptf(f, 1, 2)    # Opt.TF
root4, _ = mptfms(f, 1, 2)  # Opt.TFMS

Parameters

Parameter	Type	Default	Description
f	callable	required	Function to find root of
a	float	required	Left endpoint of bracket
b	float	required	Right endpoint of bracket
tol	float	1e-14	Absolute tolerance
max_iter	int	10000	Maximum iterations

Returns

Value	Type	Description
root	float	Approximate root
info	dict	{'iterations', 'function_calls', 'converged'}

📖 Algorithm Overview

Opt.BFMS (mpbfms)

Multi-phase Bisection-False Position-Modified Secant method:

1. Bisection step to reduce interval
2. False position step for refinement
3. Modified secant with adaptive δ

Opt.TFMS (mptfms)

Multi-phase Trisection-False Position-Modified Secant method:

1. Trisection for interval reduction
2. False position refinement
3. Modified secant with adaptive δ

Opt.BF (mpbf)

Multi-phase Bisection-False Position method. Each iteration:

1. Bisection step to reduce interval
2. False position step for refinement

Opt.TF (mptf)

Multi-phase Trisection-False Position method:

1. Trisection (divides interval into thirds)
2. False position refinement

📚 Citation

If you use this package in your research, please cite:

@article{ellithy2026hybrid,
  title={Four New Multi-Phase Hybrid Bracketing Algorithms for Numerical Root Finding},
  author={Ellithy, Abdelrahman},
  journal={Journal of the Egyptian Mathematical Society},
  volume={34},
  year={2026},
  publisher={National Information and Documentation Centre (NIDOC), Academy of Scientific Research and Technology, ASRT}
}

Running Benchmarks

pip install hybridroots[benchmark]
python -m hybridroots.benchmarks

📄 License

MIT License - see LICENSE for details.

Links

• Paper (DOI)
• Issue Tracker