HybridRoots (Rust Port)

Four Multi-Phase Hybrid Bracketing Algorithms for Numerical Root Finding

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

Why HybridRoots is Powerful

Classical root-finding algorithms face a tradeoff: methods like Bisection are 100% reliable but slow (averaging >40 iterations), while methods like Secant or Newton-Raphson are fast but can fail to converge or shoot out of bounds.

HybridRoots solves this by introducing multi-phase bracketing:

1. Guaranteed Convergence: It maintains a strict bracket [a, b] where f(a) * f(b) < 0 at all times.
2. Superior Speed: By combining Trisection, False Position, and an adaptive Modified Secant step, it drastically reduces the search space. The mptfms algorithm converges in an average of just 2.33 iterations across 48 complex benchmark functions.

The Algorithms

• Opt.BFMS (mpbfms): Bisection + False Position + Modified Secant (Avg 2.69 iterations)
• Opt.TFMS (mptfms): Trisection + False Position + Modified Secant (Avg 2.33 iterations)
• Opt.BF (mpbf): Bisection + False Position (Avg 6.58 iterations)
• Opt.TF (mptf): Trisection + False Position (Avg 5.19 iterations)

Installation

Add this to your Cargo.toml:

[dependencies]
hybridroots = "1.0.0"

Usage

use hybridroots::mptfms;

fn main() {
    // Find root of x^3 - x - 2 in interval [1.0, 2.0] using Opt.TFMS
    let f = |x: f64| x.powi(3) - x - 2.0;
    
    // mptfms(function, lower_bound, upper_bound, tolerance, max_iterations)
    let result = mptfms(&f, 1.0, 2.0, 1e-14, 10000);
    
    println!("Root: {}", result.root);           
    println!("Iterations: {}", result.iterations);
    println!("Converged: {}", result.converged);  
}

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}
}

• Paper DOI: 10.21608/joems.2026.440115.1078