ModAB Root-Finding Algorithm

This repository provides efficient implementations of the Modified Anderson-Björck (ModAB) bracketing root-finding algorithm for solving nonlinear equations of the form f(x) = 0.

Available in C, C#, Python, Julia, Zig and C++ (ROOT.CERN framework).

Features

• Guaranteed convergence - Bracketing approach ensures the root is always found.
• Great performance - Fewer function evaluations than Brent, Ridders, and other popular methods. Being simple and lightweight, it has very little computational overhead per iteration.
• Worst-case optimality - Retains the worst-case optimality of Bisection for the hard cases.
• Cross-platform - Python package that works on Windows, Linux, and macOS.
• Multiple languages - Use in your preferred environment.
• Extensive testing and benchmarking - The algorithm is benchmark against a broad set of test functions of many different types.

Algorithm Overview

The ModAB algorithm combines the reliability of bisection with the speed of the secant method:

1. Bisection phase - Starts with bisection to ensure stability
2. Linearity detection - Monitors when the function behavior is close enough to linear
3. False-position acceleration - Switches to false-position method when conditions are favorable
4. Anderson-Björck correction - Applies A&B corrections to prevent stalling
5. Adaptive fallback - Returns to bisection if progress slows

This hybrid approach achieves superlinear convergence while maintaining the worst-case optimality of bisection.

License

MIT License - see the LICENSE file for details.

Implementations in other software/libraries

Calcpad - https://www.proektsoft.bg/bg/catalog/221.html - C#
Root-Fortran - https://github.com/jacobwilliams/roots-fortran - Fortran
ROOT.CERN - https://github.com/root-project/root - C++
SCiML/NonlinearSolve.jl - https://github.com/SciML/NonlinearSolve.jl - Julia
JuliaMath/Roots.jl - https://github.com/JuliaMath/Roots.jl - Julia
MultiFloats.jl - https://github.com/dzhang314/MultiFloats.jl - Julia

References

1. Ganchovski N. Structural Analysis by Functional Modelling in the Cloud. PhD Thesis 2025, UACEG, Sofia

2. Ganchovski, N.; Traykov, A. (2023). "Modified Anderson-Björck's method for solving non-linear equations in structural mechanics." IOP Conference Series: Materials Science and Engineering, 1276(1), 012010. DOI: 10.1088/1757-899X/1276/1/012010

3. Ganchovski, N.; Smith, O.; Rackauckas, C.; Tomov, L.; Traykov, A. (2026). "Improvements of the Modified Anderson-Björck (modAB) Root-Finding Algorithm." Preprints, 2026032190. DOI: 10.20944/preprints202603.2190.v1

Contributing

Contributions are welcome. You can open an issue or submit a pull request.