pymodab 1.0.8

A fast and robust root-finding library written in C for Python, using the Modified Anderson-Bjork method: Ganchovski, N.; Smith, O.; Rackauckas, C.; Tomov, L.; Traykov, A. Improvements to the Modified Anderson–Björck(modAB) Root-Finding Algorithm. Algorithms 2026, 19, 332. https://doi.org/10.3390/a19050332 It finds the root of a single nonlinear equation f(x) = 0 within the specified interval [x1, x2].

Installation

pip install pymodab

Usage

import math
from pymodab import find_root, get_evaluation_count

# Find the root of cos(x) - x = 0 in [0, 1]
root = find_root(lambda x: math.cos(x) - x, 0, 1, 1e-3, 1e-3, 10)
print(f"Root: {root}")  # 0.7390851332086904

# Get the number of function evaluations
print(f"Evaluations: {get_evaluation_count()}")
print(f"Error:       {math.cos(root) - root}")

# Using default tolerances
root = find_root(lambda x: x**2 - 2, 1, 2)
print(f"sqrt(2) = {root}")  # 1.414213562373095

# Get the number of function evaluations
print(f"Evaluations: {get_evaluation_count()}")
print(f"Error:       {root**2 - 2}")

API

find_root(f, x1, x2, atol=1e-14, rtol=1e-14, max_iter=200)

Find the root of f(x) = 0 within the interval [x1, x2].

Parameters:

• f: A continuous function of one variable, or the address (int) of a compiled C function double f(double)
• x1, x2: Bracket interval endpoints (must satisfy f(x1) * f(x2) < 0)
• atol: Absolute tolerance (default: 1e-14)
• rtol: Relative tolerance (default: 1e-14)
• max_iter: Maximum iterations (default: 200)

Returns: The root, or NaN if not found. Exceptions raised by f are propagated to the caller.

Since version 1.0.6, find_root is a native CPython extension that calls f directly, which is about 2x faster than the previous ctypes wrapper. Version 1.0.7 ships it as a wheel for Windows, Linux (x86_64 and aarch64, glibc and musl) and macOS (Intel and Apple Silicon); one wheel serves every CPython from 3.8 up. On any other platform pymodab falls back to ctypes automatically; pymodab.NATIVE tells which one is used.

Version 1.0.8 improves the fallback from secant to bisection steps: a secant step that does not shrink the bracket fast enough is still kept if it at least halves the best residual, up to three times in a row. This avoids needless bisection steps when the secant converges from one side, and falls back sooner when it stalls. get_evaluation_count() now returns the exact number of calls to f.

For maximum speed, pass a compiled function, e.g. from numba. It is then called from C with no Python overhead:

import math
from numba import cfunc
from pymodab import find_root

@cfunc("float64(float64)")
def f(x):
    return math.cos(x) - x

root = find_root(f.address, 0, 1)

get_evaluation_count()

Returns the number of function evaluations from the last root-finding call.

Algorithm

Modified Anderson-Björck's method is a new robust and efficient bracketing root-finding algorithm. It combines bisection with Anderson-Björk's method to achieve both fast performance and worst-case optimality.

References:

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

Ganchovski, N.; Smith, O.; Rackauckas, C.; Tomov, L.; Traykov, A. Improvements to the Modified Anderson–Björck (modAB) Root-Finding Algorithm. Algorithms 2026, 19, 332. https://doi.org/10.3390/a19050332

License

MIT License

Benchmark results

The modAB algorithm is benchmarked against the available algorithms in Python/SciPy in respect to number of evaluations and execution times:

• bisect- Bisection method
• brentq - Brent’s method (van Wijngaarden–Dekker–Brent, 1973)
• brenth - Brent–Dekker variant (hyperbolic extrapolation variant, 1975)
• ridder - Ridder’s method (1979)
• toms748 - Alefeld–Potra–Shi method (1995 - TOMS Algorithm 748)
• chandr - Chandrupatla's method (1997) - scipy.optimize.elementwise.find_root
• cybrentq - Cython implementation of brentq by Gledis Caushaj
• modAB - Modified Anderson Bjork's method (Ganchovski & Traykov, 2023; improved 2026)
• modAB_ct - the old version of pymodab 1.0.5 implemented with ctypes

Function evaluations

Execution times (ms per problem, 200 iterations)

Notes:

Last Run on: 20.09.2026
Intel(R) Core(TM) i7-1065G7 CPU @ 1.30GHz (1.50 GHz) with 16.0 GB RAM
Windows 11 Home
Python Version: 3.14.7
numpy Version: 2.4.6
scipy Version: 1.18.0
cybrentq Version: 0.1.5 - by Gledis Caushaj (https://github.com/gledi-ai/cybrentq)
modAB_ct: pymodab version 1.0.5 from PyPI - the previous implementation with ctypes
modAB: pymodab version 1.0.8 - the latest implementation as native C extension