pymodab 1.0.4

Fast and robust root-finding library using the Modified Anderson-Bjork method

A fast and robust root-finding library for Python, using the Modified Anderson-Bjork method (Ganchovski & Traykov, 2023), written in C. 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
• 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.

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
• modAB - Modified Anderson Bjork's method (Ganchovski & Traykov, 2023)

Function evaluations

Func	bisect	brentq	brenth	ridder	chandr	modAB
SUM	4411	2548	2512	3158	1873	1758
AVG	48	28	27	34	20	19
MEDIAN	49	12	12	16	12	12
MIN	3	4	4	4	3	3
MAX	53	102	102	202	58	56
FACTOR	2.509x	1.449x	1.429x	1.796x	1.065x	1.000x

Execution times (ms per problem, 100 iterations)

Func	bisect	brentq	brenth	ridder	chandr	modAB
SUM	1266.30	797.21	800.98	921.14	48639.60	145.55
AVG	13.7641	8.6654	8.7063	10.0124	528.6913	1.5820
MEDIAN	13.8022	4.5878	4.3595	5.6256	306.4098	1.2803
MIN	1.4287	1.7302	1.6219	1.3884	72.9172	0.6143
MAX	21.8886	36.7684	50.2981	54.4431	1542.8365	4.3991
FACTOR	8.700x	5.477x	5.503x	6.329x	334.189x	1.000x

Notes:

Last Run on: 02.04.2026
Intel(R) Core(TM) i7-1065G7 CPU @ 1.30GHz (1.50 GHz) with 16.0 GB RAM
Windows 11 Home
numpy Version: 2.4.4
scipy Version: 1.17.1
pymodab Version: 1.0.4