pybelikov

A Python module for the high-precision computation of Fully Normalized Associated Legendre Functions (fnALFs) using the Belikov recurrence algorithm.

Why pybelikov?

Standard recursive algorithms for Legendre functions often suffer from numerical instability (underflow/overflow) at high degrees ($n > 2000$) and can be computationally slow.

pybelikov implements the stable recurrence methods originally derived by Belikov, using the formulation and evaluation presented by Lei & Li (2016). This implementation is optimized for modern Python environments and offers two execution modes:

1. JIT Mode (Default): Uses numba to compile the recursion into machine code.
   • ~38x faster than standard iterative loop implementations.
   • ~25% faster than optimized vectorized NumPy code.
2. Vectorized Mode: A pure NumPy implementation that utilizes array broadcasting.
   • ~30x faster than standard iterative loop implementations.

Benchmarks performed on N=2190 (EGM2008 standard).

Installation

pip install pybelikov

Usage

import numpy as np
from pybelikov import compute_fnALF

# Define parameters
degree = 2190
colat_rad = np.deg2rad(45.0)  # Colatitude in radians

# Compute using the fastest method (JIT)
P_nm = compute_fnALF(N=degree, theta=colat_rad, method='jit')

# Access a specific value P(n=2, m=2)
print(f"P_2,2 = {P_nm[2, 2]}")

References

The algorithm implemented in this package is based on the evaluation and formulation described in:

Lei, W., & Li, K. (2016). Evaluating Applicability of Four Recursive Algorithms for Computation of the Fully Normalized Associated Legendre Functions. Journal of Applied Geodesy, 10(4). https://doi.org/10.1515/jag-2016-0032

Citing pybelikov

If you use pybelikov in your research, please credit the software. A CITATION.cff file is included in this repository for compatibility with reference managers.

Software Citation:

Kellmayer, B. (2025). pybelikov: High-precision computation of fully normalized associated Legendre functions. https://github.com/bkellmayer6/pybelikov

Algorithm Citation:

Lei, W., & Li, K. (2016). Evaluating Applicability of Four Recursive Algorithms for Computation of the Fully Normalized Associated Legendre Functions. Journal of Applied Geodesy, 10(4).

Acknowledgements

The development of pybelikov originated as coursework for the Graduate Geodetic Science program at The Ohio State University. Special thanks to Prof. Jun-Yi Guo, Brayden Brinks, and Yushan Zhang for their help throughout the course and in the development of this code.