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).