GL – Grünwald-Letnikov Fractional Derivative

Overview

GL computes the Grünwald-Letnikov (GL) fractional derivative of a function over a specified domain, efficiently evaluating the derivative at each point using fast vectorized techniques. This method implements the classical Grünwald-Letnikov definition, which is a direct generalization of the finite-difference approximation to non-integer orders.

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Mathematical Background

The Grünwald-Letnikov derivative of order $\alpha$ for a function $f(x)$ is defined as:

$$ D^\alpha f(x) \approx h^{-\alpha} \sum_{k=0}^n (-1)^k \binom{\alpha}{k} f(x - k h) $$

where:

• $h$ is the step size between points in the domain,
• $\binom{\alpha}{k}$ is the generalized (fractional) binomial coefficient,
• the sum is taken over previous grid points.

In practice, this computation for all points in a discretized domain can be viewed as a convolution of the function values with the sequence of binomial coefficients.

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Implementation Details

• Input Flexibility: Accepts a callable function, NumPy array, or Python list. If a function is provided, it is evaluated at num_points equally spaced points between domain_start and domain_end.

• Optimized with FFT: The computation leverages the Fast Fourier Transform (FFT) to efficiently perform the convolution, making it extremely fast even for large domains. This avoids the $O(n^2)$ complexity of a naïve implementation and reduces it to $O(n \log n)$.

• Vectorized Coefficient Calculation: The binomial coefficients for the fractional derivative are precomputed using a vectorized approach for maximum efficiency.

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Function Signature

GL(
    alpha: float,
    f_name: Callable | np.ndarray | list,
    domain_start: float = 0.0,
    domain_end: float = 1.0,
    num_points: int = 100,
) -> np.ndarray

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Parameters

• alpha (float): The order of the fractional derivative.
• f_name (Callable, list, or np.ndarray): Function, lambda, or sequence of function values to be differintegrated.
• domain_start (float, optional): Left endpoint of the domain. Default is 0.0.
• domain_end (float, optional): Right endpoint of the domain. Default is 1.0.
• num_points (int, optional): Number of points in the domain. Default is 100.

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Returns

• result (np.ndarray): Array containing the GL fractional derivative at each grid point.

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Example Usage

import numpy as np
from differintP import GL

# Example 1: Fractional derivative of a quadratic polynomial
DF_poly = GL(-0.5, lambda x: x**2 - 1)

# Example 2: Fractional derivative of sqrt(x) on [0, 1]
DF_sqrt = GL(0.5, np.sqrt, 0., 1., 100)

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Notes

• The GL method is the standard reference for fractional finite-difference approximations.
• All computation is fully vectorized and highly efficient.
• For even higher-order accuracy, See the GLI page for the improved Grünwald-Letnikov method.

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