GLpoint

GLpoint – Endpoint Grünwald-Letnikov Fractional Derivative

Overview

GLpoint efficiently computes the Grünwald-Letnikov fractional derivative at the endpoint of a domain, using a fast single-pass recurrence. This method is especially useful when you only need the fractional derivative at a single point (typically the rightmost point of the discretized interval).

This implementation leverages [Numba]’s just-in-time compilation to achieve near C-level performance, even for large arrays.

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Mathematical Background

The Grünwald-Letnikov (GL) derivative at the endpoint $x_N$ is given by:

$$ D^\alpha f(x_N) \approx h^{-\alpha} \sum_{j=0}^{N} (-1)^j \binom{\alpha}{j} f(x_{N-j}) $$

where:

• $h$ is the step size,
• $\binom{\alpha}{j}$ is the generalized binomial coefficient.

Instead of computing all coefficients and performing a vectorized dot product (which is inefficient for a single point), GLpoint uses a recurrence relation to generate coefficients on-the-fly, as in fast C++ implementations.

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Implementation Details

• Input Flexibility: Accepts a callable, NumPy array, or list of function values. If a function is given, it is evaluated at num_points equally spaced points from domain_start to domain_end.

• Optimized Kernel: The core computation is performed in a Numba JIT-compiled loop for maximum speed.

• Efficient for Single Points: This method is much more efficient than computing the full array if only the endpoint derivative is required.

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Function Signature

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

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Parameters

• alpha (float): Order of the fractional derivative.
• f_name (Callable, list, or np.ndarray): The function or sequence to differentiate.
• domain_start (float, optional): Start of the domain. Default is 0.0.
• domain_end (float, optional): End of the domain. Default is 1.0.
• num_points (int, optional): Number of discretization points. Default is 100.

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Returns

• result (float): The Grünwald-Letnikov fractional derivative at the endpoint.

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Example Usage

import numpy as np
from differintP import GLpoint

# Fractional derivative of sqrt(x) at x = 1
value = GLpoint(0.5, np.sqrt, 0., 1., 1000)
print("D^0.5 sqrt(x) at x=1:", value)

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Notes

• If you require the derivative at all points in the domain, use GL for maximum efficiency.
• The function is robust for both callable functions and precomputed arrays.

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See Also

• GL – Compute the Grünwald-Letnikov derivative at all points
• GLI – Improved accuracy with Lagrange interpolation

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