RLpoint

RLpoint – Riemann-Liouville Fractional Integral/Derivative at a Point

Overview

RLpoint efficiently computes the Riemann-Liouville (RL) fractional integral or derivative at the endpoint of a domain, using an optimized vectorized approach with the trapezoidal rule. This is ideal when you only require the RL operator at a single point (typically the rightmost grid point).

This method provides an 8x–60x speedup compared to naïve implementations, thanks to its fully vectorized coefficient calculations and use of NumPy’s optimized routines.

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Mathematical Background

The Riemann-Liouville (RL) fractional differintegral of order $\alpha$ for a function $f(x)$ is classically defined as:

$$ I^\alpha f(x) = \frac{1}{\Gamma(\alpha)} \int_{a}^{x} (x-t)^{\alpha-1} f(t) dt $$

Discretizing this with the trapezoidal rule leads to a sum of the form:

$$ RL^\alpha f(x_k) \approx h^{-\alpha} \sum_{j=0}^{k} c_{j,k} f(x_j) $$

where $c_{j,k}$ are precomputed weights that depend on $\alpha$, $k$, and $j$.

This implementation calculates these weights efficiently for the endpoint and applies them in a single dot product.

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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 between domain_start and domain_end.
• Fully Vectorized: All coefficient and sum calculations are vectorized for maximum efficiency—no Python loops are used in the core computation.
• Trapezoidal Rule: Uses the trapezoidal rule to improve the accuracy of the RL integral approximation.

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Function Signature

RLpoint(
    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 differintegral to compute.
• f_name (Callable, list, or np.ndarray): The function or sequence to be differintegrated.
• domain_start (float, optional): Start of the domain. Default is 0.0.
• domain_end (float, optional): End of the domain (evaluation point). Default is 1.0.
• num_points (int, optional): Number of discretization points. Default is 100.

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Returns

• result (float): The RL fractional integral/derivative at the endpoint.

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Example Usage

import numpy as np
from differintP import RLpoint

# Fractional integral/derivative of sqrt(x) at x = 1
rl_value = RLpoint(0.5, np.sqrt, 0., 1., 100)

# Fractional integral/derivative of a polynomial
rl_poly = RLpoint(0.5, lambda x: x**2 - 4*x - 1, 0., 1., 100)

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Notes

• For the RL operator at all points, see the RL function.
• This function is especially efficient for large num_points and high-precision needs.
• Coefficient calculation is based on the discretized RL formula using the trapezoidal rule.

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See Also

• RL: Compute the RL operator at all points.
• GLpoint: Endpoint Grünwald-Letnikov derivative.
• GL: Grünwald-Letnikov derivative (all points).

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