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‎CONTRIBUTING.md‎

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Thank you for taking the time to contribute!
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The following is a set of guidelines for contributing to *differint*. These are just guidelines, not rules, so use your best judgement and feel free to propose changes to this document in a pull request.
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The following is a set of guidelines for contributing to *differintP*. These are just guidelines, not rules, so use your best judgement and feel free to propose changes to this document in a pull request.
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## Getting Started
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*differint* requires Python 3+ and NumPy. If you are new to NumPy, please head over to [this](https://docs.scipy.org/doc/numpy-dev/user/quickstart.html) guide.
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*differint* requires Python 3.10+ and NumPy. If you are new to NumPy, please head over to [this](https://docs.scipy.org/doc/numpy-dev/user/quickstart.html) guide. The gpu accelarated functions use [Cupy](), you will need cupy, cuda enabled gpu and cuda toolkit, All tests are writen with pytest
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## Community
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* The whole *differint* documentation, including installation and testing, can be read [here](https://github.com/snimpids/differint).
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* The whole *differintP* documentation, including installation and testing, can be read [here](https://github.com/iparsw/differintP).
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* If you have any questions regarding *differint*, open an [issue](https://github.com/snimpids/differint/issues/new/).
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* If you have any questions regarding *differint*, open an [issue](https://github.com/iparsw/differintP/issues/new/).
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## Issue
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Ensure the bug was not already reported by searching on GitHub under [issues](https://github.com/snimpids/differint/issues). If you're unable to find an open issue addressing the bug, open a [new issue](https://github.com/snimpids/differint/issues/new).
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Ensure the bug was not already reported by searching on GitHub under [issues](https://github.com/iparsw/differintP/issues). If you're unable to find an open issue addressing the bug, open a [new issue](https://github.com/iparsw/differintP/issues/new).
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Please consider the following points while opening an issue.
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For example:
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* The operating system.
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## Pull Requests
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Pull Requests are always welcome.
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1. When you edit the code, please run `npm run test` to check the formatting of your code before you `git commit`.
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2. Ensure the PR description clearly describes the problem and solution. It should include:
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* The operating system on which you tested.
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* The relevant issue number, if applicable.
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Pull Requests are always welcome.

‎README.md‎

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- There is also a faster version (but more limited) implemented in c++ [diffeintC](https://github.com/iparsw/differintC)
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## differint
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This package is used for numerically calculating fractional derivatives and integrals (differintegrals). Options for varying definitions of the differintegral are available, including the Grunwald-Letnikov (GL), the 'improved' Grunwald-Letnikov (GLI), the Riemann-Liouville (RL), and the Caputo (L1, L2, and L2C). Through the API, you can compute differintegrals at a point or over an array of function values.
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# differintP
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See below for an example of how to use this package, or check out the [wiki](https://github.com/differint/differint/wiki) for references, signatures, and examples for each function.
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**differintP** is a modern, pure Python library for **fractional calculus**—that is, for numerical differentiation and integration of arbitrary (non-integer) order.
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It is a high-performance fork and major modernization of the original [differint](https://github.com/differint/differint) library, with many optimizations, expanded features, and a strong emphasis on speed, completeness, and code clarity.
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## Motivation
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There is little in the way of readily available, easy-to-use code for numerical fractional calculus. What is currently available are functions that are generally either smart parts of a much larger package, or only offer one numerical algorithm. The *differint* package offers a variety of algorithms for computing differintegrals and several auxiliary functions relating to generalized binomial coefficients.
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> **Highlights**
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> - Orders of differentiation and integration: any real (fractional) value
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> - Fast, vectorized, and JIT-accelerated core routines (NumPy + Numba)
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> - GPU support (optional, via CuPy)
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> - Array and pointwise (endpoint) fractional derivative functions
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> - Covers Grünwald-Letnikov, Riemann-Liouville, Caputo, Weyl, Riesz, CRONE, and more
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> - Accurate for both smooth and non-smooth functions
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> - Extensive tests and clear, modern codebase
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> - MIT License
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## Installation
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This project requires Python 3+ and NumPy to run.
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---
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Installation from the Python Packaging index (https://pypi.python.org/pypi) is simple using pip.
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## 🚀 Quick Install
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```python
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```bash
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pip install differintP
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```
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````
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* Optional: for GPU acceleration, also install [CuPy](https://docs.cupy.dev/en/stable/install.html):
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```bash
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pip install cupy-cuda12x # or cupy-cuda11x, depending on your CUDA
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```
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---
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## ✨ Features
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* **Fast Fractional Derivatives**: Grünwald-Letnikov (`GL`, `GLpoint`), Riemann-Liouville (`RL`, `RLpoint`), Caputo (`CaputoL1point`, etc.)
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* **Modern Implementations**: All methods rewritten for clarity, speed, and extensibility
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* **Advanced/Research Methods**:
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* **Weyl** (Fourier/spectral, periodic)
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* **Riesz** (symmetric, physical applications)
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* **CRONE** (for edge detection and signal processing)
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* **GPU-accelerated GL**: `GL_gpu` (if CuPy is installed)
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* **High test coverage**: Pytest, with analytical ground truths for common functions
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## Included Files
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Core File | Description
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--------- | -----------
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differint/differint.py | Contains algorithms for fractional differentiation and integration.
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tests/test.py | Testing suite containing all unit tests.
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Both of the above files have corresponding `__init__.py` files.
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Setup File | Description
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---------- | -----------
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.gitignore | List of files to ignore during `git` push/pull requests.
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CONTRIBUTING.md | Instructions for potential contributors to the *differint* project.
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LICENSE | MIT license agreement.
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MANIFEST.in | Selects the README file for uploading to PyPi.
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README.md | This README file.
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README.rst | This README file in ReStructuredText format.
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__init__.py | `__init__` file for overall package.
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changelog.txt | List of updates to package.
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setup.py | Script for downloading package from `pip`.
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## Example Usage
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Taking a fractional derivative is easy with the *differint* package. Let's take the 1/2 derivative of the square root function on the interval [0,1], using the Riemann-Liouville definition of the fractional derivative.
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---
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## 📚 Usage Example
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```python
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import numpy as np
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import differint.differint as df
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from differintP import GL, GLpoint, Weyl, Riesz
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def f(x):
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return x**0.5
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# Fractional derivative of order 0.5 of sqrt(x) on [0, 1]
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x = np.linspace(0, 1, 100)
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df_gl = GL(0.5, np.sqrt, 0, 1, 100) # Array version
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df_point = GLpoint(0.5, np.sqrt, 0, 1, 100) # Endpoint value
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DF = df.RL(0.5, f)
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print(DF)
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# Weyl and Riesz derivatives of sin(x) on [0, 2*pi]
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x2 = np.linspace(0, 2*np.pi, 256, endpoint=False)
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df_weyl = Weyl(0.5, np.sin, 0, 2*np.pi, 256)
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df_riesz = Riesz(0.5, np.sin, 0, 2*np.pi, 256)
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```
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You can also specify the endpoints of the domain and the number of points used as follows.
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See the [wiki](https://github.com/iparsw/differintP/wiki) for detailed documentation and advanced use.
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```python
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DF = df.RL(0.5, f, 0, 1, 128)
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```
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For a description of all functions, their signatures, and more usage examples, see the project's [wiki](https://github.com/differint/differint/wiki).
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## 📑 Documentation
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## Tests
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All tests can be run with nose from the command line. Setup will automatically install nose if it is not present on your machine.
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* **Function Reference**: See the [Wiki](https://github.com/iparsw/differintP/wiki) for math, options, and examples for all supported methods.
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* **Benchmarks**: [BENCHMARK.md](https://github.com/iparsw/differintC/blob/main/BENCHMARK.md) for speed/performance notes.
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* **Importing**: Import main algorithms directly, See the [Wiki](https://github.com/iparsw/differintP/wiki/Namespaces).
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```python
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python setup.py tests
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```
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Of course! Here’s a suggested section for accuracy tests, following your README’s style:
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Alternatively, you can run the test script directly.
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```python
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cd <file_path>/differint/tests/
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python test.py
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## ✅ Accuracy Tests
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To verify the correctness and numerical accuracy of all algorithms, see the interactive **accuracy test notebook**:
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[**accuracy\_test.ipynb**](https://github.com/iparsw/differintP/blob/master/accuracy_test.ipynb)
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This notebook compares numerical results to analytical ground truth for a variety of functions and methods, and serves as a practical demonstration and benchmark reference.
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---
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## 🧑‍💻 Development and Testing
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* **Python ≥ 3.10** required
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* All code is in pure Python; C++ not required
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* **All tests use [pytest](https://pytest.org/)** for fast, expressive, and modern testing
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To run the full test suite:
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```bash
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pytest tests/
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## API Reference
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In this section we cover the usage of the various functions within the *differint* package.
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Main Function | Usage
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[GLpoint](https://github.com/differint/differint/wiki/GLpoint) | Computes the GL differintegral at a point
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[GL](https://github.com/differint/differint/wiki/GL) | Computes the GL differintegral over an entire array of function values using the Fast Fourier Transform
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[GLI](https://github.com/differint/differint/wiki/GLI) | Computes the improved GL differintegral over an entire array of function values
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[CRONE](https://github.com/differint/differint/wiki/CRONE) | Calculates the GL derivative approximation using the CRONE operator.
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[RLpoint](https://github.com/differint/differint/wiki/RLpoint) | Computes the RL differintegral at a point
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[RL](https://github.com/differint/differint/wiki/RL) | Computes the RL differintegral over an entire array of function values using matrix methods
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[CaputoL1point](https://github.com/differint/differint/wiki/CaputoL1point) | Computes the Caputo differintegral at a point using the L1 algorithm
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[CaputoL2point](https://github.com/differint/differint/wiki/CaputoL2point) | Computes the Caputo differintegral at a point using the L2 algorithm
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[CaputoL2Cpoint](https://github.com/differint/differint/wiki/CaputoL2Cpoint) | Computes the Caputo differintegral at a point using the L2C algorithm
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[PCsolver](https://github.com/differint/differint/wiki/PCsolver) | Solves IVPs for fractional ODEs of the form ${}^CD^\alpha[y(x)]=f(x,y(x))$ using the predictor-corrector method
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Auxiliary Function | Usage
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[isInteger](https://github.com/differint/differint/wiki/isInteger) | Determine if a number is an integer
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[isPositiveInteger](https://github.com/differint/differint/wiki/isPositiveInteger) | Determine if a number is an integer, and if it is greater than 0
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[checkValues](https://github.com/differint/differint/wiki/checkValues) | Used to check for valid algorithm input types
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[GLIinterpolat](https://github.com/differint/differint/wiki/GLIinterpolat) | Define interpolating coefficients for the improved GL algorithm
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[functionCheck](https://github.com/differint/differint/wiki/functionCheck) | Determines if algorithm function input is callable or an array of numbers
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[poch](https://github.com/differint/differint/wiki/poch) | Computes the Pochhammer symbol
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[Gamma](https://github.com/differint/differint/wiki/Gamma) | Computes the gamma function, an extension of the factorial to complex numbers
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[Beta](https://github.com/differint/differint/wiki/Beta) | Computes the beta function, a function related to the binomial coefficient
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[MittagLeffler](https://github.com/differint/differint/wiki/MittagLeffler) | Computes the two parameter Mittag-Leffler function, which is important in the solution of fractional ODEs
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[GLcoeffs](https://github.com/differint/differint/wiki/GLcoeffs) | Determines the convolution filter composed of generalized binomial coefficients used in the GL algorithm
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[RLcoeffs](https://github.com/differint/differint/wiki/RLcoeffs) | Calculates the coefficients used in the RLpoint and RL algorithms
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[RLmatrix](https://github.com/differint/differint/wiki/RLmatrix) | Determines the matrix used in the RL algorithm
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[PCcoeffs](https://github.com/differint/differint/wiki/PCcoeffs) | Determines the coefficients used in the PC algorithm
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## Credits
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Baleanu, D., Diethelm, K., Scalas, E., & Trujillo, J.J. (2012). Fractional Calculus: Models and Numerical Methods. World Scientific.
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Oldham, K.B. & Spanier, J. (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press Inc.
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Karniadakis, G.E.. (2019). Handbook of Fractional Calculus with Applications Volume 3: Numerical Methods. De Gruyter.
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## License
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MIT © [Matthew Adams](2018)
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---
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## 📝 License
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**MIT License**
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See [LICENSE](./LICENSE).
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---
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## 🔗 Related Projects
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* [differint](https://github.com/differint/differint) (original library, now less maintained)
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* [differintC](https://github.com/iparsw/differintC) (C/C++ accelerated version by @iparsw)
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---
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*For questions or help, open an [issue](https://github.com/iparsw/differintP/issues) or check the Discussions!*
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