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Jul 7, 2025
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Documentation update #7

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Documentation update
iparsw Jul 7, 2025
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Merge branch 'master' into Development
iparsw Jul 7, 2025
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17 changes: 6 additions & 11 deletions CONTRIBUTING.md
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Thank you for taking the time to contribute!

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.
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.

## Getting Started

*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.
*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

## Community

* The whole *differint* documentation, including installation and testing, can be read [here](https://github.com/snimpids/differint).
* The whole *differintP* documentation, including installation and testing, can be read [here](https://github.com/iparsw/differintP).

* If you have any questions regarding *differint*, open an [issue](https://github.com/snimpids/differint/issues/new/).
* If you have any questions regarding *differint*, open an [issue](https://github.com/iparsw/differintP/issues/new/).

## Issue
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).
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).

Please consider the following points while opening an issue.
For example:
Expand All @@ -24,9 +24,4 @@ For example:
* The operating system.

## Pull Requests
Pull Requests are always welcome.

1. When you edit the code, please run `npm run test` to check the formatting of your code before you `git commit`.
2. Ensure the PR description clearly describes the problem and solution. It should include:
* The operating system on which you tested.
* The relevant issue number, if applicable.
Pull Requests are always welcome.
188 changes: 94 additions & 94 deletions README.md
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- There is also a faster version (but more limited) implemented in c++ [diffeintC](https://github.com/iparsw/differintC)


## differint
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.
# differintP

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.
**differintP** is a modern, pure Python library for **fractional calculus**—that is, for numerical differentiation and integration of arbitrary (non-integer) order.
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.

## Motivation
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.
> **Highlights**
> - Orders of differentiation and integration: any real (fractional) value
> - Fast, vectorized, and JIT-accelerated core routines (NumPy + Numba)
> - GPU support (optional, via CuPy)
> - Array and pointwise (endpoint) fractional derivative functions
> - Covers Grünwald-Letnikov, Riemann-Liouville, Caputo, Weyl, Riesz, CRONE, and more
> - Accurate for both smooth and non-smooth functions
> - Extensive tests and clear, modern codebase
> - MIT License

## Installation
This project requires Python 3+ and NumPy to run.
---

Installation from the Python Packaging index (https://pypi.python.org/pypi) is simple using pip.
## 🚀 Quick Install

```python
```bash
pip install differintP
```
````

* Optional: for GPU acceleration, also install [CuPy](https://docs.cupy.dev/en/stable/install.html):

```bash
pip install cupy-cuda12x # or cupy-cuda11x, depending on your CUDA
```

---

## ✨ Features

* **Fast Fractional Derivatives**: Grünwald-Letnikov (`GL`, `GLpoint`), Riemann-Liouville (`RL`, `RLpoint`), Caputo (`CaputoL1point`, etc.)
* **Modern Implementations**: All methods rewritten for clarity, speed, and extensibility
* **Advanced/Research Methods**:

* **Weyl** (Fourier/spectral, periodic)
* **Riesz** (symmetric, physical applications)
* **CRONE** (for edge detection and signal processing)
* **GPU-accelerated GL**: `GL_gpu` (if CuPy is installed)
* **High test coverage**: Pytest, with analytical ground truths for common functions

## Included Files
Core File | Description
--------- | -----------
differint/differint.py | Contains algorithms for fractional differentiation and integration.
tests/test.py | Testing suite containing all unit tests.

Both of the above files have corresponding `__init__.py` files.

Setup File | Description
---------- | -----------
.gitignore | List of files to ignore during `git` push/pull requests.
CONTRIBUTING.md | Instructions for potential contributors to the *differint* project.
LICENSE | MIT license agreement.
MANIFEST.in | Selects the README file for uploading to PyPi.
README.md | This README file.
README.rst | This README file in ReStructuredText format.
__init__.py | `__init__` file for overall package.
changelog.txt | List of updates to package.
setup.py | Script for downloading package from `pip`.

## Example Usage
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.
---

## 📚 Usage Example

```python
import numpy as np
import differint.differint as df
from differintP import GL, GLpoint, Weyl, Riesz

def f(x):
return x**0.5
# Fractional derivative of order 0.5 of sqrt(x) on [0, 1]
x = np.linspace(0, 1, 100)
df_gl = GL(0.5, np.sqrt, 0, 1, 100) # Array version
df_point = GLpoint(0.5, np.sqrt, 0, 1, 100) # Endpoint value

DF = df.RL(0.5, f)
print(DF)
# Weyl and Riesz derivatives of sin(x) on [0, 2*pi]
x2 = np.linspace(0, 2*np.pi, 256, endpoint=False)
df_weyl = Weyl(0.5, np.sin, 0, 2*np.pi, 256)
df_riesz = Riesz(0.5, np.sin, 0, 2*np.pi, 256)
```

You can also specify the endpoints of the domain and the number of points used as follows.
See the [wiki](https://github.com/iparsw/differintP/wiki) for detailed documentation and advanced use.

```python
DF = df.RL(0.5, f, 0, 1, 128)
```
---

For a description of all functions, their signatures, and more usage examples, see the project's [wiki](https://github.com/differint/differint/wiki).
## 📑 Documentation

## Tests
All tests can be run with nose from the command line. Setup will automatically install nose if it is not present on your machine.
* **Function Reference**: See the [Wiki](https://github.com/iparsw/differintP/wiki) for math, options, and examples for all supported methods.
* **Benchmarks**: [BENCHMARK.md](https://github.com/iparsw/differintC/blob/main/BENCHMARK.md) for speed/performance notes.
* **Importing**: Import main algorithms directly, See the [Wiki](https://github.com/iparsw/differintP/wiki/Namespaces).

```python
python setup.py tests
```
Of course! Here’s a suggested section for accuracy tests, following your README’s style:

Alternatively, you can run the test script directly.
---

```python
cd <file_path>/differint/tests/
python test.py
## ✅ Accuracy Tests

To verify the correctness and numerical accuracy of all algorithms, see the interactive **accuracy test notebook**:
[**accuracy\_test.ipynb**](https://github.com/iparsw/differintP/blob/master/accuracy_test.ipynb)

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.

---

## 🧑‍💻 Development and Testing

* **Python ≥ 3.10** required
* All code is in pure Python; C++ not required
* **All tests use [pytest](https://pytest.org/)** for fast, expressive, and modern testing

To run the full test suite:

```bash
pytest tests/
```

## API Reference
In this section we cover the usage of the various functions within the *differint* package.

Main Function | Usage
------------- | -----
[GLpoint](https://github.com/differint/differint/wiki/GLpoint) | Computes the GL differintegral at a point
[GL](https://github.com/differint/differint/wiki/GL) | Computes the GL differintegral over an entire array of function values using the Fast Fourier Transform
[GLI](https://github.com/differint/differint/wiki/GLI) | Computes the improved GL differintegral over an entire array of function values
[CRONE](https://github.com/differint/differint/wiki/CRONE) | Calculates the GL derivative approximation using the CRONE operator.
[RLpoint](https://github.com/differint/differint/wiki/RLpoint) | Computes the RL differintegral at a point
[RL](https://github.com/differint/differint/wiki/RL) | Computes the RL differintegral over an entire array of function values using matrix methods
[CaputoL1point](https://github.com/differint/differint/wiki/CaputoL1point) | Computes the Caputo differintegral at a point using the L1 algorithm
[CaputoL2point](https://github.com/differint/differint/wiki/CaputoL2point) | Computes the Caputo differintegral at a point using the L2 algorithm
[CaputoL2Cpoint](https://github.com/differint/differint/wiki/CaputoL2Cpoint) | Computes the Caputo differintegral at a point using the L2C algorithm
[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

Auxiliary Function | Usage
------------------ | -----
[isInteger](https://github.com/differint/differint/wiki/isInteger) | Determine if a number is an integer
[isPositiveInteger](https://github.com/differint/differint/wiki/isPositiveInteger) | Determine if a number is an integer, and if it is greater than 0
[checkValues](https://github.com/differint/differint/wiki/checkValues) | Used to check for valid algorithm input types
[GLIinterpolat](https://github.com/differint/differint/wiki/GLIinterpolat) | Define interpolating coefficients for the improved GL algorithm
[functionCheck](https://github.com/differint/differint/wiki/functionCheck) | Determines if algorithm function input is callable or an array of numbers
[poch](https://github.com/differint/differint/wiki/poch) | Computes the Pochhammer symbol
[Gamma](https://github.com/differint/differint/wiki/Gamma) | Computes the gamma function, an extension of the factorial to complex numbers
[Beta](https://github.com/differint/differint/wiki/Beta) | Computes the beta function, a function related to the binomial coefficient
[MittagLeffler](https://github.com/differint/differint/wiki/MittagLeffler) | Computes the two parameter Mittag-Leffler function, which is important in the solution of fractional ODEs
[GLcoeffs](https://github.com/differint/differint/wiki/GLcoeffs) | Determines the convolution filter composed of generalized binomial coefficients used in the GL algorithm
[RLcoeffs](https://github.com/differint/differint/wiki/RLcoeffs) | Calculates the coefficients used in the RLpoint and RL algorithms
[RLmatrix](https://github.com/differint/differint/wiki/RLmatrix) | Determines the matrix used in the RL algorithm
[PCcoeffs](https://github.com/differint/differint/wiki/PCcoeffs) | Determines the coefficients used in the PC algorithm


## Credits
Baleanu, D., Diethelm, K., Scalas, E., & Trujillo, J.J. (2012). Fractional Calculus: Models and Numerical Methods. World Scientific.

Oldham, K.B. & Spanier, J. (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press Inc.

Karniadakis, G.E.. (2019). Handbook of Fractional Calculus with Applications Volume 3: Numerical Methods. De Gruyter.

## License

MIT © [Matthew Adams](2018)
---



## 📝 License

**MIT License**
See [LICENSE](./LICENSE).

---

## 🔗 Related Projects

* [differint](https://github.com/differint/differint) (original library, now less maintained)
* [differintC](https://github.com/iparsw/differintC) (C/C++ accelerated version by @iparsw)
---

*For questions or help, open an [issue](https://github.com/iparsw/differintP/issues) or check the Discussions!*