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Specular Differentiation

PyPI version Python 3.11 DOI License CodeFactor CodeQL Advanced Docs

The Python package specular implements specular differentiation which generalizes classical differentiation. This implementation strictly follows the definitions, notations, and results in [1] and [2].

A specular derivative (the red line) can be understood as the average of the inclination angles of the right and left derivatives. In contrast, a symmetric derivative (the purple line) is the average of the right and left derivatives. Their difference is illustrated as in the following figure.

specular-derivative-animation

Table of Contents

Installation

Requirements

specular-differentiation requires:

  • Python >= 3.11
  • numpy >= 2.0

Additional features are available through optional dependencies:

  • ode: matplotlib, pandas, tqdm
  • optimization: matplotlib, tqdm
  • numba: numba
  • jax: jax, jaxlib

User installation

Standard Installation (NumPy backend)

pip install specular-differentiation

This installs the core specular differentiation API, including A, derivative, directional_derivative, partial_derivative, gradient, and jacobian.

ODE solvers

pip install "specular-differentiation[ode]"

Optimization routines

pip install "specular-differentiation[optimization]"

Numba backend

pip install "specular-differentiation[numba]"

If Numba is installed and available, the package may use the Numba-accelerated CPU backend.

JAX backend

pip install "specular-differentiation[jax]"

See the backend documentation for JAX-specific usage.

Quick start

The following simple example calculates the specular derivative of the ReLU function \(f(x) = max(0, x)\) at the origin.

import specular

ReLU = lambda x: max(x, 0)
print(specular.derivative(ReLU, x=0))

0.41421356237309515

Backend support

The package is organized around a backend system. NumPy is the default backend, while accelerated backends are optional and may require extra dependencies.

Backend Calculation ODE Optimization
NumPy supported supported supported
Numba supported supported (recommended) not supported
JAX supported supported supported (recommended)
TensorFlow experimental experimental not supported
PyTorch experimental experimental not supported

Applications

Specular differentiation is defined in normed vector spaces, allowing for applications in higher-dimensional Euclidean spaces. The specular package includes the following applications.

Ordinary differential equation

  • Directory: examples/ode/
  • References: [1], [3], [4]

In [1], seven schemes are proposed for solving ODEs numerically:

  • the specular Euler scheme of Type 1~6
  • the specular trigonometric scheme

The following example shows that the specular Euler schemes of Type 5 and 6 yield more accurate numerical solutions than classical schemes: the explicit and implicit Euler schemes and the Crank-Nicolson scheme.

ODE-example-1

ODE-example-2

Optimization

  • Directory: examples/optimization/
  • References: [2], [5]

In [2], three methods are proposed for optimizing nonsmooth convex objective functions:

  • the specular gradient (SPEG) method
  • the stochastic specular gradient (S-SPEG) method
  • the hybrid specular gradient (H-SPEG) method

The following example compares the three proposed methods with the classical methods: gradient descent (GD), Adaptive Moment Estimation (Adam), and Broyden-Fletcher-Goldfarb-Shanno (BFGS).

optimization-example

Documentation

Getting Started

API Reference

Examples

LaTeX macro

To use the specular differentiation symbol in your LaTeX document, add the following code to your preamble (before \begin{document}):

% Required packages
\usepackage{graphicx}
\usepackage{bm}

% Definition of Specular Differentiation symbol
\newcommand\sd[1][.5]{\mathbin{\vcenter{\hbox{\scalebox{#1}{\,$\bm{\wedge}$}}}}}

Usage examples

Use the symbol in your document (after \begin{document}):

% A specular derivative in the one-dimensional Euclidean space
$f^{\sd}(x)$

% A specular directional derivative in normed vector spaces
$\partial^{\sd}_v f(x)$

Citing specular-differentiation

To cite this repository:

@software{Jung_specular-differentiation_2026,
  author = {Jung, Kiyuob},
  doi = {10.5281/zenodo.18246734},
  license = {MIT},
  month = jan,
  title = {{specular-differentiation}},
  url = {https://github.com/kyjung2357/specular-differentiation},
  version = {1.1.0},
  year = {2026},
}

References

[1] K. Jung. Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations. arXiv preprint arXiv:2601.09900, 2026.

[2] K. Jung. Specular differentiation in normed vector spaces and its applications to nonsmooth convex optimization. arXiv preprint arXiv:2601.10950, 2026.

[3] K. Jung and J. Oh. The specular derivative. arXiv preprint arXiv:2210.06062, 2022.

[4] K. Jung and J. Oh. The wave equation with specular derivatives. arXiv preprint arXiv:2210.06933, 2022.

[5] K. Jung and J. Oh. Nonsmooth convex optimization using the specular gradient method with root-linear convergence. arXiv preprint arXiv:2210.06933, 2024.