Specular Differentiation

Python 3.14

The Python package specular implements specular differentiation, which generalizes classical differentiation. This implementation strictly follows the definitions, notations, and results in [1], [2], and [3].

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 in the animation.

Animation comparing specular and symmetric derivatives

Installation

Requirements

specular-differentiation requires:

Python >= 3.14
numpy >= 2.4

Additional backends are available through optional dependencies:

numba: numba
jax: jax, jaxlib

Standard Installation

pip install specular-differentiation

The package is distributed as specular-differentiation and imported in Python as specular.

This installs scaled_mean, derivative, gradient, and jacobian, using NumPy by default, as well as the scalar ODE methods and the SPEG optimizer. Backend selection is available through set_backend, get_backend, use_backend, and available_backends.

Optional features

pip install "specular-differentiation[numba]"           # Numba backend
pip install "specular-differentiation[jax]"             # JAX backend

Developer installation

pip install -e ".[dev]"

Documentation tools can be installed separately with:

pip install -e ".[docs]"

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.

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))

0.41421356237309503

Documentation

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{amssymb}

% specular derivative symbol
\newcommand{\sd}{\mathord{\prime\mkern-2.5mu\reflectbox{$\scriptstyle\prime$}}}

% specular Gateaux derivative symbol
\newcommand{\sGd}{\widehat{\mkern-2mu d}\mkern1mu}

% specular gradient symbol
\newcommand{\sg}{%
  \mathchoice
    {\mathord{\raisebox{-0.05ex}{\rule{0pt}{1.3ex}\smash{\scalebox{1.37}[1.22]{\ensuremath{\displaystyle\blacktriangledown}}}}\mkern-1.2mu}}
    {\mathord{\raisebox{-0.05ex}{\rule{0pt}{1.3ex}\smash{\scalebox{1.37}[1.22]{\ensuremath{\textstyle\blacktriangledown}}}}\mkern-1.2mu}}
    {\mathord{\raisebox{-0.03ex}{\rule{0pt}{1.0ex}\smash{\scalebox{1.29}[1.15]{\ensuremath{\scriptstyle\blacktriangledown}}}}\mkern-0.8mu}}
    {\mathord{\raisebox{-0.02ex}{\rule{0pt}{0.8ex}\smash{\scalebox{1.18}[1.05]{\ensuremath{\scriptscriptstyle\blacktriangledown}}}}\mkern-0.5mu}}
}

Citing specular-differentiation

To cite this repository:

@software{specular_diff,
  author = {Jung, Kiyuob},
  title = {{specular-differentiation}},
  doi = {10.5281/zenodo.18246734},
  url = {https://github.com/kyjung2357/specular-differentiation},
  version = {1.3.2},
  year = {2026},
}

References

[1] K. Jung. Specular differentiation in one dimension: a quasi-mean value theorem, regularity, and discontinuities. arXiv preprint arXiv:2601.09900, 2026.

[2] K. Jung. The specular ellipse method for scalar ordinary differential equations: exactness and accuracy up to fourth order. arXiv preprint arXiv:2608.30280, 2026.

[3] K. Jung. Specular differentiation in normed vector spaces: Quasi-Mean Value and Quasi-Fermat Theorems. arXiv preprint arXiv:2601.10950, 2026.

[4] K. Jung. Specular gradient methods for nonsmooth convex optimization in Euclidean spaces: a subgradient selection strategy. arXiv preprint arXiv:2605.25490, 2026.

[5] K. Jung and J. Oh. Nonsmooth convex optimization using the specular gradient method with root-linear convergence. arXiv preprint arXiv:2412.20747, 2024.

Metadata

Release files for specular-differentiation 1.3.2

For a detailed explanation of source distributions (sdists) and built distributions (wheels), please see the package formats documentation.

Built distribution (wheel)

Table of built distributions (wheels) for specular-differentiation 1.3.2
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specular_differentiation-1.3.2-py3-none-any.whl 55.7 kB Sep 6, 2026	Python 3	none	any	Details

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