specular-differentiation 0.5.4

Specular differentiation in normed vector spaces and its applications

Specular Differentiation

This repository contains the Python package specular which implements Specular differentiation, a generalization of classical differentiation.

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

Also, specular includes the following applications:

• Nonsmooth convex optimization

  • Directory: examples/nonsmooth-convex-optimization/
  • Related reference: [2], [5]

• Initial value problems for ordinary differential equations

  • Directory: examples/numerical-ODE/
  • Related reference: [1], [3], [4]

The package specular mainly follows the definitions, notations, and results presented in [1] and [2].

Installation

Dependencies

specular-differentiation requires:

• Python 3.10 or later
• NumPy 2.2.5 or later

User installation

You can install the released version directly from PyPI:

pip install specular-differentiation

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)
>>> specular.derivative(ReLU, x=0)
0.41421356237309515

Tutorial

Detailed usage examples can be found in the tutorial document.

Applications

Specular differentiation is defined in normed vector spaces, allowing for applications in higher-dimensional Euclidean spaces. Two applications are provided in this repository.

Nonsmooth convex optimization

In [2], the specular gradient method is introduced for optimizing nonsmooth convex objective functions.

Initial value problems for ordinary differential equations

In [1], the specular Euler scheme of Type 5 is introduced for solving ODEs numerically, yielding more accurate numerical solutions than classical schemes: the explicit and implicit Euler schemes and the Crank-Nicolson scheme.

LaTeX notation

To use the symbol for specular differentiation in your LaTeX document, please refer to the following instructions.

Setup

Add the following code to your LaTeX 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
$\partial^{\sd}_v f(x)$

References

[1] K. Jung. Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations. arXiv preprint arXiv:??, 2025.

[2] K. Jung. Specular differentiation in normed vector spaces and its applications to nonsmooth convex optimization. arXiv preprint arXiv:??, 2025.

[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.