Rigorous Function Calculi in Ariadne

TL;DR

The paper introduces Ariadne, a rigorous computational framework for Euclidean functions, enabling exact, effective, and validated calculations for solving algebraic and differential equations, with applications to hybrid systems analysis.

Contribution

It presents a novel function calculus in Ariadne that separates exact, effective, validated, and approximate information, advancing computational methods in functional analysis.

Findings

Implementation of polynomial function models for concrete computations

Successful solution of algebraic and differential equations using Ariadne

Examples provided in C++ and Python demonstrating the calculus

Abstract

Almost all problems in applied mathematics, including the analysis of dynamical systems, deal with spaces of real-valued functions on Euclidean domains in their formulation and solution. In this paper, we describe the the tool Ariadne, which provides a rigorous calculus for working with Euclidean functions. We first introduce the Ariadne framework, which is based on a clean separation of objects as providing exact, effective, validated and approximate information. We then discuss the function calculus as implemented in Ariadne, including polynomial function models which are the fundamental class for concrete computations. We then consider solution of some core problems of functional analysis, namely solution of algebraic equations and differential equations, and briefly discuss their use for the analysis of hybrid systems. We will give examples of C++ and Python code for performing the various calculations. Finally, we will discuss progress on extensions, including improvements to the function calculus and extensions to more complicated classes of system.