Solving systems of inequalities in two variables with floating point arithmetic

TL;DR

This paper presents a robust method and data structure for solving systems of inequalities in two variables using floating point arithmetic, ensuring rigorous bounds despite rounding errors, with practical implementation in C++.

Contribution

It introduces an efficient data structure and algorithm for rigorous solution sets of inequalities in two variables with floating point arithmetic, addressing rounding errors.

Findings

Developed a robust algorithm for inequalities in two variables

Implemented the solution in C++ with careful testing

Code available under Mozilla Public License 2.0

Abstract

From a theoretical point of view, finding the solution set of a system of inequalities in only two variables is easy. However, if we want to get rigorous bounds on this set with floating point arithmetic, in all possible cases, then things are not so simple due to rounding errors. In this article we describe in detail an efficient data structure to represent this solution set and an efficient and robust algorithm to build it using floating point arithmetic. The data structure and the algorithm were developed as a building block for the rigorous solution of relevant practical problems. They were implemented in \texttt{C++} and the code was carefully tested. This code is available as supplementary material to the arxiv version of this article, and it is distributed under the Mozilla Public License 2.0.