Computing the exact sign of sums of products with floating point arithmetic

TL;DR

This paper introduces an efficient floating point algorithm for exactly determining the sign of sums of products, crucial for computational geometry and linear programming, offering a faster alternative to exact arithmetic.

Contribution

It presents a novel, efficient floating point algorithm for exact sign computation of sums of products, validated with proofs and tested C++ implementation.

Findings

Algorithm is correct and reliable.

Faster than traditional exact arithmetic methods.

Provides practical C++ code for implementation.

Abstract

IIn computational geometry, the construction of essential primitives like convex hulls, Voronoi diagrams and Delaunay triangulations require the evaluation of the signs of determinants, which are sums of products. The same signs are needed for the exact solution of linear programming problems and systems of linear inequalities. Computing these signs exactly with inexact floating point arithmetic is challenging, and we present yet another algorithm for this task. Our algorithm is efficient and uses only of floating point arithmetic, which is much faster than exact arithmetic. We prove that the algorithm is correct and provide efficient and tested \texttt{C++} code for it.