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

Abstract

In 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 C++ code for it.

Solving systems of inequalities in two variables with floating point arithmetic

“…That is, the numbers (9)-( 14) yield partial tests, which may be inconclusive in rare cases. In these rare cases, we resort to the technique presented in [3]. Using this technique we can evaluate exactly the sign of…”

“…If we fail then we resort to the exact expressions for the vertex coordinates given by Equations ( 7) and (8) and use the more expensive evaluation of the sign of the expression in Equation (15) with the technique described in [3].…”

“…The polygon P is the feasible region of the problem (1) with = ,1 , = ,2 . Since ∞ = 2 3 , we have that…”

“…Only recently we came to the conclusion that it is possible to perform these tasks with floating point arithmetic, provided that we use the techniques presented here and in the companion article [3].…”