Properties of polynomial bases used in a line-surface intersection algorithm

TL;DR

This paper investigates how the choice of polynomial basis affects the efficiency of the KTS algorithm for finding polynomial system zeros, comparing power, Bernstein, and Chebyshev bases.

Contribution

It provides a detailed analysis of the impact of different polynomial bases on the KTS algorithm's efficiency, including theoretical bounds and computational experiments.

Findings

Chebyshev basis has the smallest theoretical upper bound on running time.

Computational results do not show Chebyshev outperforming other bases.

The efficiency depends on basis properties and problem conditioning.

Abstract

In [5], Srijuntongsiri and Vavasis propose the "Kantorovich-Test Subdivision algorithm", or KTS, which is an algorithm for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface. The main features of KTS are that it can operate on polynomials represented in any basis that satisfies certain conditions and that its efficiency has an upper bound that depends only on the conditioning of the problem and the choice of the basis representing the polynomial system. This article explores in detail the dependence of the efficiency of the KTS algorithm on the choice of basis. Three bases are considered: the power, the Bernstein, and the Chebyshev bases. These three bases satisfy the basis properties required by KTS. Theoretically, Chebyshev case has the smallest upper bound on its running time. The computational results, however, do not show that Chebyshev case performs better than the other two.