Optimal Path Homotopy For Univariate Polynomials

TL;DR

This paper investigates the complexity and condition properties of path-following methods for univariate polynomials, focusing on geodesics, condition metrics, and the impact of approximations on the condition number.

Contribution

It introduces a detailed analysis of the condition metric and geodesics for path-following methods, including approximations and their effects on the condition number.

Findings

Analysis of the complexity of path-following methods

Properties of the condition metric along geodesics

Impact of approximation quality on the condition number

Abstract

The goal of this paper is to study the path-following method for univariate polynomials. We propose to study the complexity and condition properties when the Newton method is applied as a correction operator. Then we study the geodesics and properties of the condition metric along those curves. Last, we compute approximations of geodesics and study how the condition number varies with the quality of the approximation.