Extrapolating on Taylor Series Solutions of Homotopies with Nearby Poles

TL;DR

This paper investigates how the proximity of poles affects the effectiveness of extrapolation algorithms in accelerating the convergence of Taylor series solutions along polynomial homotopy paths, especially near singular solutions.

Contribution

It analyzes the influence of nearby poles on Taylor series convergence and provides insights into improving extrapolation methods for singular solution paths.

Findings

Pole proximity significantly impacts convergence speed.

Extrapolation algorithms can be optimized based on pole analysis.

Insights aid in developing more robust solution path algorithms.

Abstract

A polynomial homotopy is a family of polynomial systems in one parameter, which defines solution paths starting from known solutions and ending at solutions of a system that has to be solved. We consider paths leading to isolated singular solutions, to which the Taylor series converges logarithmically. Whether or not extrapolation algorithms manage to accelerate the slowly converging series depends on the proximity of poles close to the disk of convergence of the Taylor series.