Locating the Closest Singularity in a Polynomial Homotopy

TL;DR

This paper introduces a method to locate the nearest singularity in polynomial homotopies by combining the ratio theorem, Richardson extrapolation, reconditioning, and quaternion Fourier transform to improve stability and accuracy.

Contribution

It presents a novel approach that integrates multiple techniques to accurately identify the closest singularity in polynomial homotopies from a regular solution.

Findings

Effective acceleration of convergence using Richardson extrapolation.

Improved numerical stability through reconditioning.

Successful localization of the nearest singularity without numerical difficulties.

Abstract

A polynomial homotopy is a family of polynomial systems, where the systems in the family depend on one parameter. If for one value of the parameter we know a regular solution, then what is the nearest value of the parameter for which the solution in the polynomial homotopy is singular? For this problem we apply the ratio theorem of Fabry. Richardson extrapolation is effective to accelerate the convergence of the ratios of the coefficients of the series expansions of the solution paths defined by the homotopy. For numerical stability, we recondition the homotopy. To compute the coefficients of the series we propose the quaternion Fourier transform. We locate the closest singularity computing at a regular solution, avoiding numerical difficulties near a singularity.