Locating the Closest Singularity in a Polynomial Homotopy
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.
Cite
@article{arxiv.2205.07380,
title = {Locating the Closest Singularity in a Polynomial Homotopy},
author = {Jan Verschelde and Kylash Viswanathan},
journal= {arXiv preprint arXiv:2205.07380},
year = {2022}
}
Comments
Accepted for the Proceedings of the 24th International Workshop on Computer Algebra in Scientific Computing (CASC 2022)