Verified Error Bounds for Isolated Singular Solutions of Polynomial Systems: Case of Breadth One

TL;DR

This paper enhances symbolic-numeric methods for accurately computing and verifying error bounds of isolated singular solutions in polynomial systems with breadth one, ensuring solutions are within guaranteed bounds even under perturbations.

Contribution

It introduces a parameterized, deflated system with smoothing parameters to improve verification of error bounds for breadth-one multiple roots.

Findings

Improved performance of error bound verification methods.

Guaranteed bounds for slightly perturbed polynomial systems.

Extension of existing algorithms to the breadth-one case.

Abstract

In this paper we describe how to improve the performance of the symbolic-numeric method in (Li and Zhi,2009, 2011) for computing the multiplicity structure and refining approximate isolated singular solutions in the breadth one case. By introducing a parameterized and deflated system with smoothing parameters, we generalize the algorithm in (Rump and Graillat, 2009) to compute verified error bounds such that a slightly perturbed polynomial system is guaranteed to have a breadth-one multiple root within the computed bounds.