Numerical Instability of Algebraic Rootfinding Methods

TL;DR

This paper reveals that many algebraic multidimensional rootfinding algorithms are numerically unstable due to poor conditioning of subproblems, especially as the number of variables increases exponentially.

Contribution

It provides a detailed analysis of the instability in popular algebraic rootfinding methods and demonstrates how subproblem conditioning can deteriorate exponentially.

Findings

Subproblems can be exponentially worse conditioned than original problems.

Most common algebraic rootfinding algorithms are numerically unstable.

Conditioning issues grow with the number of variables.

Abstract

We demonstrate that the most popular variants of all common algebraic multidimensional rootfinding algorithms are unstable by analyzing the conditioning of subproblems that are constructed at intermediate steps. In particular, we give multidimensional polynomial systems for which the conditioning of a subproblem can be worse than the conditioning of the original problem by a factor that grows exponentially with the number of variables.