Sampling algebraic sets in local intrinsic coordinates

TL;DR

This paper improves the numerical stability and step size control in sampling algebraic sets of polynomial systems by adapting intrinsic coordinates locally, demonstrated through experiments with Maple and PHCpack.

Contribution

It introduces a local adaptation method for intrinsic coordinates to enhance numerical conditioning in sampling algebraic sets.

Findings

Improved numerical conditioning with local intrinsic coordinates

Enhanced stepsize control in sampling algorithms

Validated results on benchmark polynomial systems

Abstract

Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complimentary dimension. We may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form. These descriptions are respectively called extrinsic and intrinsic representations. While intrinsic representations lower the cost of the linear algebra operations, we observe worse condition numbers. In this paper we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets. Local intrinsic coordinates also lead to a better stepsize control. We illustrate our results with Maple experiments and computations with PHCpack on some benchmark polynomial systems.