Sampling algebraic sets in local intrinsic coordinates
For researchers working on numerical algebraic geometry, this paper offers an incremental improvement to sampling methods by adapting intrinsic coordinates locally.
The paper addresses poor conditioning in intrinsic coordinate representations for sampling algebraic sets, proposing local intrinsic coordinates that improve numerical conditioning and stepsize control, with experiments on benchmark polynomial systems.
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.