Tracking Many Solution Paths of a Polynomial Homotopy on a Graphics Processing Unit

TL;DR

This paper presents a GPU-accelerated parallel algorithm for tracking numerous solution paths of polynomial homotopies, significantly improving speed and accuracy in solving large polynomial systems.

Contribution

It introduces a massively parallel predictor-corrector algorithm leveraging data parallelism and advanced differentiation techniques for efficient solution path tracking.

Findings

Achieves significant speedups using GPU parallelism.

Handles tens of thousands of solution paths simultaneously.

Provides more accurate results with double double and quad double precision.

Abstract

Polynomial systems occur in many areas of science and engineering. Unlike general nonlinear systems, the algebraic structure enables to compute all solutions of a polynomial system. We describe our massive parallel predictor-corrector algorithms to track many solution paths of a polynomial homotopy. The data parallelism that provides the speedups stems from the evaluation and differentiation of the monomials in the same polynomial system at different data points, which are the points on the solution paths. Polynomial homotopies that have tens of thousands of solution paths can keep a sufficiently large amount of threads occupied. Our accelerated code combines the reverse mode of algorithmic differentiation with double double and quad double precision to compute more accurate results faster.