GPU-accelerated path tracker for polyhedral homotopy

TL;DR

This paper introduces a GPU-accelerated path tracker for the polyhedral homotopy method, significantly improving the efficiency of solving polynomial systems by leveraging parallel GPU architectures.

Contribution

It presents a novel GPU-based approach for evaluating multivariate Laurent polynomials and derivatives, simplifying and speeding up the path tracking process.

Findings

Achieves efficient polynomial evaluation on GPUs

Simplifies computation of Euler and Newton directions

Enhances scalability of the path tracker

Abstract

The polyhedral homotopy method of Huber and Sturmfels is a particularly efficient and robust numerical method for solving system of (Laurent) polynomial equations. A central component in an implementation of this method is an efficient and scalable path tracker. While the implementation issues in a scalable path tracker for computer clusters or multi-core CPUs have been solved thoroughly, designing good GPU-based implementations is still an active research topic. This paper addresses the core issue of efficiently evaluate a multivariate system of Laurent polynomials together with all its partial derivatives. We propose a simple approach that maps particularly well onto the parallel computing architectures of modern GPUs. As a by-product, we also simplify and accelerate the path tracker by consolidating the computation of Euler and Newton directions.