GPU acceleration of Newton's method for large systems of polynomial equations in double double and quad double arithmetic

TL;DR

This paper explores GPU acceleration of Newton's method for large polynomial systems using double double and quad double arithmetic, achieving doubled problem size and accuracy without increasing computation time.

Contribution

It introduces a GPU-based approach for accelerating Newton's method with high-precision arithmetic, enabling larger and more accurate solutions efficiently.

Findings

GPU acceleration doubles the problem size solvable within the same time

High-precision arithmetic benefits from GPU parallelism in polynomial evaluation

Memory-bound and compute-bound regimes are effectively managed on GPU

Abstract

In order to compensate for the higher cost of double double and quad double arithmetic when solving large polynomial systems, we investigate the application of NVIDIA Tesla K20C general purpose graphics processing unit. The focus on this paper is on Newton's method, which requires the evaluation of the polynomials, their derivatives, and the solution of a linear system to compute the update to the current approximation for the solution. The reverse mode of algorithmic differentiation for a product of variables is rewritten in a binary tree fashion so all threads in a block can collaborate in the computation. For double arithmetic, the evaluation and differentiation problem is memory bound, whereas for complex quad double arithmetic the problem is compute bound. With acceleration we can double the dimension and get results that are twice as accurate in about the same time.