Optimized Multivariate Polynomial Determinant on GPU

TL;DR

This paper introduces a GPU-accelerated algorithm for computing determinants of multivariate polynomial matrices, achieving significant speedups and handling complex matrices beyond traditional CPU-based tools.

Contribution

The paper presents a novel GPU-based algorithm combining modular methods, FFT, condensation, and CRT for efficient multivariate polynomial determinant calculation, solving an open problem in harmonic elimination.

Findings

Substantial speedups over Maple in determinant computation

Ability to handle complex matrices exceeding CPU-based limits

Deterministic recovery without accuracy loss during disruptions

Abstract

We present an optimized algorithm calculating determinant for multivariate polynomial matrix on GPU. The novel algorithm provides precise determinant for input multivariate polynomial matrix in controllable time. Our approach is based on modular methods and split into Fast Fourier Transformation, Condensation method and Chinese Remainder Theorem where each algorithm is paralleled on GPU. The experiment results show that our parallel method owns substantial speedups compared to Maple, allowing memory overhead and time expedition in steady increment. We are also able to deal with complex matrix which is over the threshold on Maple and constrained on CPU. In addition, calculation during the process could be recovered without losing accuracy at any point regardless of disruptions. Furthermore, we propose a time prediction for calculation of polynomial determinant according to some basic matrix attributes and we solve an open problem relating to harmonic elimination equations on the basis of our GPU implementation.