Accelerated Polynomial Evaluation and Differentiation at Power Series in Multiple Double Precision

TL;DR

This paper introduces data parallel algorithms for efficiently evaluating multivariate polynomials and their gradients at power series in multiple double precision, achieving teraflop performance on GPUs.

Contribution

It presents novel GPU-based algorithms for polynomial evaluation and differentiation at power series with multiple double precision, enabling high-performance computations.

Findings

Teraflop performance achieved in deca double precision.

Algorithms scale well with increased precision and polynomial degree.

Effective parallelization of convolutions and additions in power series computations.

Abstract

The problem is to evaluate a polynomial in several variables and its gradient at a power series truncated to some finite degree with multiple double precision arithmetic. To compensate for the cost overhead of multiple double precision and power series arithmetic, data parallel algorithms for general purpose graphics processing units are presented. The reverse mode of algorithmic differentiation is organized into a massively parallel computation of many convolutions and additions of truncated power series. Experimental results demonstrate that teraflop performance is obtained in deca double precision with power series truncated at degree 152. The algorithms scale well for increasing precision and increasing degrees.