GPU加速的牛顿法用于双双精度和四双精度算术中的大规模多项式方程组
GPU acceleration of Newton's method for large systems of polynomial equations in double double and quad double arithmetic
AI总结：
本文研究了通过NVIDIA Tesla K20C GPU加速双双精度和四双精度算术中大规模多项式方程组的牛顿法，通过算法微分的逆模式重构为二叉树结构，提升计算效率。
摘要：
为了弥补双双精度和四双精度算术在求解大规模多项式方程组时较高的计算成本，本文研究了NVIDIA Tesla K20C通用目的图形处理单元的应用。本文重点研究了牛顿法，该方法需要评估多项式、其导数以及求解线性系统以计算当前近似解的更新。对于变量乘积的算法微分逆模式被重构为二叉树结构，使得块中的所有线程都能协作计算。对于双精度算术，评估和微分问题是内存受限的，而对于复数四双精度算术问题则是计算受限的。通过加速，我们可以在大约相同的时间内将维度加倍，并得到两倍准确的结果。
英文摘要：
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.
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