A polynomial homotopy is a family of polynomial systems, typically in one parameter $t$. Our problem is to compute power series expansions of the coordinates of the solutions in the parameter $t$, accurately, using multiple double arithmetic. One application of this problem is the location of the nearest singular solution in a polynomial homotopy, via the theorem of Fabry. Power series serve as input to construct Pad\'{e} approximations. Exploiting the massive parallelism of Graphics Processing Units capable of performing several trillions floating-point operations per second, the objective is to compensate for the cost overhead caused by arithmetic with power series in multiple double precision. The application of Newton's method for this problem requires the evaluation and differentiation of polynomials, followed by solving a blocked lower triangular linear system. Experimental results are obtained on NVIDIA GPUs, in particular the RTX 2080, RTX 4080, P100, V100, and A100. Code generated by the CAMPARY software is used to obtain results in double double, quad double, and octo double precision. The programs in this study are self contained, available in a public github repository under the GPL-v3.0 License.

翻译：多项式同伦是一族多项式系统，通常包含一个参数$t$。我们的问题在于利用多倍双精度算术，精确计算解坐标关于参数$t$的幂级数展开。该问题的一个应用是通过Fabry定理定位多项式同伦中最近的奇异解。幂级数可作为构造Padé逼近的输入。借助每秒可执行数万亿次浮点运算的图形处理器的大规模并行能力，目标是弥补多倍双精度幂级数运算带来的额外开销。针对此问题应用牛顿法需要对多项式进行求值与微分，随后求解一个分块下三角线性系统。实验在NVIDIA GPU（特别是RTX 2080、RTX 4080、P100、V100和A100）上获得。通过CAMPARY软件生成的代码实现了双倍双精度、四倍双精度和八倍双精度的计算结果。本研究中的程序均为自包含模块，遵循GPL-v3.0许可协议发布于公开的github代码库。