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, P100 and V100. 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：多项式同伦是通常依赖于参数$t$的多项式系统族。我们的目标是利用多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法布里定理定位多项式同伦中的最近奇异解。幂级数可作为构造帕德逼近的输入。利用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性，旨在补偿多双精度幂级数算术带来的计算开销。应用牛顿法求解该问题需进行多项式求值与微分，随后求解块状下三角线性系统。实验在NVIDIA GPU（特别是RTX 2080、P100和V100）上完成，采用CAMPARY软件生成的代码获得双双精度、四双精度和八双精度结果。本研究的程序为独立代码，以GPL-v3.0许可协议发布于公开的github仓库。