The problem is to investigate the scalability of a new path tracker (SISC 42(6), A3610-A3637, 2020) to solve systems of polynomial equations in many variables. The many in this context is about a thousand. Path trackers repeatedly run Newton's method, evaluating and differentiating polynomials, followed by the solution of a linear system. On Taylor series, the matrices are block Toeplitz lower triangular, obtained by convolutions. Proximity to singularities requires multiple double precision arithmetic, which causes a cost overhead to be offset by acceleration with Graphics Processing Units (GPUs). In particular, GPUs capable of teraflop performance compensate for the overhead caused by quad double arithmetic. Multiple double precision is necessary to adjust the parameter representation in case the radius of convergence of the Taylor series is too small. Singularities are located efficiently via the quadratically converging Newton's method at regular points and with extrapolation on the series. While the current implementation takes polynomials on input, viewing polynomials as truncated series extends its application to analytic systems, systems of functions with well defined Taylor series developments. The software written for this investigation is licensed under GPL-3.0, available at https://github.com/janverschelde/PHCpack.
SIAM CSE 2023, 28 February 2023, Amsterdam, The Netherlands.