  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.
