  The problem is to evaluate a polynomial in several variables and its gradient
at a power series truncated to some finite degree with multiple double
precision arithmetic. To compensate for the cost overhead of multiple double
precision and power series arithmetic, data parallel algorithms for general
purpose graphics processing units are presented. The reverse mode of
algorithmic differentiation is organized into a massively parallel computation
of many convolutions and additions of truncated power series. Experimental
results demonstrate that teraflop performance is obtained in deca double
precision with power series truncated at degree 152. The algorithms scale well
for increasing precision and increasing degrees.
