Numerical continuation methods apply predictor-corrector algorithms to track a solution path defined by a family of systems, the so-called homotopy. The systems we consider are defined by polynomials in several variables with complex coefficients. For larger dimensions and degrees, the numerical conditioning worsens and hardware double precision becomes often insufficient to reach the end of the solution path. With double double and quad double arithmetic, we can solve larger problems that we could not solve with hardware double arithmetic, but at a higher computational cost. This cost overhead can be compensated by acceleration on a Graphics Processing Unit (GPU). We describe our implementation and report on computational results on two benchmark polynomial systems. In case the linear algebra dominates the total computational cost of a path tracker, the dimension needs to be of the order of at least several hundreds to offset the computational overhead caused by double double arithmetic. For general polynomials of higher degrees, when evaluating and differentiating the polynomials is computationally expensive, already in smaller dimensions, acceleration may offset the cost of higher precision arithmetic.
Key words and phrases. double double arithmetic, general purpose graphics processing unit (GPU), massively parallel algorithm, path tracking, predictor-corrector, polynomial homotopy continuation, polynomial system, quad double arithmetic.