Numerical Continuation on a Graphical Processing Unit for Kinematic Synthesis

This paper presents an implementation of a homotopy path tracking algorithm for polynomial numerical continuation on a graphical processing unit (GPU). The goal of this algorithm is to track homotopy curves from known roots to the unknown roots of a target polynomial system. The path tracker solves a set of ordinary differential equations to predict the next step and uses a Newton root finder to correct the prediction so the path stays on the homotopy solution curves. In order to benefit from the computational performance of a GPU, we organize the procedure so it is executed as a single instruction set, which means the path tracker has a fixed step size and the corrector has a fixed number iterations. This trade-off between accuracy and GPU computation speed is useful in numerical kinematic synthesis where a large number of solutions must be generated to find a few effective designs. In this paper, we show that our implementation of GPU-based numerical continuation yields 85 effective designs in 63 s, while an existing numerical continuation algorithm yields 455 effective designs in 2 h running on eight threads of a workstation.