Mixed Precision Path Tracking for Polynomial Homotopy Continuation

Abstract: This article develops a new predictor-corrector algorithm for numerical path tracking in the context of polynomial homotopy continuation. In the corrector step it uses a newly developed Newton corrector algorithm which rejects an initial guess if it is not an approximate zero. The algorithm also uses an adaptive step size control which builds on a local understanding of the region of convergence of Newton's method and the distance to the closest singularity. To handle numerically challenging situations the algorithm uses mixed precision arithmetic. The efficiency and robustness are demonstrated in several numerical examples.