  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.
