  Given a homotopy connecting two polynomial systems we provide a rigorous
algorithm for tracking a regular homotopy path connecting an approximate zero
of the start system to an approximate zero of the target system. Our method
uses recent results on the complexity of homotopy continuation rooted in the
alpha theory of Smale. Experimental results obtained with the implementation in
the numerical algebraic geometry package of Macaulay2 demonstrate the
practicality of the algorithm. In particular, we confirm the theoretical
results for random linear homotopies and illustrate the plausibility of a
conjecture by Shub and Smale on a good initial pair.
