  A path tracking algorithm that adaptively adjusts precision is presented. By
adjusting the level of precision in accordance with the numerical conditioning
of the path, the algorithm achieves high reliability with less computational
cost than would be incurred by raising precision across the board. We develop
simple rules for adjusting precision and show how to integrate these into an
algorithm that also adaptively adjusts the step size. The behavior of the
method is illustrated on several examples arising as homotopies for solving
systems of polynomial equations.
