Stepsize control for path tracking

*Dedicated to our collaborator, mentor, and friend, Andrew Sommese, by Bates, Hauenstein, and Wampler on the occasion of his sixtieth birthday.* When numerically tracking implicitly-defined paths, such as is required for homotopy continuation methods, efficiency and reliability are enhanced by using adaptive stepsize and adaptive multiprecision methods. Both efficiency and reliability can be further improved by adapting precision and stepsize simultaneously. This paper presents a strategy for adjusting precision and stepsize together to eliminate certain types of path failures that can occur when adapting the two quantities independently, while also reducing the computational effort expended per unit advance along the path.