Efficient numerical differentiation of implicitly-defined curves for sparse systems

A numerical technique is developed for the efficient numerical differentiation of regular implicitly-defined curves existing in high-dimensional real space such as those representing homotopies, where the system of equations which defines the curve implicitly is assumed to be sparse. The calculation is verified numerically through its application to the curve defined implicitly by a homotopy constructed based on a discretization of the equations governing compressible aerodynamic fluid flow. Consideration is given to computational cost, data storage, and accuracy. This method is applicable to any implicitly-defined curves or trajectories which can occur, for example, in dynamical systems analysis or control. Applications also exist in the area of homotopy continuation where implicitly-defined curves are approximately traced numerically. Such applications include the analysis of curve traceability and the construction of higher order predictors. The latter is investigated numerically and it is found that increasing the order of accuracy of the predictor can significantly improve the curve-tracing accuracy within a limited radius.