Numerical differentiation of implicitly defined space curves

In this note we develop a simple finite differencing device to calculate approximations of derivativesx′(0),x″(0),x ⁽³⁾(0), … of regular solution curvesx: ? ?s →x(s) ∈ ? ⁿ of nonlinear systems of equationsg(x)=0,g∈C ^k (? ^{n + 1}, ? ⁿ ) without having to compute points on the solution arcx(s). The derivative vectorsx′(0),x″(0),x ⁽³⁾(0),… can be used in the numerical approximation of the solution setg ^?1(0) in two ways. On one hand they can be applied to construct higher order predictors to be used in predictor-corrector branch following procedures. On the other they serve as order determining basis functions in the Reduced Basis Method. The performance of the differencing method is demonstrated by some numerical examples.