Numerical differentiation of implicitly defined curves

In this paper we develop a finite differencing device to calculate approximations of derivatives c′(0), c″(0),… of regular solution curves c : R∋s→c(s)∈Rn of nonlinear systems of equations f(x)=0, f∈Ck (Rn+1,Rn) without having to compute points on the solution curve c(s). The derivative vectors c′(0), c″(0),… can be used in the numerical approximation of the solution set f−1(0) to construct higher-order predictors to be used in the predictor–corrector continuation method.