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 There Exists s --> c(s) is an element of R-n of nonlinear systems of equations f(x) = 0, f is an element of C-k ([Rn+1,R-n) 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. (C) 1999 Elsevier Science B.V. All rights reserved.