Numerical differentiation of implicitly defined curves
In the context of numerical continuation methods, one considers curves which are implicitly defined by an undetermined system of equations \(H(u)= 0\) where \(H: \mathbb{R}^{n+1}\to \mathbb{R}^n\) is a smooth map. The authors develop a finite differencing device to calculate approximations of derivatives \(c'(0),c''(0),\dots\) of regular solution curves \(c: \mathbb{R}\ni s\to c(s)\in \mathbb{R}^n\) of systems of nonlinear equations \(f(x)= 0\), \(f\in C^k(\mathbb{R}^{n+ 1},\mathbb{R}^n)\) without having to compute points on the solution curve \(c(s)\). The derivative vectors \(c'(0),c''(0),\dots\) 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.
scientific article; zbMATH DE number 1367894