Approximation of solution curves of undetermined systems of nonlinear equations (Q1377267)

The author investigates the solutions of the underdetermined system of nonlinear equations \[ f(y)= 0,\;f: \mathbb{R}^n \to\mathbb{R}^{n-1} \text{ sufficiently smooth, } n\geq 2. \tag{1} \] The vector \(y\) consists of a vector of state variables \(\widetilde y\in \mathbb{R}^{n-1}\) and a real parameter \(\lambda\). Branching solutions of (1) may be computed using standard continuation or predictor-corrector techniques. The suggested method is based on a functional predictor-corrector principle: operators, which correct interatively the given predictor functions into the wanted neighborhood are constructed. The numerical realization of these techniques strongly depends on the different choices of the operators. Here spline collocation continuation is explained to illustrate the principle of the method.