Approximation of solution curves of underdetermined systems of nonlinear equations

Computation of solution curves of underdetermined systems of nonlinear equations is mostly performed using discrete predictor-corrector methods. Those methods calculate a discrete chain of points lying on the curve. In standard methods there is no way to guarantee that an a posteriori interpolation or other approximation of this set of points results in a curve, which lies in an E-neighborhood with an a priori prescribed tolerance epsilon and it is difficult and expendable to construct a trial and error-extension of the discrete methods based on such a posteriori information. We choose another approach to solve this enhanced task. Our methods are based on a functional predictor-corrector principle, i.e. we construct operators, which correct iteratively given predictor functions into the wanted neighborhood. The numerical realizations of these techniques depend strongly on the different choices of the operator. One possibility, the so-called Spline collocation continuation is explained in detail to illustrate the principle of the methods.