An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold

Abstract

A simplicial method is used to approximate the solution manifold to a system of nonlinear equations, $H(x) = \theta $, where $H:\mathbb{R}^{N + K} \to \mathbb{R}^N $ The method begins at a point $x_0 $ in the solution set where the derivative $DH(x_0 )$ is of full rank. Given any $\varepsilon > 0$, a piecewise linear manifold is constructed along which $\| {H(x)} \|_\infty < \varepsilon $. An algorithm is presented to carry out this construction in an efficient fashion.