Abstract
This article addresses the problem of approximating the solution manifold of a system of non-linear equations (1) of m equations with n + m variables in R^(n+m). In [8] we considered a method for approximate solution of (1) based on tensor-product piecewise linear B-spline approximation by Lagrange interpolation. In [8] were considered two algorithmic and programming implementations of this method: on vertex level (using poly¬linear patches and OpenGL), and on pixel-level (using GPU-programming in OpenGL shading language). We concluded that the pixel-level version was quite efficient, but the vertex-level implementation needed upgrading from Lagrange to Hermite interpolation to improve the geometric quality of the approximate solution of (1). Rather than following this direction of upgrading, here we consider a radically different approach which proves to be extremely efficient: the method of simplectification (or 'marching-simplex' method), which we propose in Section 2. Here we develop an algorithm for arbitrary natural n and m, and implement it with visualization for the cases (n, m) = (1,1), (2,1) and (1,2), i.e., curves in 2D and surfaces and curves in 3D. Some of the advantages of this method are: - stability and efficiency of Lagrange interpolation; - linearity of the approximating system of equations; - well-conditioned linear convex computations generated by the model; - uniform dyadic grid-refinement on one sided (outer or inner) approximation of general domains in R^(n+m); - adaptive local depth of refinement; - an elegant minimalistic method for eliminating the possibiIity of 'losing' part of the solution.
