The D2 Triangulation For Simplicial Homotopy Algorithms For Computing Solutions Of Nonlinear Equations PDF Download

A new algorithm is presented for computing vertices of a simplicial triangulation of the p-dimensional solution manifold of a parametrized equation F(x) = O, where F is a nonlinear mapping from R sub n to R sub m, p = n-m>1. An essential part of the method is a constructive algorithm for computing moving frames on the manifold; that is, of orthonormal bases of the tangent spaces that vary smoothly with their points of contact. The triangulation algorithm uses these bases, together with a chord form of the Gauss-Newton process as corrector, to compute the desired vertices. The Jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches. Several numerical examples show that the method is very efficient in computing triangulations, even around singularities such as limit points and bifurcation points. This opens up new possibilities for determining the form and special features of such solution manifolds. Keywords: Equilibrium problem.