Interpretation of a Variable Dimension Fixed Point Algorithm with an Artificial Level

In this paper two interpretations of the variable dimension algorithm to compute a fixed point of a continuous function from the product space S of simplices into itself, introduced in an earlier paper, are given. The first interpretation yields a subdivision, whereas the second one yields a triangulation of the convex hull of the set S on the natural level and some set on the artificial level. After labelling the vertices of the latter set in such a way that this set is completely labelled, a path of adjacent polyhedra (or simplices) can be generated with common completely labelled facets starting with the set on the artificial level and terminating with a completely labelled simplex on the natural level yielding an approximate fixed point. The intersection of the path with this level is the sequence of the simplices of variable dimension of the algorithm. So, the algorithm can be viewed as tracing zeroes of a piecewise linear homotopy function.