Effective Topological Degree Computation Based on Interval Arithmetic

TL;DR

This paper introduces a novel algorithm for calculating the topological degree of continuous functions using interval arithmetic, separating numerical and combinatorial steps for efficiency and applicability to arbitrary functions.

Contribution

The paper presents a new topological degree computation algorithm that does not require a Lipschitz constant and separates numerical and combinatorial computations for improved efficiency.

Findings

Algorithm effectively computes topological degree for arbitrary continuous functions.

Implementation demonstrates practical viability through computational experiments.

Method improves upon previous work by removing the need for Lipschitz constant knowledge.

Abstract

We describe a new algorithm for calculating the topological degree deg (f, B, 0) where B \subseteq Rn is a product of closed real intervals and f : B \rightarrow Rn is a real-valued continuous function given in the form of arithmetical expressions. The algorithm cleanly separates numerical from combinatorial computation. Based on this, the numerical part provably computes only the information that is strictly necessary for the following combinatorial part, and the combinatorial part may optimize its computation based on the numerical information computed before. We also present computational experiments based on an implementation of the algorithm. Also, in contrast to previous work, the algorithm does not assume knowledge of a Lipschitz constant of the function f, and works for arbitrary continuous functions for which some notion of interval arithmetic can be defined.