Complexity of computing topological degree of Lipschitz functions in n dimensions

Abstract

Journal ArticleWe find lower and upper bounds on the complexity, comp(deg), of computing t h e topological degree of functions defined on the n-dimensional unit cube Cn, f : ?Cn Rn,n ? 2, which satisfy a Lipschitz condition with constant K and whose infinity norm at each point on t h e boundary of Cn is at least d, d > 0, and such that K8d ? 1. A lower bound, complow ~ 2n (K8d)n-1(c+n) is obtained for comp(deg), assuming that each function evaluation costs c and elementary arithmetic operations and comparisons cost unity. We prove t h a t the topological degree can be computed using A = ( [ K2d + 1 ] + l ) n ?( [| K2n + 1 ] - l ) n function evaluations. It can be done by an algorithm ?* d due to Kearfott, with cost given by comp(?*) ? A(c + n2-2(n ? 1)!). Thus for small n, say n ? 5, and small say K2d ? 9, the degree can be computed in time at most 105 (c + 300). For large n and / or large K2d the problem is intractable