Complexity of computing topological degree of Lipschitz functions in n dimensions

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Publication Type Journal Article
School or College College of Engineering
Department Computing, School of
Creator Sikorski, Kris
Other Author Boult, T.
Title Complexity of computing topological degree of Lipschitz functions in n dimensions
Date 1986
Description We 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.
Type Text
Publisher University of Utah
First Page 86
Last Page 2
Subject Lipschitz functions
Subject LCSH Topological degree
Language eng
Bibliographic Citation Boult, T., & Sikorski, K. (1986). Complexity of computing topological degree of lipschitz functions in n dimensions. UUCS-86-002.
Series University of Utah Computer Science Technical Report
Relation is Part of ARPANET
Rights Management ©University of Utah
Format Medium application/pdf
Format Extent 3,986,900 bytes
Identifier ir-main,16043
ARK ark:/87278/s61z4ntm
Setname ir_uspace
ID 705295
Reference URL https://collections.lib.utah.edu/ark:/87278/s61z4ntm