A Note on the Complexity of an Algorithm for Tchebycheff Approximation

Abstract

Some remarks are given concerning the complexity of an exchange algorithm for Tchebycheff Approximation. We consider an “exchange” algorithm that constructs the best polynomial of uniform approximation to a continuous function defined on a closed interval or a finite point set of real numbers. The first, and still popular, class of methods for this problem have been called “exchange algorithms”. We shall consider the simplest method of this class, a blood relative of the dual simplex method of linear programming, and a special case of the cutting plane method. The the idea of the method was initiated by Remes, [1] and [2]. See also Cheney [3], for further developments. Klee and Minty [4], (1972) showed by example that the number of steps in a Simplex method can be exponential in the dimension of the problem. Since then considerable effort has been expended trying to explain the efficiency experienced in practice. Recently, probabilistic models have been assumed that yield expected values for the number of steps with low order monomial behaviour. See for example, Borgwardt [5], and Smale [6]. Alternatively, one might ask can one somehow classify the good problems from the bad ones. We believe that this may be possible for the exchange algorithm.Let T = [0,1], or a finite subset of distinct points of [0,1] with card T > n+1. Let A(t) = (1,t, ... , t n −1). Assume that f is in C 1(T). There exists an n-tuple x* minimizing the function F(x) = maxf{|[A(t), x]| − f(t): t ε T}, where [,] denotes the dot product. Given ε > 0 we seek x k to minimize F within a tolerance of ε. Needed in exchange algorithms is the maximization of |A(t),x] − f(t)| for fixed x. A novelty of the formulation below is that this maximization can have an error ≤ η, where η depends on ε. Most of the arguments however are borrowed from [1], [2] and [3]. The number of steps k to ensure that F(x k)−F(x*.) < ε will be shown to be proportional to log(1/ε) and to 1/ϑ, where ϑ > 0 is a number that depends on f and n. Some remarks about the behavior of ϑ will be made. At k=1 in the algorithm that follows we take t 1 i = 5(1−cos(iπ/n), O≦i≦n. See II below.KeywordsSimplex MethodUniform ApproximationFinite SubsetExchange AlgorithmBlood RelativeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.