This paper presents a numerical algorithm for the parallel computations Phi(q) in the Kolmogorov superpositions f(x) = Sigma(q=0)(m) Phi(q) circle xi(x(q)), x = (x(1),...,x(n)) and x(q) = (x(1) + q(a),...,x(n) + qa),thereby providing the final step in their numerical implementation. The first step consisting of the f-independent computation of the functions xi(x(q)) = Sigma(p=1)(n) alpha(p) psi(x(p) + qa) with a fixed psi and constants a and alpha(p) in a hidden layer in the Hecht-Nielsen feedforward neural network has been accomplished previously. The step taken in this paper is the implementation of the output layer of the network that computes an arbitrary known continuous real-valued function f defined on the unit cube E-n. Employed for the purpose is an iterative method which is intended as a basis for the possible development of adaptive methods that build on this approach. Each function Phi(q) is obtained iteratively through a series Sigma(r) Phi(q)(r) which is determined on an f and q dependent subsequence d(k1)(q),d(k2)(q),d(k3)(q)...of rational coordinates d(kr)(q) = (d(kr1)(q),...,d(krn)(q)) such that Phi(q)r is determined at the coordinate points xi((q)(k)). The paper also includes alternative constructions of the functions Phi(q)(r) and a brief discussion of the differentiability of Phi(q) circle xi(x(q)); together with a previous result it gives a constructive proof of Kolmogorov's theorem. (C) 1997 Elsevier Science Ltd.
