The Convergence of Asynchronous Monotone Newton Iterations on Distributed Computer

In this paper, according to the asynchronous iteration model presented by Baudet and Bertsekas, we present an asynchronous parallel algorithm for the non-linear system of equations F(x) = 0. In order to describe this algorithm, we need to give some notations. If x is an element of R{sup n}, we write x = (x{sub 1}, ... x{sub n}). To avoid confusion, a sequence of elements of R{sup n} will be denoted by x(t),t = 0,1,2, ... Let F: R{sup n} {r_arrow} Rn be the function defined by F(x) = (F{sub 1}(x), ... Fn(x)), for any x. Let x{sub i}(t)=value of i-th component at time t. We assume that there is a set of times T = 0, 1, 2, {hor_ellipsis} at which one or more components x{sub i} of x are updated by some processors of a distributed computing system. Let T{sub i} = set of times at which x{sub i} is updated.