Tchebychef-like method for the simultaneous finding zeros of analytic functions

Using a suitable approximation in classical Tchebychefʹs iterative method of the third order, a new method for approximating, simultaneously, all zeros of a class of analytic functions in a given simple smooth closed contour is constructed. It is proved that its order of convergence is three. The analysis of numerical stability and some computational aspects, including a numerical example, are given. Also, the asynchronous implementation of the proposed method on a distributed memory multicomputer is considered from a theoretical point of view. Assuming that the maximum delay r is bounded, a convergence analysis shows that the order of convergence of this version is the unique positive root of the equation xr+1 − 2xr − 1 = 0, belonging to the interval (2, 3).