Tchebychef-like method for the simultaneous finding zeros of analytic functions (Q1921210)

Using a suitable approximation in the classical Chebyshev 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 \(x^{r+1} - 2x^r-1 =0\), belonging to the interval (2,3].