Analysis of asynchronous polynomial root finding methods on a distributed memory multicomputer

We have studied various implementations of iterative polynomial root finding methods on a distributed memory multicomputer. These methods are based on the construction of a sequence of approximations that converge to the set of zeros. The synchronous version consists in sharing the computation of the next iterate among the processors and updating their data through a total exchange of their results. In order to decrease the communication cost, we introduce asynchronous versions. The computation of the next iterate is still shared among the processor, but the updating is done by using only nearest neighbor communications. We prove that under weak conditions, these asynchronous versions are still locally convergent, even if their convergence orders are reduced. We analyze the behavior of the asynchronous methods in function of their delay, the topology of the interconnection network, and the elementary computation and communication times. We have implemented and compared these methods on a hypercube multicomputer