Efficient methods for the inclusion of polynomial zeros

摘要

Using a suitable zero-relation and the inclusion isotonicity property, new interval iterative methods for the simultaneous inclusion of simple complex zeros of a polynomial are derived. These methods produce disks in the complex plane that contain the polynomial zeros in each iteration, providing in this manner an information about upper error bounds of approximations. Starting from the basic method of the fourth order, two accelerated methods with Newton’s and Halley’s corrections, having the order of convergence five and six respectively, are constructed. This increase of the convergence rate is obtained without any additional operations, which means that the methods with corrections are very efficient. The convergence analysis of the basic method and the methods with corrections is performed under computationally verifiable initial conditions, which is of practical importance. Two numerical examples are presented to demonstrate the convergence behavior of the proposed interval methods.