Higher Order Methods for the Inclusion of Multiple Zeros of Polynomials

Starting from a suitable fixed point relation, we derive higher order iterative methods for the simultaneous inclusion of polynomial multiple zeros in circular complex interval arithmetic. Each of the resulting disks contain one and only one zero in every iteration. This convenient inclusion property, together with very fast convergence, ranks these methods among the most powerful iterative methods for the inclusion of polynomial zeros. Using the concept of R-order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step and the single-step methods with Schröder’s and Halley’s corrections under computationally verifiable initial conditions. The proposed self-validated methods possess a great computational efficiency since the acceleration of the convergence rate from four to five and six is achieved without additional calculations. To demonstrate convergence behavior of the presented methods, two numerical examples are given.