Relationships between order and efficiency of a class of methods for multiple zeros of polynomials

The behavior of a class of high order methods for solving polynomial equations is examined. It is shown that the number of iterations for local convergence to a multiple zero, to the limits of attainable accuracy, is bounded independent of the multiplicity of the zero, and decreases as the order of the method increases. For the higher order methods, the number of iterations decreases as the multiplicity increases. Computational efficiency as a function of degree, order and multiplicity is investigated, and an effective choice of order is recommended. Numerical examples are provided.