On Halley-Like Algorithms for Simultaneous Approximation of Polynomial Complex Zeros

Abstract

Let $P(z)$ be a polynomial of degree N with n distinct zeros. Iterative methods for the simultaneous inclusion of $k( \leqq n)$ simple or multiple zeros are given. The iterative formulas presented can be regarded as an interval version of Halley’s iterative formula for multiple zeros, realized in circular arithmetic. The convergence analysis is done in the presence of multiplicity; the convergence order is three for $k < n$ and four if $k = n$. Applying the Gauss–Seidel approach to the basic iterative formula, the single-step method with accelerated convergence is stated and the lower bound of the R-order of convergence of this method is obtained. The dependence of convergence rate of the basic methods on the rounding errors is also studied. Some numerical examples illustrate the theoretical results.