On the convergence order of accelerated root iterations

Summary A Gauss-Seidel procedure for accelerating the convergence of the generalized method of the root iterations type of the (k+2)-th order (kN) for finding polynomial complex zeros, given in [7], is considered in this paper. It is shown that theR-order of convergence of the accelerated method is at leastk+1+_n(k), where _n(k)>1 is the unique positive root of the equation ⁿ--k-1 = 0 andn is the degree of the polynomial. The examples of algebraic equations in ordinary and circular arithmetic are given.