On the guaranteed convergence of new two-point root-finding methods for polynomial zeros

Abstract

We presented new two-point methods for the simultaneous approximation of all n simple (real or complex) zeros of a polynomial of degree n. We proved that the R-order of convergence of the total-step version is three. Moreover, computationally verifiable initial conditions that guarantee the convergence of one of the proposed methods are stated. These conditions are stated in the spirit of Smale's point estimation theory; they depend only on available data, the polynomial coefficients, polynomial degree n and initial approximations x 1 ( 0 ) , Â? , x n ( 0 ) $x_{1}^{(0)},\ldots ,x_{n}^{(0)}$ , which is of practical importance. Using the Gauss-Seidel approach we state the corresponding single-step version and consequently its prove that the lower bound of its R-order of convergence is at least 2 + y n > 3, where y n Â? (1, 2) is the unique positive root of the equation y n Â? y Â? 2 = 0. Two numerical examples are given to demonstrate the convergence behavior of the considered methods, including global convergence.