We study a family of high-order Ehrlich-type methods for approximating all zeros of a polynomial simultaneously. Let us denote by T(1) the famous Ehrlich method (1967). Starting from T(1) , Kjurkchiev and Andreev (1987) have introduced recursively a sequence (T (N) ) ∞ N=1 of iterative methods for simultaneous finding polynomial zeros. For given N ≥ 1, the Ehrlich-type method T (N) has the order of convergence 2N + 1. In this paper, we establish two new local convergence theorems as well as a semilocal convergence theorem (under computationally verifiable initial conditions and with an a posteriori error estimate) for the Ehrlich-type methods T (N) . Our first local convergence theorem generalizes a result of Proinov (2015) and improves the result of Kjurkchiev and Andreev (1987). The second local convergence theorem generalizes another recent result of , but only in the case of the maximum norm. Our semilocal convergence theorem is the first result in this direction. MSC: Primary 65H04; secondary 12Y05

No abstract

No abstract

Kyurkchiev and Andreev (1985) constructed an infinite sequence of Weierstrass-type iterative methods for approximating all zeros of a polynomial simultaneously. The first member of this sequence of iterative methods is the famous method of Weierstrass (1891) and the second one is the method of Nourein (1977). For a given integer N ≥ 1, the Nth method of this family has the order of convergence N + 1. Currently in the literature, there are only local convergence results for these methods. The main purpose of this paper is to present semilocal convergence results for the Weierstrass-type methods under computationally verifiable initial conditions and with computationally verifiable a posteriori error estimates.

In 1977, Nourein (Intern. J. Comput. Math. 6:3, 1977) constructed a fourth-order iterative method for finding all zeros of a polynomial simultaneously. This method is also known as Ehrlich’s method with Newton’s correction because it is obtained by combining Ehrlich’s method (Commun. ACM 10:2, 1967) and the classical Newton’s method. The paper provides a detailed local convergence analysis of a well-known but not well-studied generalization of Nourein’s method for simultaneous finding of multiple polynomial zeros. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with verifiable initial condition and a posteriori error bound) for the classical Nourein’s method. Each of the new semilocal convergence results improves the result of Petković, Petković and Rančić (J. Comput. Appl. Math. 205:1, 2007) in several directions. The paper ends with several examples that show the applicability of our semilocal convergence theorems.