Abstract
In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.
Highlights
This work deals with multi-point iterative methods for approximating all the zeros of a polynomial simultaneously
Let us recall that an iterative method for solving a nonlinear equation is called a multi-point method if it can be defined by an iteration of the form x(k+1) = φ(x(k), x(k−1), . . . , x(k−N)), k = 0, 1, 2, . . . , where N is a fixed natural number, and x(0), x(−1), . . . , x(−N) are N + 1 initial approximations
The main purpose of this paper is to provide a local and semilocal convergence analysis of the multi-point Ehrlich-type methods
Summary
This work deals with multi-point iterative methods for approximating all the zeros of a polynomial simultaneously. The main purpose of this paper is to provide a local and semilocal convergence analysis of the multi-point Ehrlich-type methods. We prove that for a given natural number N, the order of convergence of the Nth multi-point Ehrlich-type method is r = r(N), where r is the unique positive solution of the equation. The paper is structured as follows: In Section 2, we introduce the new family of multi-point iterative methods. This result contains initial conditions as well as a priori and a posteriori error estimates.
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