Abstract
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.
Highlights
This paper deals with the convergence of two iterative methods for finding all zeros of a polynomial simultaneously
The first iterative method for simultaneous finding all zeros of a polynomial was constructed by Weierstrass [3] in
Very recently in [14], it was proved that two kinds of local convergence theorems for iterative methods for simultaneous approximation of polynomial zeros can be transformed into semilocal convergence results
Summary
This paper deals with the convergence of two iterative methods for finding all zeros of a polynomial simultaneously. Very recently in [14], it was proved that two kinds of local convergence theorems for iterative methods for simultaneous approximation of polynomial zeros can be transformed into semilocal convergence results. The generalized Nourein’s method for simultaneously finding all the zeros of f is defined in Ks by the following fixed-point iteration (see, e.g., ([2], Section 20) and ([22], Section 7.2)):. To the best of authors’ knowledge, the theorems given in Sections 3–6 are the first local convergence results in the literature about both Nourein’s methods (for simple or multiple zeros).
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