Local and Semilocal Convergence of Nourein's Iterative Method for Finding All Zeros of a Polynomial Simultaneously

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 Petkovic, Petkovic and Rancic (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.