A new family of Sakurai–Torii–Sugiura type iterative methods with high order of convergence

Abstract

In this paper, we introduce a new family of iterative methods for finding simultaneously all zeros (multiple or simple) of a polynomial. The proposed family is constructed by combining the known Sakurai–Torii–Sugiura iteration function with an arbitrary iteration function. We provide a detailed convergence analysis in the following two directions: local convergence if the polynomial has multiple zeros with known multiplicity and semilocal convergence if the polynomial has only simple zeros. As an application, we study the convergence of several particular iterative methods with high order of convergence.