A new family of methods for single and multiple roots

We present a new family of iterative methods for multiple and single roots of nonlinear equations. This family contains as a special case the authors' family for finding simple roots from Herceg and Herceg (2015). Some well-known classical methods for simple roots, for example Newton, Potra-Pták, Newton-Steffensen, King and Ostrowski's methods, belong to this family, which implies that our new family contains modifications of these methods suitable for finding multiple roots. Convergence analysis shows that our family contains methods of convergence order from 2 to 4. The new methods require two function evaluations and one evaluation of the first derivative per iteration, so all our fourth order methods are optimal in terms of the Kung and Traub conjecture. Several examples are presented and compared. Through various test equations, relevant numerical experiments strongly support the claimed theory in this paper. Extraneous fixed points of the iterative maps associated with the proposed methods are also investigated. Their dynamics is explored along with illustrated basins of attraction for various polynomials.