The stability analysis of a new family of iterative methods with memory is introduced. This family, designed from Traub's method, allows to add memory through the introduction of an accelerating parameter. Hence, the speed of convergence of the iterative method increases up to 3.30, with no new functional evaluations. A dynamical study of the proposed family on quadratic polynomials is presented obtaining interesting qualitative properties. © 2019 John Wiley & Sons, Ltd.