In this paper, we present two new families of iterative methods for multiple roots of nonlinear equations. One of the families require one-function and two-derivative evaluation per step, and the other family requires two-function and one-derivative evaluation. It is shown that both are third-order convergent for multiple roots. Numerical examples suggest that each family member can be competitive to other third-order methods and Newton's method for multiple roots. In fact the second family is even better than the first.
Chun, C., Bae, H. ju, & Neta, B. (2009). New families of nonlinear third-order solvers for finding multiple roots. Computers and Mathematics with Applications, 57(9), 1574–1582. https://doi.org/10.1016/j.camwa.2008.10.070