Basins of attraction for Zhou–Chen–Song fourth order family of methods for multiple roots

There are very few optimal fourth order methods for solving nonlinear algebraic equations having roots of multiplicity mm. In a previous paper we have compared 5 such methods, two of which require the evaluation of the (m−1)(m−1)th root. We have used the basin of attraction idea to recommend the best optimal fourth order method. Here we suggest to improve on the best of those five, namely Zhou–Chen–Song method by showing how to choose the best weight function.