On a family of Laguerre methods to find multiple roots of nonlinear equations

There are several methods for solving a nonlinear algebraic equation having roots of a given multiplicity m. Here we compare a family of Laguerre methods of order three as well as two others of the same order and show that Euler–Cauchy’s method is best. We discuss the conjugacy maps and the effect of the extraneous roots on the basins of attraction.