in this paper, we present two new families of third-order and fourth-order methods for finding multiple roots of nonlinear equations. each of them requires one evaluation of the function and two of its first derivative per iteration. several numerical examples are given to illustrate the performance of the presented methods.

The pejorative manifold was defined by Kahan in 1972. While multiple roots of a polynomial are naturally ill conditioned, we have that: if we know the multiplicities of the roots, then the roots are well conditioned. Consider a polynomial p with k distinct roots zi, the ith root has multiplicity mi, for i = 1, 2, . . . , k. The degree n of p is then the sum of multiplicities: n = m1 +m2 + · · ·...

Estimating functions can have multiple roots. In such cases, the statistician must choose among the roots to estimate the parameter. Standard asymptotic theory shows that in a wide variety of cases, there exists a unique consistent root, and that this root will lie asymptotically close to other consistent (possibly inefficient) estimators for the parameter. For this reason, attention has largel...