A cubic-order variant of Newton’s method for finding multiple roots of nonlinear equations

A second‐derivative‐free iteration method is proposed below for finding a root of a nonlinear equation $f\left(x\right)=0$ with integer multiplicity $m\ge 1$ : $x_{n+1}=x_n‐\frac{f(x_n‐\mu f\left(x_n\right)/f^{\text{'}}\left(x_n\right))+\gamma f\left(x_n\right)}{f^{\text{'}}\left(x_n\right)}\text{,}n=0,1,2,\dots \text{.}$ We obtain the cubic order of convergence and the corresponding asymptotic error constant in terms of multiplicity $m$ , and parameters $\mu$ and $\gamma$ . Various numerical examples are presented to confirm the validity of the proposed scheme.