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(x)=0f(x)=0 with integer multiplicity m≥1m≥1: xn+1=xn−f(xn−μf(xn)/f′(xn))+γf(xn)f′(xn),n=0,1,2,…. We obtain the cubic order of convergence and the corresponding asymptotic error constant in terms of multiplicity mm, and parameters μμ and γγ. Various numerical examples are presented to confirm the validity of the proposed scheme.