An improved method for finding multiple roots and it's multiplicity of nonlinear equations in R

The aim of this paper is to develop an improved method for finding the multiple roots of nonlinear equations f(x)=0 in R along with an approximation to it's multiplicity when it is not known explicitly. This is done by first describing a derivative free transformation G(x) of f(x) which reduces the multiple roots of f(x)=0 to simple roots of G(x)=0. Then, a second order Newton-type method is used to compute the simple roots of G(x)=0 and the approximation to the multiplicity of the roots of f(x)=0. The method is tested on two numerical examples considered by King [R.F. King, A secant method for multiple roots, BIT 17 (1977) 321-328] and results obtained are compared. It is found that our method is more efficient than that given by King and obtain multiple roots as well as it's multiplicity much faster.