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

Abstract 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.