A cubically convergent iteration method for multiple roots of f(x)=0

Abstract

This paper describes a cubically convergent iteration method for finding the multiple roots of nonlinear equations, f(x)=0, where f:ℝ→ℝ is a continuous function. This work is the extension of our earlier work [P.K. Parida, and D.K. Gupta, An improved regula-falsi method for enclosing simple zeros of nonlinear equations, Appl. Math. Comput. 177 (2006), pp. 769–776] where we have developed a cubically convergent improved regula-falsi method for finding simple roots of f(x)=0. First, by using some suitable transformation, the given function f(x) with multiple roots is transformed to F(x) with simple roots. Then, starting with an initial point x 0 near the simple root x* of F(x)=0, the sequence of iterates {x n }, n=0, 1, … and the sequence of intervals {[a n , b n ]}, with x*∈{[a n , b n ]} for all n are generated such that the sequences {(x n −x*)} and {(b n −a n )} converges cubically to 0 simultaneously. The convergence theorems are established for the described method. The method is tested on a number of numerical examples and the results obtained are compared with those obtained by King [R.F. King, A secant method for multiple roots, BIT 17 (1977), pp. 321–328.].