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.].

Keywords:

• nonlinear equation
• multiple roots
• regula-falsi method
• newton-like method
• cubic convergence

2000 AMS Subject Classification :

• 65H05
• 90C53

Acknowledgements

The authors would like to thank the referees for their careful reading of this paper. Their comments uncovered several weaknesses in the presentation of the paper and helped us to clarify it. This work is supported by financial grant CSIR (No: 10-2(5)/2004(i)-EU II), India