New Families of Third-Order Iterative Methods for Finding Multiple Roots

R F Lin,H M Ren,Y Khan,J L Hu,Z Šmarda,Q B Wu

doi:10.1155/2014/812072

Abstract

Two families of third-order iterative methods for finding multiple roots of nonlinear equations are developed in this paper. Mild conditions are given to assure the cubic convergence of two iteration schemes (I) and (II). The presented families include many third-order methods for finding multiple roots, such as the known Dong's methods and Neta's method. Some new concrete iterative methods are provided. Each member of the two families requires two evaluations of the function and one of its first derivative per iteration. All these methods require the knowledge of the multiplicity. The obtained methods are also compared in their performance with various other iteration methods via numerical examples, and it is observed that these have better performance than the modified Newton method, and demonstrate at least equal performance to iterative methods of the same order.

Highlights

Finding the roots of nonlinear equations is one of the most important problems in numerical analysis
It is known that the modified Newton method for multiple roots is given by xn+1
We propose two new families of third-order methods for multiple roots; each of the methods requires twofunction and one-derivative evaluation per iteration, respectively

Summary

Introduction

Finding the roots of nonlinear equations is one of the most important problems in numerical analysis. We use iterative methods to find a multiple root x⋆ of multiplicity m (m > 1); that is, f(j)(x⋆) = 0, j = 0, 1, . There exists a cubically convergent method for multiple roots, presented by Hansen and Patrick [2]. Which is an extension of the classical Halley method of the third order Another cubically convergent method for multiple roots is proposed by Traub [3]. Neta [9] has developed another third-order method requiring the same information yn α f f󸀠. We propose two new families of third-order methods for multiple roots; each of the methods requires twofunction and one-derivative evaluation per iteration, respectively. By specially choosing the parameters in (12) and (13), we get two new families of third-order methods, which include methods (4)–(6), (8), (10), and (11). We use some numerical examples to compare the presented methods with the modified Newton method and some known third-order methods

Preliminaries

Development of New Families of Third-Order Methods

Some Concrete Methods

Numerical Examples

Conclusion

Conflict of Interests