Some second-derivative-free variants of Halley’s method for multiple roots.

*(English)*Zbl 1181.65070The authors discuss the problem of finding a multiple root of a nonlinear equation using iterative methods. The authors propose two new families of third-order iterative methods based on the known Halley’s method for simple roots. The first family of methods is obtained from Halley’s method by replacing the second derivative with a finite difference between the first derivatives. The second family of the proposed methods is obtained also from Halley’s method by a modification of the algorithm with a new approximation formed by the function instead of its first derivative.

Both families of the new methods are free of second derivatives. One of the families requires one evaluation of the function and two of its first derivative per iteration, and the other family requires two evaluations of the function and one of its first derivative. By introducing three new parameters, the algorithms are modified again, so that the methods are of third-order convergence in case of a multiple root. Many concret methods belonging to these families of methods are specified. Numerical examples are performed using the new methods and comparison with other iterative methods of third-order are presented.

Both families of the new methods are free of second derivatives. One of the families requires one evaluation of the function and two of its first derivative per iteration, and the other family requires two evaluations of the function and one of its first derivative. By introducing three new parameters, the algorithms are modified again, so that the methods are of third-order convergence in case of a multiple root. Many concret methods belonging to these families of methods are specified. Numerical examples are performed using the new methods and comparison with other iterative methods of third-order are presented.

Reviewer: Iulian Coroian (Baia Mare)

##### MSC:

65H05 | Numerical computation of solutions to single equations |

##### Keywords:

Nonlinear equation; Halley’s method; Multiple roots; New iterative methods; root finding; comparison of methods; algorithm; numerical examples
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\textit{S. Li} et al., Appl. Math. Comput. 215, No. 6, 2192--2198 (2009; Zbl 1181.65070)

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