## A derivative free iterative method for finding multiple roots of nonlinear equations.(English)Zbl 1205.65176

The author introduces a new method for solving equations $$f(x)=0$$ having a multiple root $$p$$ of multiplicity $$m>1$$ on an interval $$(a,b)$$ using a transformation which converts the multiple root to a simple root of $$H_\epsilon (x)=0$$. It is proven that if $$f\in \mathsf{C}^1(\alpha,\beta)$$ and $$f'(x)\neq 0$$ on $$(\alpha,\beta)\setminus\{ p\}$$ for some $$\alpha$$ and $$\beta$$ such that $$\alpha<a<b<\beta$$, then for any $$\epsilon>0$$, $$p$$ is a simple root of a transformed equation $$H_\epsilon (x)=0$$ with $$H'_\epsilon (p)=\frac{1}{m}$$. Moreover, for some $$\epsilon>0$$ such that $$x+\epsilon f(x)\in(\alpha,\beta)$$ for all $$x\in(a,b)$$, $$H_\epsilon (x)=0$$ is a continuous function having a unique simple zero $$p$$ on the interval $$(a,b)$$. The transformed function $$H_\epsilon (x)$$ of $$f(x)$$ with a small $$\epsilon >0$$ has appropriate properties in applying a derivative free iterative method to find the root. Finally, the author gives some numerical examples to show that the proposed method is superior to the existing methods.

### MSC:

 65H05 Numerical computation of solutions to single equations
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### References:

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