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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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