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.


65H05 Numerical computation of solutions to single equations
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