New high-order convergence iteration methods without employing derivatives for solving nonlinear equations. (English) Zbl 0985.65047

In order to solve a single nonlinear equation \(f(x)=0\) the authors consider a family of iterative methods that do not involve any derivative of \(f\) and are of the form \[ x_{n+1}=x_n-\frac{f^2(x_n)}{p\cdot f^2(x_n)+f(x_n)-f(x_n-f(x_n))}, \quad n=0,1,2,\cdots, \] where \(p\in \mathbb{R},|p|<\infty\) and show that, under certain assumptions, these are at least quadratically convergent.
No mention is made on the influence that the initial approximation \(x_0\) could have on the behaviour of the sequence \(\{x_n\}\). If there is no restriction on the initial value \(x_0\), it is possible that at a certain step \(m\) the iterate \(x_m\) does exceed the interval \([a, b]\) because \(x-f(x)\not\in[a, b]\) for any \(x\in [a, b]\) (\(f\) is generally not defined outside \([a, b])\).
Example. For \([a, b]=[0, 1], f(x)=e^{1/x}-e\) and \(x_0=0.5\) we obtain \(x_1<0\not\in [0, 1]\).


65H05 Numerical computation of solutions to single equations
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