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 $$x_1<0\not\in [0, 1]$$.

MSC:

 65H05 Numerical computation of solutions to single equations
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References:

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