i want to derive the following iterative scheme.

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)+f(x_n)} $$ where $n$ $=$ $0$$,$$1$$,$$.$$.$$.$

I think this scheme can be drive from Taylor's series but i didn't know how to approach. I already drive some iterative schemes using Taylor series like Householders,Halley's method and some other iterative schemes.How to approach to get this iterative scheme?


What you have looks exactly like newton's method, except with an extra $f(x)$ in the denominator. One way to derive such a form would be to choose some $g(x)$ such that $(g f)' = g f' + g f$, which is, well, a clear sign of an exponential. So choosing $h(x) = \mathrm{e}^{x} f(x)$ and applying newton's method we have: $$x_{n+1} = x_n - \frac{h(x)}{h'(x)} = x_n - \frac{\mathrm{e}^{x} f(x)}{\mathrm{e}^x f'(x) + \mathrm{e}^x f(x)} = x_n - \frac{f(x)}{f'(x) + f(x)}$$

Note that since $\mathrm{e}^x$ "adds" no additional zeroes using newton's method in this way will still converge to $f(x)$'s zeroes.


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