The Newton's method that I know is defined as follows:


$$x\_{n+1}=x\_n-\frac{f(x\_n)}{f'(x\_n)}$$


However, I've recently encountered [a paper](http://www.emis.de/journals/HOA/JAM/Volume2012/294086.pdf) that talks about a ***one-parameter family of Newton's method*** (page 4, equation 2.8), defined as follows:


$$x\_{n+1}=x\_n-\frac{f(x\_n)}{f'(x\_n)-pf(x\_n)}$$


What is this $p$ parameter in the equation above? Why is it not present in the first equation? What is this parameter useful for? What is that *one-parameter* Newton's method?



What is a one-parameter Newton's method?
"One-parameter family" simply means that one real parameter, $p$, appears in the definition. That method is simply a generalization of Newton's method exposed in the first equation; you can't derive it from the first.


The classical Newton's method has slower convergence when $f'(\alpha)=0$, $\alpha$ being the sought solution: the purpose of this generalization is obtaining a Newton-like method that does not encounter trouble when $f'(\alpha)=0$. I hope this solves your issues!


