What is a one-parameter Newton's method?

Question

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 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?

Answer 1

This is the Newton method for the function $g(x)=e^{-px}f(x)$. Since $g'(x)=e^{-px}(f'(x)-pf(x))$, the iteration $$ x_+=x-\frac{g(x)}{g'(x)}=x-\frac{f(x)}{f'(x)-pf(x)} $$ results. Note that the set of roots remains invariant, only the slopes change.

By using a complex value of $p=i\epsilon$ one can break the iteration away from the real axis in the case of polynomials with real coefficients and a real starting point.

I do not know if this trick is useful to "flatten" the function for faster convergence.