Proof of cubic convergence.

Question

I have the following iteration method $$x_{n+1}=x_n-\frac{2f(x_n)f'(x_n)}{2[f'(x_n)]^2-f(x_n)f''(x_n)}\hspace{3mm} (Halley)$$ and I want to show that it has cubic convergence (I know that the solution has a multiplicity of one and $f''$ satisfies Lipschitz's condition). I somehow needed to show that $$\exists c\in\mathbb{R}:\hspace{1mm}|x_{n+1}-a|\leq c\cdot |x_n-a|^3\hspace{1mm}$$ where $a$ is the solution. I was stuck with this problem. But then I found a proof here (Proof). I thought I finally got it, but still some things didn't add up. First, the Wikipedia proof uses functions third derivative, I don't know if I can assume that it exists? Probably not. So if I can't assume that, I'm back to square one (because I can't use Taylor's theorem in my proof). Second, I don't know where I should use the facts that the solution a multiplicity of one and $f''$ satisfies Lipschitz's condition. Other than that the Wikipedia proof seemed pretty straightforward.

So, can anyone help me with that proof and/or give insight how to use my given assumptions?

Answer 1

You can assume anything you want if it helps you get the results you want. Later on you can try to make the assumptions more restrictive, but I always assume anything I need (for example differentiable instead of just continuous).

As to assuming that the third derivative exist, I think the usual proof of Newton's method assumes the second derivative exists so I see nothing wrong in assuming the third derivative exists here.