Numerical mathematics, Newton's method

[Wingeer]

Homework Statement

"The goal of this task is to check that you understood the derivation of Newton’s method in the lecture.
1. Consider a smooth function G defined from $$\mathbb{R}^N$$ to $$\mathbb{R}^N$$. Suppose it admits a fixed point $$r \in \mathbb{R}^N$$. Write down the Taylor development of this function at the point r.
2. Consider now a function F also defined from $$\mathbb{R}^N$$ to $$\mathbb{R}^N$$. Define a function G as:
$$G(x) = x + H(x)\cdot F(x)$$ (1)
where $$H(x)$$ is now a matrix depending on x, and $$\cdot$$ denotes the matrix vector multiplication. Check that if r is a root of F (i.e., F(r) = 0), then r is also a fixed point of G.
3. G'(r) denotes the Jacobian of G at the point r. Suppose that r is a root of F, and that G takes the form (1). Write down the condition G'(r) = 0 and deduce a formula for H(r).
4. What is a natural choice of the matrix H that leads to the vector Newton method?"

Homework Equations

The Attempt at a Solution

I was not attending the lecture, so I am not sure how this was derivated.
The Taylor series is: $$G(x_1, x_2, \cdots , x_n) \equiv G(\mathbf{x}) = G(r) + \sum_{j=1}^{n} r_j \frac{\partial G(r)}{\partial x_j} + O(r^2)$$
b) Trivial by insertion. I do however not see the relationship between the function given in b, and the one in a?
c) I suppose $$G'(r)=0=1 + H'(r)F(r) + H(r)F'(r) = 1 + H(r)F'(r)$$ from which:
$$H(r)=-[F'(r)]^{-1}$$.
d) I am really not sure. I cannot see the whole picture here. And what's the point of the fixed point?

Thanks in advance.

 

[Wingeer]

Partly as a bump, an partly another question related to numerical mathematics I post in my own thread.
"Consider the formula $$\phi(h)=\frac{f(x_0 +h) - f(x_0)}{h}$$ Use Richardson extrapolation on the points h and h/2 in order to find a new formula. What should be its new order?"
What is meant by the order? And how do I use Richardson extrapolation? Am I not dependant on some error terms to use the method?