I have been given the following system of nonlinear equations F(X)=0:
$$X=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix},F(X)=\begin{bmatrix} x_1^2+2x_2-2 \\ x_1+4x_2^2-4 \end{bmatrix}$$
I was told to determine the Jacobian matrix F' for the system which I did:
$$J=\begin{bmatrix} 2x_1&2 \\ 1&8x_2 \end{bmatrix}$$
Now, I have to write the formula of Newton iteration for solving the nonlinear equations F(X)=0.
The problem is that I don't know how. Can someone help me with this? So I know the genereal formula looks like this:
$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$
But how do I apply it in my case?
Best Answer
The formula you wrote is for the one-dimensional case, that is, when $F: \mathbb{R}\rightarrow \mathbb{R}$.
Your function is $F: \mathbb{R}^2\rightarrow \mathbb{R}^2$. You need the inverse of the Jacobian, not the reciprocal of the derivative. Note that this is the case in general: in one dimension, the derivative is the Jacobian (as a one-by-one matrix), so the reciprocal of the derivative is in fact the inverse of the Jacobian in that special case.
So use $x_{n+1}=x_n-J(x_n)^{-1}F(x_n)$.
See: https://en.wikipedia.org/wiki/Newton%27s_method