Solve the zero by Jacobi Method:$f(x,y)=\begin{pmatrix}2& 1\\ 1&2 \end{pmatrix}\begin{pmatrix} x\\y\end{pmatrix}-\begin{pmatrix} 1\\ 0\end{pmatrix}$

Question

Solve by Jacobi Method $f(x,y)=\begin{pmatrix} 2 & 1\\ 1 & 2 \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}-\begin{pmatrix} 1\\ 0 \end{pmatrix}$ with starting point $(x_0,y_0)=(0,0)$

I don't want see the complete solution (it would be too long anyway, probably). But rather the scheme how it is done, the beginning.

So how do you start? You usually change this to a system of equation form but it doesn't seem to work in this case because we have $f(x,y)$

Would I be allowed to write it like that?

$$2x+y-1=0$$

$$x+2y=0$$

Now I would form the first equation to $x$ and the second equation to $y$:

$$x=\frac{1}{2}-\frac{1}{2}y$$

$$y=-\frac{1}{2}x$$

And then continue...? Is that correct?

Answer 1

Yes that is correct. Form iterations from the two equations and then approximate $(x, y)$ using $x_0=0$ and $y_0=0$ as initial values.