How would I carry out the newton method on the non-linear system below:
x_{1}^{2} - 10x_{1} + x_{2}^{2} + 8 = 0
x_{1}x_{2}^{2} + x_{1} - 10x_{2} + 8= 0
Perform one iteration using (0,0)^{T}
What are the steps i have to do?
I find it surprising that you would have a problem like this if you do not know "the steps".
In any case, the "Newton-Raphson" method for solving, say, f(x)= 0, is to start with some "test value", $\displaystyle x_0$, and approximate y= f(x) with its tangent line approximation, $\displaystyle y= f'(x_0)(x- x_0)+ f(x_0)$. Setting that equal to 0 and solving for x, giving $\displaystyle x= x_0- \frac{f(x_0)}{f'(x_0)}$, will, under reasonable conditions, give a new value of x, closer to the root. Then we can repeat to get even closer.
Here, f is "vector" function $\displaystyle f(x_1, x_2)= \begin{pmatrix}x_1^2- 10x_1+ x_2^2+ 8 \\ x_1x_2^2+ x_1- 10x_2+ 8\end{pmatrix}$ and its derivative is the matrix $\displaystyle \begin{pmatrix}2x_1- 10 & 2x_2 \\ x_2^2+ 1 & 2x_1x_2- 10\end{pmatrix}$ so that, for a given starting point $\displaystyle (x_1, x_2)$ the next point would be $\displaystyle \begin{pmatrix}x_1\\ x_2\end{pmatrix}- \begin{pmatrix}2x_1- 10 & 2x_2 \\ x_2^2+ 1 & 2x_1x_2- 10\end{pmatrix}^{-1}\begin{pmatrix}x_1^2- 10x_1+ x_2^2+ 8 \\ x_1x_2^2+ x_1- 10x_2+ 8\end{pmatrix}$.