Jacobian to do Newton's method.

Question

PICTUREI want to do a Newton's iteration using Jacobian and I did not understand how to do that. Can someone explain the steps to me please?

SOLUTION

Answer 1

Okay, you have a system of two equations, namely \begin{split} f_1(x_1,x_1) = 0\\ f_2(x_1,x_2) = 0 \end{split}

The Newton method reads $$\left[\begin{array}{c} x_1\\ x_2\end{array}\right]^{n+1} = \left[\begin{array}{c} x_1\\ x_2\end{array}\right]^{n} - \left(J^{-1}\right)^{n}\left[\begin{array}{c} f_1(x_1,x_2)\\ f_2(x_1,x_2)\end{array}\right]^n \tag1$$ Where $J^{-1}$ is the inverse of the Jacobian, that is $$J=\left[\begin{array}{cc} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2}\end{array}\right] $$ Use the equation $(1)$ recursively, starting with $n=0$ with your initial guess $$\left[\begin{array}{c} x_1\\ x_2\end{array}\right]^{0}=\left[\begin{array}{c} 1\\ 1\end{array}\right]$$