Overview
For systems of nonlinear equations \( F(x) = 0 \) (where \( x \in \mathbb{R}^n \)), Newton's Method requires the Jacobian matrix. The Jacobian of \( F(x) = (f_1(x), \dots, f_n(x)) \) is defined as:
\( J(x) = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1} & \cdots & \frac{\partial f_n}{\partial x_n} \end{pmatrix} \)
Newton's method for systems then updates the guess by:
\( x_{n+1} = x_n - J(x_n)^{-1}F(x_n) \)
Example
For a 2×2 nonlinear system:
- Write the system \( F(x,y) = (f_1(x,y), f_2(x,y)) \).
- Compute the partial derivatives to form the Jacobian \( J(x,y) \).
- Evaluate \( F(x,y) \) and \( J(x,y) \) at an initial guess.
- Solve for \( \Delta x \) using \( J(x,y)\Delta x = F(x,y) \) and update: \( x_{new} = x - \Delta x \).