Newton’s method iteratively updates the solution as:
$$ \mathbf{x}_{k+1} = \mathbf{x}_k - \mathbf{J}_k^{-1} \mathbf{F}(\mathbf{x}_k), $$where \( \mathbf{J}_k = \mathbf{J}(\mathbf{x}_k) \) is the Jacobian matrix of \( \mathbf{F} \) evaluated at \( \mathbf{x}_k \). However, computing \( \mathbf{J}_k \) and its inverse at each iteration can be computationally expensive.