Consider a system of \( n \) nonlinear equations:
$$ \mathbf{F}(\mathbf{x}) = \mathbf{0}, $$where \( \mathbf{F}: \mathbb{R}^n \to \mathbb{R}^n \) is a vector-valued function, and \( \mathbf{x} \in \mathbb{R}^n \) is the vector of unknowns. The goal is to find \( \mathbf{x}^* \) such that \( \mathbf{F}(\mathbf{x}^*) = \mathbf{0} \).