 Multivariable Newton iteration

The finite-dimensional Newton root-finding iteration for a vector equation $G(x)=0$. At a point with invertible Jacobian matrix, solve the displayed linear system. For $G=\nabla F$, its Jacobian matrix is the Hessian matrix, giving Newton's method in optimization. The update solves the affine first-order approximation $G(x_k)+DG(x_k)s_k=0$; it can converge to any nondegenerate critical point, including a saddle point.