Multivariable Newton iteration ($DG(x_k)s_k=-G(x_k),x_{k+1}=x_k+s_k$)

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.

A multivariable Newton iteration correction must solve the linear system $D\mathbf{f}(\mathbf{x})\mathbf{s}=-\mathbf{f}(\mathbf{x})$. It exists exactly when the right-hand side is in the image of the Jacobian matrix, and is unique when that Jacobian matrix is invertible. At a singular matrix, there may be either no corrections or several; singularity alone does not decide which. For finding a critical point of a scalar function $F$, use $\mathbf{f}=\nabla F$, so the Jacobian matrix is the Hessian matrix. Equal rows of the Hessian matrix with unequal corresponding gradient entries immediately make the correction equations inconsistent.