Newton correction compatibility condition ($-\mathbf{f}(\mathbf{x})\in \mathrm{im}D\mathbf{f}(\mathbf{x})$)

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.