= Newton correction compatibility condition
{c}
{title2=$-\mathbf f(\mathbf x)\in\operatorname{im}D\mathbf f(\mathbf x)$}
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.
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