= Stationary points are fixed points of Newton's method
{title2=$\nabla f(p)=0\implies N(p)=p$}
For a twice continuously differentiable real <function>, define the <Newton map in optimization> $N(p)=p-H(p)^{-1}\nabla f(p)$ wherever its <Hessian matrix> $H$ is invertible. Every <stationary point> in this domain is a <fixed point>, because its <gradient> is zero. This includes a <local maximum> and a <saddle point of a scalar function>, as well as a <local minimum>. Thus <Newton's method in optimization> needs a separate <minimum> classification or a method for ensuring descent when it is used for <minimization problem>.
For example, $f(x,y)=(x+y^2)^2+(x^2+y)^2$ has a <stationary point> at $(-1/2,-1/2)$, with <Hessian matrix>
$$
H=\begin{pmatrix}3&-4\\-4&3\end{pmatrix}.
$$
Its <eigenvalues> are $-1$ and $7$, so it is a <saddle point of a scalar function>, but starting <Newton's method in optimization> there gives a constant sequence. Its value is $1/8$, whereas the <global minimum> zero is attained at $(0,0)$ and $(-1,-1)$.
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